This book is about the subject of higher smoothness in separable real Banach spaces. It brings together several angles o

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- Michal Johanis

*Table of contents : IntroductionChapter 1. Fundamental properties of smoothness 1. Multilinear mappings and polynomials 2. Complexification 3. Fréchet smoothness 4. Taylor polynomial 5. Smoothness classes 6. Power series and their convergence 7. Complex mappings 8. Analytic mappings 9. Notes and remarksChapter 2. Basic properties of polynomials on Rn 1. Spaces of polynomials on Rn 2. Cubature formulae 3. Estimates related to Chebyshev polynomials 4. Polynomials and L_p-norms on Rn 5. Polynomial identities 6. Estimates of coefficients of polynomials 7. Notes and remarksChapter 3. Weak continuity of polynomials and estimates of coefficients 1. Tensor products and spaces of multilinear mappings 2. Weak continuity and spaces of polynomials 3. Weak continuity and _1 4. (p,q)-summing operators 5. Estimates of coefficients of multilinear mappings 6. Bohr radius 7. Notes and remarksChapter 4. Asymptotic properties of polynomials 1. Finite representability and ultraproducts 2. Spreading models 3. Polynomials and p-estimates 4. Separating polynomials. Symmetric and sub-symmetric polynomials 5. Stabilisation of polynomials 6. Sub-symmetric polynomials on Rn 7. Polynomial algebras on Banach spaces 8. Notes and remarksChapter 5. Smoothness and structure 1. Convex functions 2. Smooth bumps and structure I 3. Smooth variational principles 4. Smooth bumps and structure II 5. Local dependence on finitely many coordinates 6. Isomorphically polyhedral spaces 7. L_p spaces 8. C(K) spaces 9. Orlicz spaces 10. Notes and remarksChapter 6. Structural behaviour of smooth mappings 1. Weak uniform continuity and higher smoothness 2. Bidual extensions 3. Class ==========W 4. Uniformly smooth mappings from C(K), K scattered 5. Uniformly smooth mappings from ==========W-spaces 6. Fixing the canonical basis of c_0 7. Ranges of smooth mappings 8. Harmonic behaviour of smooth mappings 9. Notes and remarksChapter 7. Smooth approximation 1. Separation 2. Approximation by polynomials 3. Approximation by real-analytic mappings 4. Infimal convolution 5. Approximation of continuous mappings and partitions of unity 6. Non-linear embeddings into c_0() 7. Approximation of Lipschitz mappings 8. Approximation of C1-smooth mappings 9. Approximation of norms 10. Notes and remarksBibliographyNotationIndex*

Petr Hájek, Michal Johanis Smooth Analysis in Banach Spaces

De Gruyter Series in Nonlinear Analysis and Applications

Editor in Chief Jürgen Appell, Würzburg, Germany Editors Catherine Bandle, Basel, Switzerland Alain Bensoussan, Richardson, Texas, USA Avner Friedman, Columbus, Ohio, USA Karl-Heinz Hoffmann, Munich, Germany Mikio Kato, Nagano, Japan Umberto Mosco, Worcester, Massachusetts, USA Louis Nirenberg, New York, USA Boris N. Sadovsky, Voronezh, Russia Alfonso Vignoli, Rome, Italy Katrin Wendland, Freiburg, Germany

Volume 19

Petr Hájek, Michal Johanis

Smooth Analysis in Banach Spaces

Mathematics Subject Classification 2010 Primary: 46B03, 46G25, 46T20; Secondary: 46B20, 46G05 Authors Prof. Dr. Petr Hájek Academy of Sciences of the Czech Republic Institute of Mathematics Žitná 25 11567, Prague Czech Republic [email protected] and Czech Technical University in Prague Faculty of Electrical Engineering Department of Mathematics Zikova 4 16000, Prague Czech Republic Dr. Michal Johanis Charles University Faculty of Mathematics and Physics Sokolovská 83 18675, Prague Czech Republic [email protected]

ISBN 978-3-11-025898-1 e-ISBN (PDF) 978-3-11-025899-8 e-ISBN (EPUB) 978-3-11-039199-2 Set-ISBN 978-3-11-220385-9 ISSN 0941-813X Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2014 Walter de Gruyter GmbH, Berlin/Boston Printing and binding: CPI books GmbH, Leck ♾Printed on acid-free paper Printed in Germany www.degruyter.com

Contents Introduction

vii

Chapter 1. Fundamental properties of smoothness 1. Multilinear mappings and polynomials 2. Complexification 3. Fréchet smoothness 4. Taylor polynomial 5. Smoothness classes 6. Power series and their convergence 7. Complex mappings 8. Analytic mappings 9. Notes and remarks

1 2 22 26 40 54 61 68 69 79

Chapter 2. Basic properties of polynomials on Rn 1. Spaces of polynomials on Rn 2. Cubature formulae 3. Estimates related to Chebyshev polynomials 4. Polynomials and Lp -norms on Rn 5. Polynomial identities 6. Estimates of coefficients of polynomials 7. Notes and remarks

82 83 87 96 101 111 119 126

Chapter 3. Weak continuity of polynomials and estimates of coefficients 1. Tensor products and spaces of multilinear mappings 2. Weak continuity and spaces of polynomials 3. Weak continuity and `1 4. .p; q/-summing operators 5. Estimates of coefficients of multilinear mappings 6. Bohr radius 7. Notes and remarks

130 131 135 159 165 170 176 179

Chapter 4. Asymptotic properties of polynomials 1. Finite representability and ultraproducts 2. Spreading models 3. Polynomials and p-estimates 4. Separating polynomials. Symmetric and sub-symmetric polynomials

184 185 192 198 203

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5. 6. 7. 8.

Stabilisation of polynomials Sub-symmetric polynomials on Rn Polynomial algebras on Banach spaces Notes and remarks

214 223 229 235

Chapter 5. Smoothness and structure 1. Convex functions 2. Smooth bumps and structure I 3. Smooth variational principles 4. Smooth bumps and structure II 5. Local dependence on finitely many coordinates 6. Isomorphically polyhedral spaces 7. Lp spaces 8. C.K/ spaces 9. Orlicz spaces 10. Notes and remarks

239 240 257 263 270 278 290 294 304 307 321

Chapter 6. Structural behaviour of smooth mappings 1. Weak uniform continuity and higher smoothness 2. Bidual extensions 3. Class W 4. Uniformly smooth mappings from C.K/, K scattered 5. Uniformly smooth mappings from W -spaces 6. Fixing the canonical basis of c0 7. Ranges of smooth mappings 8. Harmonic behaviour of smooth mappings 9. Notes and remarks

328 329 333 338 349 357 362 365 369 378

Chapter 7. Smooth approximation 1. Separation 2. Approximation by polynomials 3. Approximation by real-analytic mappings 4. Infimal convolution 5. Approximation of continuous mappings and partitions of unity 6. Non-linear embeddings into c0 . / 7. Approximation of Lipschitz mappings 8. Approximation of C 1 -smooth mappings 9. Approximation of norms 10. Notes and remarks

382 383 391 399 407 411 422 428 452 456 462

Bibliography

467

Notation

490

Index

493

Introduction The purpose of this book is to lay down the foundations for the abstract theory of C k -smoothness in infinite-dimensional real Banach spaces, and investigate its intimate connections with the structural properties of the underlying spaces. The main objects of the theory are polynomials and C k -smooth (including real analytic) mappings. In some sense, the most important result concerning C k -smooth mappings is the Taylor formula, which takes the familiar form known from the theory of functions on Rn . This formula unveils the prominent role played by polynomials in smoothness (especially higher smoothness) questions by way of approximating smooth functions in the neighbourhood of a point. In the infinite-dimensional setting this role is brought even further, as polynomials also provide the vital link with the structure of the underlying Banach space. This explains why polynomials have received a great deal of attention in the present book. We have included plenty of results concerning polynomials with the intention to build up a supply of results and points of view that could be useful for the future development of the theory. The material is mostly organised according to the methods used. We study polynomials on Rn in connection with isometric theory of finitedimensional spaces. It turns out that homogeneous polynomials are closely related to isometric subspaces of `pn for p even. The theory of Chebyshev polynomials is used to obtain optimal estimates on the size of higher derivatives for a given polynomial. The theory of tensor products and .p; q/-summing operators is applied to obtain sharp estimates on the values of polynomial coefficients for polynomials from c0 to `p . The concept of finite representability combined with powerful finite-dimensional results concerning type and cotype of Banach spaces, spreading models, and ultrapowers lead to strong structural results for Banach spaces admitting separating polynomials. All the above results and much more are covered in the first four chapters. The breadth of the above mentioned material precludes making our presentation completely self contained. We decided, for the benefit of the reader, to include most of the needed auxiliary results (without proof) in the form of short survey paragraphs, or sometimes whole sections, which form an integral part of the text. This makes it possible for the reader to follow the text, without having to jump into an appendix or a specialised monograph, and keep track of the flow of ideas. The remaining three chapters of the book are devoted to a detailed study of smooth mappings between Banach spaces. The properties being studied can be roughly divided

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Introduction

into three main areas whose polynomial (and usually finite-dimensional) counterparts are covered in the first half of the book. The first aspect are the structural properties implied by the existence of certain polynomials, resp. separating C k -smooth functions (in the finite-dimensional setting this corresponds to the study of subspaces of `pn ). The second aspect is the supply of such mappings (this corresponds essentially to quantitative estimates on the coefficients), and finally the last aspect concerns approximation questions (this relates to the theory of algebras of sub-symmetric polynomials). The Banach space c0 plays a major role throughout the subject. From the technical point of view, this is due to the fact that its supply of polynomials (and uniformly smooth functions) is very small (they are weakly uniformly continuous), but on the other hand its supply of C 1 -smooth functions is very large (thanks to the phenomenon of functions that depend locally on finitely many coordinates). Moreover, it is a universal space for C k -smooth embeddings. There are two important classes of spaces which share some of its important features. It is the class of polyhedral spaces, which are C 1 -smooth, c0 saturated, and behave well with respect to taking subspaces. From the other side it is the class W , which contains all L1 -spaces and behaves rather well with respect to uniformly smooth mappings and quotients. The intersection of these two classes contains all isometric preduals of `1 , in particular all C.K/ spaces, K countable compact. The examples of L1 -spaces of Jean Bourgain and Freddy Delbaen do not contain c0 , so they are not polyhedral. On the other hand, Schreier’s space is a polyhedral space (isomorphic to a subspaces of C.Œ0; ! ! /) which admits a non-compact linear operator into `2 , and hence it does not belong to class W . In fact, Ioannis Gasparis constructed polyhedral spaces with quotient `2 . The existence of a separating C k -smooth function on a Banach space has strong structural consequences. For example a Banach space admitting a C 2 -smooth bump either contains c0 or it is super-reflexive. So the study of higher smoothness naturally splits into two rather distinct extreme situations, which also rely on distinct techniques. This feature is repeated with respect to smooth mappings, whose supply again depends heavily on the geometry of the underlying space. The former case corresponds to W -spaces, which admit only few uniformly smooth mappings. More precisely, if Y has no subspace isomorphic to c0 , then any uniformly smooth mappings from a W -space into Y is weakly compact. On the other hand, super-reflexive spaces admit surjective polynomials onto any separable Banach space. In the questions on approximability of functions and mappings the space c0 plays a fundamental role thanks to being a universal C k -smooth embedding space. However, for real analytic approximations the available methods again seem to distinguish between spaces with a separating polynomial or merely a separating analytic function (the c0 -type). At present it is not clear if this distinction can be overcome by improving the methods of proof. Our book is focusing on the case of real Banach spaces, using the complex case only as a tool for dealing with analytic functions. We feel that the theory has now reached a certain level of maturity, in the sense that extreme cases of higher smoothness, i.e. the

Introduction

ix

c0 -like, or finite rank phenomena, and on the other hand the separating polynomial-like phenomena have been well-understood. Our rendering of the state of the theory is rather complete and up to date, and it appears for the first time in a book form. A large part is devoted to very recent developments, sometimes in an outline form. We believe that our book may serve as a rather complete reference book for the results and techniques in the area of smoothness in separable real Banach spaces. It can also be used as a textbook by advanced graduate students and active researchers with solid background in Banach space theory, and for this purpose we pose a number of open problems for independent research. The future now lies in investigation of general polynomials, and higher order approximations. This will most probably require a new set of techniques. Let us proceed with a more detailed description of some highlighted results in the respective chapters. The first chapter contains an abstract theory of C k -smoothness in any Banach space. We introduce multilinear mappings and polynomials, as higher derivatives are defined to be these objects. The main result in this respect is the Taylor formula which provides a fundamental link between smoothness, polynomials, and the underlying Banach space structure. We also need to introduce the complexification of spaces and polynomials. This is indispensable for the development of the theory of analytic functions, but also very convenient in other situations thanks to efficient averaging methods for complex polynomials. Our treatment of (real) analytic functions was inspired by papers of Jacek Bochnak and Józef Siciak. Analytic functions are defined as functions locally admitting power series expansion. There are several important characterisations, in particular by using finite-dimensional restrictions, and by using the complexified series. These characterisations are also an important tool in developing the theory. The theory depends heavily on results from one or several complex variables, which are mentioned without proof and applied through the notion of higher Gâteaux smoothness. In brief summary, the first chapter gives a complete and self contained introduction to the subject, including the case of real analytic functions which to the best of our knowledge is for the first time in a book form. The remaining six chapters are best read consecutively. Chapters 2–4, which focus on properties of polynomials, cover all the necessary background needed for the remaining Chapters 5–7, which are dealing with general smooth mappings. Most of the material contained therein is also new to book form. In Chapter 2 we focus on the duality theory for spaces of polynomials on finitedimensional Banach spaces, in particular the cubature formulae representing positive functionals. We present a very simple and original proof of the well-known Chakalov theorem. We outline the proofs of cubature formulae related to Chebyshev polynomials on R and R2 . This topic seems to be rather special but it leads to the proof of the Skalyga-Markov-type inequality, an optimal estimate on the size of higher derivatives of real polynomials. We continue by showing that every homogeneous polynomial is a finite sum of powers of functionals, which leads to the theory of isometric embeddings of finite-dimensional spaces into `p -spaces where p is even. We present, without proof,

x

Introduction

the recent result of Vladimir Leonidovich Dol’nikov and Roman Nikolaevich Karasev, and its close (but easier) relative, the Birch theorem. According to these results, a k-homogeneous polynomial on RN can be restricted to a suitable n.N; k/-dimensional subspace E so that this restriction is equivalent to a power of the Euclidean norm on E. We cover the basic theory of identities for polynomials, which implies that a continuous function is a polynomial of degree at most n provided that its restriction to every affine one-dimensional subspace is also a polynomial of degree at most n. The last section is devoted to a relatively simple but powerful averaging method for treating polynomials and multilinear mappings, which leads to important estimates of polynomial coefficients. More sophisticated methods which combine probabilistic averaging with other techniques are treated in the subsequent chapter. Chapter 3 is devoted to the study of polynomials between Banach spaces and the way they act on sequences from the initial space. This is a classical theme in the theory of linear operators, represented by the Dunford-Pettis property and the theory of p-summing operators. We start the chapter by outlining the basics on tensor products as well as symmetric tensor products and their duality with polynomial spaces. We then outline the concepts of uniform spaces and uniform continuity which play a key role throughout the whole book. We proceed by developing in detail the basic theory of weakly, weakly sequentially, and weakly uniformly continuous mappings and in particular polynomials. This investigation was initiated in the early papers of Aleksander Pełczy´nski and more systematically developed by Richard Martin Aron and his coauthors. An important contribution of Raymond A. Ryan was the introduction of symmetric tensor products into the subject, as well as the proof that every weakly compact polynomial from a space with the Dunford-Pettis property maps weakly Cauchy sequences to norm convergent ones. In Banach spaces not containing `1 the weak sequential continuity on bounded sets coincides with full (or even uniform) continuity. This important fact results from Rosenthal’s `1 theorem and other related results of Edward Wilfred Odell and results of Jean Bourgain, David H. Fremlin, and Michel Talagrand. We introduce the language of the theory of .p; q/-summing operators, mention some connections with tensor products, and formulate some of the fundamental results of the theory, introducing also the notions of type and cotype. Using the theory of multiple .pI 1/-summing operators, we give optimal estimates on the coefficients of polynomials in P . nc0 I `p /, following the recent work of Andreas Defant and Pablo Sevilla-Peris. In Chapter 4 we apply the concept of finite representability to the study of asymptotic behaviour of polynomials and the linear structure of the underlying Banach spaces. We describe the ultrapower construction for a Banach space X, which leads to a much larger Banach space .X/U that is finitely representable in X and that is wellsuited for constructions of uniformly continuous mappings. As an application we show that uniformly smooth mappings can be extended into the bidual. By using the spreading models of X , which capture the asymptotic behaviour of infinite sequences in X, we study the upper and lower estimates of sequences, the Banach-Saks and

Introduction

xi

the weak p-Banach-Saks properties. We proceed by showing that if X has a subsymmetric basis and a separating d -homogeneous polynomial, then it is isomorphic to some `p for p even, and d D kp, k 2 N. As a corollary we obtain a fundamental result of Robert Deville stating that every Banach space with a separating polynomial contains a subspace isomorphic to `p , p even. We study in detail the restrictions of polynomials on `p to subspaces generated by suitable subsequences of the canonical basis. The main result in this direction, which goes well beyond the asymptotic but finite-dimensional results based on spreading models, claims that these restrictions are almost sub-symmetric. As a corollary to all these results we show that for an arbitrary polynomial P on X there is an infinite-dimensional subspace Y of X such that the restriction of P to Y is either separating or asymptotically zero in a strong sense. The last sections are devoted to the study of algebras An .X/ of polynomials generated by polynomials of degree at most n on a Banach space X. The main technical tool is a finite-dimensional lemma which claims that the symmetric polynomial snN .x/ D PN n N j D1 xj on R is not in the uniform closure of a suitably defined sub-symmetric subalgebra of An 1 .RN /, provided that N is large enough. We proceed by applying this result to infinite-dimensional spaces using spreading model techniques. The main result implies in particular that A1 .`p / D D An 1 .`p / ¤ An .`p / ¤ AnC1 .`p / ¤ , where n D dpe. Chapter 5 is devoted to the detailed study of Banach spaces admitting smooth separating functions. An important set of tools for obtaining structural results are the variational principles. When applied to a given lower continuous and bounded below function f , they guarantee the existence of a point x 2 X and a smooth function g such that f g attains its minimum, sometimes in a strong sense. If f itself is a smooth function, then depending on the concrete conditions we may use the Taylor formula at x in order to obtain uniformly smooth separating functions on X , or at least some structural information about X. We describe two examples of this notion. Stegall’s variational principle holds in every Banach space with the RNP and the function g can be chosen to be a functional from X . As a result, if a Banach space with the RNP admits twice Gâteaux smooth bump function, we conclude that X is super-reflexive and admits a norm with a power type 2 estimate on the modulus of smoothness. The compact variational principle of Robert Deville and Marián Fabian, which was motivated by a paper of Jaroslav Pechanec, John H. M. Whitfield, and Václav Zizler, is a key tool for studying higher smoothness in Banach spaces. Its formulation is somewhat different from the general description given above, but it leads to similar applications. This principle (we prefer to avoid the precise formulation at this point) applies to Banach spaces which do not contain c0 , and it has lead to several strong structural results. The most important structural result is that if X has no subspace isomorphic to c0 and has a C k -smooth bump, for large enough k, then X is super-reflexive, admits a separating polynomial, and contains a subspace isomorphic to `p , p even. Jaroslav Kurzweil has pioneered the field of higher smoothness by finding that the best order of smoothness of Lp -spaces is C Œp , except in the case when p is even and the space admits a separating polynomial. In the latter case he constructed real

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analytic approximations for all continuous mappings from these spaces. In the rest of the chapter we compute the best order of smoothness for several classes of Banach spaces, notably the Orlicz spaces and the class of polyhedral Banach spaces. In Chapter 6 we develop the theory of uniformly smooth mappings between Banach spaces. We study the relationship between uniform continuity and weak uniform continuity of the mapping and its higher derivatives. In particular, if a weakly uniformly continuous mapping has a uniformly continuous kth derivative, then this derivative is even weakly uniformly continuous. In Section 6.2 we introduce and study the important concept of bidual extension for C k;C -smooth mappings from the unit ball of a Banach space X into Y to a C k;C -smooth mapping from the unit ball of X into Y . This notion contains the classical bi-adjoint of a linear operator as a special case, but it is not completely canonical and in general depends on some parameters in the construction. This notion plays an important role later on in pushing some results for C.K/ spaces where K is scattered into the case of a general compact space K. In Sections 6.3–6.6 we build the theory of W -spaces, i.e. all those X such that uniformly smooth functions in BX map weakly Cauchy sequences in BX , for some 2 .0; 1, to convergent sequences. The main goal of this theory is to generalise some classical properties of linear operators from C.K/ spaces, described in Theorem 3.47, into the setting of uniformly smooth mappings. This objective is achieved in the main Theorem 6.57, which claims that weakly compact and uniformly smooth mappings from the unit ball of C.K/ spaces take weakly Cauchy sequences into norm convergent ones. The first step of the proof is to show the rigidity of `n1 with respect to uniformly smooth functions. The second step consists of showing that C.K/, for K scattered, are W1 -spaces. Next, we prove that the bidual extension of any uniformly smooth noncompact mapping in C 1;C .Bc0 I Y / has a point where the derivative is non-compact, and hence fixes a copy of c0 . This implies that Y has a subspace isomorphic to c0 . Moreover, non-compact uniformly smooth mappings from C.K/, K scattered, can always be reduced to a suitable subspace isomorphic to c0 where the restriction remains non-compact. The case of a general C.K/ space requires another ingredient, Theorem 6.56, which claims that after passing to the bidual, weakly Cauchy sequences in C.K/ spaces are uniformly close to weakly Cauchy sequences in some C.Œ0; ˛/. In Section 6.7 we give some rather general results on the ranges of smooth mappings (and derivatives of smooth functions) which illustrate that the theory of W -spaces works under nearly optimal assumptions. Indeed, according to Theorem 6.69 no structural property of the initial space is generally preserved by surjective C 1 -smooth mappings from Banach spaces with property B. In the rest of the chapter our attention shifts to smooth separating mappings from `p -spaces, proving strong structural result in this case, based on the notions of a separating mapping and harmonic behaviour. Smooth approximations in Banach spaces are studied in the last chapter of our book. Unlike the finite-dimensional case, the unit ball of an infinite-dimensional Banach space X is not a compact set and hence it is easy to find uniformly continuous functions that are not uniformly approximable by polynomials on BX . Interestingly enough, in the special case of W -spaces X not containing `1 , uniformly C k -smooth functions

Introduction

xiii

can be uniformly approximated, together with their higher derivatives, by polynomials on BX . This result underlines the extremely poor supply of uniformly smooth functions in these spaces, rather than the abundance of polynomials. Indeed, in this case all polynomials are weakly uniformly continuous on BX . This is again a situation when smoothness properties take on very different shapes for c0 -like spaces (although X considered here need not even contain c0 !), and for super-reflexive spaces. The main tool for studying C k -smooth approximations of continuous mappings are k C -smooth partitions of unity, or alternatively C k -smooth embeddings into c0 . The problem becomes more challenging if additional conditions are put on the approximants, and the usual partitions of unity cannot be employed as they destroy the character of the approximating functions. This concerns for example the problem of smooth approximations preserving the Lipschitz constants, which is the first step for obtaining approximations together with higher derivatives, a problem that remains widely open. We finish the chapter by proving that if a separable Banach space admits a C k -smooth equivalent norm, then every norm can be approximated on bounded sets by C k -smooth renormings. Let us make some remarks concerning the existing literature related to the subject of our book. Introduction to abstract smooth analysis (real or complex) forms part of Jean Dieudonné’s book [Dieu], which was a great source of inspiration for us. The complex case, where all notions of smoothness coincide, has received more attention in monographs, e.g. in [Hill], [Din], [Mu], [Na]. We have relied much on Seán Dineen’s book for its insights, historical comments, and ample references. The distinguishing feature of the real setting, as opposed to complex one, is the intricate role played by the Banach space structure and geometry for the supply of polynomials and C k -smooth functions. Concerning polynomials on finite dimension spaces, there are of course many sources, see e.g. [MMR], [BE]. Apparently the main interest in polynomials in the infinite-dimensional setting came from the study of analytic functions, see [Din] for historical comments. Finally, it was the monograph of Robert Deville, Gilles Godefroy, and Václav Zizler [DGZ], which has undertaken a systematic study of smoothness (including higher smoothness) in the context of Banach space structure. Their book treats both the separable and non-separable situation (in the latter they are still relatively up to date, so we decided for the most part to omit it). Our book overlaps only to a small extent with [DGZ], but the problems posed therein played a decisive role for the subsequent development of the subject. We mention that the first order smoothness theory is well covered (apart from [DGZ]) also in [Fab3], and the more introductory [FHHMZ]. The differentiability of Lipschitz mappings is treated in [LPT]. We are not covering any aspects of this direction of research. In a broader sense, our subject forms part of geometric non-linear analysis, a fast growing subject whose foundations are laid out in [BenLi]. Concerning prerequisites, most of the background material on the linear structural theory of Banach spaces can be found in [FHHMZ]. Deeper results from local theory,

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which play a crucial role throughout the book, can be found in [DJT]. These two references provide a solid background for the material presented in our book, but our theory draws on results from other areas as well. For the theory of Chebyshev polynomials we suggest [Ri], tensor products are treated in [DefFl] or [Ry3]. A detailed exposition of spreading models is in [BeaLa]. Structural theory of C.K/ and L1 -spaces, which is used in Chapter 6, is covered in [LiTz1], and for C.K/ spaces also in [Ros3]. Structural theory of the classical Banach spaces is developed in more detail in [AK], [Dies2], [LiTz2], and [LiTz3]. For topological results we refer mostly to [Eng]. For an additional source of open problem we refer to [FMZ].

Acknowledgement In the early 90’s the first named author started his PhD research under the supervision of Václav Zizler, then at the University of Alberta in Edmonton. At that time Václav was finishing his book [DGZ], which has played a decisive role for the future development of this area, through its thorough exposition of the known results and techniques, but also through many attractive but approachable problems. Not only that. For the benefit of his students, Václav has compiled a list of additional 50 (later expanded to about 90) problems in the areas close to [DGZ], many of them related to higher smoothness. It was precisely this abundance of nice open problems and Václav’s keen interest in any progress, that attracted the first named author to this field of research. Some of the problems have been solved, and their solutions have lead to the theory presented in this book. Many remain still open, and have been again posed in the present work in the hope that they will do a similar service to the next generation of students. We would like to thank Václav for all those years of support, interest, and friendship. We would also like to thank our closest colleagues and friends (in alphabetical order) Richard M. Aron, Robert Deville, Marián Fabian, Gilles Godefroy, Gilles Lancien, Vicente Montesinos, and Stanimir L. Troyanski for their important contribution to the present work in the form of open problems, shared knowledge, moral support, and opportunity for discussions and seminar presentations. Finally, we would like to thank our colleagues Michal Kraus and Ludˇek Zajíˇcek for reading parts of the manuscript and supplying us with errata and critical comments. Above all we thank our families for taking the burden of living with us through the long and painstaking process of writing this volume. ˇ 201/07/0394, The work on this volume has been partially supported by grants GACR ˇ 201/11/0345, GAAV A100190801, 7AMB12FR003, and RVO 67985840. GACR

Notation

xv

Notation We fix some notation for objects and notions that the reader should be familiar with. By N, Z, Q, R, and C we denote the sets of natural numbers, integers, rational numbers, reals, and complex numbers respectively. We set N0 D N [ f0g. By RC we denote the set of positive real numbers and by RC 0 the set of non-negative real numbers. By R we denote the extended real line, i.e. R D Œ 1; C1. By K we denote the scalar field R or C. We use the convention that a sum over an empty set is zero and a product over an empty set is equal to 1. Further, x 0 D 1 for any x 2 K. For x 2 R we denote by Œx the integer part of x, i.e. the unique number k 2 Z satisfying k x < k C 1, by dxe we denote the ceiling of x, i.e. the unique number k 2 Z satisfying k 1 < x k. For a set A we denote its cardinality by jAj or card A. The cardinality of the continuum is denoted by c. By abusing the notation we write fx g 2 X meaning that fx g 2 is a collection such that x 2 X for each 2 . Let .P; / be a metric space. We denote B.x; r/ D fy 2 P I .y; x/ rg and U.x; r/ D fy 2 P I .y; x/ < rg the closed, resp. open ball in P centred at x 2 P with radius r > 0. In case that it is necessary to distinguish the spaces in which the balls are taken, we will write BP .x; r/, resp. UP .x; r/. By BX and UX we denote the closed, resp. open unit ball of a normed linear space X. By SX we denote the unit sphere of a normed linear space X. An interior of a set A in a topological space is denoted by Int A, its boundary is denoted by @A. Throughout the book we use the following convention: In each statement involving multiple vector spaces we assume that all the spaces are over the same field K if not specified otherwise. Furthermore, if not specified explicitly or in the beginning of the chapter or section, then the statement holds both for K D R and K D C. When we speak of a subspace of a Banach space, we always mean a closed subspace. General subspaces will be referred to as “linear subspaces”. We define Pspan ; D f0g. If X is a normed linear space with a Schauder basis fen g and x D 1 nD1 xn en 2 X, then supp x D fn 2 NI xn ¤ 0g is called a support of x; a finitely supported vector is a vector with finite support. An algebraic dual of a vector space is denoted by X # , a topological dual of a topological vector space by X . Inner product is denoted by hx; yi, and similarly we denote the evaluation in duality by hf; xi. Let X, Y be normed linear spaces. For simplicity we say that X contains Y if X has a subspace isomorphic to Y . By C.XI Y / we denote the set of continuous mappings between topological spaces X, Y . If Y is a topological vector space, then C.X I Y / is a vector space. For functions, i.e. mappings into the scalars, we use a shortened notation C.X/ D C.X I K/; from the context it should always be clear whether K D R or K D C. For a mapping f W X ! Y , where Y is a vector space, we denote suppo f D f 1 .Y n f0g/. If X is a topological space, then we denote supp f D suppo f . An L-Lipschitz mapping is a mapping that is Lipschitz with a constant L. By A we denote the characteristic function of the set A.

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Introduction

If we say measure or Borel measure, we always mean a non-negative measure. On the other hand, Radon measure means scalar-valued Radon measure. By supp we denote the support of a Borel measure . The n-dimensional Lebesgue measure will be denoted by n , or just if the dimension is clear from the context. All topological spaces in this volume are automatically and without mention assumed to be Hausdorff.

Chapter 1

Fundamental properties of smoothness In this chapter we are going to develop the theory of C k -smoothness on normed linear spaces and the theory of analytic mappings on Banach spaces. Completing this task requires a large number of conceptual and technical ingredients. The main concept here is the notion of a Fréchet derivative. Let f W X ! Y be a mapping between normed linear spaces X and Y . The Fréchet derivative of f at x 2 X is a bounded linear operator from X to Y which approximates the mapping h 7! f .x C h/ f .x/ around 0. Higher order derivatives are defined by induction, so the second derivative is a linear operator from X into the space of bounded linear operators from X to Y . Things are getting more complicated as the order of the derivative grows and the higher derivatives at x are identified with bounded multilinear mappings from X X to Y . Higher derivatives turn out to be invariant with respect to permutations of the variables (we say that they are symmetric mappings), and this can be used to identify them with homogeneous polynomials from X into Y . This identification leads to no loss of information, but the benefit is enormous, as the domain of the differentials now remains the same space X , and we can formulate the fundamental Taylor theorem in the same way as for functions on the real line. The program outlined above is carried out in the first four sections of the chapter, together with the additional construction of the complexification of a Banach space and the corresponding canonical extensions of multilinear mappings and polynomials to mappings between the complexifications. The remaining part of the chapter is devoted to building the theory of analytic mappings. The formal definition of an analytic mapping looks rather simple, as we only require that on a neighbourhood of every point in the domain the mapping is the sum of a power series consisting of homogeneous polynomials. However, in order to establish the basic properties of analytic mappings we are forced to prove a number of non-trivial results and also work simultaneously with several equivalent conditions to being analytic. This program includes the detailed study of the modes and sets of convergence of the power series, the use of complex analytic results and methods (e.g. the Cauchy formula) from several complex variables, in tandem with the weaker notion of Gâteaux derivative and the fundamental estimate contained in the Polynomial lemma of Franciszek Leja. In this chapter all the results hold both for real and complex spaces if not indicated otherwise. Sometimes, when the statement directly involves the field, we stress again that the spaces involved are over K.

2

Chapter 1. Fundamental properties of smoothness

1. Multilinear mappings and polynomials In this section we introduce the notions of multilinear mappings and polynomials between normed linear spaces, and prove some of their main properties. In particular, homogeneous polynomials are in a canonical one-to-one correspondence with the symmetric multilinear mappings via the Polarisation formula. Polynomials are continuous mappings whenever they have at least one point of continuity. We introduce non-homogeneous polynomials of degree at most n and prove that their space is a direct sum of the homogeneous summands. Definition 1. Let X1 ; : : : ; Xn , and Y be vector spaces. We will say that a mapping M W X1 Xn ! Y is n-linear if it is linear separately in each coordinate, i.e. if x 7! M.x1 ; : : : ; xk 1 ; x; xkC1 ; : : : ; xn / is a linear mapping from Xk into Y for each x1 2 X1 ; : : : ; xn 2 Xn and each k 2 f1; : : : ; ng. By L.X1 ; : : : ; Xn I Y / we denote the vector space of all n-linear mappings from X1 Xn to Y . In the special case when Xk D X, 1 k n, we use the short notation L. nXI Y /. A mapping is called multilinear if it is n-linear for some n 2 N. A 2-linear mapping will also be called bilinear. We say that M 2 L. nX I Y / is symmetric if M.x1 ; : : : ; xn / D M.x.1/ ; : : : ; x.n/ / for every permutation of f1; : : : ; ng and every x1 ; : : : ; xn 2 X. By Ls . nXI Y / we denote the vector space of all n-linear symmetric mappings from X n to Y . By repeated applications of the definition we can see that an n-linear mapping M 2 L.X1 ; : : : ; Xn I Y / satisfies the formula ! ! m1 mn n X X X Y j 1 1 n n M ak1 xk1 ; : : : ; akn xkn D ak M.xk11 ; : : : ; xknn / (1) j

k1 D1

kn D1

j D1

1kj mj j D1;:::;n j

j

for an arbitrary choice of mj 2 N, ak 2 K, and xk 2 Xj , where kj D 1; : : : ; mj j j and j D 1; : : : ; n. The following fact also follows easily from the definition. Fact 2. Let X , Y be vector spaces, dim X n C 1, and M W X n ! Y a mapping such that M E n is n-linear for every .n C 1/-dimensional subspace E X. Then M 2 L. nXI Y /. Definition 3. Let X1 ; : : : ; Xn , and Y be normed linear spaces. We say that the mapping M 2 L.X1 ; : : : ; Xn I Y / is bounded if kM k D

sup

kM.x1 ; : : : ; xn /k < C1:

x1 2BX1;:::;xn 2BXn

It is easily checked that this defines a norm on the subspace of L.X1 ; : : : ; Xn I Y / consisting of bounded multilinear mappings. By L.X1 ; : : : ; Xn I Y /; kk , respectively L. nX I Y /; kk and Ls . nXI Y /; kk , we denote the normed linear spaces of all respective n-linear bounded mappings. For bounded n-linear forms, i.e. when the target space is the scalar field, we use a shortened notation L. nX/ D L. nX I K/.

Section 1. Multilinear mappings and polynomials

3

One of the simplest examples of an n-linear mapping belonging to L.X1 ; : : : ; Xn I Y / is given by the formula M.x1 ; : : : ; xn / D 1 .x1 /2 .x2 / n .xn /y, where j 2 Xj , j D 1; : : : ; n, y 2 Y . Note also that from (1) it follows that on finite-dimensional spaces all n-linear mappings are automatically bounded. Let M 2 L.X1 ; : : : ; Xn I Y /. Then by the homogeneity we have kM.x1 ; : : : ; xn /k kM kkx1 k kxn k

for xj 2 Xj , j D 1; : : : ; n.

(2)

Proposition 4. Let X1 ; : : : ; Xn , Y be normed linear spaces and M2 L.X1 ; : : : ; Xn I Y /. The following statements are equivalent: (i) M is bounded. (ii) M is Lipschitz on bounded sets. (iii) M is continuous. (iv) M is bounded on a neighbourhood of some point. Proof. We consider the maximum norm on the space X1 ˚ ˚ Xn . We write x D .xj /jnD1 for x 2 X1 ˚ ˚ Xn . (i))(ii) Let S X1 ˚ ˚ Xn be a given bounded set and let c > 0 be such that S .cBX1 / .cBXn /. Using (1), for any x; y 2 S we obtain X n M.y/ D M.xCy x/ D M.x/C M A .j /xj C.1 A .j //.yj xj / j D1 : A¤f1;:::;ng

Combining this with (2) we get M.x/k 2n kM k.2c/n

kM.y/

1

max kyj

1j n

xj k:

(3)

(ii))(iii))(iv) is clear, so it remains to show that (iv))(i). Using (1), for any x; h 2 X1 ˚ ˚ Xn we obtain X n M.h/ D M A .j /.xj C hj / C .1 A .j //. xj / j D1 Af1;:::;ng

D

X

. 1/n

jAj

M

A .j /.xj C hj / C .1

A .j //xj

n j D1

:

Af1;:::;ng

It follows that if M is bounded on a neighbourhood of x, then it is also bounded on a neighbourhood of zero, and hence by the homogeneity also on the unit ball. t u Theorem 5 (Uniform boundedness principle; [MO2]). Let X1 ; : : : ; Xn be Banach spaces and Y a normed linear space. Let fM˛ g˛2 L.X1 ; : : : ; Xn I Y / be a family such that sup˛2 kM˛ .x1 ; : : : ; xn /k < C1 for every .x1 ; : : : ; xn / 2 X1 Xn . Then sup˛2 kM˛ k < C1. Proof. We use induction on n. For n D 1 it is just the Banach-Steinhaus uniform boundedness principle. To prove the induction step from n to n C 1 suppose that fM˛ g˛2 L.X1 ; : : : ; Xn ; XnC1 I Y / satisfies the assumptions. By the inductive hypothesis for every ´ 2 XnC1 the family f.x1 ; : : : ; xn / 7! M˛ .x1 ; : : : ; xn ; ´/g˛2 L.X1 ; : : : ; Xn I Y / is uniformly bounded. So for every ´ 2 XnC1 there exists C´ > 0

4

Chapter 1. Fundamental properties of smoothness

such that kM˛ .x1 ; : : : ; xn ; ´/k S C´ kx1 k kxn k for every x1 2 X1 ; : : : ; xn 2 Xn , and ˛ 2 . Since XnC1 D 1 nD1 f´ 2 XnC1 I C´ ng, by the Baire category theorem there exist C > 0 and a set V XnC1 such that V has a non-empty interior and kM˛ .x1 ; : : : ; xn ; ´/k C kx1 k kxn k for every xj 2 Xj , j D 1; : : : ; n, every ˛ 2 , and every ´ 2 V . Because the mapping ´ 7! M˛ .x1 ; : : : ; xn ; ´/ is continuous, the estimate above holds for every ´ 2 V and hence for every ´ from some ball B.w; r/, where w 2 XnC1 and r > 0. Choose arbitrary xj 2 BXj for j D 1; : : : ; n C 1. We find y 2 XnC1 satisfying ky wk D r and such that xnC1 D .y w/, where D kxnC1 k=r. Because M˛ .x1 ; : : : ; xnC1 / D M˛ .x1 ; : : : ; xn ; y/ M˛ .x1 ; : : : ; xn ; w/, we conclude that 2C kM˛ .x1 ; : : : ; xnC1 /k for every ˛ 2 . r This finishes the proof. t u The following result from elementary linear algebra gives a solution to an interpolation problem. It is of fundamental importance in the theory of polynomials. Lemma 6. Let n 2 N0 and let tj 2 K, j D 0; : : : ; n, be distinct scalars. If Y is a vector space over K and ´0 ; : : : ; ´n 2 Y , then there are uniquely determined y0 ; : : : ; yn 2 Y such that ´j D tj0 y0 C C tjn yn , j D 0; : : : ; n. The vectors yk are given by the formula yk D ak0 ´0 C C ak n ´n , k D 0; : : : ; n, where akj are the elements of the inverse to the Vandermonde matrix

˙1

t0 t02 : : : t0n 1 t1 t12 : : : t1n : :: :

1 tn tn2 : : : tnn In particular, the coefficients akj are independent of ´j . We recall that a subset A of a topological space X is said to have the Baire property if A is a symmetric difference of an open set and a set of the first category. The family of all subsets of X with the Baire property is the smallest -algebra containing all open sets and all sets of the first category. In particular, every Borel set has the Baire property. The mapping f W X ! Y between topological spaces X and Y is said to be Baire measurable if f 1 .G/ X has the Baire property for every G Y open. Theorem 7. Let X1 ; : : : ; Xn be Banach spaces, Y a normed linear space, and let M 2 L.X1 ; : : : ; Xn I Y /. The following statements are equivalent: (i) M is continuous. (ii) M is separately continuous, i.e. x 7! M.x1 ; : : : ; xk 1 ; x; xkC1 ; : : : ; xn / 2 L.Xk I Y / for each x1 2 X1 ; : : : ; xn 2 Xn and each k 2 f1; : : : ; ng. (iii) M is Baire measurable. (iv) B M is Baire measurable for every 2 Y . (v) B M is separately Baire measurable for every 2 Y .

Section 1. Multilinear mappings and polynomials

5

In the proof we will use the following simple observation. Lemma 8. Let X be a topological vector space that is a Baire space, G X open, and A G residual in G. Then for each y 2 G and n 2 N there is an open neighbourhood V of y such that for each x 2 V there is h 2 X satisfying x C j h 2 A, j D 1; : : : ; n. Proof. Without loss of generality we may assume that y D 0. Let W be a balanced open neighbourhood of 0 such that W C W G and put V D n1 W . Fix x 2 V T and note that V j1 W for j D 1; : : : ; n and hence V jnD1 j1 .G x/. Each set 1 x/ is residual in j1 .G x/ and hence each V \ j1 .A x/, j D 1; : : : ; n, is j .A T residual in V . Consequently, V \ jnD1 j1 .A x/ is residual in V and in particular it is non-empty and so it suffices to take any h from this set. t u Proof of Theorem 7. (i))(iii))(iv) is obvious. To prove (iv))(i) denote X D X1 Xn . Fix S an arbitrary 2 Y . Let Ak D fx 2 X I j B M.x/j < kg, k 2 N. Then X D k2N Ak and hence there is m 2 N such that Am is of the second category. Since by our assumption Am has the Baire property, there is a non-empty open G X such that Am \ G is residual in G. By Lemma 8 there is a non-empty open V X such that for each x 2 V there is h 2 X satisfying x C j h 2 Am , j D 1; : : : ; n C 1. Let akj be the coefficients from Lemma 6 for tj D j C 1, j D 0; : : : ; n. For any x 2 V and h 2 X satisfying x C j h 2 Am , j D 1; : : : ; n C 1 we have n X X n M.x C j h/ D M.x/ C jk M .1 C .l//xl C C .l/hl lD1 kD1 C f1;:::;ng jC jDk

for j D 1; : : : ; n C 1. Lemma 6 then implies M.x/ D

n X

a0j M x C .j C 1/h

j D0

Pn

and hence j B M.x/j m j D0 ja0j j. This means that B M.V / is bounded and so B M.BX / is bounded by Proposition 4. Therefore M.BX / is weakly bounded and hence it is bounded by the Banach-Steinhaus uniform boundedness principle. Thus M is continuous by Proposition 4. (i))(ii))(v) is obvious and (v))(ii) follows from the already proved (iv))(i). (ii))(i) We use induction on n. For n D 1 there is nothing to prove. Let n > 1 and assume the statement holds for .n 1/-linear mappings. For each x1 2 X1 ; : : : ; xn 2 Xn we put Mx2 ;:::;xn .x1 / D M.x1 ; : : : ; xn / and Mx1 .x2 ; : : : ; xn / D M.x1 ; : : : ; xn /. Then Mx2 ;:::;xn 2 L.X1 I Y / by the assumption, while Mx1 2 L.X2 ; : : : ; Xn I Y / by the inductive hypothesis. Now for each x1 2 X1 we have sup kMx2 ;:::;xn .x1 /k D xj 2BXj 2j n

sup kM.x1 ; : : : ; xn /k D kMx1 k < C1; xj 2BXj 2j n

6

Chapter 1. Fundamental properties of smoothness

sup kMx2 ;:::;xn k < C1. Thus

so by the Uniform boundedness principle

xj 2BXj 2j n

sup kM.x1 ; : : : ; xn /k D xj 2BXj 1j n

sup

xj 2BXj 2j n

D

sup kM.x1 ; : : : ; xn /k

x1 2BX1

sup kMx2 ;:::;xn k < C1; xj 2BXj 2j n

which proves that M is bounded and we can use Proposition 4.

t u

The following fact is easy but important. Fact 9. Let X1 ; : : : ;Xn ;Y be normed linear spaces. Then L X1I L.X2 ; : : : ;Xn IY / is canonically (linearly) isometric to the space L.X1 ; : : : ; Xn I Y / in the following way: We identify the mapping T 2 L X1 I L.X2 ; : : : ; Xn I Y / with S 2 L.X1 ; : : : ; Xn I Y / if and only if T .x1 /.x2 ; : : : ; xn / D S.x1 ; : : : ; xn / for all xj 2 Xj , j D 1; : : : ; n. Let X, Y be vector spaces and M 2 L. nXI Y /. We define the symmetrisation of M by M s .x1 ; : : : ; xn / D

1 X M.x.1/ ; : : : ; x.n/ /; nŠ 2Sn

where Sn is the set of all permutations of f1; : : : ; ng. Then M s 2 Ls . nX I Y /. Furthermore, if X and Y are normed linear spaces and M is bounded, then so is M s and obviously kM s k kM k. If M is symmetric, then clearly M s D M . Proposition 10. Let X1 ; : : : ; Xn , and X be normed linear spaces, and let Y be a Banach space. Then L.X1 ; : : : ; Xn I Y / is a Banach space. Moreover, Ls . nXI Y / is a closed and 1-complemented subspace of L. nX I Y /. Proof. Suppose that fMk g1 L.X1 ; : : : ; Xn I Y / is a Cauchy sequence. Then kD1 1 fMk .x1 ; : : : ; xn /gkD1 is a Cauchy sequence in Y for every x1 2 X1 ; : : : ; xn 2 Xn and so we may define the mapping M.x1 ; : : : ; xn / D limk!1 Mk .x1 ; : : : ; xn /. Obviously M is n-linear. Since the sequence fkMk kg is bounded, we infer that the mapping M is bounded. Now choose any " > 0. There is k0 2 N such that sup

kMk .x1 ; : : : ; xn /

Mm .x1 ; : : : ; xn /k "

x1 2BX1 ;:::;xn 2BXn

for all k; m k0 . By passing to a limit as m ! 1 we obtain sup

kMk .x1 ; : : : ; xn /

M.x1 ; : : : ; xn /k "

x1 2BX1 ;:::;xn 2BXn

for all k k0 , i.e. kMk M k ". Thus limk!1 Mk D M in L.X1 ; : : : ; Xn I Y /. This proves that L.X1 ; : : : ; Xn I Y / is indeed a Banach space. The symmetrisation gives rise to a canonical projection Q W L. nX I Y / ! Ls . nXI Y /, Q.M / D M s . Clearly, kQk D 1. t u

Section 1. Multilinear mappings and polynomials

7

Proposition 11 (Polarisation formula). Let X , Y be vector spaces and M 2 L. nX I Y /. Then ! n n X X X 1 s M .x1 ; : : : ; xn / D n "1 "n M a C "j xj ; : : : ; a C "j xj 2 nŠ j D1

"j D˙1

j D1

for every a; x1 ; : : : ; xn 2 X . In particular, if M is symmetric, then it is uniquely determined by its values M.x; : : : ; x/, x 2 X, along the diagonal. Proof. For convenience we put x0 D a and "0 D 1. Using (1) we obtain ! n n X X X 1 "1 "n M "j xj ; : : : ; "j xj 2n nŠ j D0

"j D˙1 j D1;:::;n

D

1 n 2 nŠ

1 D n 2 nŠ

X

j D0

X

"1 "n

"j D˙1 j D1;:::;n

"j1 "jn M.xj1 ; : : : ; xjn /

j1 ;:::;jn 2f0;:::;ng

! X

X

j1 ;:::;jn 2f0;:::;ng

"j D˙1 j D1;:::;n

"1 "n "j1 "jn M.xj1 ; : : : ; xjn /:

If there is k 2 f1; : : : ; ng n fj1 ; : : : ; jn g, then X "1 "n "j1 "jn "j D˙1 j D1;:::;n

D

X

"j1 "jn

"j D˙1 j ¤k

Y

"j C . 1/

X

"j1 "jn

"j D˙1 j ¤k

j ¤k

Y

(4) "j D 0:

j ¤k

It follows that 1 n 2 nŠ

! X

X

j1 ;:::;jn 2f0;:::;ng

"j D˙1 j D1;:::;n

"1 "n "j1 "jn M.xj1 ; : : : ; xjn /

1 X D n 2 nŠ

2Sn

! X

"21 "2n M.x.1/ ; : : : ; x.n/ /

"j D˙1

1 X M.x.1/ ; : : : ; x.n/ / D M s .x1 ; : : : ; xn /; D nŠ 2Sn

where Sn is the set of all permutations of f1; : : : ; ng. This finishes the proof.

t u

Definition 12. Let X , Y be vector spaces and n 2 N. A mapping P W X ! Y is called an n-homogeneous polynomial if there is an n-linear mapping M 2 L. nX I Y / such . For convenience we also that P .x/ D M.x; : : : ; x/. We use the notation P D M

8

Chapter 1. Fundamental properties of smoothness

define 0-homogeneous polynomials as constant mappings from X to Y . We denote by P . nXI Y /, n 2 N0 , the vector space of all n-homogeneous polynomials from X into Y . Suppose X , Y are normed linear spaces, n 2 N0 . We say that P 2 P . nX I Y / is a bounded polynomial, if kP k D sup kP .x/k < C1: x2BX

We denote by P . nXI Y /; kk the normed linear space of all n-homogeneous bounded polynomials from X into Y . When the target space is the scalar field, we use a shortened notation P . nX/ D P . nXI K/.

Let P 2 P . nXI Y /. From the definition it immediately follows that P .cx/ D c n P .x/ for every x 2 X and c 2 K. This implies that for P 2 P . nX I Y / kP .x/k kP kkxkn for every x 2 X . This in turn gives P .x/ D o.kxkp /; x ! 0 for every p 2 R, p < n. For a given n-homogeneous polynomial P the n-linear mapping M that gives rise to P is not determined uniquely. In particular, the symmetrised n-linear mapping leads D M s . However, the to the same polynomial: for every M 2 L. nX I Y / we have M following fundamental result holds.

b

Proposition 13 (Polarisation formula; [BH], [MO1]). Let X, Y be vector spaces and n 2 N. For every P 2 P . nX I Y / there exists a unique symmetric n-linear mapping } 2 Ls . nXI Y / such that P .x/ D P }.x; : : : ; x/. It satisfies the formula P ! n X X 1 }.x1 ; : : : ; xn / D "j xj ; (5) P "1 "n P a C 2n nŠ j D1

"j D˙1

where a 2 X can be chosen arbitrarily. Moreover, if X, Y are normed linear spaces } is also bounded and we have and P is bounded, then P n }k n kP k: kP k kP nŠ

(6)

On the other hand, for every m > n and a; x1 ; : : : ; xm 2 X the following holds: ! m X X "1 "m P a C "j xj D 0: (7) "j D˙1

j D1

Section 1. Multilinear mappings and polynomials

9

b

. Put P } D M s . Obviously, P D M s Proof. Let M 2 L. nX I Y / be such that P D M and so using Proposition 11 we obtain }.x1 ; : : : ; xn / D M s .x1 ; : : : ; xn / P ! n n X X X 1 s D n "1 "n M a C "j xj ; : : : ; a C "j xj 2 nŠ j D1 j D1 "j D˙1 ! n X X 1 "1 "n P a C "j xj : D n 2 nŠ "j D˙1

j D1

The uniqueness follows from the fact that a symmetric n-linear mapping is uniquely determined by its values along the diagonal (Proposition 11). The estimate (6) follows readily from the formula (5). To prove (7) it is enough to follow the proof of Proposition 11 and notice that for each summand there is always k 2 f1; : : : ; mg n fj1 ; : : : ; jn g and so (4) holds. t u Consequently, a homogeneous polynomial is continuous if and only if it is bounded. As another immediate corollary we obtain the next proposition. Proposition 14. Let X , Y be normed linear spaces and n 2 N. Then P . nXI Y / is canonically isomorphic to Ls . nX I Y /. In particular, if Y is a Banach space, then so is P . nXI Y /. The following example shows that in general the right-hand side estimate in (6) cannot be improved. Example 15. Consider X D `1 with the canonical basis f.ej I fj /gj1D1 . Choose n 2 N and define P 2 P . nX/ by P .x/ D f1 .x/ fn .x/. It is indeed an n-homogene , where M.x1 ; : : : ; xn / D f1 .x1 / fn .xn /. Then ous polynomial, since P D M 1 } } kP k P .e1 ; : : : ; en / D nŠ , while jP .x/j attains its maximum n1n over BX at the point x D . n1 ; : : : ; n1 ; 0; : : : /. However, interestingly if X is an inner product space, then the spaces P . nXI Y / and Ls . nXI Y / are isometric: Theorem 16 ([Kel], [Bana]). Let X be an inner product space, Y a normed linear k. space, n 2 N, and M 2 Ls . nXI Y /. Then kM k D kM Proof. (Stanisław Łojasiewicz) By considering the real versions of the spaces we may without loss of generality assume that K D R. First suppose that dim X < 1. Then there are y1 ; : : : ; yn 2 BX such that kM k D kM.y1 ; : : : ; yn /k. Pick a 2 X such ˚that ha; yj i ¤ 0 for all j 2 f1; : : : ; ng. Clearly there is ı > 0 such that the set K D .x1 ; : : : ; xn / 2 .BX /n I kM.x1 ; : : : ; xn /k D kM k; ha; xj i ı; j D 1; : : : ; n is P non-empty. Put '.x1 ; : : : ; xn / D jnD1 ha; xj i. As K is compact, the function ' attains its maximum on K, say at .´1 ; : : : ; ´n / 2 K. We claim that ´1 D ´2 D D ´n , .´1 /k kM k and this finishes which means that kM k D kM.´1 ; : : : ; ´n /k D kM the proof in the finite-dimensional case.

10

Chapter 1. Fundamental properties of smoothness

Assuming the contrary, by the symmetry of M we may without loss of generality assume that ´1 ¤ ´2 . Since ha; ´1 i ı and ha; ´2 i ı, it follows that also ´1 ¤ ´2 . C´2 Put u D k´´11 C´ and v D k´´11 ´´22 k . We show that in this case .u; u; ´3 ; : : : ; ´n / 2 K. 2k Clearly, ha; ui ı, as k´1 C ´2 k 2. Let us define the mapping T W X 2 ! Y by T .x; y/ D M.x; y; ´3 ; : : : ; ´n /. Then T is bilinear and symmetric and therefore T .´1 ; ´2 / D 14 T .´1 C´2 ; ´1 C´2 / T .´1 ´2 ; ´1 ´2 / . Now if kT .u; u/k < kM k, then 1 kM k D kT .´1 ; ´2 /k k´1 C ´2 k2 kT .u; u/k C k´1 ´2 k2 kT .v; v/k 4 kM k kM k k´1 C ´2 k2 C k´1 ´2 k2 D k´1 k2 C k´2 k2 kM k; < 4 2 a contradiction. Thus .u; u; ´3 ; : : : ; ´n / 2 K. But since k´1 ´2 k > 0, by the rotundity k´1 C ´2 k < 2, and so n

'.u; u; ´3 ; : : : ; ´n / D

X 2 ha; ´1 i C ha; ´2 i C ha; ´j i > '.´1 ; : : : ; ´n /; k´1 C ´2 k j D3

which contradicts the fact that ' attains its maximum on K at .´1 ; : : : ; ´n /. Finally, we deal with the infinite-dimensional case. Fix x1 ; : : : ; xn 2 BX and denote Z k kM k by Z D spanfx1 ; : : : ; xn g. Then kM.x1 ; : : : ; xn /k kM Z n k kM the first part of the proof. This means that kM k kM k, while the reverse inequality is obvious. t u To shorten the notation we will use the following convention: for k 2 N0 we denote k

˜

x D x; : : : ; x : k times

The following result holds also for complex spaces, but for those we show much stronger (and optimal) estimate in Corollary 2.70. Corollary 17 ([Harr2]). Let X, Y be real normed linear spaces and M 2 Ls . nX I Y /. Then s

n

nn n

M. 1x1 ; : : : ; kxk / k kM n1 n1 nnkk for all x1 ; : : : ; xk 2 BX , n1 ; : : : ; nk 2 N with n1 C C nk D n, 1 k n. Proof. Let 1 k n, x1 ; : : : ; xk 2 BX , and n1 ; : : : ; nk 2 N with n1 C C nk D n be given. Fix for a while r D .r1 ; : : : ; rk / 2 .RC /k . Define T 2 L.`k2 I X/ by the P formula T .t1 ; : : : ; tk / D jkD1 tj rj xj . Note that the Cauchy inequality implies

T .t1 ; : : : ; tk / Pk jtj jrj kxj k k.t1 ; : : : ; tk /k2 krk2 and hence kT k krk2 . j D1 Further, let Q.y1 ; : : : ; yn / D M T .y1 /; : : : ; T .yn / for y1 ; : : : ; yn 2 `k2 . Clearly,

Section 1. Multilinear mappings and polynomials

11

D M B T , and kQk kM kkrkn . Denote by e1 ; : : : ; ek the Q 2 Ls . n`k2 I Y /, Q 2 canonical basis in `k2 . Then by Theorem 16

r1n1 rknk M.n1x1 ; : : : ; nkxk / D M n1.r1 x1 /; : : : ; nk.rk xk /

D Q.n1e1 ; : : : ; nkek / kM kkrkn2 : kQk D kQk Since the function r 7!

krkn 2

n

attains its minimum on .RC /k for rj D

n

r1 1 rk k

q

j D 1; : : : ; k, the estimate follows.

nj n

,

t u

Fact 18. Let X, Y be vector spaces, n 2 N0 , P 2 subspace. Then P Z 2 P . nZI Y /.

P . nXI Y /,

and let Z X be a

1

}Z n . Proof. For n D 0 it is obvious. Otherwise clearly P Z D P

t u

Fact 19. Let X , Y be vector spaces, n 2 N0 , dim X n C 1, and let P W X ! Y be a mapping such that P E 2 P . nEI Y / for every subspace E X of dimension n C 1. Then P 2 P . nXI Y /. Proof. For n D 0 it is obvious. Otherwise define the mapping M W X n ! Y formally by formula (5) with a D 0. By Proposition 13, M E n 2 L. nEI Y / for every subspace , as E X of dimension n C 1. Thus M 2 L. nXI Y / by Fact 2. Clearly, P D M E for every subspace E X of dimension n C 1. P E D M t u n Definition Pn 20. Let n 2 N. Given a multi-index ˛ 2 N0 we denote its order by j˛j D j D1 ˛j . Further, we denote the set of multi-indices of order d 2 N0 by ˚ I.n; d / D ˛ 2 f0; : : : ; d gn I j˛j D d :

We extend the definition also to the case when n D 1, setting 1 X I.1; d / D ˛ 2 f0; : : : ; d gN I j˛j D ˛j D d : j D1 nCd 1 n 1

For n 2 N we have jI.n; d /j D (indeed, from the combinatorial point of view it represents the number of distributions of d identical balls into n distinct boxes). A given .kj /jdD1 2 f1; : : : ; ngd determines a unique ˛ 2 I.n; d / by the relation ˛ D jfj I kj D 1gj; jfj I kj D 2gj; : : : ; jfj I kj D ngj : (8) Conversely, a given ˛ 2 I.n; d / determines a unique k.˛/ D k1 .˛/; : : : ; kd .˛/ , k1 .˛/ kd .˛/, such that (8) holds. Given x D .x1 ; : : : ; xn / 2 Kn and ˛ D .˛1 ; : : : ; ˛n / 2 I.n; d / we use the standard multi-index notation x˛ D

n Y lD1

xl˛l D

d Y

xkj .˛/ :

j D1

The case n D 1 is similar and corresponds to multi-indices whose domain is N.

12

Chapter 1. Fundamental properties of smoothness

Note that x 7! x ˛ 2 P . d Kn / for any ˛ 2 I.n; d /. Indeed, denote by l 2 .Kn / the coordinate functional l .x/ D xl . Let M 2 L. d Kn I K/ be given by the formula Q .x/. M.y1 ; : : : ; yd / D jdD1 kj .˛/ .yj /. Then we have x ˛ D M Given ˛ D .˛1 ; : : : ; ˛n / 2 I.n; d /, we denote the corresponding multinomial coefficient by d d dŠ dŠ D D : D ˛ ˛1 ; : : : ; ˛n ˛Š ˛1 Š ˛n Š Lemma 21. Let X, Y be vector spaces, n 2 N, P 2 P . nXI Y /, and x; y 2 X . Then n X n }j n j P .x C y/ D P . x; y/ and j j D0

P .x/

P .y/ D

n X1 j D0

n }j n j P y; .x j

y/ :

}. The Proof. To obtain the first formula, it suffices to apply (1) and the symmetry of P second formula follows easily from the first one. t u Proposition 22 (Multinomial formula). Let X and Y be vector spaces, d 2 N, let P 2 P . dXI Y /, and x1 ; : : : ; xn 2 X . Then X d }.˛1x1 ; : : : ; ˛nxn /: P .x1 C C xn / D P ˛ ˛2I.n;d /

Proof. Using (1) and then collecting the terms according to the value of ˛ using the } we obtain symmetry of P ! n X X X d } }.˛1x1 ; : : : ; ˛nxn /: P xj D P .xk1 ; : : : ; xkd / D P ˛ j D1

1kj n j D1;:::;d

˛2I.n;d /

t u The next proposition asserts that the abstract definition of homogeneous polynomials coincides on Kn with the classical definition that uses coordinates. Note that in this case all homogeneous polynomials are automatically bounded. Proposition 23. Let n; d 2 N and let Y be a vector space over K. A mapping P W Kn ! Y is a d -homogeneous polynomial if and only if there exists a collection P fy˛ g˛2I.n;d / Y such that P .x/ D ˛2I.n;d / x ˛ y˛ . Moreover, each y˛ is uniquely determined by d } ˛1 y˛ D P . e1 ; : : : ; ˛n en /; ˛ where fej gjnD1 is the canonical basis of Kn .

Section 1. Multilinear mappings and polynomials

13

Proof. Let P 2 P . d Kn I Y /. By Proposition 22 we have ! n X X ˛ d } ˛1 P xj ej D x P . e1 ; : : : ; ˛nen /: ˛ j D1

˛2I.n;d /

The converse is clear from the fact that x 7! x ˛ 2 P . d Kn /. P To prove the uniqueness we show a bit more: if Q.x1 ; : : : ; xn / D ˛2A x ˛ ´˛ D 0 for each x 2 Kn , then ´˛ D 0 for each ˛ 2 A, where A D f˛ 2 N0n I j˛j d g. First notice that it suffices to show this only for Y D K, the general case follows by using composition with linear functionals from Y . We use induction on n. For n D 1 it is a standard fact that can be proved for example by factorisation. To prove the induction step from n 1 to n put Ak D f˛ 2 AI ˛1 D kg, k D 0; : : : ; d . Then Pd P ˛2 ˛n k Q.x1 ; : : : ; xn / D ˛ D 0 for each x1 ; : : : ; xn 2 K. kD0 x1 ˛2Ak x2 xn ´P From the first step of the induction it follows that ˛2Ak x2˛2 xn˛n ´˛ D 0 for each x2 ; : : : ; xn 2 K and k 2 f0; : : : ; d g. Hence by the inductive hypothesis ´˛ D 0 for S every ˛ 2 dkD0 Ak D A. t u P In case that Y D K this reduces to the familiar formula P .x/ D ˛2I.n;d / a˛ x ˛ , where the coefficients a˛ 2 K. While working with polynomials P 2 P . d Kn I Y / we are often going to use freely the classical notation using the coordinates and multinomial expressions, e.g. we will sometimes write P .x1 ; : : : ; xn / instead of the formally correct P .x1 ; : : : ; xn / , xj 2 K, j D 1; : : : ; n. Proposition 24. Let X be a normed linear space with a Schauder basis fej gj1D1 , Y a vector space, d 2 N, and P 2 P . dXI Y /. Denote X0 D spanfej gj1D1 . Then there is a unique collection of vectors fy˛ g˛2I.1;d / Y such that the formula X P .x/ D x ˛ y˛ (9) ˛2I.1;d /

holds for every x D

P

xj ej 2 X0 . The coefficients y˛ are given by d } ˛1 ˛2 y˛ D P . e1 ; e2 ; : : : /: ˛

Conversely, any collection fy˛ g˛2I.1;d / Y uniquely determines a polynomial P 2 P . dX0 I Y / by the formula (9). Proof. By Fact 18 the restrictions of P to spanfej gjnD1 are d -homogeneous polynomials, so we can use Proposition 23. To prove the converse statement define P W X0 ! Y by (9). Each restriction of P to a finite-dimensional subspace of X0 is a d -homogeneous polynomial, since it is a finite sum of d -homogeneous polynomials x 7! x ˛ y˛ . Thus P 2 P . dX0 I Y / by Fact 19. t u

14

Chapter 1. Fundamental properties of smoothness

Even if X and Y are Banach spaces (or Y D K) and P 2 P . dX I Y /, the sum in (9) may not be convergent (absolutely) for infinitely supported vectors. However, the following statement holds. Proposition 25. Let X be a normed linear space with a Schauder basis fej gj1D1 , Y a Banach space, and d 2 N. If there is a collection of vectors fy˛ g˛2I.1;d / Y such that the expression X x ˛ y˛ ˛2I.1;d /

is uniformly bounded for all finitely supported vectors x 2 BX , then there is a unique polynomial P 2 P . dXI Y / such that the formula (9) holds for every finitely supported vector x 2 X. Proof. By Proposition 24 the formula (9) defines a polynomial P 2 P . dX0 I Y / on X0 D spanfej gj1D1 . Moreover, from our hypothesis it follows that P is bounded. } is bounded and hence Lipschitz on bounded sets by Proposition 4. Consequently, P } Therefore P can be uniquely extended to X n D .X0 /n D X0n . Denote this extension by M . Using the continuity of M , the d -linearity of M passes from X0n to X n . Since 2 P . dX I Y / is a desired extension M is obviously also bounded, we conclude that M of P . The uniqueness is clear from the density of X0 in X. t u Lemma 26. Let X, Y1 ; : : : ; Yk , Z be vector spaces, let M 2 L.Y1 ; : : : ; Yk I Z/, and Pj 2 P . njXI Yj /, nj 2 N0 , for j D 1; : : : ; k. Then the composite mapping x 7! M P1 .x/; : : : ; Pk .x/ belongs to P . n1 CCnkX I Z/. Proof. It is clear that M P}1 . /; : : : ; P}k . / 2 L. n1 CCnkXI Z/. t u As an immediate corollary we obtain the following two facts. Fact 27. Let X, Y be vector spaces over K, m; n 2 N0 , and let P 2 P . mX I K/ and Q 2 P . nXI Y /. Then P Q 2 P . mCnXI Y /. Proof. The mapping .t; y/ 7! t y is a bilinear mapping from K Y to Y .

t u

Fact 28. Let X , Y , Z be vector spaces, m; n 2 N0 , and let P 2 P . mY I Z/ and Q 2 P . nXI Y /. Then P B Q 2 P . mnX I Z/. } Q.x/; : : : ; Q.x/ . Proof. We have P B Q.x/ D P t u Theorem 29. Let X be a Banach space, Y a normed linear space, and n 2 N0 . If fPk g1 P . nXI Y / is such that limk!1 Pk .x/ D P .x/ exists for every x 2 X, kD1 then P 2 P . nXI Y /. If fP g 2 is a net in P . nX I Y / such that lim P .x/ D P .x/ exists for every x 2 X, then P 2 P . nX I Y /. If moreover fP .x/g 2 is bounded for every x 2 X, then P 2 P . nXI Y /. Proof. The statement is obvious for n D 0 so let n > 0. Then lim P} .x1 ; : : : ; xn / exists for every x1 ; : : : ; xn 2 X by Proposition 13. Denote this limit by M.x1 ; : : : ; xn /

Section 1. Multilinear mappings and polynomials

15

.x/ for all x 2 X and notice that M 2 Ls . nX I Y /. It follows that lim P .x/ D M n and hence P 2 P . X I Y /. Note that in the case of the sequence, fPk .x/g1 is kD1 automatically bounded for each x 2 X . Since again by the Polarisation formula } .x1 ; : : : ; xn /g 2 is bounded for every x1 ; : : : ; xn 2 X, by Theorem 5 there is fP } k C for every 2 . Thus kP .x/k D lim kP .x/k C for C 2 R such that kP x 2 BX , which shows that P 2 P . nXI Y /. t u Definition 30. Let X, Y be vector spaces and n 2 N0 . A mapping P W X ! Y is k called a polynomial Pn of degree at most n if there are Pk 2 P . XI Y /, k D 0; : : : ; n, such that P D kD0 Pk . If Pn ¤ 0, we say that P has degree n and we use the n .X I Y / the space of all polynomials of degree notation deg P D n. We denote by P S n at most n. We denote by P .X I Y / D 1 nD0 P .XI Y / the space of all polynomials. The fact that deg P is well-defined will be apparent from Corollary 32 below. The interpolation Lemma 6 has the following consequence: Corollary 31. For every n 2 N0 and every ı > 0 there exist numbers akj 2 R, k; j D 0; : : : ; n, such that whenever X , Y are vector spaces over K, P 2 P n .XI Y /, P P and Pk 2 P . kXI Y / satisfy P D nkD0 Pk , then Pk .x/ D jnD0 akj P .1 jn ı/x for every x 2 X . P k Proof. It suffices to notice that P .1 jn ı/x D nkD0 1 jn ı Pk .x/ and apply Lemma 6 with tj D 1 jn ı. t u 32. Let X, Y be vector spaces, n 2 N0 , and let P 2 P n .X I Y / be such Corollary P that P D nkD0 Pk , where Pk 2 P . kXI Y /. Then the homogeneous summands Pk of P are uniquely determined. The next lemma shows that the Polarisation formula applied to a non-homogeneous polynomial extracts its “leading term”. n Lemma Pn33. Let X , Y be kvector spaces, n 2 N, and let P 2 P .XI Y / be such that P D kD0 Pk , Pk 2 P . XI Y /. Then ! n X X 1 } "1 "n P a C "j xj Pn .x1 ; : : : ; xn / D n 2 nŠ j D1

"j D˙1

for every a; x1 ; : : : ; xn 2 X. Proof. By (7) in Proposition 13 X "j D˙1

"1 "n Pk a C

n X

! "j xj

D0

j D1

whenever 0 < k < n, and it is of course true also for k D 0. Hence the result follows from the Polarisation formula (5). t u Lemma 34. Let X, Y be vector spaces, n 2 N0 , P 2 P n .XI Y /, and ´ 2 X. Then R.x/ D P .x C ´/ 2 P n .XI Y /.

16

Chapter 1. Fundamental properties of smoothness

P Proof. Assume that P D nkD0 Pk , Pk 2 P . kXI Y /. The statement follows from Lemma 21 and the obvious fact that x 7! P}k .jx; k j´/ is a j -homogeneous polynomial. t u The following fact immediately follows from Fact 18. Fact 35. Let X, Y be vector spaces, n 2 N0 , P 2 P n .XI Y /, and let Z X be a subspace. Then P Z 2 P n .ZI Y /. Fact 36. Let X, Y be vector spaces, n 2 N0 , dim X n C 1, and let P W X ! Y be a mapping such that for every subspace E X of dimension n C 1 the mapping P E is a polynomial of degree at most n. Then P 2 P n .XI Y /. Proof. For every subspace E X of dimension n C 1 there are PkE 2 P . kEI Y /, Pn E 0 k n, such that P E D kD0 Pk . By Fact 35 and Corollary 32 for any subspaces E X, F X of dimension n C 1 we have PkE E \F D PkF E \F , 0 k n. So we may define Pk .x/ D PkE .x/, 0 k n, where E X is any k subspace Pn of dimension n C 1 containing x. Then Pk 2 P . XI Y / by Fact 19. Clearly, P D kD0 Pk . t u Later we show that it is actually enough to consider only restrictions to one-dimensional affine subspaces of X (Corollary 55, Theorem 2.49). The polynomials behave as expected with regards to various operations: Fact 37. Let X, Y be vector spaces over K, m; n 2 N0 , and let P 2 P m .XI K/ and Q 2 P n .X I Y /. Then P Q 2 P mCn .XI Y /. Fact 38. Let X , Y , Z be vector spaces, m; n 2 N0 , and let P 2 P m .Y I Z/ and Q 2 P n .X I Y /. Then P B Q 2 P mn .X I Z/. The first fact follows from Fact 27 and the second one is proved using the Multinomial formula (Proposition 22) and Lemma 26. Fact 39. Let X , Y be normed linear spaces and let P; Q 2 P .X I Y / be such that P and Q are equal on some ball. Then P D Q. Proof. Clearly it suffices to prove that if a polynomial P 2 P .X I Y / vanishes on some ball B.a; r/, then P D 0. Assume that P ¤ 0. Let n D deg P and denote the n-homogeneous summand of P by Pn . Lemma 33 implies that Pn .x/ D 0 for x 2 B.0; nr /, and so by homogeneity Pn D 0, a contradiction. t u Proposition 40. Let X, Y be normed linear spaces, n 2 N0 , and P 2 P n .XI Y /, Pn P D kD0 Pk , where Pk 2 P . kXI Y /. The following statements are equivalent: (i) Pk , k D 0; : : : ; n, are Lipschitz on bounded sets. (ii) P is Lipschitz on bounded sets. (iii) P is continuous. (iv) P is bounded on a neighbourhood of some point.

Section 1. Multilinear mappings and polynomials

17

Proof. (i))(ii))(iii))(iv) are immediate, so it remains to prove (iv))(i). Suppose that P is bounded on B.a; 2r/ for some a 2 X , r > 0. Then by Corollary 31 (letting r ı D rCkak ) each Pk , k D 0; : : : ; n, is bounded on B.a; r/. Consequently, by the Polarisation formula (Proposition 13) each P}k is bounded on B.0; r /k and thus is k

Lipschitz on bounded sets by Proposition 4. Since the mapping x 7! .kx/ is Lipschitz, the polynomials Pk are Lipschitz on bounded sets. t u Definition 41. Let X and Y be normed linear spaces over K and n 2 N0 . Set kP k D supx2BX kP .x/k whenever P 2 P n .X I Y / is continuous. We denote by P n .XI Y /; kk (resp. P .XI Y /; kk ) the normed linear space of all continuous polynomials of degree at most n (resp. all continuous polynomials) from X to Y . When the target space is the scalar field, we use a shortened notation P n .X/ D P n .X I K/, resp. P .X / D P .XI K/. The following fact follows at once from Corollary 31. Fact 42. Let X, Y be normed linear spaces and n; j 2 N0 , j P n. The mapping Hjn W P n .XI Y / ! P . jX I Y / given by Hjn .P / D Pj for P D nkD0 Pk , where Pk 2 P . kXI Y /, is a bounded linear projection. The norm of Hjn depends on n and j , but not on X or Y . As an immediate consequence we obtain the next result. Proposition 43. Let X be a normed linear space, Y a Banach space, and n 2 N0 . Then the space P n .XI Y / is a Banach space. Moreover, the canonical decomposition P 7! H0n .P /; : : : ; Hnn .P / for P 2 P n .XI Y / is a canonical isomorphism of Banach spaces P n .XI Y / and P . 0XI Y / ˚ P . 1X I Y / ˚ ˚ P . nXI Y /. By combining Corollary 31, the Polarisation formula (Proposition 13), and condition (v) in Theorem 7 we obtain the following corollary: Corollary 44. Let X be a Banach space, Y a normed linear space, n 2 N0 , and P 2 P n .XI Y /. If B P is Baire measurable for every 2 Y , then P is continuous. Theorem 45 (Uniform boundedness principle; [MO2]). Let X be a Banach space, Y a normed linear space, and n 2 N0 . Let fP˛ g˛2 P n .XI Y / be a family such that sup˛2 kP˛ .x/k < C1 for every x 2 X . Then sup˛2 kP˛ k < C1. Proof. Corollary 31 entails that for a fixed j n, sup˛2 kHjn .P˛ /.x/k < C1 for every x 2 X. The result now follows from Proposition 13 and Theorem 5. t u Theorem 46. Let X be a Banach space, Y a normed linear space, and n 2 N0 . If fPk g1 P n .X I Y / is such that lim Pk .x/ D P .x/ exists for every x 2 X, then kD1 k!1 n P 2 P .XI Y /. Proof. Fix j 2 f0; : : : ; ng and denote Qk D Hjn .Pk /. Then limk!1 Qk .x/ exists for every x 2 X by Corollary 31. By Theorem 29, limk!1 Qk 2 P . jX I Y /. Therefore P 2 P n .X I Y /. t u

18

Chapter 1. Fundamental properties of smoothness

The following lemma is useful for estimating the norm of a homogeneous polynomial using its values on an arbitrary ball. Lemma 47. linear spaces, n 2 N, let P 2 P n .X I Y / be such PnLet X , Y be normed k that P D kD0 Pk , Pk 2 P . XI Y /, and let a 2 X. Then kPn .x/k

nn nŠ

sup kP .a C tx/k t 2Œ 1;1

for every x 2 X. In particular, for any r > 0 sup kPn .x/k x2B.0;r/

nn nŠ

sup kP .x/k: x2B.a;r/

Proof. By Lemma 33,

! n

x n X X n x

"j kPn .x/k D nn Pn

n

P a C

n 2 nŠ n j D1

"j D˙1

nn nŠ

sup kP .a C tx/k:

t u

t 2Œ 1;1

Fact 48. Let X , Y be normed linear spaces, n 2 N0 , and P 2 P n .XI Y /. Suppose that P .x/ D o.kxkn /; x ! 0. Then P D 0. P Proof. Assume that P D jnDk Pj , Pj 2 P . jX I Y /, Pk ¤ 0. Then there is y 2 X such that Pk .y/ ¤ 0. Hence we have Pk .y/ C tPkC1 .y/ C C t n P .ty/ D lim t !0C t !0C ktykn t n k kykn

0 D lim

k P .y/ n

; t u

a contradiction.

Fact 49. Let X, Y be normed linear spaces, n 2 N0 , and let P W X ! Y . Then P 2 P n .XI Y / if and only if B P is a polynomial of degree at most n for every 2 ˚ , where ˚ Y separates the points of Y . If moreover B P is continuous for every 2 Y , then so is P . Proof. ) This is straightforward. ( We use induction on n. The case n D 0 follows easily from the fact that ˚ separates the points of Y . Now we prove the induction step from n 1 to n. By Lemma 33 for every 2 ˚ there exists an n-linear form M 2 L. nXI K/ such that ! n X X 1 M .x1 ; : : : ; xn / D n "j xj "1 "n . B P / 2 nŠ "j D˙1

X 1 "1 "n P D n 2 nŠ "j D˙1

j D1

n X j D1

! "j xj

:

Section 1. Multilinear mappings and polynomials

19

Set

n X

X 1 M.x1 ; : : : ; xn / D n "1 "n P 2 nŠ

! "j xj :

j D1

"j D˙1

As M.x1 ; : : : ; ˛xCˇy; : : : ; xn / D M .x1 ; : : : ; ˛x C ˇy; : : : ; xn / D ˛M .x1 ; : : : ; x; : : : ; xn / C ˇM .x1 ; : : : ; y; : : : ; xn / D ˛M.x1 ; : : : ; x; : : : ; xn / C ˇM.x1 ; : : : ; y; : : : ; xn / for any 2 ˚ and ˚ separates the points of Y , it follows that M 2 L. nXI Y /. Letting , we see that by Lemma 33 the mapping P Pn satisfies the assumption of Pn D M the theorem with n 1 instead of n, which finishes the induction step. If B P is continuous for every 2 Y , then P .BX / is weakly bounded and hence it is bounded by the Banach-Steinhaus uniform boundedness principle. Thus P is continuous by Proposition 40. t u The above fact has the following immediate consequence (notice that the set of functionals ff B R I f 2 Y ; 2 g `1 . I Y / separates the points of `1 . I Y /): Fact 50. Let X , Y be normed linear spaces, let be an arbitrary set, n 2 N0 , and let P W X ! `1 . I Y /. Denote by R˛ , ˛ 2 the canonical projections R˛ W `1 . I Y / ! Y given by R˛ ..y // D y˛ and by P D R B P the component mappings of P . Then P 2 P n .XI `1 . I Y // if and only if fP g 2 is a bounded set in P n .XI Y /. To prove that on real spaces it suffices to test the polynomiality of a mapping only on affine lines we need to introduce several technical tools. Definition 51. Let X, Y be vector spaces, U X, and f W U ! Y . For a fixed h 2 X we define the first difference by 1f .xI h/ D f .x C h/

f .x/

for all x 2 U such that the right-hand side is defined. Further, we inductively define the differences of higher order by nf .xI h1 ; : : : ; hn / D n 1f .x C h1 I h2 ; : : : ; hn /

n 1f .xI h2 ; : : : ; hn /

for all x 2 U such that the right-hand side is defined. Lemma 52. Let X , Y be vector spaces, U X, f W U ! Y , and n 2 N. Then X X n n jAj f .xI h1 ; : : : ; hn / D . 1/ f xC hj j 2A

Af1;:::;ng

if either of the sides is defined. Hence nf .xI h

1 ; : : : ; hn /

D

nf .xI h.1/ ; : : : ; h.n/ /

for every permutation of f1; : : : ; ng and n

f .xI h; : : : ; h/ D

n X kD0

n k

. 1/

n f .x C kh/: k

20

Chapter 1. Fundamental properties of smoothness

Proof. We use induction on n. For n D 1 it is just the definition. So let n > 1 and suppose that the formula holds for n 1 . Then nf .xI h1 ; : : : ; hn / X X X X n 1 jAj n 1 jAj D . 1/ f x C h1 C hj . 1/ f xC hj j 2A

Af2;:::;ng

D

X

. 1/n

j 2A

Af1;:::;ng 12A

j 2A

Af2;:::;ng

X X jAj f xC hj C . 1/n

X jAj f xC hj ; j 2A

Af1;:::;ng 1…A

t u

from which the formula follows.

Lemma 53 ([Kuc]). Let X, Y be vector spaces, U X, f W U ! Y , and n 2 N. Then X X X X 1 1 : : ; nf .xI h1 ; : : : ; hn / D . 1/jAj nf x C hj I h ; : h j j j j j 2A

Af1;:::;ng

j 2A

j 2A

for all x; h1 ; : : : ; hn 2 X such that the right-hand side is defined. Proof. First note that nf .xI u1 ; : : : ; un / D 0 if uj D 0 for some j 2 f1; : : : ; ng, which follows from the definition and the symmetry of n (Lemma 52). Therefore using Lemma 52 twice we obtain X X X X jAj n 1 1 . 1/ f x C hj I j hj ; : : : ; j hj j 2A

Af1;:::;ng

D

X

. 1/

jAj

Af1;:::;ng

D

n X

. 1/

D

n

. 1/n

kD0

. 1/

k n k

j 2A

j 2A n

X k n f xC hj k

X

k

j 2A

kD0

kD0 n X

n X

Af1;:::;ng

j 2A

X X jAj . 1/ f x C 1

k n . 1/n nf xI .1 k

1 j hj

k j

hj

j 2A

k/h1 ; 1

k 2

h2 ; : : : ; 1

k n

hn

D nf .xI h1 ; : : : ; hn /:

t u

Let X1 ; : : : ; Xn , Y be vector spaces. We say that a mapping M W X1 Xn ! Y is n-additive if it is additive separately in each coordinate, i.e. if M.x1 ; : : : ; xk 1 ; x C y; xkC1 ; : : : ; xn / D M.x1 ; : : : ; xk

1 ; x; xkC1 ; : : : ; xn /

C M.x1 ; : : : ; xk

1 ; y; xkC1 ; : : : ; xn /

for each x1 2 X1 ; : : : ; xn 2 Xn , x; y 2 Xk , and each k 2 f1; : : : ; ng. It is easy to see that if M W X1 Xn ! Y is an n-additive mapping, then M.x1 ; : : : ; xk 1 ; txk ; xkC1 ; : : : ; xn / D tM.x1 ; : : : ; xn / for each x1 2 X1 ; : : : ; xn 2 Xn , t 2 Q, and each k 2 f1; : : : ; ng.

Section 1. Multilinear mappings and polynomials

21

By inspecting the proofs it is easy to see that Proposition 4, Theorem 7, and the Polarisation formula (Proposition 11, Proposition 13) hold also for n-additive mappings. In particular, if X1 ; : : : ; Xn , Y are real normed linear spaces and M W X1 Xn ! Y is an n-additive mapping that is bounded on a neighbourhood of some point, then M 2 L.X1 ; : : : ; Xn I Y /. Theorem 54 ([MO1]). Let X , Y be vector spaces, f W X ! Y , and n 2 N0 . The following statements are equivalent: (i) nC1 f .xI h; : : : ; h/ D 0 for all x; h 2 X, i.e. f satisfies the Fréchet formula nC1 X nC1 k n C 1 . 1/ f .x C kh/ D 0: (10) k kD0

nC1 f .xI h

(ii) 1 ; : : : ; hnC1 / D 0 for all x; h1 ; : : : ; hnC1 2 X . (iii) There are symmetric k-additive mappings Mk W X k ! Y , k D 1; : : : ; n such that P f .x/ D f .0/ C nkD1 Mk .x/ for every x 2 X, where Mk .x/ D Mk .x; : : : ; x/.

b

b

Proof. (i))(ii) follows from Lemma 53. (iii))(i) follows from (7) in Proposition 13 by setting a D x C .n C 1/ h2 and xj D h2 . (ii))(iii) We use induction on n. For n D 0 it is clear, so assume that the implication 1 n holds for n 1 in place of n. Set Mn .x1 ; : : : ; xn / D nŠ f .0I x1 ; : : : ; xn /. It is easy to n check using the definition that f .0I x C y; x2 ; : : : ; xn / D nf .0I x; x2 ; : : : ; xn / C nf .xI y; x2 ; : : : ; xn / for every x; y; x2 ; : : : ; xn 2 X . Hence Mn .x C y; x2 ; : : : ; xn / Mn .x; x2 ; : : : ; xn / Mn .y; x2 ; : : : ; xn / 1 n f .0I x Cy; x2 ; : : : ; xn / nf .0I x; x2 ; : : : ; xn / nf .0I y; x2 ; : : : ; xn / D nŠ 1 n D f .xI y; x2 ; : : : ; xn / nf .0I y; x2 ; : : : ; xn / nŠ 1 D nC1f .0I x; y; x2 ; : : : ; xn / D 0 nŠ by the assumption. So Mn is a symmetric n-additive mapping. Using the substitution P h h a D x C jnD1 2j and xj D 2j the Polarisation formula (Proposition 11) yields

b

1 n Mn .xI h1 ; : : : ; hn / for every x; hj 2 X. Finally, we put Mn .h1 ; : : : ; hn / D nŠ g.x/ D f .x/ Mn .x/ for x 2 X. Then

b

ng.xI h1 ; : : : ; hn / D nf .xI h1 ; : : : ; hn / n

D f .xI h1 ; : : : ; hn /

nŠMn .h1 ; : : : ; hn / nf .0I h1 ; : : : ; hn /

D nC1f .0I x; h1 ; : : : ; hn / D 0: Thus we may use the inductive hypothesis on g and the proof is finished.

t u

Combining the previous theorem with Corollary 31, the Polarisation formula (Proposition 13), and condition (v) in Theorem 7 we obtain the following corollary:

22

Chapter 1. Fundamental properties of smoothness

Corollary 55. Let X be a real Banach space, Y a real normed linear space, n 2 N0 , and let P W X ! Y be such that B P is Baire measurable for each 2 Y . The following statements are equivalent: (i) (ii) (iii) (iv)

nC1 P .xI h; : : : ; h/ D 0 for all x; h 2 X. nC1 P .xI h1 ; : : : ; hnC1 / D 0 for all x; h1 ; : : : ; hnC1 2 X. The restriction of P to each line in X is a polynomial of degree at most n. P 2 P n .XI Y /.

We end this section by presenting the Lagrange interpolation formula for vectorvalued polynomials on scalars which, although it belongs in spirit to Chapter 2, cannot be postponed, as it will be useful later on in this chapter. Proposition 56 (Lagrange interpolation formula). Let Y be a normed linear space over K, let x0 ; : : : ; xn be distinct points in K, and let y0 ; : : : ; yn 2 Y . Then there exists a unique polynomial P 2 P n .KI Y / satisfying P .xj / D yj , j D 0; : : : ; n. The polynomial P can be expressed by the formula n X wj .x/ yj ; P .x/ D wj .xj / j D0

where wj .x/ D

n Y

.x

xk /:

kD0 k¤j

Proof. The formula for P of course gives the existence of such polynomial. The uniqueness follows from the fact that Y separates the points of Y , so we may apply the scalar case. t u

2. Complexification Complex Banach spaces are an indispensable tool in the theory of real analytic mappings. The complexification of a given real Banach space X can be described as X CiX , a space which is real-isomorphic to X ˚ X. The complex operations are obtained naturally. In general, the complex norm can be introduced in many non-isometric (but of course real-isomorphic) ways. This fact plays an inessential role for the most part of our theory, except perhaps for the precise estimates of norms obtained by passing to the complexification, and we choose to work with a complexification norm due to Angus Ellis Taylor. Here we focus on proving some basic extension inequalities to be used in later sections and chapters.

Section 2. Complexification

23

Definition 57. By a complexification of a real normed linear space .X; kk/ we mean the complex normed linear space XQ D f.x; y/I x; y 2 X g with operations given by .x; y/ C .u; v/ D .x C u; y C v/; .˛ C iˇ/.x; y/ D .˛x

ˇy; ˛y C ˇx/;

and a norm kkC defined as k.x; y/kC D sup

q

.x/2 C .y/2 :

2BX

Q kkC / is a complex normed linear space real-isomorphic to X ˚X Checking that .X; is easy. The space X is canonically isometrically embedded in XQ by x 7! .x; 0/ and if X is a real Banach space, then XQ is also a Banach space. In some situations it is convenient to use this equivalent formula for the Taylor norm: k.x; y/kC D sup sup cos.t/.x/ C sin.t/.y/ 2BX t 2Œ0;2

D sup kcos.t/x C sin.t/yk:

(11)

t 2Œ0;2

We will use an alternative notation for XQ D X C iX using the correspondence .x; y/ D x C iy and assuming implicitly that x; y 2 X. We will also use the usual projections Re.x C iy/ D x and Im.x C iy/ D y. For ´ D x C iy 2 XQ we denote ´ D x iy. From the definition of the Taylor norm it is clear that k´kC D k´kC and kRe ´k k´kC , kIm ´k k´kC . We are going to omit the subscript C in order to simplify the notation later on in the text. Let us make a few remarks. It is clear from the formula (11) that our complexification procedure respects subspaces, i.e. if T W Z ! X is an isometric embedding, then kx C iykZ;C D kT .x/ C iT .y/kX;C whenever x; y 2 Z. Also, it is not difficult to check using (11) that the space X ˚ Y is isometric to the space XQ ˚ YQ if we take the maximum norm on both direct sums. In case of the Banach spaces C.K/ Taylor’s norm is natural and coincides with the usual definition of the norm of the respective complex spaces. However, for the spaces `p , p ¤ 1, the complexification norm does not coincide with the norm of the complex version of the sequence spaces. Taylor’s norm also does not in general pass to quotients and it does not respect duality. We are not going to discuss the various possible alternative definitions of complexification, as they do not play a significant role for our purposes. Let us only remark that defining the complexification norm by p the formula kxk2 C kyk2 is not possible, as the complex homogeneity of the norm would be violated. Given real normed linear spaces X, Y and T 2 L.X I Y /, it is easily checked that Q YQ /, there is a uniquely determined (by complex linearity) complex extension TQ 2 L.XI Q defined by T .x C iy/ D T .x/ C iT .y/.

B

Proposition 58. Let X, Y be real normed linear spaces and T 2 L.X I Y /. Then kTQ kC D kT k.

24

Chapter 1. Fundamental properties of smoothness

Proof. We only need to verify that kTQ kC kT k: kTQ .x C iy/kC D kT .x/ C iT .y/k D

sup kcos.t/T .x/ C sin.t/T .y/k t 2Œ0;2

D

sup kT .cos.t/x C sin.t/y/k t 2Œ0;2

kT k sup kcos.t/x C sin.t/yk D kT kkx C iykC :

t u

t 2Œ0;2

Obviously, T is one-to-one if and only if TQ is one-to-one, and T is onto if and only if TQ is onto. Q Then there is 2 SX such Fact 59. Let X be a real normed linear space and ´ 2 X. Q that j.´/j D k´k. Proof. p Let ´ D x C iy. By the w -compactness of BX there is 2 SX such that Q k´k D .x/2 C .y/2 D j.´/j. t u Proposition 60 ([BoSi]). Let X , Y be real normed linear spaces, n 2 N, and let z 2 L. nXI Q YQ / such that M z X n D M . M 2 L. nXI Y /. There is a unique extension M It is given by the formula Pn X z .x10 C ix11 ; : : : ; xn0 C ixn1 / D M i j D1 "j M.x1"1 ; : : : ; xn"n /: (12) "1 ;:::;"n 2f0;1g

z is bounded if and only if M is bounded. Moreover, M z is n-linear as a real multilinear mapping and also that every Proof. It is clear that M complex extension must satisfy (12). It remains to prove that it is coordinatewise complex homogeneous. Let a D ˛ C iˇ 2 C. From (12) we obtain for every u2 ; : : : ; un 2 XQ , x0 ; x1 2 X, z .x0 C ix1 ; u2 ; : : : ; un / D M z .x0 ; u2 ; : : : ; un / C i M z .x1 ; u2 ; : : : ; un /: M So z a.x0 C ix1 /; u2 ; : : : ; un D M z .˛x0 M z .x0 ; u2 ; : : : ; un / D ˛M

ˇx1 / C i.ˇx0 C ˛x1 /; u2 ; : : : ; un

z .x1 ; u2 ; : : : ; un / ˇM

z .x0 ; u2 ; : : : ; un / C i˛ M z .x1 ; u2 ; : : : ; un / C iˇ M z .x0 ; u2 ; : : : ; un / C .i˛ D .˛ C iˇ/M

z .x1 ; u2 ; : : : ; un / ˇ/M

z .x0 C ix1 ; u2 ; : : : ; un /: D aM The assertion about boundedness is clear.

t u

Section 2. Complexification

25

Proposition 61 ([Tay1]). Let X, Y be real normed linear spaces, n 2 N, and let P 2 P . nX I Y /. Then there is a unique extension PQ 2 P . nXQ I YQ / such that PQ X D P . It is given by the formula n

PQ .x C iy/ D

Œ2 X

. 1/k

kD0

n }n P. 2k

2k

x; 2ky/

Œ .n 2 1/

Ci

X

k

. 1/

kD0

n }.n P 2k C 1

.2kC1/

x; 2kC1y/:

Moreover, PQ is bounded if and only if P is bounded. t u

Proof. Use Lemma 21 combined with Proposition 60. P . 0XI Y /

For P 2 the extension is trivial. Naturally, if P .x/ D P k Pk 2 P . XI Y /, then we denote PQ .x/ D nkD0 Pzk .x/. The following is an immediate corollary of Proposition 61.

Pn

kD0 Pk .x/,

Fact 62. Let X, Y be real normed linear spaces, n 2 N0 , and P 2 P n .XI Y /. Then Q PQ .´/ D PQ .´/ for every ´ 2 X. The next theorem gives in particular a norm estimate of the complexified homogeneous polynomials. For the general non-homogeneous case see Theorem 2.26. Theorem 63 ([MST]).PLet X, Y be real normed linear spaces, n 2 N, and let P 2 P n .XI Y /, P D nkD0 Pk , where Pk 2 P . kX I Y /. Then kPzn kC 2n kPn 1 kC 2

1

n 2

kP k; kP k

(for n 2).

Proof. Let ´ D x C iy 2 XQ , k´k D 1 (hence also k´k D 1). For t 2 R define it e ei t f .t/ D PQ ´ D P .cos.t/x C sin.t/y/: ´C 2 2 From (11) it follows that kf .t/kC kP k;

t 2 R:

(13)

Using Lemma 21 we obtain f .t / D P0 .0/ C

n X k .k X k e kD1 lD0

l

2l/it

2k

zk .l´; k l ´/ D P}

n X

e ik t ak

kD n 1 z 2n Pn .´/

for suitable ak 2 YQ . We can easily see that a n D and an D 21n Pzn .´/. Furthermore, exploiting the fact that the inner sum for k D n does not contain summands with factors e .n 1/it , e .n 1/it , we obtain also a .n 1/ D 2n1 1 Pn 1 .´/ and an 1 D 2n1 1 Pn 1 .´/. Note also that the homogeneous summands of PQ of lower degree are not expressed as easily.

26

Chapter 1. Fundamental properties of smoothness

Next, we use the following formula (a sum of a finite geometric series): 2n 1 ikp 1 X . 1/p e n D 2n pD0

(

1 0

for k n .mod 2n/, otherwise.

(14)

Thus 2n 1 2n 1 n X p 1 X 1 X p p D . 1/ f t C p . 1/ e ik.t C n / ak 2n n 2n pD0 pD0 kD n ! 2n 1 n X 1 X p i kp ik t . 1/ e n ak D e 2n pD0

kD n

De

i nt

a

n

C e i nt an D

e

i nt

2n

e i nt Pzn .´/ C n Pzn .´/: 2

Combining this with (13) we get

sup e

i nt

t 2Œ0;2

Pzn .´/ C e i nt Pzn .´/ C 2n kP k:

Now let Pzn .´/ D u C iv. By Fact 62 we have Pzn .´/ D u

sup e t2Œ0;2

i nt

Pzn .´/ C e i nt Pzn .´/ C D

iv and so by (11)

sup k2 cos.nt/u C 2 sin.nt/vkC t 2Œ0;2

D 2kPzn .´/kC ; which finishes the proof of the first estimate. The second estimate is proved analogously, using n 1 in place of n in (14). t u Pn Corollary 64. Let X, Y be real normed linear spaces, n 2 N, and P D kD0 Pk , where Pk 2 P . kX I Y /. Then kPn k 2n 1 kP k and kPn 1 k 2n 2 kP k (for n 2).

3. Fréchet smoothness We begin by introducing the directional and the Gâteaux derivative. The stronger concept of Fréchet derivative is by far the most important notion of differentiation. We develop its basic properties including the Implicit function theorem. We introduce higher derivatives inductively as multilinear mappings and prove some of their standard properties.

Section 3. Fréchet smoothness

27

Definition 65. Let X, Y be normed linear spaces over K, A X, f W A ! Y , and x 2 A. The directional derivative of f at x in the direction h 2 X is defined as @f f .x C th/ .x/ D lim t !0 @h t

f .x/

(15)

t 2K

if the limit exists in Y . If @f .x/ exists for all h 2 X and the mapping L W X ! Y , @h @f L.h/ D @h .x/, is a continuous linear operator, then we say that f is Gâteaux differentiable at x. The operator L is then called the Gâteaux derivative of f at x and is denoted by ıf .x/. For the evaluation of the Gâteaux derivative we use the notation ıf .x/Œh D L.h/. @f .x/ exists, then @.C .x/ exists for any C 2 K Clearly, if the directional derivative @f @h h/ @f @f and @.C h/ .x/ D C @h .x/. For mappings defined in Rn the directional derivatives along the canonical basis vectors are just the usual partial derivatives. For a fixed h 2 X the natural domain of the mapping @f (i.e. of the mapping x 7! @f .x/) is the set of all @h @h @f points x 2 A at which @h .x/ exists. Similarly for the mapping ıf .

Definition 66. Let X, Y be normed linear spaces over K, A X, f W A ! Y , and x 2 A. We say that f is Fréchet differentiable at x if f is Gâteaux differentiable at x and the limit (15) is uniform with respect to h 2 SX . In this case we call the linear operator ıf .x/ the Fréchet derivative of f at x and denote it by Df .x/. For the evaluation we again use the notation Df .x/Œh D ıf .x/Œh. Note that from the definition it follows that if f is Fréchet differentiable at x, then x is an interior point of A. If f W A ! Y is Fréchet differentiable at every point of B A, we say that f is Fréchet differentiable in B. Denoting by C the set of all x 2 A at which Df .x/ exists, Df W C ! L.XI Y / is then the mapping x 7! Df .x/. For convenience we also use the notation D 1f D Df and D 0f D f , and analogously for the Gâteaux derivative. Let us make a few remarks. It is clear that f is Fréchet differentiable at x if and only if there exists L 2 L.X I Y / satisfying lim

h!0

kf .x C h/

f .x/ khk

L.h/k

D 0:

(16)

The linear operator L is then the Fréchet derivative of f at x. It follows that if f is Fréchet differentiable at x, then it is continuous at x. This is not necessarily true for Gâteaux differentiable mappings, even for functions on R2 – consider the function ˚ 3 3 f .x; y/ D max xy2 2 xy2 ; 0 for x; y 2 R, x ¤ 0, and f .0; y/ D 0, y 2 R, which is Gâteaux differentiable at every point of R2 but is discontinuous at the origin. It is however not difficult (using the compactness of the unit sphere) to show that for locally Lipschitz mappings defined on finite-dimensional spaces the notions of Fréchet and Gâteaux differentiability coincide. Note also that on one-dimensional spaces the existence of directional (for h ¤ 0), Gâteaux, and Fréchet derivatives are all equivalent. Finally, unlike the finite-dimensional case, if X is infinite-dimensional, then even the

28

Chapter 1. Fundamental properties of smoothness

˚ equi-continuity of x 7! @f .x/ h2SX at a 2 X does not ensure the existence of the @h Gâteaux derivative ıf .a/ (consider a discontinuous linear functional). From (16) we immediately obtain the following two facts. Fact 67. Let X, Y be normed linear spaces and L 2 L.X I Y /. Then DL.x/ D L for every x 2 X. Proposition 68. Let X, Y be normed linear spaces over K, U X an open set, let f; g W U ! Y be Fréchet differentiable at a 2 U , and c 2 K. Then f C g is Fréchet differentiable at a and D.f C g/.a/ D Df .a/ C Dg.a/. Further, c f is Fréchet differentiable at a and D.cf /.a/ D cDf .a/. An equivalent reformulation of (16) is the following: f .x C h/

f .x/

Df .x/Œh D o.khk/; h ! 0:

This form is useful for proving the Chain rule. Theorem 69 (Chain rule). Let X , Y , Z be normed linear spaces, U X open, and let g W U ! Y be Fréchet differentiable at a 2 U . Suppose further that g.U / V , where V Y is an open set, and f W V ! Z is Fréchet differentiable at g.a/. Then f B g W U ! Z is Fréchet differentiable at a and D.f B g/.a/ D Df g.a/ B Dg.a/: Proof. We denote b D g.a/ and put !.h/ D g.a C h/ g.a/ Dg.a/Œh and .k/ D f .b C k/ f .b/ Df .b/Œk. These mappings are defined on suitable neighbourhoods of the origins. Choose an arbitrary " 2 .0; 1/. There is > 0 such that for h 2 X, khk < and k 2 Y , kkk < we have k!.h/k "khk and k.k/k "kkk. Put ı D =.1 C kDg.a/k/. Then for h 2 X , khk < ı we have kDg.a/Œh C !.h/k .kDg.a/k C "/khk < and hence

f g.a C h/ f g.a/ Df g.a/ Dg.a/Œh

D f b C Dg.a/Œh C !.h/ f .b/ Df .b/ Dg.a/Œh

D Df .b/Œ!.h/ C Dg.a/Œh C !.h/ kDf .b/k"khk C " kDg.a/kkhk C "khk < kDf .b/k C kDg.a/k C 1 "khk; which proves the theorem.

t u

For Gâteaux differentiability we have the following version: Theorem 70. Let X, Y , Z be normed linear spaces, A X, and let g W A ! Y be Gâteaux differentiable at a 2 A. Suppose further that g.A/ V , where V Y is an open set, and f W V ! Z. If either f is Fréchet differentiable at g.a/, or f is Gâteaux differentiable at g.a/ and f is Lipschitz on V , then f B g W A ! Z is Gâteaux differentiable at a and ı.f B g/.a/ D ıf g.a/ B ıg.a/:

Section 3. Fréchet smoothness

29

Proof. The case when f is Fréchet differentiable at g.a/ follows easily from Theorem 69 used on f and t 7! g.a C th/ for any fixed h 2 X. So suppose that f is Gâteaux differentiable at g.a/ and f is L-Lipschitz on V . Fix h 2 X . Then for all t 2 K n f0g small enough such that a C th 2 A and g.a/ C tıg.a/Œh 2 V we have 1 1 f g.a/ D f g.a/ C tıg.a/Œh f g.a C th/ f g.a C th/ t t 1 C f g.a/ C tıg.a/Œh f g.a/ : t Now the second summand tends to ıf g.a/ ıg.a/Œh as t ! 0, while the first summand tends to 0. Indeed, from the Lipschitz property of f we get

1

f g.a C th/ f g.a/ C tıg.a/Œh

t

1

t u L g.a C th/ g.a/ tıg.a/Œh

: t We remark that if the mapping g is Féchet differentiable at a and f is Gâteaux differentiable at g.a/, then the composition f B g may not be Gâteaux differentiable at a, and even if it is, the chain rule formula may not hold. Consider the mapping g W R ! R2 , g.x/ D .x; x 2 /, and f W R2 ! R defined as f .x; y/ D 0 for y ¤ x 2 , f .x; x 2 / D sgn x (or f .x; x 2 / D x). Proposition 71. Let X , Y be normed linear spaces, U X , f W U ! Y , and C U convex. Suppose that f is Gâteaux differentiable at each point of C and supx2C kıf .x/k D K < C1. Then f is K-Lipschitz on C . Proof. In case that X and Y are complex spaces, we may consider their real versions XR and YR , and the real Gâteaux derivative of the mapping f considered as a mapping from XR to YR . This change of structure has no impact on the differentiability: ıR f .x/ 2 L.XR I YR / and ıR f .x/ D ıf .x/. Thus we may assume without loss of generality that the spaces X, Y are real. Fix any a; b 2 C , put h D b a, and define the function g W Œ0; 1 ! Y by g.t / D f .a C th/. Let 2 SY be such that kf .b/ f .a/k D f .b/ f .a/ . We have . B g/0 .t/ D ıf .a C th/Œh for t 2 Œ0; 1. Thus by the Mean value 0 theorem there is 2 .0; 1/ such that B0 g.1/ B g.0/ D . B g/ ./. Hence kf .b/ f .a/k D g.1/ g.0/ D . Bg/ ./ kkkıf .a Ch/kkhk Kkb ak. t u The following variant of the previous proposition, where the convexity is slightly relaxed, can be useful in certain situations. Proposition 72. Let X, Y be normed linear spaces, U X , f W U ! Y , C X convex, and let D C be such that C n D U and for every x; y 2 C n D there is ´ 2 C such that Z D spanfy x; ´ xg is two-dimensional and D \ .x C Z/ is countable. (This holds in particular if C is not a subset of a line and D is countable or discrete.) Suppose that f is Gâteaux differentiable at each point of C n D and supx2C nD kıf .x/k D K < C1. Then f is K-Lipschitz on C n D.

30

Chapter 1. Fundamental properties of smoothness

Proof. Fix x; y 2 C n D and let ´ 2 C be such that Z D spanfy x; ´ xg is two-dimensional and D \ .x C Z/ is countable. Set u D 21 .x C y/ and v D k´´ uuk . Let " > 0 be arbitrary. Then the sets B1 D ft 2 Œ0; "I Œx; u C tv \ D ¤ ;g and B2 D ft 2 Œ0; "I Œy; u C tv \ D ¤ ;g are countable. Therefore there is t0 2 Œ0; "n.B1 [B2 /. Set w D uCt0 v. The line segments Œx; w and Œy; w lie in C nD and so by Proposition 71 we get kf .x/ f .y/k kf .x/ f .w/kCkf .w/ f .y/k K.kx wk C kw yk/ K.kx uk C t0 C ku yk C t0 / K.kx yk C 2"/. Since " can be chosen arbitrarily small, the assertion follows. t u Fact 73. Let X , Y be normed linear spaces, U X open, and let f W U ! Y be a Gâteaux differentiable mapping such that its Gâteaux derivative ıf W U ! L.X I Y / is continuous at x 2 U . Then f is Fréchet differentiable at x. Proof. Define a mapping g W U ! Y by g.y/ D f .y/ ıf .x/Œy x. Notice that (using Fact 67) ıg.y/ D ıf .y/ ıf .x/. Choose " > 0 and find > 0 such that kıf .y/ ıf .x/k < " whenever y 2 U satisfies ky xk < . Then, by Proposition 71, g is "-Lipschitz on U.x; /. Thus, for an arbitrary h 2 U.0; /,

f .x C h/ f .x/ ıf .x/Œh D kg.x C h/ g.x/k "khk: t u Definition 74. Let X, Y be normed linear spaces over K, U X an open set, let f W U ! Y , a 2 U , and k 2 N. We say that f is twice Fréchet differentiable at a if Df .x/ exists for x in some neighbourhood of a and the mapping x 7! Df .x/ is Fréchet differentiable at a. We define the second Fréchet derivative of f at a by D 2f .a/ D D.Df /.a/. Then D 2f .a/ 2 L XI L.X I Y / , but using the natural identification from Fact 9 we may consider D 2f .a/ to be an element of L. 2XI Y /. Further, we define inductively the kth Fréchet derivative D kf .a/ of f at a. We say that f is k-times Fréchet differentiable at a if D k 1f .x/ 2 L. k 1XI Y / exists for x in some neighbourhood of a and the mapping x 7! D k 1f .x/ is Fréchet differentiable at a. We put D kf .a/ D D.D k 1f /.a/. Using the identification from Fact 9 we may consider D kf .a/ to be an element of L. kXI Y /. We denote by D kf .a/Œx1 ; : : : ; xk 2 Y the evaluation of D kf .a/ at .x1 ; : : : ; xk / 2 X k . We say that f is C k -smooth if D kf is continuous in the domain. We say that f is C 1 -smooth if D kf is continuous in the domain for every k 2 N. We denote by C k .U I Y / the set of all C k -smooth mappings from U into Y , k 2 N [ f1g. For convenience we put C 0 .U I Y / D C.U I Y /. As usual, for functions, i.e. mappings into the scalars, we use the shortened notation C k .U / D C k .U I K/. By abusing the notation slightly we write f 2 C k .U I M /, where M is a subset of a normed linear space Y , as a short for f 2 C k .U I Y /, f .U / M . The following fact follows immediately by induction. Fact 75. Let X, Y be normed linear spaces over K, U X open, k 2 N, let f; g W U ! Y be k-times Fréchet differentiable at a 2 U , and c 2 K. Then f C g is k-times Fréchet differentiable at a and D k .f C g/.a/ D D kf .a/ C D kg.a/. Further, c f is k-times Fréchet differentiable at a and D k .cf /.a/ D cD kf .a/. In particular, C k .U I Y / is a linear subspace of C.U I Y /.

Section 3. Fréchet smoothness

31

Theorem 76. Let X, Y be normed linear spaces, U X an open set, f W U ! Y , k 2 N, and a 2 U . Suppose that D kf .a/ exists. Then D kf .a/ 2 Ls . kX I Y /. Proof. Notice that for g.x/ D f .a C x/ we have D kg.0/ D D kf .a/ and so we may without loss of generality assume that a D 0. Assume first that k D 2. Given " > 0, choose ı > 0 so that

Df .x/ Df .0/ D.Df /.0/Œx "kxk for x 2 U.0; 2ı/. For a fixed v 2 U.0; ı/ we define gv .u/ W U.0; ı/ ! Y by gv .u/ D f .u C v/

f .u/

Observe that (using Fact 67) Dgv .u/ D Df .u C v/ D Df .u C v/

f .v/ C f .0/

Df .u/ Df .0/

D.Df /.0/Œv Œu:

D.Df /.0/Œv D.Df /.0/Œu C v

Df .u/

Df .0/

D.Df /.0/Œu :

Thus kDgv .u/k "ku C vk C "kuk 2".kuk C kvk/: By Proposition 71, gv is 2".Ckvk/-Lipschitz on B.0; /, < ı, and since gv .0/ D 0 we conclude that kgv .u/k 2".kuk C kvk/kuk 2".kuk C kvk/2 . It follows that for any w; ´ 2 X, .w; ´/ ¤ .0; 0/, we have

f .tw C t´/ f .tw/ f .t´/ C f .0/ D 2f .0/Œtw; t´ D0 lim t !0 .ktwk C kt´k/2 and so f .tw C t´/ f .tw/ f .t´/ C f .0/ D D 2f .0/Œw; ´: lim t !0 t2 Interchanging the roles of w and ´ we see that D 2f .0/Œ´; w is given by the same limit and hence D 2f .0/ is a symmetric 2-linear mapping. We proceed by induction. Assume that k > 2 and D k 1f .x/ 2 Ls . k 1X I Y / for each x from some neighbourhood of a. Recall that each permutation is a composition of adjacent transpositions, therefore it suffices to show that given u1 ; : : : ; uk 2 X and p 2 f1; : : : ; k 1g we have D kf .0/Œu1 ; : : : ; uk D D kf .0/Œu1 ; : : : ; up

1 ; upC1 ; up ; upC2 ; : : : ; uk :

For p D 1 this follows from the case k D 2 used on the mapping D k 2f : we obtain D 2 .D k 2f /.0/Œu1 ; u2 Œu3 ; : : : ; uk D D 2 .D k 2f /.0/Œu2 ; u1 Œu3 ; : : : ; uk : For p 2 we use the fact that D.D k 1f /.0/Œu1 Œu2 ; : : : ; uk D D D k 1f ./Œu2 ; : : : ; uk .0/Œu1

(17)

and the inductive hypothesis. The formula (17) can be proved for example by using the evaluation operator "u2 ;:::;uk W L. k 1X I Y / ! Y , "u2 ;:::;uk .M / D M.u2 ; : : : ; uk /, which is a bounded linear operator, together with the Chain rule (Theorem 69) and Fact 67. t u

32

Chapter 1. Fundamental properties of smoothness

Suppose that X, Y are normed linear spaces and f W U ! Y , where U is an open subset of X. If W D a C Z is an affine subspace of X, where a 2 X and Z X is a subspace of X, then we will usually identify f W \U with the mapping g W Z \.U a/ ! Y , g.x/ D f .x Ca/. Using this identification we say that f W \U is C k -smooth meaning actually that g is C k -smooth, and similarly we write (formally incorrectly) D k .f W \U /.x/ meaning D kg.x a/. (Note that under this identification D k .f W \U /.x/ does not depend on the choice of a 2 W .) The next statement is easy, but important. Fact 77. Let X, Y be normed linear spaces, U X an open set, f 2 C k .U I Y / for some k 2 N, let Z X be a subspace, and a 2 X. Denote W D a C Z. Then D k .f W \U /.x/ D D kf .x/ Z k 2 Ls . kZI Y / for every x 2 W \ U . Fact 78. Let X, Y , Z be normed linear spaces, L 2 L.Y I Z/, U X an open set, and k 2 N. Let f W U ! Y be k-times Fréchet differentiable at a 2 U . Then D k .L B f /.a/ D L B D kf .a/. Proof. We use induction on k. For k D 1 we use Theorem 69 and Fact 67. For the induction step from k 1 to k we compute

k

D

1

.L B f /.a C h/

Dk

D L B D k 1f .a C h/

kLk D k 1f .a C h/

1

L B D kf .a/ Œh; ; : : : ;

L B D k 1f .a/ L B D kf .a/ Œh; ; : : : ;

D k 1f .a/ D kf .a/Œh; ; : : : ; D o.khk/; h ! 0: .L B f /.a/

t u Proposition 79. Let X1 ; : : : ; Xn , and Y be normed linear spaces and further let M 2 L.X1 ; : : : ; Xn I Y /. Put X D X1 ˚ ˚ Xn . Then M 2 C 1 .XI Y /. For k n and all x D .x1 ; : : : ; xn / 2 X we have D kM.x/Œ.h11 ; : : : ; h1n /; : : : ; .hk1 ; : : : ; hkn / D

X

M.´ 1 ; : : : ; ´n /;

where the summation is over all one-to-one mappings W f1; : : : ; kg ! f1; : : : ; ng and where ( hl if j D .l/ for some l 2 f1; : : : ; kg, ´j D j xj otherwise. For k > n we have D kM.x/ D 0 for all x 2 X.

Section 3. Fréchet smoothness

33

Proof. Without loss of generality we may consider X with the maximum norm. Fix x D .x1 ; : : : ; xn / 2 X. For h D .h1 ; : : : ; hn / 2 X we have by (1)

n X

M.x C h/ M.x/ M.x1 ; : : : ; xj 1 ; hj ; xj C1 ; : : : ; xn /

j D1

M A .j /hj C .1

X

A .j //xj

j D1

n

Af1;:::;ng jAj2

X

kM kkhkjAj kxkn

jAj

D o.khk/; h ! 0:

Af1;:::;ng jAj2

P Since clearly h 7! jnD1 M.x1 ; : : : ; xj 1 ; hj ; xj C1 ; : : : ; xn / 2 L.X I Y /, it follows that M is Fréchet differentiable at x and the formula for the first derivative holds. We proceed by induction. Let 1 k < n and suppose that M is k-times Fréchet differentiable on X and the formula for D kM holds. Fix a one-to-one mapping W f1; : : : ; kg ! f1; : : : ; ng. Let p D n k and j1 < < jp be the enumeration of f1; : : : ; ng n .f1; : : : ; kg/. Set M .xj1 ; : : : ; xjp /Œh1 ; : : : ; hk D M.´ 1 ; : : : ; ´n /. k k Then M .xj1 ; : : : ; xjp / 2 L. XI Y / and M 2 L Xj1 ; : : : ; Xjp I L. X I Y / . Thus by the first step of the proof M is Fréchet differentiable on Xj1 ˚ ˚ Xjp . Now if T W X ! Xj1 ˚ ˚ Xjp is the canonical projection, then M B T is Fréchet P differentiable on X by Theorem 69. As D kM D M B T , it follows that M is .k C 1/-times differentiable on X. The formula for D kC1M follows from the formula for DM from the first step. When k D n we thus obtain X D nM.x/Œ.h11 ; : : : ; h1n /; : : : ; .hn1 ; : : : ; hnn / D M.h1.1/ ; : : : ; hn.n/ /; 2Sn

where Sn is the set of all permutations of f1; : : : ; ng. It follows that x 7! D nM.x/ is a constant mapping, and hence D nC1M.x/ D 0 for every x 2 X . t u Next, we are going to introduce the generalised partial derivatives. Definition 80. Let X1 ; : : : ; Xn , Y be normed linear spaces, X D X1 ˚ ˚ Xn , A X, and let f W A ! Y . We say that the mapping f is Fréchet differentiable at a point a D .a1 ; : : : ; an / 2 A with respect to the kth variable if the mapping g W Xk \ .A a/ ! Y defined as g.y/ D f .a1 ; : : : ; ak 1 ; ak C y; akC1 ; : : : ; an / is Fréchet differentiable at 0 2 Xk . The partial Fréchet derivative will be denoted by Dk f .a1 ; : : : ; an / D Dg.0/ 2 L.Xk I Y /. Proposition 81. Let X1 ; : : : ; Xn , Y be normed linear spaces, X D X1 ˚ ˚ Xn , A X , and let f W A ! Y be Fréchet differentiable at a 2 A. Then Dk f .a/, 1 k n, exist and n X Df .a/Œh D Dk f .a/Œhk for each h D .h1 ; : : : ; hn / 2 X. kD1

34

Chapter 1. Fundamental properties of smoothness

Proof. The fact that Dk f .a/ exist is immediate and the formula follows from the fact that Dk f .a/Œhk D Df .a/Œ.0; : : : ; 0; hk ; 0; : : : ; 0/; hk 2 Xk ; which follows immediately from the definition. t u Theorem 82. Let X1 ; : : : ; Xn , Y be normed linear spaces, X D X1 ˚ ˚Xn , a 2 X, and suppose that Dk f , k D 1; : : : ; n exist in a neighbourhood of a and are continuous at a. Then f is Fréchet differentiable at a. Proof. Consider X with the maximum norm. Choose an arbitrary " > 0. Let ı > 0 be such that kDk f .a C h/ Dk f .a/k < n" , k D 1; : : : ; n, whenever h 2 X, khk < ı. Let Pk W X ! X, 1 k n, be the projection onto the first k “coordinates”, i.e. Pk .h/ D .h1 ; : : : ; hk ; 0; : : : ; 0/ for h D .h1 ; : : : ; hn /, and put P0 .h/ D 0. Then f .a C h/

f .a/

n X

Dk f .a/Œhk

kD1

D

n X

(18)

f a C Pk .h/

f a C Pk

1 .h/

Dk f .a/Œhk

kD1

for all h 2 X , khk < ı. Given h 2 X , khk < ı, by Proposition 71 used on a mapping gk .y/ D f a C Pk 1 .h/ C y Dk f .a/Œy for y 2 Xk , kyk < ı, we obtain

f a C Pk .h/ f a C Pk 1 .h/ Dk f .a/Œhk D kgk .hk / gk .0/k

" sup Dk f a C Pk 1 .h/ C y Dk f .a/ khk k khk k: n kykkhk k Combining this with (18) gives

n X

"

f .a C h/ f .a/

kh1 k C C khn k "khk; D f .a/Œh k k

n kD1

t u

which proves the claim.

Theorem 83. Let X, Y1 ; : : : ; Yn , Z be normed linear spaces, Y D Y1 ˚ ˚ Yn , U X open, and let gj W U ! Yj be Fréchet differentiable at a 2 U for each j 2 f1; : : : ; ng. Suppose further that g1 .x/; : : : ; gn .x/ 2 V for all x 2 U , where V Y is an open set, and f W V ! Z is Fréchet differentiable at g1 .a/; : : : ; gn .a/ . Then h.x/ D f g1 .x/; : : : ; gn .x/ is Fréchet differentiable at a and Dh.a/ D

n X

Dj f g1 .a/; : : : ; gn .a/ B Dgj .a/:

(19)

j D1

Moreover, if f , gj , j D 1; : : : ; n, are C k -smooth, k 2 N [ f1g, then h is C k -smooth as well. Note that Dgj .a/ 2 L.X I Yj /, Dj f g1 .a/; : : : ; gn .a/ 2 L.Yj I Z/, and B represents the usual composition of linear operators.

Section 3. Fréchet smoothness

35

Proof. Let W U ! Y be defined as .x/ D g1 .x/; : : : ; gn .x/ . Then it is easily seen that D.a/ D Dg1 .a/; : : : ;Dgn .a/ . Since h D f B , by the Chain rule (Theorem 69) Dh.a/ D Df .a/ B D.a/ from which the formula (19) follows using Proposition 81. To show the C k -smoothness case we use induction on k. The case k D 1 is easily proved using the formula (19) so we move on to the induction step from k to k C 1, assuming that f , gj , j D 1; : : : ; n, are C kC1 -smooth. By (19), Dh is a finite sum of expressions of the type Dj f g1 ./; : : : ; gn ./ B Dgj ./, where Dj f W V ! L.Yj I Z/, Dgj W U ! L.XI Yj / are C k -smooth. By the inductive assumption, Dj f g1 ./; : : : ; gn ./ W U ! L.Yj I Z/ is C k -smooth as well. The composition operation B W L.Yj I Z/ L.X I Yj / ! L.X I Z/ is bilinear and bounded, and so C 1 -smooth by Proposition 79. By the inductive hypothesis (applied to B), Dj f g1 ./; : : : ; gn ./ B Dgj ./ W U ! L.X I Z/ is C k -smooth, so Dh is C k -smooth and consequently h is C kC1 -smooth as stated. t u The closed formulae for higher derivatives of compositions can be deduced from Corollary 117 and the Polarisation formula. The following is an easy application of Theorem 83. Corollary 84. Let X , Y be normed linear spaces over K, U X an open set, and k 2 N [ f1g. Let f 2 C k .U / and g 2 C k .U I Y /. Then f g 2 C k .U I Y /. If moreover f ¤ 0 on U , then f1 2 C k .U /. Proof. Define h W K Y ! Y by h.p; q/ D p q. Then f g D h.f; g/ and since h is a bounded bilinear mapping, the result follows using Theorem 83 and Proposition 79. Analogously for f1 . t u Theorem 85. Let X be a normed linear space, Y a Banach space, U X an open convex bounded set, k 2 N, and let fn W U ! Y be a sequence of k-times Fréchet differentiable mappings. If the sequence fD kfn g1 nD1 converges uniformly j on U and there are x0 ; : : : ; xk 1 2 U such that fD fn .xj /g1 nD1 converges for each j 2 f0; : : : ; k 1g, then there is a k-times Fréchet differentiable mapping f W U ! Y such that D jfn ! D jf uniformly on U for each j 2 f0; : : : ; kg. Proof. It suffices to prove the theorem for k D 1, the general case then follows by a straightforward induction. So assume that Dfn ! g uniformly on U and let R D diam U . For a given " > 0 there is n0 2 N such that kfn .x0 / fm .x0 /k 2" " and kDfn .y/ Dfm .y/k 2R for each y 2 U , whenever m; n n0 . It follows from " Proposition 71 that the mappings fn fm are 2R -Lipschitz on U . Thus kfn .x/

fm .x/k kfn .x/ fm .x/ fn .x0 / C fm .x0 /k C kfn .x0 / fm .x0 /k " " kx x0 k C " 2R 2 for every x 2 U and m; n n0 . Hence there is f W U ! Y such that fn ! f uniformly on U .

36

Chapter 1. Fundamental properties of smoothness

Further, fix x 2 U , and let V be a punctured neighbourhood of 0 such that xCV U . 1 fn .x C h/ fn .x/ Dfn .x/Œh for h 2 V . Similarly as above Define n .h/ D khk we get km .h/

1

.fn khk

n .h/k

fm /.x C h/

fm /.x/

.fn

1

Dfn .x/ Dfm .x/ Œh " khk R for every h 2 V and m; n n0 . Hence n ! uniformly on h 2 V , where 1 .h/ D khk f .x C h/ f .x/ g.x/Œh . By the Moore-Osgood theorem we have limh!0 .h/ D limn!1 limh!0 n .h/ D 0, which implies that f is Fréchet differentiable at x and Df .x/ D g.x/. t u C

In the next lemma the completeness is crucial. Lemma 86. Let X, Y be isomorphic Banach spaces and let U L.XI Y / be the set consisting of all isomorphisms. Then U is open and ˚ W U ! L.Y I X/, ˚.T / D T 1 , is a C 1 -smooth mapping. 1k 1.

Proof. Let T 2 U, S 2 L.X I Y /, kS k < kT .T C S/

1

D

1 X

. 1/n .T

1

Then S/n T

1

:

(20)

nD0

Indeed, since kT .T C S /

1 X

1S k

< 1, the series is absolutely convergent. We have

. 1/n .T

1

1

C

S/n T

1

nD0

D Id C ST

1 X

. 1/n S.T

1

S/n

1

1

T

1

C S.T

S/n T

1

D Id:

nD1

A similar calculation works when composing the sum with T C S the other way round, which shows that (20) holds. It follows that U is open. If kS k < kT 1 k 1 , then we have

1

X

kT 1 k3 kSk2

n 1 n 1 . 1/ .T S/ T :

1 kT 1 kkS k nD2

Thus .T C S/

1

T

1

D

T

1

T

1

ST

1

ST

1

C

1 X

. 1/n .T

1

S/n T

1

nD2

D

C o.kS k/; S ! 0;

which implies that ˚ is Fréchet differentiable at T and D˚.T /ŒS D

T

1

ST

1

;

D˚.T / 2 L L.X I Y /I L.Y I X/ :

Section 3. Fréchet smoothness

37

To show that the mapping ˚ is C 1 -smooth consider the bounded bilinear mapping W L.Y I X / L.Y I X/ ! L L.X I Y /I L.Y I X/ , .P; Q/.S/ D P B S B Q. By Proposition 79, is C 1 -smooth. We have D˚.T / D .T 1 ; T 1 / D ˚.T /; ˚.T / : This formula shows that ˚ is C 1 -smooth. Further, by Theorem 83 it is clear that if ˚ is C k -smooth, then it is also C kC1 -smooth, which proves the claim via induction on k. t u Theorem 87 (Implicit function theorem). Let X be a normed linear space, Y and Z Banach spaces, G X ˚ Y an open set, and f 2 C 1 .GI Z/. Let .x0 ; y0 / 2 G and let D2 f .x0 ; y0 / be an isomorphism of Y onto Z. Then there are open neighbourhoods U of x0 in X and V of y0 in Y such that there is a unique mapping u W U ! V satisfying f x; u.x/ D f .x0 ; y0 / for all x 2 U , this mapping u is C 1 -smooth on U , and 1 Du.x/ D D2 f x; u.x/ B D1 f x; u.x/ (21) for all x 2 U . If f is moreover C k -smooth for some k 2 N [f1g, then u 2 C k .U I Y /. Proof. We may assume without loss of generality that f .x0 ; y0 / D 0. We denote S D D2 f .x0 ; y0 / 2 L.Y I Z/ and define g W G ! Y by g.x; y/ D y

S

1

B f .x; y/:

Notice that f .x; y/ D 0 if and only if g.x; y/ D y, and that D2 g.x0 ; y0 / D 0. Since f and g are C 1 -smooth, there exist an open neighbourhood U of x0 and r > 0 such that U B.y0 ; r/ G and for x 2 U , y 2 B.y0 ; r/ the following holds: D2 f .x; y/ is an isomorphism, kD2 g.x; y/k 12 , and kg.x; y0 / y0 k < 2r . Put V D U.y0 ; r/. By Proposition 71, y 7! g.x; y/ is 12 -Lipschitz on B.y0 ; r/ for every x 2 U . It follows that kg.x; y/ y0 k kg.x; y/ g.x; y0 /k C kg.x; y0 / y0 k < 1 y0 k C 2r r whenever x 2 U , y 2 B.y0 ; r/. So, if x 2 U , then y 7! g.x; y/ 2 ky is a contraction of B.y0 ; r/ into V B.y0 ; r/ and thus by the Banach contraction principle there is a unique mapping u W U ! V such that g x; u.x/ D u.x/, or equivalently f x; u.x/ D 0. Further, the mapping u is continuous. Indeed, choose any a 2 U and " > 0. Since by the choice of U the mapping D2 f a; u.a/ is an isomorphism, by the just proved part of the theorem applied to the mapping f restricted to U U.u.a/; "/ \ V and a point a; u.a/ , there is a neighbourhood W of a such that x; u.x/ 2 U U.u.a/; "/ whenever x 2 W . Note that here we substantially used the uniqueness of the mapping u. Next we prove that u is Fréchet differentiable on U . For a fixed x 2 U choose ı > 0 such that B.x; ı/ U . Let h W B.0; ı/ ! Y be the continuous mapping h.´/ D u.x C ´/ u.x/. We have f x C ´; u.x/ C h.´/ D 0, so by Proposition 81

D1 f x; u.x/ Œ´ C D2 f x; u.x/ Œh.´/

D f x C ´; u.x/ C h.´/ f x; u.x/ Df x; u.x/ ´; h.´/ D o ´; h.´/ ; ´ ! 0:

38

Chapter 1. Fundamental properties of smoothness

Recall that T D D2 f x; u.x/ is an isomorphism. We deduce

u.x C ´/ u.x/ C T 1 B D1 f x; u.x/ Œ´

D h.´/ C T 1 B D1 f x; u.x/ Œ´ (22)

kT 1 k D2 f x; u.x/ Œh.´/ C D1 f x; u.x/ Œ´ D o k´k C kh.´/k ; ´ ! 0:

In particular, for ´ sufficiently small we obtain h.´/ C T 1 B D1 f x; u.x/ Œ´ 1 2 k´k C kh.´/k , so by the triangle inequality

kh.´/k k´kC2 T 1 B D1 f x; u.x/ Œ´ 1C2 T 1 B D1 f x; u.x/ k´k: Plugging this back into (22) we finally obtain

u.x C ´/ u.x/ C T 1 B D1 f x; u.x/ Œ´ D o.k´k/; ´ ! 0; which proves that u is Fréchet differentiable at x and (21) holds. The C k -smoothness of the mapping u follows by a similar bootstrapping argument as in the proof of Lemma 86. Denote by ˚ the mapping from Lemma 86 and by W L.ZI Y / L.XI Z/ ! L.XI Y / the composition mapping .P; Q/ D P B Q, and note that both ˚ and are C 1 -smooth. Then the formula (21) can be rewritten as Du.x/ D ˚ D2 f x; u.x/ ; D1 f x; u.x/ : Since u is continuous on U , this formula immediately implies that u 2 C 1 .U I Y /. Now suppose that f is C k -smooth. If u is C j -smooth on U for some j < k, then from Theorem 83 and the above formula it follows that Du is C j -smooth on U and hence u is C j C1 -smooth on U . Thus u 2 C k .U I Y / by induction. t u Definition 88. Let Y be a normed linear space over K. Then every M 2 L. kKI Y / is of the form .t1 ; : : : ; tk / 7! t1 tk M.1; : : : ; 1/ and so there is a canonical isometry L. kKI Y / Š Y given by M 7! M.1; : : : ; 1/. Thus if f W U ! Y , where U K, is k-times Fréchet differentiable at a 2 U , then each derivative D jf .a/, j D 1; : : : ; k, can be identified with an element of Y . We will use the traditional notation f 0 , f 00 , f .j / for this identification, i.e. D jf .a/Œt1 ; : : : ; tj D t1 tj f .j / .a/, which corresponds to the classical notion of derivative in one dimension. Later, we will use the fundamental theorem of calculus in the Banach space setting. The integral below is the Bochner integral. Theorem 89. Let Y be a real Banach space, Œa; b R an interval, and f W Œa; b ! Y a continuous mapping which is C 1 -smooth on .a; b/. Then Z b f .b/ f .a/ D f 0 .t/ dt a

whenever the right-hand side is defined (in particular whenever f 0 can be continuously extended to Œa; b).

Section 3. Fréchet smoothness

39

Proof. For any 2 Y we have f .b/

f .a/ D B f .b/ b

Z D

a

Z B f .a/ D

f 0 .t/ dt D

a b

Z

b

. B f /0 .t/ dt

f 0 .t/ dt ;

a

where we used the real valued fundamental theorem of calculus (see e.g. [RudinW, Theorem 7.21]) and the fact that the Bochner integral commutes with continuous linear operators. Since Y separates the points of Y , the assertion follows. t u We close this section with a theorem on differentiation under the integral sign. By a slight abuse of notation in what follows we will denote by D1 f .x; t/ the derivative of the mapping ´ 7! f .´; t/ at x, although f will not be defined on an open subset of a product of two normed linear spaces (as in Definition 80). Theorem 90. Let .˝; / be a measure space, X a normed linear space, Y a Banach space, U X an open connected set, and f W U ˝ ! Y . Suppose that t 7! f .x; t/ is (strongly) -measurable for each x 2 U and x 7! f .x; t/ is Fréchet differentiable on U for -almost all t 2 ˝. Further, suppose that there is g 2 L1 ./ such that for -almost all t 2 ˝ we have kD1 f .x; t/k g.t/ for all x 2 U , and that t 7! f .x0 ; t/ is Bochner integrable for some x0 2 U . ThenR t 7! f .x; t/ is Bochner integrable for all x 2 U , the mapping FR W U ! Y , F .x/ D ˝ f .x; t/ d.t/ is Fréchet differentiable on U , and DF .x/ D ˝ D1 f .x; t/ d.t/ for all x 2 U . If x 7! f .x; t/ is moreover C 1 -smooth on U for almost all t 2 ˝, then so is F . Proof. Fix any x 2 U . Since U is connected and open, there is a polygonal path Œx0 ; x1 ; : : : ; xn in U , where xn D x. For almost all t 2 ˝ the mapping y 7! f .y; t/ is g.t /-Lipschitz on each of thePline segments Œxj 1 ; xj (Proposition 71). Hence kf .x; t /k kf .x0 ; t/k C g.t/ jnD1 kxj xj 1 k for almost all t 2 ˝. Since by the assumptions kf .x0 ; /k 2 L1 ./ and g 2 L1 ./, the mapping t 7! f .x; t/ is Bochner integrable. Further, find ı > 0 such that U.x; ı/ U . Note that the mapping y 7! f .y; t/ is g.t /-Lipschitz on U.x; ı/ for almost all t 2 ˝ again by Proposition 71. Define r W U.0; ı/ ˝ ! Y by r.h; t/ D f .x C h; t/ f .x; t/ D1 f .x; t/Œh whenever D1 f .x; t / is defined, r.h; t/ D 0 otherwise (this occurs for t in a set of measure zero). Then for almost all t 2 ˝ we have r.h; t/ D o.khk/; h ! 0 and also kr.h; t /k 2g.t/khk. Using the fact that the evaluation operator "h W L.X I Y / ! Y defined by "h .L/ D L.h/ is a bounded linear operator and hence commutes with the Bochner integration, we obtain

R

F .x C h/ F .x/

˝ D1 f .x; t/ d.t/ Œh khk

Z

R

f .x; t/ D1 f .x; t/Œh d.t/ kr.h; t/k ˝ f .x C h; t/ d.t /: D khk khk ˝

40

Chapter 1. Fundamental properties of smoothness

By the Lebesgue dominated convergence theorem the right-hand side tends to zero as h ! 0, which shows the Fréchet differentiability. The C 1 -smoothness follows by another application of the Lebesgue dominated convergence theorem. t u Corollary 91. Let be a finite Borel measure on a compact space K, X a normed linear space, Y a Banach space, U X open, f W U K ! Y , and k 2 N [ f1g. Suppose that t 7! f .x; t/ is Bochner integrable for all x 2 U . Put u t .x/ D f .x; t/. Suppose that u t 2 C k .U I Y / for all t 2 K and .x; t/ 7! D ju t .x/ is locally bounded j on U K for all 1 j k (this holds R in particular if .x; kt/ 7! D u t .x/ are continuous). Then the mapping F .x/ D K f .x; t/ d.t/ is C -smooth on U and R D jF .x/ D K D ju t .x/ d.t/ for 1 j k. Proof. Fix x 2 U . By the compactness there is ı > 0 and M > 0 such that kDu t .y/k M for all y 2 U.x; ı/ and t 2 K. Thus we may apply Theorem 90 on U.x; ı/ K for g.t/ D M . The higher order smoothness follows by an easy induction. t u

4. Taylor polynomial Since higher derivatives are symmetric multilinear mappings, they are in one-to-one correspondence with homogeneous polynomials. So using differentials, i.e. homogeneous polynomials, as objects representing derivatives leads to no loss of information. Moreover, we benefit from the fact that the domain of the differentials considered as polynomials will remain the same underlying space. In fact, the most important method for applying higher smoothness is through using the Taylor formula, which locally approximates the mapping by a polynomial. Definition 92. Let X, Y be normed linear spaces, U X open, f W U ! Y , k 2 N, and a 2 U . Suppose that D kf .a/ exists. The kth differential of f at a is defined

2

by d kf .a/ D D kf .a/, i.e. it is the k-homogeneous polynomial which corresponds to the kth derivative of f at a. For the evaluation at h 2 X we use the notation d kf .a/Œh D D kf .a/Œkh. For convenience we denote d 0f D f , df D d 1f . Lemma 93. Let X, Y be normed linear spaces, U X open, f W U ! Y , k 2 N, k 1 and a 2 U . Then d kf .a/ exists if and only if D.d f /.a/ exists. In this case k k 1 d f .a/Œh D D.d f /.a/Œh Œh for every h 2 X. Proof. Let us denote by I W P . k 1X I Y / ! Ls . k 1XI Y / the canonical isomorphism } from the Polarisation formula (Proposition 13). Then D k 1f D I B d k 1f I.P / D P k 1 and d f D I 1 B D k 1f and so the equivalence follows from the Chain rule (Theorem 69). Also, D.D k 1f /.a/ D I B D.d k 1f /.a/ (Fact 67), and so d kf .a/Œh D D kf .a/Œkh D D.D k 1f /.a/Œh Œk 1h D I D.d k 1f /.a/Œh Œk 1h D D.d k 1f /.a/Œh Œh: t u

Section 4. Taylor polynomial

41

Notice that by the Polarisation formula (Proposition 13) f 2 C k .U I Y / if and only if the mappings x 7! d jf .x/, j D 1; : : : ; k, are continuous on U . Before going further and proving the main results of this section we have to make a slight digression and define the Gâteaux and directional derivatives of higher order. Although these notions are not as important for the general theory as the Fréchet derivative, they are indispensable for the computation of concrete derivatives (see Fact 98). Definition 94. Let X, Y be normed linear spaces, U X open, f W U ! Y , and x 2 U . We denote the (first) Gâteaux derivative of f by ı 1f D ıf and define inductively the kth Gâteaux derivative ı kf .x/ 2 Ls . kX I Y / at x provided that ı k 1f exists in some neighbourhood of x and the limit ı k 1f .x C th1 /Œh2 ; : : : ; hk t !0 t

ı kf .x/Œh1 ; : : : ; hk D lim

ı k 1f .x/Œh2 ; : : : ; hk

t 2K

exists for all h1 ; : : : ; hk 2 X and is a symmetric bounded k-linear mapping. We say that a mapping is G k -smooth in the domain if it has the kth Gâteaux derivative at each point therein. We say that a mapping is G 1 -smooth if it is G k -smooth for each k 2 N. By induction it is easy to see that if a mapping f is k-times Fréchet differentiable at a point x, then it is also k-times Gâteaux differentiable at x and ı kf .x/ D D kf .x/. The converse is false in general. Definition 95. Let X , Y be normed linear spaces, A X , f W A ! Y , and x 2 A. 1 We denote the directional derivative of f in the direction h 2 X by @@hf D @f and we @h define inductively the directional derivatives of higher order by ! @ @k 1f @kf .x/ D .x/ 2 Y; @hk @hk 1 : : : @h1 @hk @hk 1 : : : @h1 where h1 ; : : : ; hk 2 X . The order of hj s in taking the directional derivatives in general matters, but under various assumptions hj s can be reordered not affecting the final value. In particular, k

f .y/ are continuous on a this is the case whenever the mappings y 7! @h @:::@h .1/ .k/ neighbourhood of x for every permutation of f1; : : : ; kg, or whenever the mappings kf y 7! @h @:::@h .y/ are continuous at x for all 2 f1; : : : ; kgk . This follows from .k/ .1/ the classical theorems on differentiation in finite dimensions; the former fact follows by induction from the theorem on interchangeability of partial derivatives of the second order, the latter fact follows from the next theorem combined with Theorem 76 and Fact 98 below.

Theorem 96. Let U Rn be open, Y a normed linear space, f W U ! Y , and suppose that all the partial derivatives of f of order k 2 N exist on U and are continuous at x 2 U . Then f is k-times Fréchet differentiable at x.

42

Chapter 1. Fundamental properties of smoothness

Sketch of proof. By shrinking the U if necessary we may assume that all the partial derivatives of order k are bounded on U . Without loss of generality we may assume that U is convex and hence by Proposition 71 the partial derivatives of order up to k 1 are coordinatewise Lipschitz on U and thus continuous on U . The statement now follows by using inductively Theorem 82 and Fact 98 below. t u The following fact follows from the definition using induction. Fact 97. Let X, Y be normed linear spaces over K, A X, f W A ! Y , x 2 A, @kf h 2 X , and k 2 N. Then @h:::@h .x/ exists if and only if the mapping g.t/ D f .x C th/ (defined for t 2 K satisfying x C th 2 A) is k-times differentiable at 0. In this case @kf .x/ D g .k/ .0/ D .gR /.k/ .0/. @h:::@h The following instrumental fact also follows by an easy induction. Fact 98. Let X, Y be normed linear spaces, U X open, f W U ! Y , x 2 U , and suppose that ı kf .x/ exists. Then @kf .x/: ı kf .x/Œh1 ; : : : ; hk D @h1 @h2 : : : @hk This fact may be used in order to obtain the values of derivatives of k-times differentiable mappings. For example we can easily compute the derivatives of a polynomial. Lemma 99. Let X and Y be normed linear spaces and let P 2 P . nXI Y /. Then P 2 C 1 .XI Y /, @P }.n 1x; h/; and .x/ D nP @h }.n 1x; h/ DP .x/Œh D nP for every x; h 2 X . More generally, for k n we have D kP .x/Œh1 ; : : : ; hk D n.n

1/ .n

}.n kx; h1 ; : : : ; hk /; k C 1/P

d kP .x/Œh D n.n

1/ .n

}.n kx; kh/: k C 1/P

Thus D kP W X ! Ls . kXI Y / is a polynomial mapping from P n kXI Ls . kXI Y / , while d kP W X ! P . kXI Y / is a polynomial mapping from P n kXI P . kXI Y / . In particular x 7! d nP .x/ is a constant mapping whose value is a polynomial nŠP , and so d nC1P D 0. Proof. The polynomial P is C 1 -smooth by Proposition 79 and Theorem 83. By Lemma 21 we have ! n 1 1 X n n j}j n j 1 P .x C th/ P .x/ D t P . x; h/ : t t j j D0

So limiting for t ! 0 we immediately obtain the first order formulae. Note that in the first formula for a fixed direction h the resulting mapping x 7! @P .x/ belongs to @h P . n 1XI Y /, whereas in the second formula DP 2 P n 1X I L.X I Y / . The higher order formulae follow by a similar argument using induction and Fact 98. t u

Section 4. Taylor polynomial

43

By an easy induction we obtain the following lemma. Lemma 100. Let X , Y be normed linear spaces, U X , f W U ! Y , x 2 U , and @kf h1 ; : : : ; hk 2 X. Suppose that @h1 @h .x/ exists. Then 2 :::@hk 1 @kf k .x/ D lim lim : : : lim f .xI t1 h1 ; : : : ; tk hk / : tk !0 t1 tk t1 !0 t2 !0 @h1 @h2 : : : @hk A version of the following theorem was proved by Jacek Bochnak in [Boc] under the additional assumption that the finite-dimensional restrictions are C k -smooth. Theorem 101. Let X be a Banach space, Y a normed linear space, U X open, f W U ! Y , and let k 2 N be such that dim X k C 1. Suppose further that B f is Baire measurable for every 2 Y . Then f is G k -smooth on U if and only if for every .k C 1/-dimensional affine subspace E X the mapping f E \U is G k -smooth. In this case ı kf .x/F k D ı k.f .xCF /\U /.x/ (23) for every x 2 U and every subspace F X satisfying dim F k C 1. Proof. ) is clear. ( We prove this implication by induction on k. Because the proof of the case k D 1 is almost identical to the induction step, we show only the induction step from k 1 to k. Fix x 2 U and let M W X k ! Y be defined as M.h1 ; : : : ; hk / D ı k.f .xCE /\U /.x/Œh1 ; : : : ; hk ;

where E D spanfhj gjkD1 .

It is easy to check that M 2 Ls . kX I Y / using an analogue of Fact 77 for Gâteaux derivatives. Next, we show that M is continuous. Let ftn g be a decreasing sequence of reals such that tn ! 0 and B.x; kt1 / U , and let 1 kf .xI tn1 h1 ; : : : ; tnk hk /; hj 2 BX : fn1 ;:::;nk .h1 ; : : : ; hk / D tn1 tnk From Lemma 100 we get M.h1 ; : : : ; hk / D lim lim : : : lim fn1 ;:::;nk .h1 ; : : : ; hk / n1 !1

n2 !1

nk !1

whenever hj 2 BX . Choose any 2 The functions B fn1 ;:::;nk W BXk ! K are separately Baire measurable (i.e. Baire measurable in each variable) by Lemma 52. Thus B M B k is separately Baire measurable. Similarly we obtain that B M nB k X X is separately Baire measurable for any n 2 N and consequently B M is separately Baire measurable. By Theorem 7 we can conclude that M is continuous. Now choose arbitrary h1 ; : : : ; hk 2 X and put E D spanfhj gjkD1 . Then by the inductive hypothesis ı k 1f .y/Œh2 ; : : : ; hk D ı k 1.f .xCE /\U /.y/Œh2 ; : : : ; hk for every y 2 .x C E/ \ U . Hence @ k 1 ı f ./Œh2 ; : : : ; hk .x/ D ı k.f .xCE /\U /.x/Œh1 ; : : : ; hk D M.h1 ; : : : ; hk /; @h1 Y .

44

Chapter 1. Fundamental properties of smoothness

which shows that f is k-times Gâteaux differentiable at x and ı kf .x/ D M . The formula (23) follows again from an analogue of Fact 77 for Gâteaux derivatives. u t We note that the assumption on the dimension of the affine subspaces in the previous 2y theorem is in general necessary: the function f .x; y/ D x 2xCy 2 , f .0; 0/ D 0, is C 1 -smooth on every line in R2 , but it is not Gâteaux differentiable at the origin. However, the complex case is dramatically different, see Theorem 160. Similarly, the measurability assumption is not superfluous, as can be seen by considering any discontinuous linear functional. Corollary 102. Let X be a Banach space, Y a normed linear space, U X open, and f W U ! Y . Suppose further that B f is Baire measurable for every 2 Y . Then f is G 1 -smooth on U if and only if for every finite-dimensional subspace Z X the mapping f Z\U is G 1 -smooth. Proof. If X is finite-dimensional, then the statement is obvious. Otherwise just apply Theorem 101. t u Armed with the computational tools we can return to our main theme. The following result is essential for the theory. Theorem 103 (Peano’s form of Taylor’s theorem). Let X, Y be normed linear spaces, U X an open set, f W U ! Y , a 2 U , k 2 N, and suppose that D kf .a/ exists. Then

k

X 1 j

d f .a/Œx a D o.kx akk /; x ! a:

f .x/

jŠ j D0

Proof. We use induction on k. For k D 1 the statement follows from the definition of the Fréchet derivative. Now suppose that k > 1 and the theorem holds for k 1. Put Pk 1 j a. Then R is differentiable on a neighbourhood R.x/ D f .x/ j D0 j Š d f .a/Œx of a with DR continuous at a, so there is an open ball V D U.a; ı/, ı > 0, such that for every x 2 V kR.x/k D kR.x/

R.a/k kx

ak

kDR.y/k

sup y2B.a;kx ak/

by Proposition 71. To compute DR denote Pj .y/ D DPj .y/Œh D

1 .j

1/Š

D jf .a/Œj 1y; h D

for every y; h 2 X, and therefore DPj .y/ D j D 1; : : : ; k. It follows that for every y 2 V DR.y/ D Df .y/

k X1 j D0

1 j j Š d f .a/Œy.

1 .j

1/Š

Dj

1

.Df /.a/Œj 1y Œh

1 d j 1 .Df .j 1/Š

1 j d .Df /.a/Œy jŠ

Then by Lemma 99

/.a/Œy 2 L.X I Y /,

a:

Section 4. Taylor polynomial

45

Using the inductive hypothesis on the mapping Df we conclude that kDR.y/k D o.ky akk 1 /; y ! a. Therefore kR.x/k kx

ak

sup

kDR.y/k D o.kx

akk /; x ! a:

t u

y2B.a;kx ak/

P The polynomial jkD0 j1Š d jf .a/ 2 P k .X I Y / is called the Taylor polynomial of order k of the mapping f at a. Proposition 104. Let X, Y be normed linear spaces, U X open, f W U ! Y , and a 2 U . Suppose that there are Pj 2 P . jXI Y /, j D 0; : : : ; k, such that

k

X

Pj .x a/ D o.kx akk /; x ! a:

f .x/

j D0

Then the polynomials Pj are uniquely determined. In particular, if D kf .a/ exists, then Pj D j1Š d jf .a/, j D 0; : : : ; k. Proof. Suppose that there are polynomials Qj 2 P . jXI Y /, j D 0; : : : ; k, such that

f .x/ Pk Qj .x a/ D o.kx akk /; x ! a. Then j D0

k

X

.Qj Pj /.x a/ D o.kx akk /; x ! a:

j D0

Since the estimated mapping is a polynomial of degree at most k, it must be identically zero (Fact 48), and so by Corollary 32 we get Qj D Pj . The final statement follows from Theorem 103. t u The above proposition says that the Taylor polynomial of order k of a C k -smooth mapping is uniquely determined among all polynomials of degree at most k by the property of giving the (best) estimate of order o.kx akk /. This fact can be used conveniently for obtaining various formulae for higher derivatives, see e.g. Corollary 116, Corollary 117. Next we show that if f is C k -smooth, then the remainder in Taylor’s theorem can be given by an integral formula. Lemma 105. Let I R be an open interval, let X , Y , Z be real normed linear spaces, k 2 N, f 2 C k .I I X/, g 2 C k .I I Y /, and 2 L.X; Y I Z/. Then !0 k X1 . 1/j .f .j / ; g .k j 1/ / D .f; g .k/ / . 1/k .f .k/ ; g/ on I . (24) j D0

0 Proof. Using Theorem 83 and Fact 67 we get .f0 ; g0 / D .f00 ; g0 / C .f0 ; g00 / on I for any f0 2 C 1 .I I X/, g0 2 C 1 .I I Y /. Applying this formula to compute the derivative on the left-hand side of (24), the sum telescopes to the right-hand side of (24). t u

46

Chapter 1. Fundamental properties of smoothness

Proposition 106. Let I R be an open interval, Y a real Banach space, k 2 N, and f 2 C k .I I Y /. Then for any pair of points a; x 2 I we have f .x/ D

k X1 j D0

.x

a/j .j / f .a/ C jŠ

x

Z a

.x t/k 1 .k/ f .t/ dt: .k 1/Š

Proof. By applying Lemma 105 to the bilinear mapping W Y R ! Y defined k 1 by .p; q/ D q p, the function g.t/ D .x.k t/1/Š , and the mapping f , using g .k

j 1/ .t / k X1

j 1 .x t/j jŠ

D . 1/k

. 1/

k 1 .x

j D0

we obtain

t/j .j / f .t/ jŠ

!0 D . 1/kC1

.x t/k 1 .k/ f .t/; .k 1/Š

t 2 I:

Integrating both sides in t between a and x in the Bochner sense and using Theorem 89 we obtain the desired formula. t u Theorem 107 (Taylor formula). Let X be a normed linear space, Y a Banach space, U X an open convex set, k 2 N, and f 2 C k .U I Y /. Then for any x 2 U and h 2 X satisfying x C h 2 U we have f .x C h/ D

k X1 j D0

1 j d f .x/Œh C jŠ

1

Z 0

! .1 t/k 1 k d f .x C th/ dt Œh: .k 1/Š

Proof. We apply Proposition 106 to the mapping g W . ı; 1 C ı/ ! YR , where YR is the real version of the space Y in case that K D C, defined for a sufficiently small ı > 0 by the formula g.t/ D f .x C th/. Using Fact 98 and Fact 97 we obtain g .j / .t / D D jf .x C th/Œjh D d jf .x C th/Œh. Thus f .x C h/ D

k X1 j D0

1 j d f .x/Œh C jŠ

1

Z 0

.1 t/k 1 k d f .x C th/Œh dt: .k 1/Š

To finish, consider the evaluation operator "h W P . kX I Y / ! Y , "h .P / D P .h/. Then obviously "h 2 L.P . kXI Y /I Y / and hence the evaluation commutes with Bochner integration. t u Corollary 108. Let X , Y be normed linear spaces, U X an open convex set, k 2 N, and f 2 C k .U I Y /. Then for any x 2 U and h 2 X satisfying x C h 2 U we have

f .x C h/

k

X

1 j 1

d f .x/Œh sup d kf .x C th/

kŠ t2Œ0;1 jŠ

j D0

d kf .x/ khkk :

Section 4. Taylor polynomial

47

Proof. Working with the completion of Y if necessary we may use Theorem 107 to obtain

k

X 1

d jf .x/Œh

f .x C h/

jŠ j D0

Z

1 .1 t/k 1

1 k

k d f .x C th/ dt d f .x/ khkk

0 .k 1/Š

kŠ ! Z 1

.1 t/k 1 k k

d f .x C th/ d f .x/ dt khkk ; .k 1/Š 0 t u

from which the estimate follows.

Let us spell out the connections between our abstract formalism and the classical theory of partial derivatives. In case f is defined on a subset of Kn , Pnthat the mapping i.e. f .x/ D f .x1 ; : : : ; xn / D f j D1 xj ej , we will use the classical notation for the operator of partial derivatives (derivatives along the canonical basis vectors ej ) @j˛jf .x0 /: : : : @xn˛n

@x1˛1

Let U Kn be an open set and f W U ! Y , where Y is a normed linear space over K. If f is m-times Fréchet differentiable at a 2 U , then combining Proposition 23 and Fact 98 we obtain the classical equality X k @kf .a/x ˛ d kf .a/Œx D ˛ @x1˛1 : : : @xn˛n ˛2I.n;k/

for k m. Combining this with Theorem 103 we obtain the classical multivariate Taylor formula f .a C x/ D

m X

X

kD0 ˛2I.n;k/

1 @kf .a/x ˛ C o.kxkm /; x ! 0: ˛1 Š ˛n Š @x1˛1 : : : @xn˛n

Definition 109. Let X, Y be normed linear spaces, U X open, f W U ! Y , and p 2 Œ0; C1/. We say that f is T p -smooth at x 2 U if there exists a polynomial P 2 P Œp .XI Y / satisfying P .0/ D f .x/ and f .x C h/

P .h/ D o.khkp /; h ! 0:

(25)

We say that f is T p -smooth on U if it is T p -smooth at every point x 2 U . Notice a few things. First, the polynomial in (25) is uniquely determined by Proposition 104. Next, it is easy to see that T p -smoothness implies T q -smoothness for q < p. Finally, if f is T 1 -smooth at x, then f is Fréchet differentiable at x with Df .x/ D P1 , the 1-homogeneous term of P .

48

Chapter 1. Fundamental properties of smoothness

Theorem 103 implies that a C k -smooth mapping is also T k -smooth. The converse is not true in general: consider f W R ! R, f .x/ D x kC1 sin x1k , f .0/ D 0. Then f is T k -smooth but not even C 1 -smooth. Nevertheless, under certain uniformity assumptions the converse does hold. Theorem 110 (Converse Taylor theorem). Let X, Y be normed linear spaces, U X an open set, f W U ! Y , and k 2 N0 . Then f 2 C k .U I Y / if and only if f is a T k -smooth mapping satisfying lim

.y;h/!.x;0/ h¤0

kR.y; h/k D0 khkk

(26)

for every x 2 U , where R.x; h/ D f .x C h/ P x .h/ and where the polynomials P x 2 P k .XI Y / come from the definition of T k -smoothness of f at x. In this case P P x D jkD0 j1Š d jf .x/. P Proof. ) If f 2 C k .U I Y /, then f is T k -smooth, and P x D jkD0 j1Š d jf .x/ by Proposition 104. Fix x 2 U and choose any " > 0. Let ı > 0 be such that B.x; 2ı/ U and kd kf .´/ d kf .x/k < " for ´ 2 B.x; 2ı/. By Corollary 108

k

2" 1 k

sup d f .´/ d f .y/ khkk khkk kR.y; h/k kŠ ´2B.y;ı/ kŠ whenever y 2 B.x; ı/ and h 2 B.0; ı/, from which (26) follows. ( We use induction on k. For k D 0 the assertion is obvious, since both T 0 -smoothness and C 0 -smoothness mean just the continuity of f . So assume that k 2 N and the theorem holds for k 1. Fix x 2 U and let ı > 0 be such that U.x; 2ı/ U . We have f .x C h C y/ D P x .h C y/ C R.x; h C y/; f .x C h C y/ D P xCh .y/ C R.x C h; y/

(27)

for all h; y 2 U.0; ı/. Set q.h; y/ D P xCh .y/ P x .h C y/. Denote by Pj´ the j -homogeneous summands of P ´ for j D 0; : : : ; k. By Lemma 21 we can write P q.h; y/ D jkD0 qj .h; y/, where qj .h; y/ D

PjxCh .y/

z

k X l Plx .l j

j

h; jy/:

lDj

Note that q.h; / 2 P k .X I Y / and qj .h; / 2 P . jXI Y /, j D 0; : : : ; k, for h 2 U.0; ı/. By subtracting the equalities (27) we obtain q.h; y/ D R.x; h C y/ R.x C h; y/. Thus for any h; y 2 U.0; ı/ such that kyk khk, y ¤ 0, and y ¤ h we obtain kq.h; y/k kR.x; h C y/k C kR.x C h; y/k kR.x C h; y/k k kR.x; h C y/k 2 C khkk : kh C ykk kykk

Section 4. Taylor polynomial

49

It follows (using also simpler versions of the above estimate if y D 0 or y D

h) that

k

kq.h; y/k D o.khk /; .h; y/ ! .0; 0/; kyk khk: Recall that by Corollary 31 there are constants Kn;j such that

kQj .x/k Kn;j max Q nl x 0ln

P n .X I Y /,

whenever Q 2 x 2 X, and j 2 f0; : : : ; ng, where Qj is the j -homogeneous summand of Q. Applying this we get kqj .h; y/k Kk;j max kq.h; kl y/k for 0lk all h 2 U.0; ı/, y 2 Y , and j 2 f0; : : : ; kg. Therefore kqj .h; y/k D o.khkk /; .h; y/ ! .0; 0/; kyk khk: So finally by taking the supremum over y 2 B.0; khk/ and using the j -homogeneity of qj .h; / we obtain kqj .h; /k D

1 khkj

D o.khk

sup kqj .h; y/k D kykkhk k j

kqj .h; y/k khkj kykkhk sup

(28)

/; h ! 0

for each j 2 f0; : : : ; kg. Since qk .h; / D PkxCh Pkx , it follows that the mapping x 7! Pkx is continuous on U . Further, since for h ¤ 0

y

f .y C h/ Pk 1 P y .h/ kR.y; h/k C kPk .h/k j D0 j khkk 1 khkk 1 kR.y; h/k y C kPk k khk; khkk the continuity of x 7! Pkx implies

lim

.y;h/!.x;0/ h¤0

1 j D0 k 1 khk

kf .yCh/

Pk

y

Pj .h/k

D 0 and so by

the inductive hypothesis f is C k 1 -smooth and Pjx D j1Š d jf .x/, j D 0; : : : ; k Thus 1 d k 1f .x C h/Œy d k 1f .x/Œy k Pkx .h; k 1y/ qk 1 .h; y/ D .k 1/Š and from (28) we get

k 1

d f .x C h/ d k 1f .x/ kŠP x .h; ; : : : ; / D o.khk/; h ! 0: k

1.

By Lemma 93 this means that d kf .x/ D kŠPkx , which finishes the proof.

t u

z

z

Definition 111. Let X and Y be normed linear spaces, A X, f W A ! Y , and p 2 Œ0; C1/. We say that f is weakly T p -smooth at x 2 A if there exists a polynomial P 2 P Œp .XI Y / satisfying P .0/ D f .x/ and f .x C th/ P .th/ D o.jtjp /; t ! 0 for each h 2 X. We say that f is weakly T p -smooth on A if it is weakly T p -smooth at every x 2 A.

50

Chapter 1. Fundamental properties of smoothness

We say that f is directionally T p -smooth at x 2 A if the mapping t 7! f .x C th/ is T p -smooth at 0 for each h 2 X , i.e. for each h 2 X there is P x;h 2 P Œp .KI Y / satisfying P x;h .0/ D f .x/ and f .x C th/ P x;h .t/ D o.jtjp /; t ! 0. Finally, we say that f is directionally T p -smooth on A if it is directionally T p -smooth at every x 2 A. Note that again all the polynomials in the above definitions are uniquely determined by Proposition 104. In particular, if the mapping f is directionally T p -smooth at x, then P x;h .t / D P x;th .1/ for every h 2 X and t 2 R. Obviously, if f is T p -smooth at x, then it is weakly T p -smooth at x, and if f is weakly T p -smooth at x, then it is directionally T p -smooth at x. Also, f is weakly T 1 -smooth at x if and only if f is Gâteaux differentiable at x. Proposition 112. Let X, Y be normed linear spaces, U X an open set, f W U ! Y , and k 2 N. If f is G k -smooth at x 2 U , then f is weakly T k -smooth at x with P P D jkD0 j1Š ı jf .x/.

2

Proof. Follows from Fact 98 and Fact 97 combined with Theorem 103.

t u

The following fact is immediate. Fact 113. Let X , Y be normed linear spaces over K, A X , p 2 Œ0; C1/, let f; g W A ! Y be (weakly; directionally) T p -smooth at a 2 A, and c 2 K. Then f C g and c f are (weakly; directionally) T p -smooth at a. Proposition 114. Let X , Y be normed linear spaces over K, U X an open set, f W U ! K, g W U ! Y , x 2 U , and p 2 Œ0; C1/. Suppose that f and g are T p -smooth at x with f .x C h/ P .h/ D o.khkp / and g.x C h/ Q.h/ D o.khkp /, Pk Pk n h ! 0, where P D nD0 Pn , Pn 2 P . X/, P .0/ D f .x/, Q D nD0 Qn , Qn 2 P . nXI Y /, Q.0/ D g.x/, and k D Œp. Then the mapping f g is T p -smooth P at x with fg.x C h/ R.h/ D o.khkp /; h ! 0 for R D knD0 Rn , Rn 2 P . nX I Y /, where n X Rn D Pn j Qj ; n D 0; : : : ; k: j D0

Proof. Using Fact 27 we collect the n-homogeneous terms up to order k and put the rest into the remainder to obtain ! k ! k X X p p Ql .h/ C o.khk / f .x C h/g.x C h/ D Pj .h/ C o.khk / j D0

D

k X X

lD0

Pj .h/Ql .h/ C o.khkp /; h ! 0:

nD0 j ClDn

It is easy to see that a similar statement holds also in the weak T p -smooth case.

t u

Section 4. Taylor polynomial

51

Proposition 115. Let X , Y , and Z be normed linear spaces, U X and V Y open sets, and p 2 Œ0; C1/. Further, let g W U ! V be T p -smooth at x 2 U and let f W V ! Z be T p -smooth at y D g.x/ with g.x C h/ Q.h/ D o.khkp / P and f .y C h/ P .h/ D o.khkp /; h ! 0, where P D f .y/ C knD1 Pn with P Pn 2 P . nY I Z/ and Q D g.x/ C knD1 Qn with Qn 2 P . nXI Y / for k D Œp. Then the mapping f B g is T p -smooth at x and f B g.x C h/ R.h/ D o.khkp /; h ! 0 P for R D knD0 Rn , Rn 2 P . nXI Z/, where R0 .h/ D f B g.x/ and Rn .h/ D

n X j D1

X

P}j Ql1 .h/; : : : ; Qlj .h/ ;

n D 1; : : : ; k:

1l n Pj s sD1 ls Dn

Proof. Put !.h/ D f .y C h/ P .h/ and .h/ D g.x C h/ neighbourhoods of the origins. Then ! k X f g.x C h/ D f g.x/ C Ql .h/ C .h/

Q.h/ on suitable

lD1

D f .y/ C

k X

k X

Pj

j D1

! Ql .h/ C .h/ C !

lD1

k X

! Ql .h/ C .h/ :

lD1

Using the fact that !.0/ D 0 it is not difficult to check that the last summand is P o.khkp /; h ! 0. Further, P}j j n klD1 Ql .h/ ; n .h/ D o.khknp /; h ! 0 for j D 1; : : : ; k, n D 1; : : : ; j . Hence by Lemma 21 ! k k X X f B g.x C h/ D f .y/ C Pj Ql .h/ C o.khkp /; h ! 0: j D1

lD1

We expand this using (1) into f B g.x C h/ D f .y/ C

k X

X

P}j Ql1 .h/; : : : ; Qlj .h/ C o.khkp /:

j D1 1ls k sD1;:::;j

The formulae for Rn now follow from Lemma 26.

t u

We note that similarly if g is assumed to be only weakly T p -smooth at x, then f B g is weakly T p -smooth at x and the same formula for the composition holds. Corollary 116 (Leibniz formula). Let X , Y be normed linear spaces, U X an open set, f 2 C k .U /, g 2 C k .U I Y /, and a 2 U . Then k X k k j d k .f g/.a/ D d f .a/ d jg.a/: j j D0

52

Chapter 1. Fundamental properties of smoothness

Proof. By Corollary 84 f g 2 C k .U I Y /. By Theorem 103, Proposition 114, and Proposition 104 we have k X 1 k 1 1 d .f g/.a/ D d k jf .a/ d jg.a/: kŠ .k j /Š jŠ

t u

j D0

Corollary 117 (Chain rule). Let X, Y , Z be normed linear spaces, U X and V Y open sets, f 2 C k .V I Z/, g 2 C k .U I Y /, g.U / V , and a 2 U . Then k X X 1 k d k .f B g/.a/ D D jf g.a/ Œd l1g.a/; : : : ; d ljg.a/: jŠ l1 ; : : : ; lj j D1

1l k Pj s sD1 ls Dk

Proof. By Theorem 83, f B g 2 C k .U I Z/. By Theorem 103, Proposition 115, and Proposition 104 we have k X X 1 l 1 k 1 j 1 d .f B g/.a/ D D f g.a/ d 1g.a/; : : : ; d ljg.a/ ; kŠ jŠ l1 Š lj Š j D1

1l k Pj s sD1 ls Dk

t u

from which the formula follows.

Expressing d k .f B g/ using only d jf and d lg is in principle possible by using the Polarisation formula, however the resulting formula will not be a sum of homogeneous polynomials. Theorem 118 ([JohaZa]). Let X be a real Banach space, Y a real normed linear space, U X open, f W U ! Y , and a 2 U . Suppose that B f is Baire measurable for each 2 Y , f is directionally T k -smooth on U , and further R.a C ty; th/ D o.t k /; t ! 0

for each y; h 2 X,

(29)

where R.x; h/ D f .x C h/ P x;h .1/ and where the polynomials P x;h 2 P k .RI Y / come from the definition of the directional T k -smoothness of f at x. Then f is weakly T k -smooth at a. Proof. Without loss of generality we may assume that a D 0. Set P .h/ D P 0;h .1/ for each h 2 X . We show that P 2 P k .XI Y / using Corollary 55. First we show that B P is Baire measurable for each 2 Y . So fix 2 Y . For each h 2 X there are P g0 .h/; : : : ; gk .h/ 2 R such that B P 0;h .t/ D jkD0 gj .h/t j for t 2 R. Note that P B P .h/ D B P 0;h .1/ D jkD0 gj .h/ and so it suffices to show that the functions g0 ; : : : ; gk W X ! R are Baire measurable. Clearly g0 .h/ D B f .0/ for each h 2 X. We proceed by induction, assuming that g0 ; : : : ; gm 1 are Baire measurable for some 1 m k. Since for a fixed h 2 X we have ! k X f .th/ P 0;h .t/ 1 j gj .h/t D lim lim m B f .th/ D 0; t !0 t !0 t tm j D0

Section 4. Taylor polynomial

53

it follows that gm .h/ D lim t !0

1 tm

B f .th/ m

gm .h/ D lim n n!1

Pm

1 j j D0 gj .h/t .

h Bf n

m X1 j D0

In particular,

!

gj .h/ : nj

Hence on each bounded set gm is a pointwise limit of a sequence of Baire measurable functions and so it is Baire measurable. Now we show that P satisfies the formula (10). Fix x; h 2 X and define kC1 X kC1 q.t/ D . 1/kC1 j P .tx C jth/: j j D0

P k .RI Y /. Let r

Clearly q 2 and jtj r. We have

kC1

X

kq.t /k D . 1/kC1

j D0

j

> 0 be such that t.xCj h/ 2 U for all j 2 f0; : : : ; kC1g kC1 P .tx C jth/ j

f .tx C jth/

C f .tx C jth/ P tx;th .j / C P tx;th .j /

kC1 X k C 1 kR.0; tx C jth/k C kR.tx; jth/k j j D0

P kC1 j kC1 P tx;th .j / D 0 by Corollary 55. This for all jt j r, since jkC1 D0 . 1/ j estimate combined with the assumption (29) implies that q.t/ D o.t k /; t ! 0. This means that q is a zero polynomial (Fact 48), and in particular q.1/ D 0, which is what we wanted to prove. t u As a consequence of the above theorem we obtain that the Converse Taylor theorem holds even if we assume only directional T k -smoothness: Theorem 119 ([JohaZa]). Let X be a real Banach space, Y a real normed linear space, U X an open set, f W U ! Y , and k 2 N0 . Then f 2 C k .U I Y / if and only if f is a directionally T k -smooth mapping such that B f is Baire measurable for each 2 Y and kR.y; h/k D0 (30) lim .y;h/!.x;0/ khkk h¤0

for every x 2 U , where R.x; h/ D f .x C h/ P x;h .1/ and where the polynomials P x;h 2 P k .XI Y / come from the definition of the directional T k -smoothness of f P at x. In this case P x;h .t/ D jkD0 j1Š d jf .x/Œth. Proof. It suffices to notice that the assumption (30) implies (29) in Theorem 118 and it also promotes the weak T k -smoothness to T k -smoothness. Hence we can apply Theorem 110. t u

54

Chapter 1. Fundamental properties of smoothness

5. Smoothness classes In this section we lay down definitions of various classes of smooth mappings which will be used throughout the book. We show some relations between them, including the uniform version of the Converse Taylor theorem, and prove some basic arithmetic and composition rules. Let .P; /, .Q; / be metric spaces. The minimal modulus of continuity of a uniformly continuous mapping f W P ! Q is defined as ˚ !f .ı/ D sup .f .x/; f .y//I x; y 2 P; .x; y/ ı for ı 2 RC 0. Clearly, !f is continuous at 0. A modulus is a non-decreasing function ! W Œ0; C1/ ! Œ0; C1 continuous at 0 with !.0/ D 0. The set of all moduli will be denoted by M. We say that f W P ! Q is uniformly continuous with modulus of continuity ! 2 M if !f !. An important subset of all moduli consists of the sub-additive moduli, which we shall denote by Ms . A nice feature of ! 2 Ms is that it is real-valued and uniformly continuous with modulus of continuity !. It is easy to check that the minimal modulus of continuity of a uniformly continuous mapping defined on a convex subset of a normed linear space is sub-additive. Note also that both the sets M and Ms are stable under composition. 0 .0/ D sup t>0 !.t/ Fact 120. Let ! 2 Ms . Then !C t . In particular, if ! is not identically zero, then for each a > 0 there is L > 0 such that !.t/ Lt for t 2 Œ0; a.

Note that it means that Lipschitz mappings on convex bounded sets have the best modulus of continuity. 0 > 0. Proof. If ! is identically zero, then !C .0/ D 0. Otherwise q D sup t>0 !.t/ t !.s/ C Assume that q 2 R and choose any " > 0. Let s 2 R be such that s > q " and let ı > 0 be such that !.ı/ < s". Choose any t 2 .0; ı/ and put n D st . Then s ı < nt s and hence using the sub-additivity of ! we obtain

q

"

0 such that for each x 2 U there is a 2 U satisfying conv fxg [ B.a; r/ U . Fix any x 2 U and h 2 X n f0g r 1 such that x C h 2 U . Put ˛ D diam U and s D ˛khk, and note that ˛ 2 and s r. Let a 2 U be such that conv fx C hg [ B.a; r/ U . Set u D a if h khk otherwise. Note that B.u; s/ U . ka x hk khk and u D x C h C kaa xx hk Indeed, if u D a, then we use the fact that s r. Otherwise, every ´ 2 B.u; s/ can be expressed as a convex combination ´ D 1 ka khk .x C h/ C ka khk w, where x hk x hk x hk w D a C .´ u/ ka khk 2 B.a; r/. Now for any y 2 B.u x h; s/ we have x C h C y 2 B.u; s/ U and kyk ku x hk C s khk C s D .1 C ˛/khk < 2khk. Hence, similarly as in (31), kq.h; y/k !.kh C yk/kh C ykk C !.kyk/kykk !.3khk/.2 C ˛/k khkk C !.2khk/.1 C ˛/k khkk 3.2 C ˛/k C 2.1 C ˛/k !.khk/khkk 5.2 C ˛/k !.khk/khkk : Thus using Lemma 47 we obtain

k

d f .x C h/

1 d kf .x/ D k s D

sup d kf .x C h/Œy

d kf .x/Œy

y2B.0;s/

kŠ kk kq .h; y/k sup k s k y2B.0;s/ sk

sup

kq.h; y/k

y2B.u x h;s/

kk 2 k k k k !.khk/: k 5.2 C ˛/ !.khk/khk D 5k 1 C ˛ s t u

58

Chapter 1. Fundamental properties of smoothness

In the convex case the assumption on the sub-additivity of the modulus ! can be dropped: Corollary 126. Let X, Y be normed linear spaces and let U X be an open convex set that is either bounded or has the property that there are a 2 X, r > 0, and fun gn2N X , kun k D n such that B.a C un ; nr/ U for each n 2 N (this holds in particular if U contains an unbounded cone). If f W U ! Y is U T k -smooth, k 2 N, and ! is the modulus from the definition of U T k -smoothness, then f 2 C k;˝ .U I Y /, where ˝ is the convex cone generated by !. More precisely, in the bounded case d kf U k is uniformly continuous on U with modulus ck eU !, where eU D supfrIdiam and B.a;r/U g ck > 0 is a constant depending only on k; in the unbounded case d kf is uniformly k continuous on U with modulus ck 1 C 1r !.

Proof. First suppose that U is bounded and B.a; r/ U . By the convexity the assumption of Theorem 125 is satisfied and from the proof it is easily seen that we k

obtain d kf .x C h/ d kf .x/ 2k k 1 C 2 diam U !.3khk/ for any x; h 2 U . r

k Thus !1 .t / ck eU !.3t/, where !1 is the minimal modulus of continuity of d kf k on U . But since !1 is sub-additive, we get !1 .t/ 3!1 . 13 t/ 3ck eU !.t/. In the unbounded case choose any n 2 N and put V D U \ U.a; n C nr/. Then U.aCun ; nr/ V and so by the previous part f 2 C k;˝ .V I Y / and d kf is uniformly k k continuous on V with modulus m!, where m D ck 2.nCnr/ D 2k ck 1C 1r . Since nr this constant is independent of n, it follows that f 2 C k;˝ .U I Y / and d kf is uniformly continuous on the whole of U with modulus m!. t u

The next theorem asserts that it actually suffices to test the C k;C -smoothness only on every line. Compare with similar characterisations in Corollary 55 and Theorem 160; see also the example after Theorem 101. Theorem 127. Let X be a real Banach space, Y a real normed linear space, U X an open convex bounded set or U D X, f W U ! Y , and k 2 N. Suppose that B f is Baire measurable for each 2 Y and there is a modulus ! such that the restriction of f to every line in U is C k;C -smooth with its kth differential having modulus of continuity !. Then f 2 C k;˝ .U I Y /, where ˝ is the convex cone generated by !. k More precisely, d kf is uniformly continuous with modulus ck eU ! (or just ck ! if diam U U D X ), where eU D supfrI B.a;r/U g and ck > 0 is a constant depending only on k. Proof. Taylor’s theorem (Corollary 108) implies that f is directionally T k -smooth 1 with kR.x; h/k kŠ !.khk/khkk , where R.x; h/ D f .xCh/ P x;h .1/ and where the x;h polynomials P 2 P k .RI Y / come from the definition of the directional T k -smoothness. Now it suffices to combine Theorem 118 and Corollary 126. t u Next we prove several statements that deal with operations with C k;C -smooth mappings.

Section 5. Smoothness classes

59

Proposition 128. Let X , Y , Z be normed linear spaces, U X and V Y open sets, k 2 N, f 2 C k .V I Z/, g 2 C k .U I Y /, g.U / V , and A U . Assume further that one of the following conditions is satisfied: (i) There is ! 2 Ms such that d jg is bounded and uniformly continuous on A with modulus ! and d jf is bounded and uniformly continuous on g.A/ with modulus ! for all j 2 f1; : : : ; kg, and g is Lipschitz on A. (ii) There is ! 2 M such that d jg is bounded and uniformly continuous on A with modulus ! and d jf is bounded and Lipschitz on g.A/ for all j 2 f1; : : : ; kg, and g is uniformly continuous on A with modulus !. Then there is C > 0 such that d j .f B g/ is bounded and uniformly continuous on A with modulus C ! for all j 2 f1; : : : ; kg. Note that the condition (i) above is automatically satisfied if the mappings f and g are C k;˝ -smooth and the sets A and V are convex and bounded (Proposition 71, Fact 120). Proof. Let M > 0 be such that d jf is bounded by M on g.A/ and d jg is bounded by M on A for all j 2 f1; : : : ; kg. Fix j 2 f1; : : : ; kg and l1 ; : : : ; lj 2 f1; : : : ; kg. Let L 2 N be such that g is L-Lipschitz on A in case (i), resp. L > 0 be such that j d on g.A/

fj is L-Lipschitz in case (ii). Notice that in both of these cases we obtain j

d f g.x/ d f g.y/ L!.kx yk/ for any x; y 2 A. Using this together with Proposition 13 and (3) from the proof of Proposition 4 we can estimate

j

D f g.x/ Œd l1g.x/; : : : ; d ljg.x/ D jf g.y/ Œd l1g.y/; : : : ; d ljg.y/

D jf g.x/ D jf g.y/ Œd l1g.x/; : : : ; d ljg.x/

C D jf g.y/ Œd l1g.x/; : : : ; d ljg.x/ D jf g.y/ Œd l1g.y/; : : : ; d ljg.y/

jj

d jf g.x/ jŠ

d jf g.y/ kd l1g.x/k kd ljg.x/k

C 2j .2M /j 1 D jf g.y/ max kd lsg.x/ d lsg.y/k sD1;:::;j

j j 2j 1 j jj j M !.kx yk/ M L!.kx yk/ C 2 jŠ jŠ jj j M .L C 22j 1 /!.kx yk/ jŠ for any x; y 2 A. Corollary 117 now finishes the proof.

t u

Proposition 129. Let X, Y be normed linear spaces, U X an open set, k 2 N, f 2 C k .U /, and g 2 C k .U I Y /. Assume further that there are A U and ! 2 M such that d jf , d jg are bounded and uniformly continuous on A with modulus ! for all j 2 f0; : : : ; kg. Then there is C > 0 such that d j .f g/ is bounded and uniformly continuous on A with modulus C ! for all j 2 f0; : : : ; kg. If moreover infA jf j > 0, then there is C > 0 such that d j f1 is bounded and uniformly continuous on A with modulus C ! for all j 2 f0; : : : ; kg.

60

Chapter 1. Fundamental properties of smoothness

Proof. The first statement follows immediately from Corollary 116. The second statement follows from Proposition 128. t u Proposition 130. Let X, Y be normed linear spaces, U X an open set, an arbitrary set, and let f W U ! `1 . I Y / be locally bounded. Denote by R˛ , ˛ 2 the canonical projections R˛ W `1 . I Y / ! Y given by R˛ ..y // D y˛ and by f D R B f the component mappings of f . Then f is T p -smooth at x 2 U if and only if all f , 2 are equi-T p -smooth at x, i.e. if and only if there are ! 2 M, a neighbourhood V of the origin in X, and polynomials P 2 P Œp .X I Y / satisfying kf .x C h/ P .h/k !.khk/khkp for all h 2 V and all 2 . Further, the mapping f is C k -smooth with d kf uniformly continuous with modulus ! 2 M if and only if each f , 2 , is C k -smooth with d kf uniformly continuous with modulus !. Proof. First we deal with the first part of the theorem. To see ) it suffices to notice that if P is the approximating polynomial from the definition of T p -smoothness of f P .h/k at x, then it suffices to put !.ı/ D suph2B.0;ı/nf0g kf .xCh/ and P D R B P . khkp To prove ( let K > 0 and ı 2 .0; 1 be such that !.ı/ K and kf .x C h/k K for h 2 B.0; ı/. Then kP .h/k kf .x C h/k C !.khk/khkp 2K for h 2 B.0; ı/ and 2 . By Corollary 31 the set fP g 2 is bounded in the space P Œp .XI Y /. If we set P .h/ D .P .h// 2 , then P 2 P Œp .X I `1 . I Y // by Fact 50. Clearly, kf .x C h/ P .h/k !.khk/khkp for h 2 V . As for the second part of the theorem, the implication ) follows from Fact 78. ( Given x 2 U , let r > 0 be such that U.x; r/ U . Corollary 108 implies that the collection ff g 2 is equi-U T k -smooth on U.x; r/. Thus by the first part of the proof f is U T k -smooth on U.x; r/. By Theorem 110 the mapping f is C k -smooth on U.x; r/. Finally, using Fact 78 we obtain

k

d f .x/Œh d kf .y/Œh D sup R d kf .x/Œh R d kf .y/Œh

2

D sup d kf .x/Œh

d kf .y/Œh !.kx

yk/

2

for any x; y 2 U , h 2 BX .

t u

Corollary 131. Let X , Y be normed linear spaces, U X an open set, f W U ! Y a locally bounded mapping, k 2 N, ! 2 M, and A BY a -norming set. If B f 2 C k;C .U / with d k . B f / uniformly continuous with modulus ! for each 2 A, then f 2 C k;C .U I Y / and d kf is uniformly continuous with modulus !. Proof. Let T W Y ! `1 .A/ be given by T .y/ D .y/ 2A . Then T is clearly a linear isomorphism of Y onto Z D T .Y / satisfying kT k 1 and kT 1 k . By Proposition 130 the mapping T Bf is C k -smooth with d k .T Bf / uniformly continuous with modulus !. Since f D T 1 B .T B f /, Fact 78 finishes the proof. t u

Section 6. Power series and their convergence

61

6. Power series and their convergence In this section we will investigate the convergence of power series. We prove the Polynomial lemma of Leja and apply it to obtain that a power series convergent on some translate of an absorbing set has a positive radius of convergence. We begin with some auxiliary notions. Definition 132. Let X be a vector space over K. A subset A X is called balanced if A A for every 2 K, jj 1. Definition 133. Let X be a vector space over K. A subset A X is called absorbing if for every x 2 X there is rx > 0 such that rx A contains the line segment joining x and 0, i.e. the set ftxI t 2 Œ0; 1g. Equivalently, for every x 2 X there is x > 0 such that tx 2 A for every t 2 Œ0; x . Obviously, any superset of an absorbing set is also absorbing. Lemma 134. An F absorbing set A in a Banach space X has a non-empty interior. S Proof. S Suppose that S A D k2N Ak , where each Ak is closed. Obviously we have X D n2N nA D k;n2N nAk , where each nAk is closed. Hence by the Baire category theorem some nAk (and thus also A) has a non-empty interior. t u Definition 135. Let X be a normed linear space over K, Y a Banach space over K, and Pn 2 P . nX I Y /. Consider the power series 1 X Pn : (32) nD0

˚ P The set Int x 2 XI 1 nD0 Pn .x/ converges is called the domain of convergence of the power series (32). The radius of convergence of the real power series 1 X kPn kn nD0

is called the radius of norm convergence of the power series (32). We start with some simple observations. If the power series (32) converges in x 2 X (orP just if the sequence fPn .x/g is bounded), then it converges absolutely in x (i.e. 1 nD0 kP .x/k converges) for every 2 K, jj < 1. Denote by G the domain of convergence of (32). Since for every x 2 G there is > 1 such that x 2 G, it follows that G is a balanced set and the power series (32) converges absolutely on G. Further, if R > 0 is the radius of norm convergence of (32), then the series is absolutely uniformly convergent on B.0; r/ for each r < R. On the other hand, if the power series (32) converges uniformly on U.0; r/, then its radius of convergence is at least r. Indeed, in this case there is N 2 N such that

Pnorm

1 Pn .x/ 1 for x 2 U.0; r/ and m N . Hence if m N and x 2 U.0; r/, nDm

P P1 m

then kPm .x/k 1 nDm Pn .x/ nDmC1 Pn .x/ 2 and so kPm kr 2, from which the statement follows.

62

Chapter 1. Fundamental properties of smoothness

We note that some authors use the term radius of uniform convergence rather than the radius of norm convergence, but we find it somewhat misleading, since there may be an abundance of open sets outside the ball B.0; R/, R being the radius of norm convergence, on which the series converges uniformly; see e.g. Example 137. Recall the well-known Cauchy-Hadamard formula: Proposition 136. The radius R of norm convergence of the power series (32) satisfies the formula 1 RD : 1 lim supkPn k n n!1

(Here we put

1 0

D C1.)

Example 137. Let X D c0 (real or complex). Denote the coordinate functionals by en n and put Pn .x/ D en .x/ , n 2 N, P0 D 0. Then Pn 2 P . nX/ (Fact 38). Since for each fixed x 2 X we have jen .x/j 21 for n sufficiently large, the power series (32) converges absolutely for every x 2 X. Moreover, it is not difficult to check that for every a 2 X it converges absolutely uniformly on B.a; r/ for every 0 < r < 1. However, since kPn k D 1 for each n 2 N, by Proposition 136 the radius of norm convergence of (32) is equal to 1. The following important lemma has powerful applications in the theory of power series and analytic mappings. It can be used for example to prove the Hartogs theorem or to give a constructive proof of the Riemann conformal mapping theorem. Lemma 138 (Polynomial lemma; Franciszek Leja, [Lej1]). For every N 2 N, d > 0, and every q > 1 there is ı D ı.N; d; q/ > 0 such that if Y is a normed linear space over K, Aj K are connected compact sets with diam Aj d , j D 1; : : : ; N , and ˚ P .KN I Y / n f0g is a family of polynomials pointwise bounded on the set A D A1 AN KN , then there is M > 0 such that for every P 2 ˚ we have sup kP .x/k M q deg P ; x2Aı

where Aı D fx 2 KN I dist.x; A/ < ıg. To prove this result we need first a technical lemma which asserts that given any subset of an interval with a positive Lebesgue measure, we can always find in this set any number of nodes for the Lagrange interpolation formula which are suitably spaced apart. Lemma 139. Let 0 < " < r and E Œ0; r be such that .E/ r 2" . Further, let W Œ0; 1 ! Œ0; 1 be a non-decreasing function. Then for each n 2 N there are points x0 ; : : : ; xn 2 E satisfying jxj xk j j. jn / . kn /j.r "/ for all j; k 2 f0; : : : ; ng. Proof. Fix n 2 N. Define a function W Œ0; r ! RC 0 by .x/ D .E \ Œ0; x/. We find the points x0 ; : : : ; xn 2 E by an inductive construction so that they satisfy the

Section 6. Power series and their convergence

63

following two conditions: j 1 j .r "/; j D 1; : : : ; n; C 1 n n j j C1" .r "/ C ; j D 0; : : : ; n: .xj / < n nC14

xj xj

(33)

The assertion of the lemma will be satisfied thanks to the first condition. 1 " To start, choose x0 2 E \ inf E; inf E C nC1 4 . Note that since E \.0; inf E/ D ;, we have .inf E/ D 0 and the condition (33) is satisfied for j D 0. Now assume that the points x0 ; : : : ; xj 1 have been chosen. Put a D xj 1 C . jn / . j n 1 / .r "/. As j 1 j .r "/ .a/ .xj 1 / C n n (34) j j " " .r "/ C 1 is odd, then the set k X j C1 l 1 n . 1/ ´j xj < 0 M D x 2 R I h.x/ j D1

is open and non-empty. We distinguish two cases. First, suppose that there exist i; j 2 f1; : : : ; kg such that . 1/i ¤ . 1/j and j´i j > j´j j. Then there exist suitable 0 < xi < xj such that xil xjl < 0 while ´li 1 xi ´jl 1 xj > 0, and so M ¤ ;. In the opposite case j´1 j D j´2 j D D j´k j ¤ 0. But then .1; 32 ; 1; 0; : : : ; 0/ 2 M , since l > 1. As for all x 2 M again t 7! h.x C t´/ changes sign on Œ0; 1/, we can finish the proof as above. t u Proof of Theorem 47. By Lemma 48 and the separation theorem there is P 2 P . p Rn / such that ŒP; Q > 0 for all Q 2 Pn;p .k 1/ n f0g and ŒP; R < 0 for some R 2 Pn;p .k/. In particular, P .x/ D ŒP; hx; ip > 0 for each x 2 Rn n f0g. Let E Rn be a subspace of dimension k 1. For any Q 2 Pn;p .E/ we have ŒP E ; QE D ŒP; Q 0 and hence P E 2 Qk 1;p on E (Proposition 36). Thus 1 .P E / p is a p-polynomial norm on E and E is isometric to a subspace of `pN for

Section 5. Polynomial identities

111

some N jI.k 1; p/j (Proposition 37). Since this holds for any subspace E Rn of 1 1 n p is a norm on Rn and we put X p dimension k 1 2, the function n;k D .R ; P /. PP On the other hand, let R D jmD1 Rj , where 0 ¤ Rj 2 Pn;p .Ej /, dim Ej k. P Since 0 > ŒP; R D jmD1 ŒP; Rj and ŒP; Rj > 0 whenever dim Ej < k, it follows that there is l 2 f1; : : : ; mg such that ŒP; Rl < 0 and dim El D k. Thus P El … Qk;p 1 on El (Proposition 36) and hence .El ; P p / is not isometric to a subspace of Lp .Œ0; 1/ (Proposition 39). t u

5. Polynomial identities The section is devoted to the study of polynomial identities, a subject inspired by Corollary 1.55 and the characterisation of the Hilbert space by the parallelogram identity. We are interested in general polynomials in connection with linear dependence of the sets of nodes and the Lagrange interpolation formula. In this section all spaces are real unless stated otherwise. We begin by proving a local version of Corollary 1.55 (and also a version for complex spaces). We remark that the proof below is analytic rather than algebraic and completely different from that of Corollary 1.55. Theorem 49. Let X, Y be real normed linear spaces, U X a convex set with non-empty interior, and let f W U ! Y be continuous. The following statements are equivalent: (i) There is P 2 P n .XI Y / such that f D P U . (ii) nC1 f .xI h1 ; : : : ; hnC1 / D 0 for all x; hj 2 X such that the difference is defined. (iii) nC1 f .xI h; : : : ; h/ D 0 for all x; h 2 X such that the difference is defined. (iv) f E \U is a restriction of a polynomial of degree not exceeding n for every affine one-dimensional subspace E of X. (v) f is G nC1 -smooth and ı nC1f D 0 on Int U . If X and Y are complex spaces, then all the statements are equivalent if we additionally assume in (ii) and (iii) that f is Gâteaux differentiable on Int U . The additional assumptions in the complex case cannot be dropped, as witnessed by the complex function f .´/ D Re ´. The main step of the proof of the above theorem is contained in the following result. Theorem 50. Let X , Y be real normed linear spaces, U X a convex set with nonempty interior, let f W U ! Y be continuous, and n 2 N0 . There is P 2 P n .XI Y / such that f D P U if and only if nC1 X nC1 . 1/nC1 k f .x C kh/ D 0 (18) k kD0

for all x; h 2 X such that x C kh 2 U , k D 0; : : : ; n C 1.

Chapter 2. Basic properties of polynomials on Rn

112

Proof. The case n D 0 is trivial, so we may assume that n 1. ) This implication follows from the Polarisation formula (Proposition 1.13) by h setting a D x C nC1 2 h and xj D 2 . ( Without loss of generality we may assume that U is open and 0 2 U . Further, it suffices to prove this implication only for X D RN , N n C 1. The general case then follows: For any subspace E X with dim E D n C 1 let PE 2 P n .EI Y / be such that PE E \U D f E \U . If E; F X are any two subspaces of dimension n C 1, then PE E \F \U D f E \F \U D PF E \F \U and hence PE E \F D PF E \F by Fact 1.39. So we may define P .x/ D PE .x/, where E X is any subspace of dimension n C 1 containing x. Then P 2 P n .XI Y / by Fact 1.36, and since f is continuous, so is P (Proposition 1.40). First consider the case N D 1. Let M 2 N be large enough so that f 2jM gjnD0 U . For each m M let Pm be the unique polynomial of degree at most n satisfying Pm . 2jm / D f . 2jm / for j D 0; : : : ; n (Proposition 1.56). Assume that m is large 1 enough so that nC1 2m 2 U . Since both Pm and f satisfy (18), using x D 0 and h D 2m nC1 we conclude that Pm . nC1 2m / D f . 2m /. Repeating this argument inductively we deduce j j that f . 2m / D Pm . 2m / for all j 2 Z such that 2jm 2 U . For any m; r M the set f 2jm gj 2Z \ f 2jr gj 2Z \ U has at least n C 1 elements. Thus Pm and Pr agree on a set of cardinality at least nC1 and hence Pm D Pr on the whole R. Since f is assumed to be continuous and f 2jm gm2N;j 2Z contains a dense subset of U , the mapping f D PM U is a restriction of a continuous polynomial of degree at most n. Let us pass to the case N > 1. Denote by W the vector space of all continuous mappings from U into Y satisfying (18). Fix g 2 W and h 2 RN . By the onedimensional part of the proof, for any x 2 U the mapping t 7! g.x C th/ defined .x/ exists. We on a neighbourhood of zero is a restriction of a polynomial and so @g @h @g show that @g . Clearly, satisfies (18). It remains to show that it is a continuous 2 W @h @h mapping. Fix x 2 U and " > 0. Without loss of generality

we" may assume that khk 1. There @g

is > 0 such that g.x C th/ g.x/ t @h .x/ < 2n2 jt j whenever jt j . Since g is uniformly continuous on B.x; 2/, there is 0 < ı such that kg.y/ g.´/k < 2n" 2 whenever y; ´ 2 B.x; 2/, ky ´k < ı. Choose an arbitrary y 2 U.x; ı/ and define '.t / D g.y C th/ g.x/ t @g .x/ for t 2 Œ ; . Since by the first part of the proof @h ' is a polynomial of degree at most n, by the Markov inequality (Theorem 80; note that it holds also for vector-valued polynomials, which can be easily shown by composing with functionals from Y ; see also more involved Theorem 28)

2

@g

.y/ @g .x/ D k' 0 .0/k n max k'.t/k

@h @h t 2Œ ;

2

n

g.y C th/ g.x C th/ C g.x C th/ g.x/ t @g .x/ < ": max

t 2Œ ; @h This proves that

@g @h

is continuous and so

@g @h

2W.

Section 5. Polynomial identities

113

It follows that any mapping in W (and in particular f ) has continuous directional derivatives of all orders in all directions. Thus by Theorem 1.96 the mapping f is C 1 -smooth on U . Since t 7! f .th/ is a polynomial of degree at most n, using the Taylor formula (Theorem 1.107) and the uniqueness of the Taylor polynomial (Proposition 1.104) we obtain n X tk k f .th/ D d P .0/Œh kŠ kD0 P 1 k for each h 2 U and t 2 Œ0; 1. Therefore f D nkD0 kŠ d f .0/ on U and so f is a restriction of a polynomial of degree at most n. u t

Proof of Theorem 49. (i))(ii) follows from Lemma 1.52 and the Polarisation forP hj hj mula (Proposition 1.13) by setting a D x C jnC1 D1 2 and xj D 2 . (ii))(iii) is trivial. (iii))(i) and (iv))(i) in the real case follow from Lemma 1.52 and Theorem 50. (i))(iv) is clear. (i))(v) follows from Lemma 1.99. (v))(i) Fix any x 2 Int U . Given h 2 X such that x C h 2 Int U we define g.t / D f .x C th/. By Fact 1.97, Fact 1.98 and the Taylor formula (Theorem 1.107) P used on g we obtain f .x C h/ D jnD0 j1Š ı jf .x/Œh. We finish by using Lemma 1.34. Finally, assume that the spaces X and Y are complex and either (iii) or (iv) holds. Then f is C 1 -smooth on Int U by Theorem 1.160. From the real case we already know that f is a restriction of a polynomial from XR into YR (the real versions of the spaces) of degree at most n. Thus for each x 2 Int U and h 2 X we have d nC1f .x/Œh D 0 by Fact 1.97 and so (v) holds. t u

2

In the rest of this section we indicate a general approach to linear identities which characterise polynomials of prescribed degree. It turns out that these identities are independent of the underlying normed linear spaces and are closely related to the Lagrange formulae. linear spaces and f 2 C.X I Y /. We say that f is Definition 51. Let X, Y be normedP compatible with 2 F .Rm /, D nkD1 ak ıxk , if hf B L; i D

n X

ak f .L.xk // D 0 for all L 2 L.Rm I X/.

kD1

P Remark 52. Clearly, f 2 C.X I Y / is compatible with D nkD1 ak ıxk 2 F .Rm /, where xk D .xk1 ; : : : ; xkm /, if and only if ! n m X X j ak f xk ´j D 0 for every ´1 ; : : : ; ´m 2 X. kD1

j D1

In particular, Fréchet’s theorem (Theorem 50) is equivalent to saying that f is a polynomial of degree at most n if and only if f is compatible with the element

Chapter 2. Basic properties of polynomials on Rn

114

nF 2 F .R2 / given by nF

D

nC1 X

. 1/

nC1 k

kD0

nC1 ı.1;k/ : k

(19)

The following fact is a simple consequence of the definition. P Fact 53. Let 2 F .Rm /, D jkD1 aj ıxj . Further, let L 2 L.Rm I Rn / and P 2 F .Rn /, D jkD1 aj ıL.xj / . Finally, let X, Y be normed linear spaces and f 2 C.X I Y /. If f is compatible with , then f is compatible with as well. Easy examples show that the implication in the previous fact cannot be reversed, see [HK]. Lemma 54. Let 2 F .Rm /. The following statements are equivalent: (i) For all normed linear spaces X, Y each P 2 P . dX I Y / (resp. P 2 P d .XI Y /) is compatible with . (ii) Every P 2 P . d Rm / (resp. every P 2 P d .Rm /) is compatible with . (iii) hP; i D 0 for every P 2 P . d Rm / (resp. for every P 2 P d .Rm /). Proof. (i))(ii))(iii) are clear. (iii))(ii) Let P 2 P . d Rm /. If L 2 L.Rm I Rm /, then P B L 2 P . d Rm /, hence hP B L; i D 0, and therefore P is compatible with . (ii))(i) Let X, Y be normed linear spaces and P 2 P . dXI Y /. Let L 2 L.Rm I X/ and choose an arbitrary f 2 Y . Then f B P B L 2 P . d Rm /, and therefore we have 0 D hf B P B L; i D f hP B L; i . Since f was arbitrary, we conclude that hP B L; i D 0. The proof of the non-homogeneous case is identical. t u Pd Lemma 55. Let X, Y be normed linear spaces and let P D kD0 Pk 2 P d .X I Y /, where Pk 2 P . kX I Y /. If P is compatible with 2 F .Rm /, then each Pk is compatible with . Proof. Fix L 2 L.Rm I X/. For each t 2 R the mapping g t .´/ D tL.´/ belongs to L.Rm I X / and hence 0 D hP B g t ; i D

d X

t k hPk B L; i:

kD0

The right-hand side is a Y -valued polynomial in t. Thus each hPk B L; i D 0.

t u

Lemma 56 ([Rez1]). Let X and Y be normed linear spaces, 2 F .Rm /, and let 0 ¤ P 2 P . dXI Y /. Then P is compatible with if and only if the polynomial t 7! t d from P . d R/ is compatible with . Proof. ) There exists a one-dimensional subspace E X such that P E D at d , a ¤ 0. For every L W Rm ! E we have hP B L; i D 0. Consequently, t 7! t d is compatible with .

Section 5. Polynomial identities

115

( If t 7! t d is compatible with , then so is every hy; id , where y 2 Rm . Indeed, hy; id is a composition of a linear projection of Rm onto R with t 7! t d . Since fhy; id I y 2 Rm g forms a basis of P . d Rm /, each Q 2 P . d Rm / is compatible with . Lemma 54 then finishes the proof. t u Pn Corollary 57. An element D kD1 ak ıxk 2 F .Rn / is compatible with t 7! t d (or any other non-zero d -homogeneous polynomial) if and only if hP; i D 0 for all P 2 P . d Rn /. The following result can be viewed as a further abstraction of the cubature formulae treated in Section 2, where the functional is represented by the polynomial Q, and the evaluation at the nodes have been replaced by evaluation at a given element P and its linear translates. Corollary 58. Let 0 ¤ P 2 P . d Rn /. Then for any Q 2 P . d Rn / there exist a finite collection P of Lk 2 L.Rn I Rn / and "k 2 f 1; 1g, k D 1; : : : ; r, r jI.n; d /j, such that Q D rkD1 "k P B Lk . Proof. By contradiction, suppose that H D spanfP B LI L 2 L.Rn I Rn /g is a proper subspace of P . d Rn /, i.e. there exist some Q 2 P . d Rn / n H and a linear functional 2 F .Rn / which is zero on H and hQ; i ¤ 0. Then P is compatible with but Q is not. This contradicts Lemma 56. To obtain the estimate r jI.n; d /j use Theorem 8. t u The above corollary is analogous to Theorem 41. Indeed, choose 0 ¤ y 2 Rn , put p P D hy; ip and Q.x/ D kxk`2 , and apply Corollary 58. The difference is that here we do not have any additional assumptions because we allow linear combinations with both signs. Next, we investigate the properties of 2 F .Rn / which lead to compatibility (Theorem 60). From this, the theorem of Pascual Jordan and John von Neumann (Corollary 61) follows immediately. The result was proved by Bruce Reznick under the assumption that the continuous mapping f is homogeneous. In the proof of Theorem 60 we will use the following result of William Harold Wilson. Theorem 59 ([Wi]). Let X , Y be normed linear spaces, f 2 C.X I Y /, n 2 N0 , and P 2 let D nC1 kD0 ak ıxk 2 F .R /. Suppose that a0 ¤ 0 and for every 0 < k n C 1 the vectors x0 and xk are linearly independent. Let p C 2 be the number of distinct (unoriented) directions determined by the vectors x0 ; : : : ; xnC1 . If f is compatible with , then f is compatible with pF from (19). Proof. Suppose that xj D .rj ; sj /, j D 0; : : : ; n C 1. By Fact 53 we may assume without loss of generality that r0 D 0 and s0 D 1. Then rj ¤ 0 for 1 j n C 1. Moreover, we may also assume that the vectors x0 ; : : : ; xpC1 are pairwise linearly independent. Put Ik D f1 j n C 1I xj 2 spanfxk gg for k D 1; : : : ; n C 1 and Jk D f1; : : : ; nC1gn.I1 [ [Ik / for k D 0; : : : ; pC1. Finally, set k;j D sj rj rskk for k; j 2 f1; : : : ; n C 1g and note that k;j D 0 if and only if xk and xj are linearly dependent which is if and only if j 2 Ik .

Chapter 2. Basic properties of polynomials on Rn

116

Fix an arbitrary ´ 2 X and define inductively 0 .x; y/

D a0 f .y/ C

nC1 X

aj f .rj x C sj y/

j D1

s ´; y C ´ and mC1 .x; y/ D m x rmC1 m .x; y/ for x; y 2 X , m D 0; : : : ; p. mC1 By Remark 52 we have 0 .x; y/ D 0 for every x; y 2 X and thus also m .x; y/ D 0 for every x; y 2 X, m D 1; : : : ; p C 1. Using the fact that mC1;j D 0 if and only if j 2 ImC1 it is easy to check by induction that m X m k m . 1/ f .y C k´/ m .x; y/ D a0 k kD0 X X X C aj . 1/mCjAj f rj x C sj y C k;j ´ : j 2Jm

Af1;:::;mg

k2A

Since JpC1 D ;, we obtain a0

pC1 X

. 1/

pC1 k

kD0

pC1 f .y C k´/ D 0 k

for every y; ´ 2 X, and therefore f is compatible with pF .

t u

Theorem 60 ([Rez1], [HK]). Let X, Y be normed linear spaces, f 2 C.XI Y /, and P m D nC1 kD0 ak ıxk 2 F .R /, m 2, n 2 N0 . Suppose that anC1 ¤ 0 and for every k ¤ n C 1 the vectors xk and xnC1 are linearly independent. Let p C 2 be the number of distinct directions determined by the vectors x0 ; : : : ; xnC1 . If f is compatible with , then f is a polynomial of degree at most p. Proof. First suppose that m D 2. Since f is compatible with , it is compatible with pF by Theorem 59. By Remark 52 the mapping f is a polynomial of degree at most p. Now we consider the case m > 2. By Fact 53 it is enough to find a T 2 L.Rm I R2 / P such that D nC1 kD0 ak ıT .xk / satisfies the assumptions of the previous case, i.e. such that the vectors T .xk / and T .xnC1 / are linearly independent for each k ¤ n C 1. (The number of distinct directions determined by T .x0 /; : : : ; T .xnC1 / is clearly less than or equal to p C 2.) This is easily done as follows: Let Ek D spanfxk ; xnC1 g Rm , k 2 f0; : : : ; ng, be a system of two-dimensional subspaces of Rm . There exists an .m 2/-dimensional subspace F Rm such that F \ Ek D f0g, k 2 f0; : : : ; ng. (Equivalently, F C Ek D Rm ). Then the orthogonal projection T in Rm with kernel F and the two-dimensional range F ? Rm clearly satisfies the requirements. t u Corollary 61 (Pascual Jordan, John von Neumann; [JN]). Let X be a real Banach space such that kx C yk2 C kx

yk2 D 2kxk2 C kyk2

Then X is isometric to a Hilbert space.

for all x; y 2 X .

Section 5. Polynomial identities

117

Proof. The function f .x/ D kxk2 satisfies f .x Ch/Cf .x h/ 2f .x/ 2f .h/ D 0, i.e. f is compatible with D ı.1;1/ C ı.1; 1/ 2ı.1;0/ 2ı.0;1/ 2 F .R2 /. By p Theorem 60, f is a polynomial of degree 2. Thus kxk D M.x; x/, where M is a bounded bilinear form on X . t u If 2 F .Rm / and X, Y are normed linear spaces, then the set of all mappings in C.X I Y / which are compatible with is a subspace of the space of polynomials (Theorem 60). We are now ready to describe this space more precisely. P m Theorem 62 ([HK]). Suppose D nC1 kD0 ak ıxk 2 F .R /, where x0 ; : : : ; xnC1 are pairwise linearly independent vectors. Then there exists A f0; : : : ; ng such that if X, Y are normed P linear spaces and f 2 C.X I Y /, then f is compatible with if and only if f D k2A Pk for some Pk 2 P . kXI Y /. Proof. Let A be the set of all k 2 f0; : : : ; ng for which there exist normed linear spaces X, Y and a non-zero polynomial from P . kX I Y / which is compatible with . By Lemma 56, if k 2 A, then for all normed linear spaces X and Y each polynomial from P . kXI Y / is compatible with and the same holds also for their linear combinations. Let now X , Y be normed linear spaces and P W X ! Y be a continuous mapping compatibleP with . By Theorem 60 the mapping P is a polynomial of degree at most n, say P D nkD0 Pk , Pk 2 P . kXI Y /. If Pk ¤ P 0 for some k 2 f0; : : : ; ng, then it follows from Lemma 55 that k 2 A. Hence P D k2A Pk . t u More can be said if the points x0 ; : : : ; xnC1 lie in an affine hyperplane not containing 0. P m Lemma 63. Suppose D nC1 kD0 ak ıxk 2 F .R /, where x0 ; : : : ; xnC1 lie in an affine hyperplane not containing 0. If every polynomial from P . d Rm / is compatible with , then the same holds for every polynomial from P d .Rm /. Proof. Let H be an affine hyperplane in Rm which contains x0 ; : : : ; xnC1 and does not contain 0. Suppose that every polynomial from P . d Rm / is compatible with . If P 2 P d .Rm /, then there exists Q 2 P . d Rm / such that QH D P H (see Fact 2). Since Q is compatible with , we see that hP; i D hQ; i D 0. By Lemma 54 every polynomial from P d .Rm / is compatible with . t u PnC1 m Theorem 64 ([HK]). Suppose D kD0 ak ıxkP2 F .R /, where x0 ; : : : ; xnC1 nC1 lie in an affine hyperplane not containing 0. If kD0 ak D 0, then there exists p 2 N0 , p n, such that if X, Y are normed linear spaces and f 2 C.X I Y /, then f is compatible with if and only if f is a polynomial of degree at most p. If PnC1 kD0 ak ¤ 0, then there is no non-zero mapping compatible with . Proof. Since x0 ; : : : ; xnC1 are pairwise linearly independent, the assumptions of TheP orem 62 are satisfied. Let A f0; : : : ; ng be the set from Theorem 62. If nC1 kD0 ak D 0, then t 7! 1; t 2 R, is compatible with and therefore A is non-empty. Let p D max A. Since every polynomial from P . p Rm / is compatible with , by Lemma 63 so is every polynomial from P p .Rm /. Hence A D f0; : : : ; pg. This argument also shows that if

118

Chapter 2. Basic properties of polynomials on Rn

P A is non-empty, then t 7! 1 is compatible with and consequently nC1 kD0 ak D 0. PnC1 Hence if kD0 ak ¤ 0, then there is no non-zero mapping compatible with . u t In order to generate linear identities we rely on Theorem 7. In fact, the Lagrange formula is an expression of linear dependence of functionals in the dual of P d .Rm /. Let fxk grkD1 Rm be a basic set of nodes for P d .Rm / and let fpk grkD1 P d .Rm / be biorthogonal to fxk grkD1 . For any fixed ´ 2 Rm n fxk grkD1 Theorem 7 implies P P .´/ D rkD1 pk .´/P .xk / for every P 2 P d .Rm /. It follows that every polynomial P P 2 P d .Rm / is compatible with D ı´ C rkD1 pk .´/ıxk (Lemma 54). Lemma 65. Suppose that fxk grkD1 Rm is a basic set of nodes for P d .Rm / and fpk grkD1 P d .Rm / is biorthogonal to fxk grkD1 . Further, let ´ 2 Rm n fxk grkD1 and P D ı´ C rkD1 pk .´/ıxk . If every P 2 P l .Rm / is compatible with , then l d . Proof. Assume without loss of generality that p1 .´/ ¤ 0. Clearly, there exists R 2 P 1 .Rm / (an affine function on Rm ) such that R.x1 / ¤ 0 and R.´/ D 0. Set P D p1 R 2 P d C1 .Rm /. Then P .x1 / ¤ 0, P .xk / D 0 for k D 2; : : : ; r, P .´/ D 0, and so hP; i D p1 .´/P .x1 / ¤ 0. Hence P is not compatible with , and thus l d . t u The following theorem describes a method of generating linear identities which characterise polynomials of degree at most d . Theorem 66 ([HK]). Let fxk grkD1 Rm be a basic set of nodes for P d .Rm /, and let fpk grkD1 P d .Rm / be biorthogonal to fxk grkD1 . Further, let ´ 2 Rm n fxk grkD1 P and D ı´ C rkD1 pk .´/ıxk . Let T W Rm ! Rn , n > m, be an affine one-to-one mapping such that 0 … T .Rm /. Then a1 D p1 .´/; : : : ; ar P D pr .´/ are the unique coefficients with the following property: Let D ıT .´/ C rkD1 ak ıT .xk / . If X, Y are normed linear spaces and f 2 C.XI Y /, then f is compatible with if and only if f is a polynomial of degree at most d . Proof. Since T .x1 /; : : : ; T .xr /; T .´/ lie in an affine hyperplane not containing 0, the assumptions of Theorem 64 are satisfied. Thus it suffices to prove the theorem for X D Rn and Y D R, and it also follows that the space of those continuous f W Rn ! R which are compatible with is either P l .Rn / for some l 2 N0 , or a trivial space. If P 2 P d .Rn /, then P B T 2 P d .Rm /, so P B T is compatible with , and therefore hP; i D 0. By Lemma 54 every member of P d .Rn / is compatible with . Hence the space of compatible functions is non-trivial and l d . On the other hand, if P 2 P l .Rm /, then P B T 1 W T .Rm / ! R can be extended to a member of P l .Rn /, which is compatible with by the definition of l. It follows from Lemma 54 that every polynomial from P l .Rm / is compatible with . By Lemma 65 we conclude that l d . Theorem 7 then yields the uniqueness part. t u

Section 6. Estimates of coefficients of polynomials

119

6. Estimates of coefficients of polynomials In this section we show some applications of a simple complex averaging technique which leads to many useful and sharp coefficient estimates. Our goal is to find estimates which are independent of the number of variables, i.e. of the dimension of the domain, as these estimates can be easily applied to polynomials defined on the classical Banach spaces c0 or `p . We begin with a few result for complex polynomials analogous to the Polarisation formula. Let us define the generalised Rademacher system. For given n; k 2 N let m r1n;k .l/ D e i 2 n for l 2 N, mnk 1 .l 1/ mod nk < .mC1/nk 1 , 0 m n 1, and put rjn;k .l/ D r1n;k .nj 1 l/, l 2 f1; : : : ; nk g, j D 2; : : : ; k. We note that the system can be defined also on Œ0; 1 or more generally on a suitable probability space, but we chose to work with the discrete domain to underline the discrete nature of the methods used here. Lemma 67 ([ALRT], [Din]). Let n; k 2 N. The generalised Rademacher system has the following property: If m1 ; : : : ; mk 2 N0 , then the average ( nk 1 X n;k m1 1 if mj 0 .mod n/, j D 1; : : : ; k, n;k mk r1 .l/ rk .l/ D k 0 otherwise. n lD1

Proof. The first case is clear, since rjn;k .l/n D 1 for every l 2 f1; : : : ; nk g and j D 1; : : : ; k. To prove the other case let s be the biggest number in f1; : : : ; kg such that ms ¥ 0 .mod n/. Then k

n X

r1n;k .l/m1 rkn;k .l/mk

lD1

D

s 1 nX

r1n;k .q nk sC1 /m1

rsn;k1 .q nk sC1 /ms

qD1

since

1 i 2 jn ms j D0 e

Pn

1

sC1 nkX

! rsn;k .l/ms

D0

lD1

D 0.

t u

The following two results should be compared with the Polarisation formula (Proposition 1.13, Lemma 1.33), where k D n and "j D rj2;n . Proposition 68 ([ALRT]). Let X, Y be complex vector spaces, n 2 N, and let P 2 P . nX I Y /. Then }.n1x1 ; : : : ; nkxk / P k

n n1 Š nk Š 1 X n;k n r1 .l/ D nŠ nk lD1

n1

rkn;k .l/n nk P

k X

! rjn;k .l/xj

j D1

for all x1 ; : : : ; xk 2 X, n1 ; : : : ; nk 2 N with n1 C C nk D n, 1 k n.

Chapter 2. Basic properties of polynomials on Rn

120

Proof. Without loss of generality we may assume that k 2. Using the Multinomial formula (Proposition 1.22) and Lemma 67 we obtain ! nk k X 1 X n;k n n1 n;k n;k r1 .l/ rk .l/n nk P rj .l/xj nk j D1 lD1

k

n 1 X D k n

X

lD1 ˛2I.k;n/

n n;k n r .l/ ˛ 1

n1 C˛1

rkn;k .l/n

nk n } ˛1 1 X n;k n ˛k r1 .l/ D P . x1 ; : : : ; xk / k ˛ n lD1 ˛2I.k;n/ n }.n1x1 ; : : : ; nkxk / D P n1 ; : : : ; n k X

nk C˛k

}.˛1x1 ; : : : ; ˛kxk / P !

n1 C˛1

rkn;k .l/n nk C˛k

t u

from which the formula follows.

Lemma 69. LetPX, Y be complex vector spaces, n 2 N, and let P 2 P n .X I Y / be such that P D nmD0 Pm , Pm 2 P . mX I Y /. Then for every x1 ; : : : ; xk 2 X ! k k nk X X 1 X n;k P0 .0/ C Pn .xj / D k P rj .l/xj : n j D1 j D1 lD1

Proof. By Lemma 67 k

n 1 X P nk lD1

k X

! rjn;k .l/xj

j D1 k

n n X 1 X D P0 .0/ C nk mD1

X

.l/P}m .xj1 ; : : : ; xjm / rjn;k .l/ rjn;k m 1

lD1 1js k sD1;:::;m

D P0 .0/ C

k X

Pn .xj /:

t u

j D1

The next results improves the estimate from Corollary 1.17 for complex multilinear mappings. Corollary 70 ([Harr1]). Let X and Y be complex normed linear spaces and let M 2 Ls . nXI Y /. Then n

n

M. 1x1 ; : : : ; nkxk / n1 Š nk Š n kM k nn1 1 nnkk nŠ for all x1 ; : : : ; xk 2 BX , n1 ; : : : ; nk 2 N with n1 C C nk D n, 1 k n.

Section 6. Estimates of coefficients of polynomials

121

Proof. Let x1 ; : : : ; xk 2 BX and n 1 ; : : : ; nk 2 N with n1 C C nk D n. Put P P P n n ´j D nj xj . Then jkD1 rjn;k .l/´j jkD1 k´j k D jkD1 nj kxj k 1. Hence by Proposition 68

n

n

nn nk 1

M. 1x1 ; : : : ; nkxk / D nk M. ´1 ; : : : ; ´k / n1 n1 nk

! nk k

nn n1 Š nk Š 1 X

X n;k

n1 M r .l/´

j j

nŠ nk n1 nnkk j D1

lD1

nn

n1 Š nk Š k: kM nn1 1 nnkk nŠ

t u

Theorem 71 ([AG], [ABE]). Let p 2 P n .C N / and N X

p.x1 ; : : : ; xN / D

aj xjn C

j D1

Then

N X

jaj j

j D1

max

´j 2C;j´j jD1

X

aˇ x ˇ :

ˇ 2J.N;n/ ˇ ¤.0;:::;0;n;0;:::;0/

jp.´1 ; : : : ; ´N /j kpk1 :

In particular, in the complex scalar case, if P 2 P . nc0 /, then

P1

j D1 jP .ej /j

kP k1 .

Proof. Assume without loss of generality that the constant term C of p is real and non-negative. Let j 2 C be such that jj j D 1 and aj jn D jaj j, j D 1; : : : ; N . Using Lemma 69 we obtain ! N N N nN N X X X X 1 X n;N n jaj j C C aj j D C C pn .j ej / D N p rj .l/j ej ; n j D1

j D1

j D1

lD1

j D1

where pn denotes the n-homogeneous summand of p and fej gjND1 is the canonical basis of C N . This finishes the proof of the finite-dimensional case. The infinite-dimensional case follows immediately. t u The following result is an optimal generalisation of the Chebyshev theorem (Theorem 22), which corresponds to N D 1. Theorem 72 ([MST]). Let p 2 P n .RN / and N X

p.x1 ; : : : ; xN / D

aj xjn C

j D1

Then

and the constant

N X

jaj j 2n

j D1 n 1 2 is the

1

X

aˇ x ˇ :

ˇ 2J.N;n/ ˇ ¤.0;:::;0;n;0;:::;0/

max jp.x1 ; : : : ; xN /j

xj 2Œ 1;1

best possible.

Chapter 2. Basic properties of polynomials on Rn

122

Proof. Denote by pn the n-homogeneous term of p and apply Theorem 71 on pzn . Note that Taylor’s norm on C N which arises from the supremum norm on RN is the canonical supremum norm on C N . Thus N X

jaj j kpzn k1;C 2n

1

kpk1

j D1

by Theorem 1.63. Finally, the polynomial q.x1 ; : : : ; xN / D Tn .x1 / C C Tn .xN / shows that the estimate is sharp. t u Theorem 73 ([ABE]). Let p 2 P . n RN / and N X

p.x1 ; : : : ; xN / D

aj xjn C

j D1

X

aˇ x ˇ :

ˇ 2I.N;n/ ˇ ¤.0;:::;0;n;0;:::;0/

Then N X

jaj j 4n2 max jp.x1 ; : : : ; xN /j: xj 2Œ0;1

j D1

In particular, in the real scalar case, if P 2 P . nc0 /, then

P1

j D1 jP .ej /j

4n2 kP k1 .

Proof. First, choose a subset L f1; : : : ; N g such that aj , j 2 L have the same ˇP ˇ P sign and ˇ j 2L aj ˇ 12 jND1 jaj j. Without loss of generality we may assume that L D f1; : : : ; mg. From now on consider p restricted to the first m variables only. Fix t 2 .0; 1/ and let fvj gjmD1 be independent random variables on Œ0; 1 taking values 1 with probability t and 0 with probability 1 t. (We can take for example non-symmetric Rademacher-like functions, where v1 D Œ0;t/ , v2 D Œ0;t 2 /[Œt;t Ct.1 t// and so on.) Note that vjk D vj for each k 2 N, j 2 f1; : : : ; mg. Let us denote by lˇ the number P of non-zero components in ˇ 2 I.m; n/ and put Aj D ˇ W lˇ Dj aˇ , j D 2; : : : ; n. Note that in our notation A1 D a1 C : : : C am . Then Z 1 Z 1 Z 1 Z 1 X X ˇ ˇ ˇm ˇm vm p v1 .s/; : : : ; vm .s/ ds D aˇ v1 1 vm D aˇ v1 1 0

ˇ 2I.m;n/

0

ˇ 2I.m;n/

0

0

D A1 t C A2 t 2 C C An t n : Let us denote q.t/ D A1 t C A2 t 2 C C An t n and put r.t/ D q. t C1 2 /. Using Markov’s inequality (Theorem 80) we get 1 max jq.t/j D max jr.t/j 2 max jr 0 .t/j n t 2Œ 1;1 t 2Œ0;1 t 2Œ 1;1 1 1 1 D 2 max jq 0 .t/j 2 jq 0 .0/j D 2 jA1 j: 2n t 2Œ0;1 2n 2n It follows that for every " > 0 there is t 2 .0; 1/ such that jq.t/j > 2n1 2 jA1 j " and considering fvj g corresponding to this t there must be s 2 Œ0; 1 such that

Section 6. Estimates of coefficients of polynomials

p v1 .s/; : : : ; vm .s/ > N X

1 jA1 j 2n2

123

". Hence

jaj j 2jA1 j < 4n2 max jp.x1 ; : : : ; xm /j C 4n2 "; xj 2Œ0;1

j D1

t u

which proves the claim.

The best multiplicative constant in Theorem 73 appears to be unknown, even asymptotically. A lower estimate follows from the next example. Example 74. Define by induction a sequence of polynomials by p0 .x1 / D x1 ; pnC1 .x1 ; : : : ; x2nC1 / D pn .x1 ; : : : ; x2n /2

pn .x2n C1 ; : : : ; x2nC1 /2 :

Then for every n 2 N0 , pn is a 2n -homogeneous polynomial in 2n variables satisfying n jpn .x1 ; : : : ; x2n /j 1 for x1 ; : : : ; x2n 2 Œ 1; 1. The coefficients of xj2 are equal to ˙1 and hence the sum of their absolute values is 2n . Thus in Theorem 73 the constant grows at least linearly. Proposition 75 ([Za2]). Consider the complex scalar field. Let 1 < p < 1, n 2 N, n < p, and P 2 P . n`p /. Then ! pp n 1 X p jP .ej /j p n k.P .ej //k p D kP k: p n

j D1

Proof. ([ALRT], [Din, Proposition 1.58]) For each j 2 N let j 2 C be such that p p jn P .ej / D jP .ej /j p n and j D 0 if P .ej / D 0. Note that jj jp D jP .ej /j p n . For a fixed k 2 N we have by Lemma 69 ! k k k nk k X X X X X p 1 jn P .ej / D jP .ej /j p n D P .j ej / D k P rjn;k .l/j ej n j D1 j D1 j D1 j D1 lD1 n ! pn ! k k p X X p D kP k jj jp jP .ej /j p n kP k j D1

from which the inequality follows.

j D1

t u

For the real scalar field, passing to the complexification using Theorem 1.63 we obtain an extra multiplicative constant 2n 1 C in Proposition 75, where C is the constant of equivalence of the canonical norm of the complex `p and the Taylor norm on `Qp . The optimal result again seems to be unknown. Another application of the averaging procedure will be useful later on when dealing with the cotype of Banach spaces.

Chapter 2. Basic properties of polynomials on Rn

124

Theorem 76 (Kahane’s inequality, [DJT, Theorem 11.1]). Let 0 < p; q < 1. There is a constant Kp;q > 0 such that for any normed linear space X and any x1 ; : : : ; xn 2 X

n

q ! 1

n

p ! 1

q

p 1 X 1 X

X

X

" x " x K :

j j j j p;q

2n 2n "j D˙1 j D1

"j D˙1 j D1

Notice that for p q this is a consequence of the Hölder inequality and in this case p may take Kp;q D 1. We also remark for later use that K1;2 D 2, [DJT, p. 227]. Lemma 77. Let X, Y be vector spaces over K and P 2 P n .X I Y /, n 2. Then for every x1 ; : : : ; xk 2 X k X

P .xj / D

j D1

1 2.n

X 1/k

} P

"jm D˙1 1j k 1mn 1

k X j D1

k X

"j1 xj ;

"j1 "j2 xj ;

j D1

:::;

k X

"j2 "j3 xj ; : : :

j D1

k X

"jn 2 "jn 1 xj ;

j D1

k X

! "jn 1 xj

:

j D1

Proof. The right-hand side can be expanded into 1

X

2.n 1/k

X

}.xj1 ; : : : ; xjn /: "j11 "j12 "j22 "j23 "j33 "jnn 21 "jnn 11 "jnn 1 P

"jm D˙1 1js k 1j k sD1;:::;n 1mn 1

By interchanging the sums and applying Lemma 67 (where n of the lemma is 2 and k of the lemma is .n 1/k) we can see that the only non-zero summands of the outer sum are those, for which j1 D j2 D D jn . This implies the desired formula. u t Proposition 78. Let n 2 N and p 2 .0; C1/. There is a constant Cp;n > 0 such that if X, Y are normed linear spaces, P 2 P . nX I Y /, and x1 ; : : : ; xk 2 X , then

k

k

p ! n

X

p 1 X

X

"j xj P .xj / Cp;n kP k k :

2 " D˙1 j D1 j D1 j

Proof. It is easy to see that 1 2.n

X 1/k

"jm D˙1 1j k 1mn 1

k

n

n

k

X

X X 1

"jl "jlC1 xj D k "j xj

2 " D˙1 j D1 j D1 j

1j k

Section 6. Estimates of coefficients of polynomials

125

Thus using Lemma 77, the (generalised) Hölder inequality [HLP, Theorem 11], and the Polarisation formula (Proposition 1.13) we obtain

k

k

k

X

X

X X 1

1 1 2 }k " x " " x P .xj / kP

j j j j j

2.n 1/k m "j D˙1 j D1

j D1

j D1

k

X

"jn

j D1

}k kP

1 2.n

1/k

n !1

k

n X X

"j1 xj

m

"j D˙1 j D1

k

X

2 n 1

"j xj

"jn 1 xj

j D1

k

n !1

n X X

"j1 "j2 xj

m

"j D˙1 j D1

k X

X n "j

m

"j D˙1 j D1

D

1 nn kP k k nŠ 2

1

n ! 1

n

xj

k

n

X

X

"j xj :

"j D˙1 j D1

Now it suffices to apply Kahane’s inequality (Theorem 76).

t u

We finish this section with another important application of the averaging technique, this time applied to the structural properties of spaces of polynomials. Proposition 79 ([DiD], [DiGo]). Let X be a Banach space over K with an unconditional Schauder basis and let P 2 P . nX/ be a polynomial such that jP .xj /j a > 0, j 2 N, for some normalised block basic sequence fxj g. Then P . nX/ contains `1 . Proof. Let fej g be the normalised unconditional Schauder basis of X. There exist sequences of integers n1 < m1 < n2 < m2 < such that supp xj Œnj ; mj . P1 Pmj Denote by j W X ! X the projection j a e and define lD1 al el D lDnj l l n Pj 2 P . X / by Pj D P B j . Assume first that K D C. Fix .aj / 2 B`1 and x 2 X n f0g and find j 2 C such that jn D

Pj .x/ , jPj .x/j

j 2 N. For any k 2 N we obtain using Lemma 69

! nk k X 1 X n;k P rj .l/j j .x/ jaj Pj .x/j jPj .x/j D Pj .j x/ D k n j D1 j D1 j D1 j D1 lD1

k

n

X

n;k rj .l/j j .x/ .2K/n kP kkxkn ; kP k max k

1ln j D1 P where K is the unconditional basis constant of fej g. It follows that the sum j1D1 aj Pj converges pointwise to a polynomial Q 2 P . nX/ with kQk .2K/n kP k (Theorem 1.29). k X

k X

k X

Chapter 2. Basic properties of polynomials on Rn

126

On the other hand, choosing m 2 N so that jam j 12 k.aj /k1 and 2 K with n D jaam , we have mj ˇ1 ˇ ˇ1 ˇ ˇX ˇ ˇX ˇ a ˇ ˇ ˇ ˇ kQk D sup ˇ aj Pj .x/ˇ ˇ aj Pj .xm /ˇ D jam jjP .xm /j k.aj /k1 : ˇ ˇ ˇ 2 x2BX ˇ j D1

j D1

The real case follows immediately by passing to the complexification.

t u

7. Notes and remarks Section 1. A thorough introduction into the spaces of polynomials on Rn can be found in the paper [Rez3] of Bruce Reznick. If n D 1 the basic sets of nodes for P d .R/ are characterised by the classical one-dimensional Lagrange theorem (Proposition 1.56) as any sets of distinct points of cardinality d C 1. The problem of finding simple and general geometrical conditions forcing the sets of nodes to be independent (or basic) for P . d Rn / and P d .Rn / when n 2 appears to be open. On one hand, the Lebesgue measure of the point configurations of a given cardinality (which does not exceed the dimension of the respective space of polynomials) which are linearly dependent as nodes equals to zero. Thus “randomly chosen” sets of nodes will be linearly independent. However, finding concrete examples of independent nodes and their biorthogonal polynomials (and associated Lagrange interpolation) seems to be a difficult task, see [GaSa] for a recent survey of available techniques. One of the main goals is finding good explicit examples of sets of nodes and their biorthogonal polynomials so that the norm of the projection in (1) will be as small as possible, leading to efficient polynomial approximations for given classes of continuous functions. In the one-dimensional case, by a classical result of Viktor Fedorovich Nikolaev ([Kor, pp. 150–153]) there is a lower estimate: there is c > 0 such that kLn k > c log n, where Ln is any bounded linear projection from C.Œ0; 1/ onto P n .R/. It turns out that the projections based on the Lagrange interpolation using Chebyshev polynomials satisfy an upper estimate of the same order of growth log n. For more on the projection constants in several variables see e.g. John Charles Mason’s paper [Mas1] or the survey [Mas2] which contains many further references. Pavel Petrovich Korovkin’s book [Kor] serves as a nice introduction to approximation theory respecting the functional analytic point of view. Multivariate approximation theory, in particular the Lagrange interpolation, is studied in [Lor] or [BDVX], where some estimates of the norms of the projections are given. For example, for the square Œ 1; 12 the best order of the norm of the projection is O..log n/2 /, and it is attained by the so called Padua basic set of nodes. Section 2. The topic of cubature formulae was historically motivated in part by the need for efficient numerical formulae for evaluation of integrals, see [Boj] for a nice historical account. Corollary 13 is a celebrated theorem of Vladimir L. Chakalov [Cha]. Many proofs have been published, see [CF] and references therein. Our proof

Section 7. Notes and remarks

127

is original and perhaps one of the shortest in the literature. There is a large body of literature on the subject of the location of the nodes in the representation formula, as well as on the minimal N in (2) in the multivariate case. We point out that determining whether 2 P d .X/ admits a cubature formula, i.e. corresponds to an integral with respect to positive measure, based on the knowledge of the values .x ˛ /, ˛ 2 J.n; d / is the well-known moment (or more precisely truncated moment) problem. In [Rez3] several topics closely related to the moment problems in several variables, and the problems of representation of positive semi-definite forms as sums of powers of forms of lower degree are studied (see also [El1]). We refer to [Ri], [MH], or [BE] for information on Chebyshev polynomials. We refer to [X1] and references therein for the bivariate (or multivariate) situation, which is much more complicated, but which still admits a theory using the same basic approach of orthogonal polynomials. See [GaSa] for a survey on Lagrange interpolation. Section 3. This section is devoted to various estimates concerning polynomials and their derivatives on R. There are numerous books devoted to this broad subject, e.g. [BE], [MMR], [Kl1], [Dur]. There is a vast body of literature on the subject of Chebyshev polynomials and Markov inequalities, see e.g. [RS], [Ri], [BE]. It is interesting to point out that the inequality of A. A. Markov was first conjectured by the famous Russian chemist Dmitri Ivanovich Mendeleev. We refer to [Sk] for a thorough historical account. Theorem 80 (Andrei Andreevich Markov, [MarkoA]). Let p 2 P n .R/. Then max jp 0 .x/j n2 max jp.x/j:

x2Œ 1;1

x2Œ 1;1

The equality holds if and only if p D ˙Tn . Theorem 81 (Vladimir Andreevich Markov, [MarkoV]). Let p 2 P n .R/. Then ˇ ˇ max ˇp .k/ .x/ˇ Tn.k/ .1/ max jp.x/j x2Œ 1;1

x2Œ 1;1

D

n2 .n2

1/ .n2 .k 1/2 / max jp.x/j: 1 3 .2k 1/ x2Œ 1;1

The equality holds if and only if p D ˙Tn . At the beginning of the section we followed an elementary approach of Werner Wolfgang Rogosinski in [Rog] and Pál Erd˝os in [Er], in order to expose the basic idea of finding extremal polynomials for the given functional on the space of polynomials. The same basic idea is also used later in the proof of Harris’s theorem. The main Theorem 28 was first proved by Valentin Ivanovich Skalyga in [Sk]. In the complex setting there is the following optimal result of Maciej Klimek [Kl2]: If P 2 P n .C N /, then p n sup jP .t1 ; : : : ; tN /j: sup jP .´1 ; : : : ; ´N /j 1 C 2 j´j jD1 j D1;:::;N

tj 2Œ 1;1

128

Chapter 2. Basic properties of polynomials on Rn

The next problem is posed in [MST]. Problem 82. Is it possible to improve the estimate in Theorem 26, for arbitrary real n Banach spaces, by leaving out the factor 2 2 ? Section 4. The homogeneous cubature formulae have been studied e.g. in [GoSe], [LV], [Rez3] and the references therein. The problem consists of finding concrete representing nodes and weights for a positive functional defined on homogeneous polynomials of a fixed degree. In the case when the nodes are placed on the unit sphere S n 1 and the weights are all equal we speak of the spherical design. These are rather rare objects, see e.g. [Bann] and the references therein. The presentation of the closely related theory of isometric embeddings of `n2 into `pN comes from [LV], where some of these results were proven for the first time, see also [JL, Chapter 21] and references therein. Theorem 38 was proved many times independently, e.g. [El1], [Rez3], and [DJP]. The existence of an isometric imbedding of `n2 into `pN for large enough N D N.n; p/ goes back to David Hilbert [Hilb], in the course of his positive solution of the famous Waring’s problem. D. Hilbert’s result in number theory claims that for every given k 2 N even there exists an integer N.k/ such that every integer n 2 N can be expressed as a sum of at most N.k/ of kth powers of integers. For more on this and the proof of Hilbert’s theorem see [El2]. There is a large amount of literature on the closely inter-related subjects of Lagrange interpolation and its use in numerical mathematics, isometric embeddings of Banach spaces, error-correcting codes, and optimal designs; for references see e.g. [JL, Chapter 21]. The remarkable Theorem 44 was proved in [DoKar] by Vladimir Leonidovich Dol’nikov and Roman Nikolaevich Karasev, solving a problem of Mikhail Leonidovich Gromov and Vitali Davidovich Milman from [Mi]. It was conjectured in [Mi] that n.p; k/ k d . This result is an isometric analogue of the celebrated Dvoretzky theorem. The suggested approach to the problem was via the so called Knaster conjecture. We recall that the Knaster conjecture claimed (in particular) that for any configuration of n C 1 points fxj gjnD0 on S n and any continuous real valued function f on S n there exists an orthogonal transformation T such that f .T .x0 // D D f .T .xn //. This claim seems to be quite reasonable, by the heuristic argument of “counting the dimension”. Unfortunately, counterexamples to the Knaster conjecture have been found in [KS] by Boris Sergeevich Kashin and Stanisław J. Szarek. It is still possible though that the Knaster conjecture holds true for sets containing less than n C 1 points, or sets with particular geometrical properties. This would lead to an alternative proof of Theorem 44, and perhaps also to some explicit estimates. Problem 83. Find a reasonable explicit estimate for n.p; k/ in Theorem 44 in the case of even p. We point out that the related problem of characterising isometric subspaces of Lp -spaces for non-integer p has been treated e.g. in [Kri], [DJP], and references therein. The following problem arises naturally as a finite-dimensional reformulation of a certain renorming question that we will encounter in later chapters.

Section 7. Notes and remarks

129

Problem 84. Let r; R 2 R, k 2 N be given. Are there some s; S 2 R, K 2 N with the following property: Let X be an n-dimensional Banach space, P 2 P . kX/ be a polynomial such that r < infBX P supBX P < R. Then there is a convex polynomial Q 2 P . KX/ such that s < infBX Q supBX Q < S. There seem to be no analogous results for good polynomial behaviour of quotient spaces. Problem 85. Let X be a quotient of `pn , p 2 N even. Is the quotient norm on X again a polynomial norm? Section 5. The theory of functional equations and identities in the complex plane was initiated by Maurice Fréchet [Fré1], [Fré2], William Harold Wilson [Wi], and Stanisław Mazur and Władysław Orlicz [MO1], [MO2]. The proof of Theorem 50 is new. A nice application of the theory of linear identities to the structure of Banach spaces, Corollary 61, was made by Pascual Jordan and John von Neumann. This theorem has served as a model for results characterising classes of Banach spaces by means of polynomial norms, e.g. [Gi], [Koe], [Rez1], [Rez2], and the references therein. This theory is closely related to the isometric Banach space theory, see e.g. [JL, Chapter 21] and references therein. Theorem 60 was first formulated by B. Reznick [Rez1] for polynomial norms on Banach spaces. A related approach to characterising subspaces of Lp -spaces using systems of inequalities can be found e.g. in [Kri] and references therein. Section 6. Theorem 71 was shown in [AG] for homogeneous polynomials and in general in [ABE]. We have mostly followed the presentation in [ALRT] in this section. Proposition 78 is in [DGZ, Lemma V.4.5].

Chapter 3

Weak continuity of polynomials and estimates of coefficients The present chapter is devoted to the study of polynomials between Banach spaces with the emphasis laid on their various weak continuity properties and coefficient estimates. The quantitative results, especially in Section 5, are formulated in a completely elementary language, but their study relies on several advanced tools; in particular tensor products and .p; q/-summing operators, which are outlined in the respective sections in order to keep the flow of the theory uninterrupted. These topics are covered with full proofs by the well-known books [DefFl], [DJT] devoted to these areas. The first section deals with some rather basic results concerning the theory of projective and injective tensor products of Banach spaces. We discuss, mostly without detailed proofs, the duality theory for these spaces and the related universality properties for these spaces as well as the symmetric tensor products and their duality with polynomial spaces. In the second section we first give a brief summary of the theory of uniform spaces and their topologies. We use the language of uniformities in order to study the weakly uniformly (or weakly sequentially) continuous polynomials. The class of Banach spaces with the Dunford-Pettis property plays an important role here. We apply the tensor product theory to prove the result of Raymond A. Ryan, showing that every weakly compact polynomial from a space with the DPP maps weakly Cauchy sequences to norm convergent ones. In Section 3 we study in detail the weak continuity properties in Banach spaces not containing `1 . Using the classical Rosenthal’s `1 -theorem and its relatives we show that weak sequential continuity on bounded sets coincides with full (resp. uniform) continuity for continuous mappings. We also give some results on quotients of Banach spaces which contain `1 in the spirit of the well-known fact that such spaces have a quotient isomorphic to `2 . In Section 4 we introduce the language of the theory of .p; q/-summing operators and mention some connections with tensor products. This abstract theory provides precise quantitative estimates concerning the properties of sequences of vectors in some classical Banach spaces (such as Lp , C.K/) and their images under linear operators. We formulate, without proof, some of the fundamental and deep results of the theory, introducing also the notions of type and cotype.

Section 1. Tensor products and spaces of multilinear mappings

131

In Section 5 we introduce the theory of multiple .pI q/-summing operators, which is a generalisation of the .p; q/-summing operators into the multilinear and polynomial setting. Drawing on the results from Sections 1, 4, and some other analytic tools such as the complex interpolation we outline the proof of a recent theorem of Andreas Defant and Pablo Sevilla-Peris, which gives optimal estimates on the coefficients of polynomials in P . nc0 I `p / and contains several classical inequalities as special cases. In the last section the so-called Bohr radius of absolute convergence of power series in finite-dimensional Banach spaces is studied. In this chapter except for Sections 3 and 6 all the spaces are over K, i.e. we consider both real and complex scalars.

1. Tensor products and spaces of multilinear mappings The goal of this section is to collect some definitions and elementary facts concerning tensor products and symmetric tensor products. Tensor products offer an important point of view of polynomials and multilinear mappings. Moreover, they are an indispensable tool for handling polynomial duality and .p; q/-summing operators. Given vector spaces X1 ; : : : ; Xn over K we let be the vector space of all formal P k k k linear combinations N kD1 ak .x1 ˝ ˝ xn /, ak 2 K, xj 2 Xj , where we identify PN PN .k/ .k/ k k a.k/ .x1 ˝ ˝xn / for any permutation kD1 ak .x1 ˝ ˝xn / and PNkD1 P C1 k k of f1; : : : ; N g, and similarly kD1 ak .x1k ˝ ˝ xnk / and N kD1 ak .x1 ˝ ˝ xn / for aN C1 D 0. By 0 we denote the linear subspace of spanned by the vectors .x1 ˝ ˝ .axk / ˝ ˝ xn /

a.x1 ˝ ˝ xn /;

.x1 ˝ ˝ .xk C yk / ˝ ˝ xn / .x1 ˝ ˝ xk ˝ ˝ xn /

.x1 ˝ ˝ yk ˝ ˝ xn /;

where k 2 f1; : : : ; ng, xj ; yj 2 Xj , N a 2 K. Then the quotient space =0 is called the tensor product X1 ˝ ˝ Xn D jnD1 Xj . It is easily verified that tensor products are associative, so they can be built up inductively starting from the tensor product of a pair of vector spaces. Note also that by the definition of 0 each ´ 2 X1 ˝ ˝ Xn P j j has a representation ´ D jkD1 x1 ˝ ˝ xn . An element of X1 ˝ ˝ Xn that admits a representation x1 ˝ ˝ xn is called an elementary tensor. The following is a fundamental observation: Given j 2 Xj# , j D 1; : : : ; n, the function k X j D1

j aj .x1

˝ ˝

xnj /

7!

k X

j

aj 1 .x1 / n .xnj /

(1)

j D1

is a linear form on the vector space . Thus we obtain a useful criterion for distinguishing the vectors in a tensor product:

132

Chapter 3. Weak continuity of polynomials and estimates of coefficients

Proposition 1. Let X1 ; : : : ; Xn be vector spaces and let Aj Xj# be subsets that P j j separate the points of Xj , j D 1; : : : ; n. Then jkD1 aj x1 ˝ ˝ xn D 0 in the space X1 ˝ ˝ Xn if and only if k X

j

aj 1 .x1 / n .xnj / D 0

j D1

for every choice of j 2 Aj . Proof. ) The form (1) clearly vanishes on the generating vectors of 0 and thus also on 0 . ( By the definition of the space 0 it is easily seen that we may assume that the j vectors fxl gjkD1 are linearly independent for each l D 1; : : : ; n. Using an elementary j

j

linear algebra we obtain functionals fl gjkD1 span Al biorthogonal to fxl gjkD1 . P j j Hence al D jkD1 aj 1l .x1 / nl .xn / D 0 for each l D 1; : : : ; k. t u Nn Further, we define an n-linear mapping ˝ W X1 Xn ! j D1 Xj by the formula ˝.x1 ; : : : ; xn / D x1 ˝ ˝ xn . Theorem 2 (Universality of the tensor product – algebraic case). Let X1 ; : : : ; Xn , Y be vector spaces. For every n-linear mapping M 2 L.X1 ; : : : ; Xn I Y / there exists a unique linear operator LM 2 L.X1 ˝ ˝ Xn I Y / such that M D LM B ˝: M

X1 Xn ˝

/8 Y

LM

X1 ˝ ˝ Xn The operator LM satisfies LM .x1 ˝ ˝ xn / D M.x1 ; : : : ; xn /:

(2)

Proof. Using Proposition 1 it is easily seen that ˝ maps products of linearly independent sets to linearly independent sets. Let Aj be the basis of Xj . We define LM by the formula (2) on ˝.A1 An / and extend it linearly onto the whole tensor product. Using the multilinearity it can be checked that the formula (2) still holds and it clearly uniquely determines LM . t u The operator LM corresponding to M in the above theorem is called the linearisation of M . The following result is immediate. Theorem 3. Let X1 ; : : : ; Xn be vector spaces. Given M 2 L.X1 ; : : : ; Xn I K/ and P j j ´ D jkD1 x1 ˝ ˝ xn 2 X1 ˝ ˝ Xn we put hM; ´i D

k X j D1

j

M.x1 ; : : : ; xnj / D

k X

j

LM .x1 ˝ ˝ xnj / D LM .´/:

j D1

Then hL.X1 ; : : : ; Xn I K/; X1 ˝ ˝ Xn i forms a dual pair.

Section 1. Tensor products and spaces of multilinear mappings

133

Tensor products of Banach spaces admit many non-equivalent natural norms. We will describe two important examples, namely the projective and the injective tensor norm. These norms can be shown to be in some sense extreme cases, as every “reasonable” tensor norm is bounded below by the injective norm and bounded above by the projective norm, [Ry3, Proposition 6.1]. Definition 4. Let X1 ; : : : ; Xn be normed linear spaces. The projective tensor norm on X1 ˝ ˝ Xn is defined by the formula ˚ .´/ D sup hM; ´iI M 2 L.X1 ; : : : ; Xn I K/; kM k 1 ; ´ 2 X1 ˝ ˝ Xn : The projective tensor product, denoted by X1 ˝ ˝ Xn , is the completion of the normed linear space .X1 ˝ ˝ Xn ; /. A very useful alternative description of the projective tensor product and its norm is the following, see [Ry3, Proposition 2.8]. Proposition 5. Let X1 ; : : : ; Xn be normed linear spaces. Then for every vector j ´ 2 X1 ˝ ˝ Xn there exist bounded sequences fxl gj1D1 Xl , l D 1; : : : ; n, P1 j j such that ´ D j D1 x1 ˝ ˝ xn is an absolutely convergent series and ( 1 ) 1 X j X j .´/ D inf kx1 k kxnj kI ´ D x1 ˝ ˝ xnj : j D1

j D1

Furthermore, .x1 ˝ ˝ xn / D kx1 k kxn k for every xj 2 Xj , j D 1; : : : ; n. This can be translated into a simple geometrical description of the unit ball of the projective tensor product: BX1 ˝ ˝ Xn D conv ˝.BX1 BXn /:

(3)

In particular, ˝ W X1 Xn ! X1 ˝ ˝ Xn is a bounded n-linear mapping of norm 1. From the above it follows that the projective norm is defined so that the universality property of the tensor product remains valid also in the topological sense: Theorem 6 (Universality of the tensor product – continuous case). Let X1 ; : : : ; Xn , and Y be normed linear spaces. For every M 2 L.X1 ; : : : ; Xn I Y / there exists a unique LM 2 L.X1 ˝ ˝ Xn I Y / such that M D LM B ˝. The operator LM satisfies (2) and the mapping M 7! LM is an isometry of the spaces L.X1 ; : : : ; Xn I Y / and L.X1 ˝ ˝ Xn I Y /. In particular, if Y D K, then we get a simple but important duality relation. Theorem 7. Let X1 ; : : : ; Xn be normed linear spaces. Then .X1 ˝ ˝ Xn / D L.X1 ; : : : ; Xn I K/; where the evaluation is given by hM; x1 ˝ ˝ xn i D M.x1 ; : : : ; xn /. If n D 2, then the canonical identification L.X1 ; X2 I K/ D L.X1 I X2 / of Fact 1.9 leads to an equivalent dual representation:

134

Chapter 3. Weak continuity of polynomials and estimates of coefficients

Fact 8. Let X , Y be normed linear spaces. Then .X ˝ Y / D L.XI Y /, where the evaluation is given by hL; x ˝ yi D L.x/.y/. Let us pass to the description of another, non-equivalent, natural norm on the tensor product space. Definition 9. Let X1 ; : : : ; Xn be normed linear spaces. The injective tensor norm " on X1 ˝ ˝ Xn is defined as follows: ˇ k ˇ ! k ˇX ˇ X ˇ ˇ j j " x1 ˝ ˝ xnj D sup ˇ x1 .x1 / xn .xnj /ˇ: ˇ x 2B ˇ j D1

j

X j

j D1

j D1;:::;n

By X1 ˝" ˝" Xn we denote the injective tensor product, that is the completion of the normed linear space .X1 ˝ ˝ Xn ; "/. It is easy to show that both the injective and the projective tensor products are associative, i.e. .X ˝" Y /˝" Z D X ˝" .Y ˝" Z/, resp. .X ˝ Y /˝ Z D X ˝ .Y ˝ Z/, and this holds by induction for any number of factors. In the finite-dimensional case we have the following identification. Proposition 10 ([FHHMZ, Proposition 16.8]). Let X be a finite-dimensional Banach space and Y a Banach space. Then X ˝" ˝" X ˝" Y is canonically isometric

œ n-times

to L. nXI Y / via the identification which maps every .x1 ˝ ˝ xn / ˝ y to an n-linear Qn mapping j D1 xj ./ y. Next, we look at the tensor products of linear operators. Proposition 11. Let Xj , Yj , j D 1; : : : ; n, be vector spaces and Tj 2 L.Xj I Yj /. Then there is a unique linear operator S 2 L.X1 ˝ ˝ Xn I Y1 ˝ ˝ Yn / with the property S.x1 ˝ ˝ xn / D T1 .x1 / ˝ ˝ Tn .xn /: The operator is denoted by S D T1 ˝ ˝ Tn . Proof. This follows from Theorem 2 used on an n-linear mapping M.x1 ; : : : ; xn / D T1 .x1 / ˝ ˝ Tn .xn /. t u Proposition 12. Let Xj , Yj , j D 1; : : : ; n, be normed linear spaces and Tj 2 L.Xj I Yj /. Then

T1 ˝ ˝ Tn W .X1 ˝ ˝ Xn / ! .Y1 ˝ ˝ Yn / D kT1 k kTn k;

T1 ˝ ˝ Tn W .X1 ˝" ˝" Xn / ! .Y1 ˝" ˝" Yn / D kT1 k kTn k: Proof. The first equality follows from Proposition 5. The second equality follows from the fact that x1 .T1 .x1 // xn .Tn .xn // D T1 .x1 /.x1 / Tn .xn /.xn /. t u

Section 2. Weak continuity and spaces of polynomials

135

We continue by collecting some facts on symmetric tensor products and their close relationship with polynomials. Recall that the symmetrisation of the tensor product mapping, ˝s W X X ! X ˝ ˝ X , is a symmetric n-linear mapping given by 1 X 1 X ˝.x.1/ ; : : : ; x.n/ / D x.1/ ˝ ˝ x.n/ ; ˝s .x1 ; : : : ; xn / D nŠ nŠ 2Sn

2Sn

where Sn is the set of all permutations of f1; : : : ; ng. We will also use the notation ˝s .x1 ; : : : ; xn / D x1 ˝s ˝s xn and ˝n x D ˝.nx/ D x ˝ ˝ x. The Polarisation formula (Proposition 1.13) yields that ˝ns X D spanf˝n xI x 2 Xg. The space ˝ns X is called a symmetric tensor product and the elements of ˝ns X are called symmetric tensors. When ˝ns X is equipped with the projective norm inherited from its superspace ˝n X, its completion becomes a closed subspace ˝n;s X of ˝n X . Then the linearisation Xn W ˝n X ! ˝n;s X of ˝s from Theorem 6 is a projection of norm 1. Thus we obtain the following: Theorem 13 (Universality of the symmetric tensor product). Let X, Y be normed linear spaces. For every symmetric M 2 Ls . nXI Y / there exists a unique linear operator LM 2 L.˝n;s X I Y / such that M D LM B ˝s D LM B Xn B ˝. The mapping M 7! LM is an isometry of the spaces Ls . nXI Y / and L.˝n;s X I Y /. These facts are expressed by the following commutative diagram: Xn ˝

˝n X

{ n X

˝s

M

/ ˝n X sO

LM

˝n

!

/Y =

M

X The following is immediate, using also Fact 8. Corollary 14. Let X, Y be normed linear spaces. Then the spaces P . nX I Y / and L.˝n;s X I Y / are canonically isomorphic. In particular, .˝n;s X/ D P . nX/ in the isomorphic sense, where the evaluation is given by hP; ˝n xi D P .x/. More generally, .˝n;s X/ ˝ Y D L.˝n;s X I Y / D P . nX I Y / in the isomorphic sense, where the evaluation is given by hP; ˝n x ˝ yi D P .x/.y/.

2. Weak continuity and spaces of polynomials In this section we introduce the weak continuity properties for continuous mappings. For the most part of this section we will be interested in the special case of polynomials.

136

Chapter 3. Weak continuity of polynomials and estimates of coefficients

We apply tensor products to the study of spaces of polynomials and their properties which are reflected by the weak continuity properties of individual polynomials. We remind the reader that all topological spaces in this book are assumed to be Hausdorff. Parallel to the concept of topology there is the notion of uniformity and of uniform spaces, see [Eng, Chapter 8] or [Cho]. Since this concept is perhaps slightly less known we recall some of its basic properties. All the general results below can be found in [Eng, Chapter 8]. Let S be a non-empty set. Given a subset U of S S, denote U D f.y; x/I .x; y/ 2 U g, and if V is another subset of S S , let ˚ U C V D .x; ´/I there exists y 2 S such that .x; y/ 2 U , .y; ´/ 2 V : A uniformity U in S is a filter consisting of subsets of S S that satisfy the following: (U1) (U2) (U3) (U4)

U for all U 2 U, where D f.x; x/I x 2 S g; if U 2 U, then U 2 U; for each U 2 U there exists V 2 U such that V C V U ; T U 2U U D .

The pair .S; U/ is called a uniform space. Every uniform space .S; U/ becomes canonically a topological space with the basis of the topology consisting of all sets U.x/ D fy 2 S I .y; x/ 2 U g for x 2 S , U 2 U. The topology so defined is called the topology induced by the uniformity. Every metric space .P; / is a uniform space. Indeed, it suffices to consider the family U of all supersets of sets of the form f.x; y/ 2 P P I .x; y/ < 1=ng, n 2 N. Then U is a uniformity on P such that the induced topology is the topology defined by the metric. Every topological vector space .X; / has a unique translation-invariant uniformity defined on X that induces the topology . The uniformity U is defined as follows: U X X belongs to U if and only if U D f.x; y/ 2 X XI x y 2 V g, where V is a neighbourhood of 0. The case we are going to deal with mostly is the weak topology on a Banach space .X; w/ or its restriction to some subset of X. We will call the canonical uniformity on .X; w/ the weak uniformity on X (or on its subsets). Topological spaces whose topology arises from a suitably chosen uniformity are called uniformisable. Uniformisable topological spaces are characterised as being Tychonov (i.e. completely regular) spaces. Equivalently, they are homeomorphic to subsets of Œ0; 1 . In general, the uniformity on a uniformisable space is not uniquely determined by the topology, but every compact topological space admits a unique uniformity. A mapping f from a uniform space .S; U/ into another uniform space .T; V / is called uniformly continuous if for every V 2 V there exists U 2 U such that .f .x/; f .y// 2 V whenever .x; y/ 2 U . Obviously, every uniformly continuous mapping is continuous when S and T are endowed with their induced topologies. If S T is a subset of a uniform space .T; U/, then S is naturally a uniform space by using the restricted uniformity V D fV I V D U \ S S; U 2 Ug. The identity mapping Id W S ! T is then uniformly continuous. Note that any continuous linear mapping between topological vector spaces is w–w uniformly continuous.

Section 2. Weak continuity and spaces of polynomials

137

A net fx g 2 in a uniform space .S; U/ is said to be Cauchy if given U 2 U there exists 0 2 such that .x˛ ; xˇ / 2 U for all ˛; ˇ 2 , ˛ 0 , ˇ 0 . It is easy to see that a subnet of a Cauchy net is again a Cauchy net. Note that a net fx g 2 in a topological vector space X is weakly Cauchy if and only if for each f 2 X the net ff .x /g 2 is Cauchy (or equivalently convergent). A uniform space .S; U/ is said to be complete if every Cauchy net in S converges. For every uniform space .S; U/ there exists a unique (up to a uniform isomorphism) Q i.e. a complete uniform space containing S as a dense subset. completion .SQ ; U/, A uniformly continuous mapping has a unique extension to a uniformly continuous mapping between the respective completions. Let .S; U/ be a uniform space and U 2 U. A subset A of S is said to be U-dense in S if for every x 2 S there is y 2 A such that x 2 U.y/, i.e. .x; y/ 2 U . The space .S; U/ is called totally bounded if for every U 2 U there exists a finite U-dense subset of S. The space .S; U/ is totally bounded if and only if every net in S has a Cauchy subnet ([Cho, Proposition 5.23]). A uniform space is compact if and only if it is totally bounded and complete. Consequently, if B X is a bounded subset of a Banach space equipped with the uniformity inherited from .X; w/, then its completion coincides with the w

w -closure of B in X equipped with the w -uniformity .B ; w /. The completion of a totally bounded space is totally bounded and hence compact. A continuous mapping from a compact topological space .X; / is automatically uniformly continuous in the unique uniformity on X and the unique uniformity on its compact range. A mapping between uniform spaces is called Cauchy-continuous (resp. sequentially Cauchy-continuous) if it maps Cauchy nets to Cauchy nets (resp. Cauchy sequences to Cauchy sequences). In particular, a function (or more generally a mapping into a complete space) is (sequentially) Cauchy-continuous if and only if it maps Cauchy nets (sequences) to convergent ones. It is easy to see that a uniformly continuous mapping is Cauchy-continuous (and hence also sequentially Cauchy-continuous). Further, a (sequentially) Cauchy-continuous mapping f W .X; U/ ! .Y; V/ is (sequentially) continuous. Indeed, given a net fx g 2 X converging to x, consider the directed set D f0; 1g with the lexicographic ordering and set x ;0 D x and x ;1 D x. Then the net fx˛ g˛2 is Cauchy and hence also ff .x˛ /g˛2 is Cauchy. Now if V 2 V, then there is 0 2 such that .f .x ;0 /; f .x ;1 // 2 V whenever 0 , i.e. f .x / 2 V .f .x//. A Cauchy-continuous mapping from a uniform space X to a complete uniform space can be extended to a continuous mapping on the completion of X ([Schec, Theorem 19.27]). The following observation is of great importance. Proposition 15. Let X be a totally bounded uniform space, Y a uniform space, and f W X ! Y a Cauchy-continuous mapping. Then f is uniformly continuous. Proof. The mapping f can be extended to a continuous mapping F W XQ ! YQ , where Q YQ are the completions of X, Y . Since XQ is compact, F is uniformly continuous, X, and thus so is f . t u A closed, convex, and bounded subset of a normed linear space X will be abbreviated as a CCB set. Of great importance is the fact that a bounded set B X equipped with the uniformity inherited from .X; w/ is totally bounded, which follows from the

138

Chapter 3. Weak continuity of polynomials and estimates of coefficients w

fact that B X is w -compact. In particular, a mapping from B into a Banach space Y is w–kk uniformly continuous if and only if it maps weakly Cauchy nets to convergent nets (Proposition 15). We continue by giving a list of various notions of weak continuity that will be used in our investigations. Some of these classes have been introduced and studied by Richard Martin Aron and his co-authors, e.g. in [AHV], [AP], [Aro1]; see also [Din]. Definition 16. Let X be a normed linear space, Y a Banach space, and U X a convex set. By C.U I Y / we denote the space C.U I Y / endowed with the locally convex topology b of uniform convergence on CCB subsets of U . By Cw .U I Y / we denote the linear subspace of C.U I Y / consisting of all mappings that are w–kk continuous on CCB subsets of U . By Cwu .U I Y / we denote the linear subspace of C.U I Y / consisting of all mappings that are w–kk uniformly continuous on CCB subsets of U . (In other words, f 2 Cwu .U I Y / if and only if for any CCB set V and any " > 0 there are ı > 0 and 1 ; : : : ; k 2 BX such that kf .x/ f .y/k < " whenever x; y 2 V are such that jj .x y/j < ı for j D 1; : : : ; k.) By Cwsc .U I Y / we denote the linear subspace of C.U I Y / consisting of all mappings that are w–kk sequentially continuous on CCB subsets of U , i.e. that map weakly convergent sequences in CCB subsets of U to convergent sequences in Y . By CwsC .U I Y / we denote the linear subspace of C.U I Y / consisting of all mappings that are w–kk sequentially Cauchy-continuous on CCB subsets of U , i.e. that map weakly Cauchy sequences in CCB subsets of U to convergent sequences in Y . By CK .U I Y / we denote the linear subspace of C.U I Y / consisting of all mappings that map CCB subsets of U to relatively compact sets in Y . By CwK .U I Y / we denote the linear subspace of C.U I Y / consisting of all mappings that map CCB subsets of U to relatively weakly compact sets in Y . We use the usual convention for the notation when the range space is scalars, e.g. Cwsc .U / D Cwsc .U I K/. We remark that if U is closed (in particular if U D X), then the topology on C.U I Y / is the topology of uniform convergence on bounded subsets of U ; moreover, we may replace the CCB sets in the definitions above by bounded sets and Cwsc .U I Y / (resp. CwsC .U I Y /) are just mappings w–kk sequentially continuous on U (resp. w–kk sequentially Cauchy-continuous). Also, if X is separable, thenPit is easy to see that 1 y/j the weak uniformity on BX is induced by the metric .x; y/ D 1 nD1 2n jfn .x for some sequence ffn g dense in BX . Thus in this case Cwsc .U I Y / D Cw .U I Y / and CwsC .U I Y / D Cwu .U I Y / (Proposition 15). It is easy to prove that C .U I Y /, where is one of the properties w, wu, wsc, wsC, K, is a closed subspace of C.U I Y /. In the case of CwK .U I Y / we additionally have to employ the following lemma. Lemma 17 (Alexander Grothendieck; [FHHMZ, Lemma 13.32]). Let X be a Banach space and A X . If for every " > 0 there is a weakly compact set A" X such that A A" C "BX , then A is relatively weakly compact.

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139

The following inclusions hold for any Banach space Y and any convex subset U of a normed linear space: C .U I Y / CwK .U I Y / K Cwu .U I Y / Cw .U I Y / Cwsc .U I Y / CwsC .U I Y / All the inclusions are obvious except for the one from the following lemma. Lemma 18 ([AP]). Let X be a normed linear space, Y a Banach space, and U X a convex subset. Then Cwu .U I Y / CK .U I Y /. Proof. Let f 2 Cwu .U I Y /, B U a CCB subset, and " > 0. There exist ı > 0 and 1 ; : : : ; n 2 X such that kf .x/ f .y/k < " if x; y 2 B and maxj jj .x y/j < ı. Choose a finite set fxj gjkD1 B such that f.1 .xj /; : : : ; n .xj //I j kg forms a ı-dense set in the totally bounded set f.1 .x/; : : : ; n .x//I x 2 Bg. Then clearly ff .xj /I j kg is a finite "-dense set in f .B/, so the latter set is relatively compact. t u Proposition 19. Let X be a normed linear space, Y a Banach space, U X a convex set, and f 2 CK .U I Y /. Let be one of the properties w, wsc, wsC, wu. Then f 2 C .U I Y / if and only if B f 2 C .U / for every 2 Y , i.e. if and only if f is w–w (sequentially/Cauchy/uniformly) continuous on CCB subsets of U . Proof. ) is clear. ( We give a proof in the case of Cwu .U I Y /, as the other cases are similar. Assuming the contrary, there are a CCB set B U and " > 0 such that for each finite subset F BX and for each ı > 0 there exist x.F ;ı/ ; y.F ;ı/ 2 B such that maxg2F jg.x.F ;ı/ y.F ;ı/ /j < ı and kf .x.F ;ı/ / f .y.F ;ı/ /k ". Consider a directed set D f.F; /I F BX ; jF j < 1; > 0g endowed with the partial ordering .F; / .E; ı/ provided that F E and ı. Because f .B/ is relatively compact, there exists a cofinal subset such that the nets ff .x /g 2 and ff .y /g 2 are convergent. Let v D lim 2 f .x / and w D lim 2 f .y /. Then kv wk ". Choose a functional 2 BY with .v w/ ". By the assumption there are a finite F BX and ı > 0 such that j B f .x/ B f .y/j < 2" whenever x; y 2 B and maxg2F jg.x y/j < ı. Since is cofinal in , there is ˛ 2 such that ˛ .F; ı/, kf .x˛ / vk < 4" , and kf .y˛ / wk < 4" . It follows that j B f .x˛ / B f .y˛ /j > 2" , which is a contradiction. t u Similarly we have the following. Proposition 20. Let X be a normed linear space, Y a Banach space, U X, and f W U ! Y . Then f is w–kk sequentially continuous if and only if f is w–w sequentially continuous and maps weakly compact subsets of U to compact sets. In particular, if U is a convex subset of a Banach space, then f 2 Cwsc .U I Y / if and only if f is w–w sequentially continuous on CCB subsets of U and maps weakly compact CCB subsets of U to compact sets.

140

Chapter 3. Weak continuity of polynomials and estimates of coefficients

Proof. ) The mapping f is clearly w–w sequentially continuous. Let K U be weakly compact and fxn g a sequence in K. There is a subsequence fxnk g weakly convergent to some x 2 K. Then f .xnk / ! f .x/. It follows that f .K/ is compact. w

( By contradiction, suppose that there exist " > 0 and xn ! x such that kf .xn / f .x/k " for each n 2 N. The set fxn I n 2 Ng [ fxg is weakly compact, hence ff .xn /I n 2 Ng [ ff .x/g is compact. So there is a subsequence fxnk g such w

that f .xnk / ! ´ 2 Y . By the assumption f .xnk / ! f .x/, and so ´ D f .x/, a contradiction. t u Having defined the general concepts, we introduce the corresponding classes of polynomials in a natural way. Let X be a normed linear space, Y a Banach space, and let be one of the properties w, wu, wsc, wsC, K, wK. We define P .nX I Y / D P . nXI Y / \ C .X I Y /; P n .XI Y / D P n .X I Y / \ C .X I Y /; and P .XI Y / D P .XI Y / \ C .X I Y /: Finally, we introduce the multilinear versions, using the product topology on the Cartesian power X n , which can be naturally considered as a normed linear space X ˚ ˚ X: L .nX I Y / D L. nXI Y / \ C .X ˚ ˚ XI Y /: The spaces P . nXI Y / and P n .XI Y / are closed in C.X I Y / (i.e. in the topology of uniform convergence on bounded sets; see the proofs of Theorems 1.29 and 1.46 and notice that because of the uniform convergence we do not need to use the Uniform boundedness principle) and similarly L. nXI Y / is closed in C .X ˚ ˚ XI Y /. Therefore P .nX I Y / is a closed subspace of P . nXI Y /, P n .XI Y / is a closed subspace of P n .X I Y /, and finally L .nX I Y / is a closed subspace of L. nX I Y /, where is one of the properties w, wu, wsc, wsC, K, wK. The Polarisation formula (Proposition 1.13) gives readily the following useful fact. Fact 21. Let X be a normed linear space, Y a Banach space, and let be one of 2 P .nXI Y /. If the properties w, wu, wsc, wsC, K, wK. If M 2 L .nX I Y /, then M } 2 L .nX I Y /. P 2 P . nX I Y /, then P 2 P .nXI Y / if and only if P Proposition 22. Let X be a normed linear space, Y a Banach space, andP let be n one of the properties w, wu, wsc, wsC, K, wK. Let P 2 P .XI Y /, P D jnD0 Pj , Pj 2 P . jX I Y /. The following statements are equivalent: (i) P B 2 C .BI Y / for some ball B X . (ii) Pj 2 P .jXI Y / for each j D 0; : : : ; n. (iii) P 2 P .XI Y /.

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141

Proof. (ii))(iii))(i) is obvious. To prove (i))(ii), suppose that B D B.a; r/ and define Q.x/ D P .a C x/. Then QB.0;r/ 2 C .B.0; r/I Y /. Let Qj be the j -homogeneous summand of Q. Then Qj B.0;r/ 2 C .B.0; r/I Y /, j D 0; : : : ; n, by Corollary 1.31. Hence Qj 2 P .jX I Y / by the homogeneity. It follows that Q 2 P .XI Y / and thus also P 2 P .XI Y /. t u We note that the operators in Lwsc .XI Y / are also called completely continuous operators. From Proposition 20 we obtain the following corollary. Corollary 23. Let X be a normed linear space and Y be a Banach space. Then T 2 Lwsc .XI Y / if and only if T maps weakly compact sets to compact sets. Combining Proposition 19 and Lemma 18 with the obvious fact that X Cwu .X/ we obtain the following corollary. Corollary 24. Let X be a normed linear space and let Y be a Banach space. Then LK .XI Y / D Lwu .XI Y /. Fact 25. Let X be a normed linear space and Y a Banach space. Then M 2 LK . nX I Y / M 2 LwK . nXI Y / P 2 PK . nX I Y /

if and only if LM 2 LK .˝n XI Y /; if and only if LM 2 LwK .˝n XI Y /; if and only if LP} 2 LK .˝n;s XI Y /;

P 2 PwK . nXI Y / if and only if

LP} 2 LwK .˝n;s XI Y /:

Proof. Clearly, the statements on the right-hand side imply the statements on the left-hand side. The reverse implications follow by (3), Theorems 6 and 13, the fact that in every locally convex space the convex hull of a totally bounded set is a totally bounded set, and Fact 21. t u Theorem 26 ([AHV]). Let X1 ; : : : ; Xn , and Y be normed linear spaces and let M 2 L.X1 ; : : : ; Xn I Y /. The following statements are equivalent: (i) The mapping M is w–kk continuous on bounded sets, resp. w–kk sequentially continuous. (ii) lim 2 M.x 1 ; : : : ; x n / D 0 whenever fx k g 2 Xk , k D 1; : : : ; n are weakly convergent bounded nets and at least one of them is weakly null, and is any directed set, resp. D .N; /. (iii) lim 2 M.x 1 ; : : : ; x n / D 0 whenever fx k g 2 Xk , k D 1; : : : ; n are weakly Cauchy bounded nets and at least one of them is weakly null, and is any directed set, resp. D .N; /. (iv) M maps weakly Cauchy bounded nets to Cauchy nets, resp. weakly Cauchy sequences to Cauchy sequences. Proof. (i))(ii), (iii))(ii), and (iv))(i) are clear.

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Chapter 3. Weak continuity of polynomials and estimates of coefficients

(ii))(i) Let fx k g 2 kM.x 1 ; : : : ; x n /

w

Xk be bounded nets such that x k ! x k . Then

M.x 1 ; : : : ; x n /k

n X

M.x 1 ; : : : ; x j

1

; x j

x j ; x j C1 ; : : : ; x n / :

j D1

Each of the summands on the right-hand side has a bounded weakly null net in one of the coordinates, so by (ii) the right-hand side tends to zero in limit with respect to

2 . (iii))(iv) By contradiction, assume that there are weakly Cauchy bounded nets fx k g 2 Xk and an " > 0 such that for every 2 there are ˛. / and 1 n 1 n ˇ. / with kM.x˛. / ; : : : ; x˛. / / M.xˇ. / ; : : : ; xˇ. / /k ". But 1 n kM.x˛. / ; : : : ; x˛. / /

1 n M.xˇ. / ; : : : ; xˇ. / /k

n X

M.x 1 ; : : : ; x j 1 ; x j ˇ. / ˇ. / ˛. /

j j C1 n xˇ. / ; x˛. / ; : : : ; x˛. / / :

j D1

Each of the summands on the right-hand side has a bounded weakly null net in one of the coordinates, so by (iii) the right-hand side tends to zero in limit with respect to

2 , which is a contradiction. (i))(iii) We use induction on n. If n D 1, then the statement is trivially satisfied. The inductive step from n 1 to n is proved by contradiction. Assume that fx k g 2 Xk are bounded weakly Cauchy nets for all k D 1; : : : ; n, fx 1 g 2 is weakly null, and kM.x 1 ; : : : ; x n /k > " > 0 for all 2 1 , where 1 is some cofinal subset. For any x 2 Xn the mapping .´1 ; : : : ; ´n 1 / 7! M.´1 ; : : : ; ´n 1 ; x/ 2 L.X1 ; : : : ; Xn 1 I Y / satisfies (i). Therefore for every ˛ 2 1 there is ˇ 2 1 , ˇ D ˇ.˛/ ˛ such that kM.xˇ1 ; : : : ; xˇn 1 ; x˛n /k < 2" . Whence " kM.xˇ1 ; : : : ; xˇn / M.xˇ1 ; : : : ; xˇn 1 ; x˛n /k D kM.xˇ1 ; : : : ; xˇn 1 ; xˇn x˛n /k: 2 We replace the original collection of weakly Cauchy nets by a new collection of k , k < n, ˛ 2 1 , weakly Cauchy nets fy˛1 g˛2 1 ; : : : ; fy˛n g˛2 1 , where y˛k D xˇ.˛/ n n n and y˛ D xˇ .˛/ x˛ . For k < n these new nets are subnets of the original ones ( 1 is cofinal in ). The net fy˛n g is a bounded weakly null net. This reduces the problem to the case when at least two of the nets fy˛k g˛2 1 , k D 1; : : : ; n, are weakly null. By repeating this argument .n 2/-times (passing to cofinal subsets of the index set n 1 1 ) we arrive at a collection of bounded weakly null nets f´k˛ g˛2 n 1 , k D 1; : : : ; n, such that kM.´1˛ ; : : : ; ´n˛ /k 2n" 1 for all ˛ 2 n 1 . This is a contradiction, because n 1 is a cofinal subset. t u Combining this with Proposition 15 and Fact 21 we obtain the following corollary.

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143

Corollary 27 ([AHV]). Let X be a normed linear space, Y a Banach space, and n 2 N. Then Lw . nX I Y / D Lwu . nXI Y /; Lwsc . nX I Y / D LwsC . nXI Y /; Pw . nX I Y / D Pwu . nXI Y /; Pwsc . nX I Y / D PwsC . nXI Y /: From this and the relations shown earlier we obtain the following inclusions: P .XI Y / PwK .X I Y / K Pwu .XI Y / D Pw .XI Y / Pwsc .X I Y / D PwsC .XI Y / L .X I Y / wK Lwu .XI Y / D LK .XI Y / D Lw .X I Y / Lwsc .X I Y / D LwsC .XI Y / We remark that unlike the kk–kk continuity, generally it is not sufficient to check the w–kk continuity of polynomials only at the origin. P 2 Example 28 ([Aro2]). Let P 2 P . 3 `2 / be defined as P .x/ D x1 1 nD2 xn . Then the restriction of P to any bounded set is weakly continuous at the origin, but P is w not weakly sequentially continuous. Indeed, e1 C en ! e1 , but P .e1 C en / D 1 and P .e1 / D 0. Proposition 29. Let X be a normed linear space, Y a Banach space, and let be one of the properties w, wu, wsc, wsC, K, wK. If n 2 N is such that P . nX I Y / D P .nX I Y /, then P n .XI Y / D P n .X I Y /. Proof. First we prove the statement for wsc polynomials. The proof will be in two steps. First we show that any polynomial in P n .XI Y / is weakly sequentially continuous at every point except at zero. Let k 2 N, k < n, and P 2 P . kXI Y /. Further, let w fxj g X be such that xj ! ´ ¤ 0. Let f 2 X be such that f .´/ D 1. Put Q.x/ D f .x/n k P .x/ for x 2 X. Then by our assumption Q 2 Pwsc . nXI Y /. Since P .xj / D Q.xj /=f .xj /n k for j large enough, we conclude that P .xj / ! P .´/. w

Now let P 2 P n .XI Y / and assume that xj ! 0. Let y 2 X n f0g and define Q.x/ D P .x y/ for x 2 X . Then Q 2 P n .X I Y / (Lemma 1.34). Thus by the first part of the proof P .xj / D Q.y C xj / ! Q.y/ D P .0/. The case of Pwn .X I Y / is identical using bounded nets, the cases of wu and wsC are clear from Corollary 27. The proof for compact (resp. weakly compact) polynomials is similar: First we show that P .V / is relatively compact for any P 2 P n .XI Y / and any CCB set V X not containing the origin. Fix such a set V and let k 2 N, k < n, and P 2 P . kXI Y /. Let f 2 X be such that inf f .V / D ı > 0. Put Q.x/ D f .x/n k P .x/ for x 2 X. Then by our assumption Q 2 PK . nX I Y /. Since P .x/ D Q.x/=f .x/n k for x 2 V and 0 f n1 k .V / ı n1 k , it is easy to see that P .V / is relatively compact in Y .

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Finally, if P 2 P n .XI Y / and V is a CCB set with 0 2 V , then there is y 2 X satisfying y … V . Define Q 2 P n .X I Y / by Q.x/ D P .x y/ for x 2 X. Then by the first part of the proof P .V / D Q.y C V / is relatively compact in Y . t u The next two lemmata have numerous applications as they allow one to conclude, under suitable assumptions, that the origin is the point of weak discontinuity. Lemma 30 ([CHL]). Let X be a normed linear space and Y a Banach space such that P . n 1XI Y / D Pwsc . n 1X I Y /. If P 2 P . nXI Y / n Pwsc . nX I Y /, then there is a weakly null sequence fyk g1 SX such that infk kP .yk /k > 0. kD1 X is weakly convergent to x 2 X and satisfies Proof. Suppose that fxk g1 kD1 kP .xk / P .x/k " > 0 for all k 2 N. Put yk D xk x. By Lemma 1.21 n X1 n }.jx; n jyk / P .xk / P .x/ D P .yk / C P j j D1

}.jx; n jyk / D 0 for each k 2 N. The assumption and Proposition 29 imply limk!1 P for each j 2 f1; : : : ; n 1g. It follows that there is k0 2 N such that kP .yk /k 2" for k k0 . Finally, since fyk g1 is bounded and non-zero, we can pass to the kDk0 normalised sequence. t u Lemma 31 ([D’H]). Let X , Y be Banach spaces such that X does not contain `1 and P . n 1XI Y / D PK . n 1XI Y /. If P 2 P . nXI Y / n PK . nXI Y /, then there is a weakly null sequence fyk g1 SX such that fP .yk /g1 is not relatively compact. kD1 kD1 Proof. By Rosenthal’s `1 -theorem (Theorem 71) there are an " > 0 and a weakly BX such that kP .xk / P .xl /k " for all k; l 2 N, Cauchy sequence fxk g1 kD1 k ¤ l. By Lemma 1.21 n X1 n }.jxl ; n jxk / C . 1/n P .xl /: P .xk xl / D P .xk / C . 1/j P j j D1

Pn

1 n j D1 j .

}.jx; n jxk / are compact by the assumption The polynomials x 7! 1/j P and Proposition 29, and so by passing to subsequences and then diagonalising there is an infinite M N such that the limits n X1 n }.jxl ; n jxk / . 1/j P yk D lim j l!1 l2M j D1

exist for each k 2 N. Thus for each

k "2 N we can choose mk k such that Pn 1 n j j n j

yk } xk / < 8 whenever l 2 M , l mk . Then j D1 j . 1/ P . xl ; kP .xk

xl /

P .xp xr /k

> P .xk / C yk C . 1/n P .xl /

P .xp /

yp

. 1/n P .xr /

" 4

Section 2. Weak continuity and spaces of polynomials

145

whenever k; p 2 N, l; r 2 M , l mk , and r mp . Suppose that k; l; p 2 N are given and let ´ D P .xl /C. 1/n .P .xk /Cyk P .xp / yp /. Then there is rk;l;p 2 N such that kP .xr / ´k 2" for all r rk;l;p . Whence kP .xk xl / P .xp xr /k > 4" whenever k; p 2 N, l; r 2 M , l mk , and r maxfmp ; rk;l;p g. Now it suffices to find lk 2 M such that lk maxfmk ; r1;l1 ;k ; : : : ; rk 1;lk 1 ;k g and put ´k D xk xlk . Then f´k g is weakly null and fP .´k /g is an 4" -separated sequence. Finally, by passing to a subsequence we may assume that k´k k ! c > 0. We set yk D k´´k k . Then fyk g is weakly null and k

1

1

kP .yk / P .yl /k D P .´k / P .´l /

n n k´k k k´l k ˇ ˇ ˇ ˇ ˇ ˇ ˇ1 ˇ1 1 1 1 ˇ kP .´l /k ˇ ˇ> " n kP .´k / P .´l /k kP .´k /k ˇˇ n ˇ ˇ n n n c c k´k k c k´l k ˇ 8c n for all k; l large enough, k ¤ l. t u We recall a fundamental result from the theory of Schauder bases. For the proof see [LiTz2, Proposition 1.a.11] and the remark after. Proposition 32 (Bessaga-Pełczy´nski selection principle). Suppose that X is a Banach space with a Schauder basis f.en I fn /g. If fxn g X is a sequence that satisfies limn!1 fk .xn / D 0 for each k 2 N (in particular if fxn g is weakly null) and lim supkxn k > 0, and if f"k g RC , then there are a block basic sequence fuk g X and a subsequence fxnk g such that kxnk uk k "k for each k 2 N. In particular, for a suitably chosen f"k g the sequence fxnk g is a basic sequence equivalent to fuk g. The following folklore result will be useful several times. Proposition 33. Let X be a Banach space, Y D `p , 1 p < 1, or Y D c0 , and suppose there is a non-compact operator T 2 L.XI Y /. Then there are S 2 L.XI Y / and a normalised basic sequence fxn g X such that S.xn / D en , n 2 N, where fen g is the canonical basis of Y . If X does not contain `1 , then fxn g may be chosen to be weakly null. If Y D `1 , then S is in fact onto. Proof. Denote by ffn g the functionals biorthogonal to fen g. Let fyn g1 nD1 BX be such that fT .yn /g is "-separated for some " > 0. By passing to subsequences and then diagonalising we may assume that for each k 2 N the sequence ffk .T .yn //g1 nD1 is convergent. Further, we may assume that spanfyn g is a subspace of a Banach space with a Schauder basis (e.g. C.Œ0; 1/) with coordinate functionals fn g1 nD1 . Hence similarly by passing to further subsequence we may assume that for each k 2 N the sequence fk .yn /g1 yn , n 2 N. Then nD1 is convergent. Put ´n D ynC1 " kT .´n /k kT kk´n k and so kT" k k´n k 2. Thus we may set xn D k´´nn k , n 2 N. We have limn!1 fk .T .xn // D 0 and limn!1 k .xn / D 0 for each k 2 N. Thus from Proposition 32 it follows that by passing to a subsequence we may assume that fxn g is a basic sequence. Further, from Proposition 32 combined with [FHHMZ, Theorem 4.23] and [FHHMZ, Proposition 4.45] it follows that by passing to another

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Chapter 3. Weak continuity of polynomials and estimates of coefficients

subsequence we may assume that fT .xn /g is a basic sequence equivalent to a block basis of fen g (and consequently equivalent to fen g) and moreover Z D spanfT .xn /g is complemented in Y . Let P W Y ! Z be a projection and R W Z ! Y be an isomorphism satisfying R.T .xn // D en . Then S D R B P B T W X ! Y satisfies S.xn / D en . If X does not contain `1 , then by Rosenthal’s `1 -theorem (Theorem 71) fyn g can be chosen P1 to be weakly Cauchy and consequently fxn g is weakly null. If Y D `1 , then P1 nD1 an x n clearly P1 converges absolutely in X whenever .an / 2 `1 and hence S t u nD1 an xn D nD1 an en . We will also use a non-linear version of the previous proposition (typically it is used by putting g D T B f for some non-linear non-compact mapping f ). Proposition 34. Let 1 < p < 1 and let fyn g `p be a bounded sequence that is not relatively compact. Then there are an affine transformation T 2 P 1 .`p I `p / and a subsequence fynk g such that T .ynk / D ek , k 2 N, where fek g is the canonical basis of `p . Proof. We may assume that fyn g is "-separated for some " > 0. Using the weak w compactness and separability we may assume that yn ! y 2 `p . Thus from Proposition 32 combined with [FHHMZ, Theorem 4.23] and [FHHMZ, Proposition 4.45] it follows that there is a subsequence fynk g such that fynk yg is a basic sequence equivalent to a block basis of fek g (and consequently equivalent to the basis fek g) and moreover Z D spanfynk yg is complemented in `p . Let P W `p ! Z be a projection and R W Z ! `p be an isomorphism satisfying R.ynk y/ D ek . Then T .x/ D R B P .x y/ is the desired affine transformation. t u For `1 we have a stronger statement: the transformation T can be even linear. For the proof we need the following folklore result (of course the interesting case is p D 1). Proposition 35. Let 1 p < 1 and let fxn g `p be a semi-normalised basic sequence. Then there is a subsequence fxnk g equivalent to the canonical basis of `p such that spanfxnk g is complemented in `p . Proof. By passing to a subsequence we may assume that there is y 2 `p such that w w xn ! y. Put yn D xn y. Then yn ! 0 and by Proposition 32 combined with [FHHMZ, Theorem 4.23] and [FHHMZ, Proposition 4.45] it follows that by passing to a subsequence we may assume that fyn g is a basic sequence equivalent to the canonical basis of `p and Y D spanfyn g is complemented in `p by the projection Q W `p ! Y . w

If y D 0 (as is the case when p > 1, since by the reflexivity xn ! y), then the proof is finished. So assume that p D 1 and y ¤ 0. Since fyn g is a basic sequence, by removing one of the vectors from fyn g if necessary we may assume that y … Y . Similarly, since fxn g is also a basic sequence, we may assume that y … Z D spanfxn g. Notice that Z Y ˚ spanfyg. Let f; g 2 `1 be such that f .y/ D g.y/ D 1, Y ker f , and Z ker g. Then R W `1 ! Y ˚ spanfyg, R.x/ D f .x/ y Q.y/ C Q.x/ and S W Y ˚ spanfyg ! Z, S.x/ D x g.x/y are projections, and hence we may define a projection P W `1 ! Z by P D S B R.

Section 2. Weak continuity and spaces of polynomials

147

P1 P1 P Finally, assuming 1 xn converges we get f nD1 an xn D nD1 an 2 K. nD1 anP P1 P 1 a x y a Consequently nD1 an yn D 1 converges and hence fxn g is nD1 n n nD1 n equivalent to the canonical basis of `1 . t u Proposition 36. Let fyn g `1 be a bounded sequence that is not relatively compact. Then there are T 2 L.`1 I `1 / and a subsequence fynk g such that T .ynk / D ek , k 2 N, where fek g is the canonical basis of `1 . Proof. By passing to a subsequence we may assume that fyn g is ı-separated for some ı > 0. By Schur’s theorem ([FHHMZ, Theorem 5.36]) fyn g does not contain weakly Cauchy subsequences and so by Rosenthal’s `1 -theorem we may assume that fyn g is a basic sequence equivalent to fen g. By Proposition 35 (passing to a further subsequence) there is a projection P W `1 ! spanfyn g. Let S W spanfyn g ! `1 be the isomorphism satisfying S.yn / D en . It suffices to put T D S B P . t u Recall a result of Czesław Bessaga and Aleksander Pełczy´nski, [FHHMZ, Theorem 4.44]: Let X be a Banach space such that X contains c0 . Then X contains a complemented subspace isomorphic to `1 (and hence X actually contains a complemented subspace isomorphic to `1 ). Applying this result to the duality relation in Corollary 14 we get the next result. Theorem 37 ([D’H]). Suppose that X is a normed linear space and n 2 N. Then P . nXI `1 / D PK . nXI `1 / if and only if the space P . nX/ does not contain c0 . Proof. ) Assume that P . nX/ contains c0 . By Corollary 14 the space P . nX/ is isomorphic to .˝n;s X/ , and so ˝n;s X contains a complemented subspace isomorphic to `1 by the Bessaga-Pełczy´nski theorem. Hence there is a T 2 L.˝n;s X I `1 / which is non-compact. Consequently T B ˝s … PK . nX I `1 / by Fact 25. ( If P . nXI `1 / ¤ PK . nXI `1 /, then by Fact 25 there is a non-compact operator T 2 L.˝n;s XI `1 /. Thus `1 is a quotient of ˝n;s X by Proposition 33. By the duality and Corollary 14 the space P . nX/ then contains `1 , which is a contradiction. t u

2

We recall some classical concepts and results concerning the weak sequential continuity properties of linear operators. Definition 38. We say that a normed linear space X has the Schur property (or that X is a Schur space) if every weakly convergent sequence in X is norm convergent. The space `1 is a Schur space [FHHMZ, Theorem 5.36]. It is clear that the Schur property passes to subspaces. Further, in a Schur space X every weakly Cauchy sequence is norm Cauchy. Indeed, assuming the contrary, there are a weakly Cauchy sequence fxn g X, " > 0, and increasing sequences fnk g N, fmk g N such that kxnk xmk k " for all k 2 N. Then fxnk xmk g1 is weakly null and kD1 hence also norm null, a contradiction. Consequently, P .XI Y / D PwsC .X I Y / for every Schur space X and every Banach space Y , and more generally if U is a convex subset of a Banach space with the Schur property and Y is a Banach space, then C.U I Y / D CwsC .U I Y /. We also remark that every infinite-dimensional Banach space X with the Schur property is `1 -saturated: Assume that X does not contain `1 .

148

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Then by Rosenthal’s `1 -theorem any sequence in BX has a subsequence that is weakly Cauchy and hence also norm Cauchy. This means that X is finite-dimensional. Definition 39. We say that a normed linear space X has the Dunford-Pettis property (DPP for short) if for every Banach space Y every T 2 LwK .X I Y / maps weakly compact sets to compact sets. Every infinite-dimensional reflexive Banach space clearly fails the DPP. It follows from Corollary 23 that a normed linear space X has the DPP if and only if LwK .XI Y / Lwsc .X I Y / for every Banach space Y (which is by Corollary 27 equivalent to LwK .XI Y / LwsC .X I Y /). Thus every Schur space has the DPP. There are many characterisations of the DPP. We are going to use the following one for Banach spaces. Proposition 40 (Alexander Grothendieck). Let X be a Banach space. The following statements are equivalent: (i) X has the Dunford-Pettis property. 1 (ii) lim fn .xn / D 0 whenever fxn g1 nD1 X, ffn gnD1 X are both weakly null. n!1

1 (iii) lim fn .xn / D 0 whenever fxn g1 nD1 X is weakly Cauchy and ffn gnD1 X n!1 is weakly null. 1 (iv) lim fn .xn / D 0 whenever fxn g1 nD1 X is weakly null and ffn gnD1 X is n!1 weakly Cauchy.

Proof. For (i) , (ii) see [FHHMZ, Proposition 13.42]. (ii))(iii) Assume the contrary. Then there are " > 0, a weakly Cauchy fxn g X, and a weakly null ffn g X such that fn .xn / " for every n 2 N. Since ffn g is weakly null, there is a subsequence ffnk g that satisfies fnk .xk / < 2" for each k 2 N. We put yk D xnk xk . Then fyk g is weakly null and so fnk .yk / ! 0. But fnk .yk / D fnk .xnk / fnk .xk / > 2" for each k 2 N, a contradiction. The rest of the implications is obvious or similar. t u From this it is clear that a Banach space X has the DPP if X has the DPP, in particular c0 has the DPP. The converse is false in general. The DPP also does not in general pass to subspaces or quotients. Polynomials on spaces with the DPP will be investigated later. The following theorem collects some classical results on the DPP, see [AK], [Dies1], [BourgJ2, Corollary 1.30]. Theorem 41. (i) L1 -spaces and L1 -spaces have the DPP. In particular L1 ./ and C.K/, K compact, have the DPP. (ii) Subspaces of c0 . / have the DPP. (iii) Schur spaces have the DPP. (iv) Let X be a Banach space. X is a Schur space if and only if X has the DPP and does not contain `1 . (v) If a Banach space X has the DPP and Y X is a subspace that does not contain `1 , then X=Y has the DPP.

Section 2. Weak continuity and spaces of polynomials

149

P1 Recall that a series P nD1 xn in a Banach space is called unconditionally convergent provided that the series 1 nD1 x.n/ converges P1 regardless of the permutation of the set N. It is not hard to show that a series nD1 xn is unconditionally convergent if P and only if 1 converges regardless of the scalar values jan j 1 (see e.g. a x nD1 n n [FHHMZ, Exercises 1.37–39]). In connection with this let us also recall the following simple fact. Fact 42 ([FHHMZ, Exercise 1.37]). Suppose that X is a normed linear space and x1 ; : : : ; xn 2 X . Then

n

n

n

X

X

X

max "j xj sup aj xj D sup j.xj /j:

jaj j1

2BX "j D˙1 j D1

j D1

j D1

In the case of real scalars all quantities are equal. The terminology for the following notion is traditional, albeit somewhat misleading. Definition 43. A sequence fxn g1 nD1 in a Banach space is called weakly unconditionally Cauchy (meaning weakly unconditionally Cauchy summable; wuC for short) provided P that the series 1 is weakly Cauchy regardless of the permutation of the x .n/ nD1 set N. In view of the remarks preceding the definition it is easy to see that a sequence fxn g in a Banach space X is weakly unconditionally Cauchy if and only if the series P1 nD1 .xn / converges absolutely for every P1 2 X . It follows that fxn g is weakly unconditionally Cauchy if and only if nD1 an xn is weakly Cauchy regardless of the scalar values jan j 1. Thus a subsequence of a weakly unconditionally Cauchy sequence is again a weakly unconditionally Cauchy sequence. Also note that a weakly unconditionally Cauchy sequence is weakly null and thus bounded. Furthermore, the P convergence of the series 1 .x n / is uniform in the following sense: nD1 Fact 44. A sequence fxn g1 nD1 in a Banach space X is weakly unconditionally Cauchy if and only if there is K > 0 such that for every 2 X 1 X

j.xn /j Kkk:

nD1

P Proof. Define an operator T W X ! `1 by T ./ D 1 nD1 .xn /en . Then T is clearly linear an has a closed graph, so it is bounded, from which the inequality follows. u t Bounded linear operators obviously map wuC sequences to wuC sequences. The canonical basis fej g of the space c0 is intimately connected with wuC sequences. Clearly, fej g is wuC, and fxj g is wuC in a Banach space X if and only if there is a bounded linear operator T W c0 ! X, T .ej / D xj (Facts 42 and 44). The fundamental role of the canonical basis of c0 for wuC sequences is apparent from the following two results.

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Theorem P 45 (Władysław Orlicz and Billy James Pettis, [DJT, Theorem 1.8]). Suppose that 1 nD1 xn is a series in a Banach space. Then the following statements are equivalent: P1 (i) xn converges unconditionally. PnD1 1 (ii) xnk converges for every increasing sequence fnk g N. PkD1 1 (iii) kD1 xnk converges weakly for every increasing sequence fnk g N. Note however, that the conditions above are not equivalent to the statement that P the series 1 x nD1 n converges weakly unconditionally, as the example x1 D e1 , xn D en en 1 in c0 shows. In view of the next theorem, this is essentially the only counterexample. Theorem 46 (Czesław Bessaga and Aleksander Pełczy´nski, [FHHMZ, Corollary 4.52]). 1 Let X be a Banach space Cauchy P1and let fxn gnD1 X be a weakly unconditionally sequence. Then either nD1 xn is unconditionally convergent, or fxn g1 has a block nD1 sequence equivalent to the canonical basis of c0 . In particular, the following statements are equivalent: (i) X does not contain c0 . ˚Pn (ii) If fxn g X is a sequence such that the set "j xj I "j D ˙1; n 2 N is j D1 P bounded, then 1 nD1 xn is unconditionally convergent. ˚Pn (iii) If fxn g X is a sequence such that the set S D j D1 "j xj I "j D ˙1; n 2 N is bounded, then S is relatively compact. These concepts will be studied in more detail in Section 4, now we restrict ourselves to stating a classical result which motivates much of the theory of smooth mappings and which will be generalised in the course of the present chapter and the book. For the proof see [DU, Theorem VI.2.15, Corollary VI.2.17] and [HMVZ, Theorem 7.6]. Note that the implication (i))(ii) characterises the DPP. Theorem 47. Let Y be a Banach space, K a compact space, and T 2 L.C.K/I Y /. The following statements are equivalent: (i) T 2 LwK .C.K/I Y /. (ii) T 2 LwsC .C.K/I Y /. P (iii) If fxn g C.K/ is weakly unconditionally Cauchy, then 1 nD1 T .xn / converges unconditionally. (iv) If a subspace Z C.K/ is isomorphic to c0 , then T Z is not an isomorphism. Moreover, if K is a scattered compact space, then the above statements are equivalent to T 2 LK .C.K/I Y /. We continue by introducing the notions of upper and lower estimate for sequences, which lead to rather precise results on the weak sequential continuity of polynomials. These notions will also play an important role in the asymptotic approach to polynomials in Chapter 4.

Section 2. Weak continuity and spaces of polynomials

151

Definition 48. Let 1 p; q 1. We say that a sequence fxj gj1D1 in a Banach space over K has an upper p-estimate (resp. lower q-estimate) if there exists C > 0 such that for every n 2 N and every a1 ; : : : ; an 2 K

n

! p1 n

X

X

aj xj C jaj jp ; (4)

j D1

j D1

respectively

n

X

aj xj C

j D1

n X

! q1 jaj j

q

;

j D1

where the right-hand side is replaced by max jaj j if p D 1, resp. q D 1. j D1;:::;n

Clearly a sequence has an upper 1-estimate if and only if it is bounded and an upper p-estimate implies an upper r-estimate for every r < p. Similarly, a lower q-estimate implies a lower r-estimate for every r > q. A sequence fxj g with a lower 1-estimate satisfies infj kxj k > 0 and if fxj g is a basic sequence with infj kxj k > 0, then it has a lower 1-estimate. Obviously a sequence has both upper and lower p-estimates if and only if it is a basic sequence equivalent to the canonical basis of `p . Thus if a sequence has both an upper p-estimate and a lower q-estimate, then necessarily q p. Suppose that the sequence fxj g has an upper p-estimate with Pthe constant C . It is easy to see that if .aj / 2 `p (or c0 if p D 1), then the series j1D1 aj xj converges

P

and j1D1 aj xj C k.aj /k`p . The following fact is clear. Fact 49. Let X be a Banach space and 1 p; q < 1. A sequence fxj gj1D1 X has an upper p-estimate if and only if the linear operator T W `p ! X, T .ej / D xj , is bounded. A sequence fxj gj1D1 X has a lower q-estimate if and only if the linear operator T W spanfxj g ! `q , T .xj / D ej , is bounded. In case p D 1 we replace `p by c0 and analogously for q D 1. The next fact easily follows from the above using duality. Fact 50. Let X be a Banach space and f.xj I xj /gj1D1 X X be a biorthogonal sequence. Let 1 p; q 1, p1 C q1 D 1. If fxj g has an upper p-estimate, then fxj g has a lower q-estimate. If fxj g has a lower q-estimate, then fxj g has an upper p-estimate in spanfxj g , i.e. in a quotient of X . We will use the following observation. Fact 51. A sequence fxj g in Banach space X has an upper 1-estimate if and only if it is weakly unconditionally Cauchy. Proof. If fxj g is wuC, then it has an upper 1-estimate by Fact 44. On the other ˛j 2 K, j˛ jP j D 1 such that .˛j xj / D j.xj /j. Then hand, Pn given 2 X Pchoose n

n ˛j xj C kk for any n 2 N. j.x /j D ˛ x kk t u j j j j D1 j D1 j D1

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Theorem 52 ([Peł1], [ALRT], [GJ1]). Let X , Y be Banach spaces and P 2 P . nX I Y /. If n < p < 1, then P maps sequences with an upper p-estimate to sequences with an upper pn -estimate. Proof. Let PQ 2 P . nXQ I YQ / be the complex extension of P in the case of real scalars. Suppose that fxj gj1D1 X has the upper p-estimate with the constant C > 0 and let a1 ; : : : ; ak 2 K. Combining Lemma 2.69 and Theorem 1.63 we obtain

k

k

nk ! k

X

X

X X 1 1 1

n;k n n aj P .xj / D PQ aj xj D k PQ rj .l/aj xj

n j D1

j D1

lD1

j D1

k

n

1 1 X Q

X n;k

n kP k rj .l/aj xj C n 2n k

n j D1 nk

lD1

1

kP k

k X

jaj j

p n

! pn :

j D1

t u As we have seen above, wuC sequences coincide with sequences having an upper 1-estimate, so the next theorem is a natural extension of the previous theorem and it is proved by an identical argument. Theorem 53 ([GGu3], [GJ1]). Let X and Y be Banach spaces. Then any polynomial P 2 P .X I Y / satisfying P .0/ D 0 maps wuC sequences to wuC sequences. Definition 54 ([KO2], [GJ1]). Let 1 p 1. We say that a Banach space has property Sp (resp. Tq ) if every normalised weakly null sequence has a subsequence with an upper p-estimate (resp. lower q-estimate). Obviously every Banach space has property S1 and T1 . If a Banach space has property Sp , then it has also property Sr for every r < p, and if it has property Tq , then it has also property Tr for every r > q. Both properties pass to subspaces. The space `p . / has both property Sp and Tp , the space c0 . / has both property S1 and T1 , Proposition 32 and [FHHMZ, Proposition 4.45]. The property S1 is equivalent to saying that every normalised weakly null sequence contains a subsequence equivalent to the canonical basis of c0 . If a Banach space has both property Sp and Tq , then necessarily q p. A normed linear space X is said to have the hereditary Dunford-Pettis property (hereditary DPP) if every closed subspace of X has the Dunford-Pettis property. We mention that Schur spaces, the subspaces of c0 . / (Theorem 41), as well as of the Hagler tree space [Hag2], have the hereditary DPP. It was noted in [Cem] that property S1 for a Banach space coincides with the hereditary DPP. The difficult implication depends on some deep work of John Hancock Elton (see [O] and for more recent exposition [LóTo]) and it was observed in [Dies1]. We refer to [KO2] (see also [KO1] for the case p D 1) for the proof of the next closely related theorem. Theorem 55 ([KO2]). Let 1 p 1. If a Banach space X has property Sp , then the constant C in the estimate (4) for the subsequence may be chosen independently of the normalised weakly null sequence fxj gj1D1 .

Section 2. Weak continuity and spaces of polynomials

153

We remark that an analogous theorem for property Tq fails to be true. It was observed in [Gon1] how to use spaces X constructed in [JO] to obtain a counterexample. Theorem 56. Let X be a Banach space and let ffn g X be a bounded sequence. The following statements are equivalent: (i) ffn g is not a relatively compact set. (ii) There are a subsequence fgn g of ffn g and an (infinite-dimensional) subspace Y X such that fgn Y g Y is a semi-normalised w -null sequence. (iii) There is a semi-normalised basic sequence fxn g X which is biorthogonal to a subsequence of ffn g. Moreover, we may assume in addition that fxn g is either weakly null or equivalent to the canonical basis of `1 . Proof. (iii))(i) Let fgn g be the of ffn g biorthogonal to fxn g. Then subsequence xm 1 kgm gn k .gm gn / kxm k D kxm k for m; n 2 N, m ¤ n. (i))(ii) Since ffn g is not totally bounded, there is an " > 0 such that ffn g is not contained in a union of finitely many balls of radius 2". Hence by passing to a subsequence we may assume that dist.spanff1 ; : : : ; fn g; fnC1 / ". Using [FHHMZ, Proposition 2.7] together with Helly’s theorem we can find a semi-normalised sequence fyn g1 nD1 X such that fk .yn / D 0 for 1 k < n and fn .yn / D 1, n 2 N. By passing to further subsequences and then to a diagonal sequence we may assume that limk!1 fk .yn / D tn 2 K for all n 2 N. If ftn g contains infinitely many zeros, then we are done, so from now on we assume that tn ¤ 0 for all n 2 N. Let A; B > 0 be such that kyn k A and kfn k B for all n 2 N. We distinguish two cases. If tn there is ı > 0 such that jtn j ı for all n 2 N, then we set ´n D yn tnC1 ynC1 and AB gn D fn . Then k´n k A C ı A and g .´ / D 1. In the opposite case by passing to ˇ tnC1nˇ n 1 tnC1 ˇ ˇ a subsequence we may assume that tn 2AB . We set ´n D ynC1 tn yn and 1 and jgn .´n /j 21 . gn D fnC1 . Then k´n k A C 2B Finally, we put Y D spanf´n g. Clearly, limk!1 gk .´n / D 0 for all n 2 N and kgn Y k k´1n k jgn .´n /j. Hence fgn Y g is semi-normalised and w -null. (ii))(iii) The set fgn Y g is not relatively compact. Indeed, otherwise there would be a subsequence converging to some g 2 Y . This contradicts the fact that this subsequence is w -null (and so g D 0) and semi-normalised. As in the previous part of the proof we can then find a semi-normalised sequence fyj gj1D1 Y such that gk .yj / D 0 for 1 k < j and gj .yj / D 1, j 2 N. Let a; A > 0 be such that 2a kyj k A for all j 2 N. We construct a sequence f´n g1 nD1 Y by induction, along with increasing sen 1 quences fkj gj D1 N, n 2 N0 , such that fkjn gj1D1 is a subsequence of fkjn 1 gj1D1 and k1n > k1n 1 for each n 2 N. Put kj0 D j , j 2 N. Suppose that ´l and fkjl gj1D1 , 1 l < n are already defined for some n 2 N. Since fgk n 1 Y gj1D1 is w -null, there j

a 1 is a subsequence fkjn gj1D1 of fkjn 1 gj1D1 such that k1n > k1n 1 , jgkjn .yk n 1 /j A 2j C1 1 and jgkjn .ykln /j 2j1C1 for all 1 l < j , j 2 N. Put a1 D gk1n .yk n 1 / and 1

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Chapter 3. Weak continuity of polynomials and estimates of coefficients

Pj 1 a 1 aj D gkjn .yk n 1 / a g n .ykln /, and note that jaj j A . Finally, set lD1 l kj 2j 1 P1 ´n D yk n 1 j D1 aj ykjn . Then a k´n k A C a, gk1n 1 .´n / D 1, gk j .´n / D 0 1 1 for 0 j < n 1, and gkjn .´n / D 0 for all j 2 N. Put hn D gk n 1 . By the construction, f.´n I hn /g1 nD1 is a biorthogonal system. To 1 finish the proof, by Rosenthal’s `1 -theorem we may assume by passing to a subsequence that either f´n g is equivalent to the canonical basis of `1 (in this case we set xn D ´n ) or f´n g is weakly Cauchy. In the latter case we set xn D ´2n ´2n 1 noting that f.xn I h2n /g1 nD1 is a biorthogonal system. Then fxn g is weakly null and we can pass to a final subsequence of fxn g which will be basic (Proposition 32). t u Corollary 57 ([KO1], [GJ1]). Let X be a Banach space which does not contain `1 and 1 p 1. If X has property Sp , then X has property Tq , where p1 C q1 D 1. Proof. Let fxn g1 nD1 be a normalised weakly null sequence in X . By Theorem 56 there 1 are a semi-normalised weakly null sequence fyk gkD1 and a subsequence fyk g1 of kD1 1 1 fxn g1 which is biorthogonal to fy g . By the assumption, fy g has a further k kD1 k kD1 nD1 subsequence fykj gj1D1 with an upper p-estimate. By Fact 50, fyk gj1D1 has a lower j q-estimate as required. t u

We remark that the dual statement that property Tq of X implies property Sp of X does not hold: It is observed in [GJ1] that a Banach space constructed in [JO] is a counterexample. Corollary 58 ([GJ1]). Let X be a Banach space with property Sp , 1 < p 1. If n n < p, then P n .X/ D PwsC .X/. Proof. By Corollary 27 and Lemma 30 it suffices to show that limk!1 jP .xk /j D 0 for any weakly null sequence fxk g X and any P 2 P . nX/, 1 n < p. By contradiction, suppose that jP .xk /j > ı > 0, k 2 N, for some weakly null sequence fxk g. By passing to a subsequence we may assume that fxk g has an upper p-estimate. By Theorem 52 (resp. Theorem 53) we conclude that fP .xk /g has an upper pn -estimate. P By considering the sums jmD1 tk1 tkj it is easy to see that any sequence ftk g1 in C kD1 j

with an upper r-estimate, r > 1, is convergent to 0, which is a contradiction.

t u

Since `p . / has property Sp and c0 . / has property S1 , we obtain the next result. (Recall also that in case of the separable dual the weak uniformity on bounded sets is metrisable.) Corollary 59 ([Bog], [Peł1]). Let be any set, 1 < p < 1, and n 2 N, n < p. Then n P n .`p . // D PwsC .`p . //; P .c0 . // D PwsC .c0 . //: In particular, in the separable case n P n .`p / D Pwu .`p /; P .c0 / D Pwu .c0 /: P On the other hand, if n p, then x 7! j1D1 xjn 2 P . n`p / n Pwsc . n`p /.

Section 2. Weak continuity and spaces of polynomials

155

Corollary 60 ([AAD]). Let T denote the (original) Tsirelson space. Then we have P .T / D Pwu .T /. Proof. By [CasaSh, Proposition V.10] each bounded block basis of the canonical basis of T has an upper p-estimate for every p < 1. Thus T has property Sp for every p < 1 by Proposition 32 and we may apply Corollary 58. t u Corollary 61. Let Q 2 P n .`p /, n < p < 1, and " > 0. Then there exists N 2 N such that kQ Q B PN k ", where PN W `p ! `p is the projection onto the first N coordinates. Proof. By contradiction, suppose that there exists a sequence fxk g B`p such that jQ.xk / Q.Pk .xk //j > ". By the weak compactness there is a subsequence fxkj g w that converges weakly to x. Thus also Pkj .xkj / ! x (check the convergence on the coordinate functionals), which contradicts the fact that Q is weakly sequentially continuous (Corollary 59). t u be an uncountable set and P 2 P . nc0 . //, n 2 N, resp. Corollary 62. Let n P 2 P . `p . //, 1 n < p < 1. Then P is countably supported in the sense that the set f 2 I P .e / ¤ 0g is countable. Proof. Assume the contrary. Then there are " > 0 and an infinite set 0 such that w jP .e /j > " for all 2 0 . But then there is a sequence f n g 0 such that e n ! 0, a contradiction with Corollary 59. t u Theorem 63 ([Peł1], [ALRT],[GGu3], [GJ1]). Let X be a Banach space with property Sp , 1 < p < 1, and Y a Banach space with property Tq , 1 q < 1. Then n P n .XI Y / D PwsC .X I Y / for n < pq . If X has property S1 and Y has no subspace isomorphic to c0 , then P .X I Y / D PwsC .XI Y /. Proof. By Lemma 30 it suffices to show that limk!1 kP .xk /k D 0 for any weakly in X and any P 2 P . nXI Y /, 1 n < pq (resp. any n 2 N). null sequence fxk g1 kD1 Assume the contrary. Then there are n < pq (resp. n 2 N), P 2 P . nX I Y /, " > 0, and a normalised weakly null sequence fxk g satisfying kP .xk /k " for all k 2 N. Consider the case p < 1 first. By the assumptions we may assume without loss of generality that fxk g has an upper p-estimate. By Theorem 52, fP .xk /g has an upper p n -estimate. Note that by Corollary 58 the sequence fP .xk /g is also weakly null. Hence some subsequence of it has a lower q-estimate, which is not possible as q < pn . Suppose now that p D 1. By the assumptions and Fact 51 we may assume without loss P1 of generality that fxk g is wuC. By Theorem 53, fP .xk /g is also wuC. Thus nski theorem (Theorem 46), which kD1 P .xk / is convergent by the Bessaga-Pełczy´ is a contradiction. t u Corollary 64 ([AAD], [GJ1]). Let T be the (original) Tsirelson space and Y a Banach space with property Tq , 1 q < 1. Then P .T I Y / D Pwu .T I Y /. Proof. Combine Theorem 63 with the fact that T has property Sp for every p < 1, [CasaSh, Proposition V.10], and use the separability of .T / . u t

156

Chapter 3. Weak continuity of polynomials and estimates of coefficients

Note that by Proposition 4.36 the previous result applies in particular to any Y of a non-trivial cotype. If Y is of a non-trivial type, then even much stronger results hold, see Theorem 6.54 and Theorem 6.32. We will use the following classical theorem. For the proof see e.g. [Con, Theorem VI.5.5]. Theorem 65 (Vera Ruvimovna Gantmacher). Let X , Y be Banach spaces and let T 2 L.XI Y /. The following statements are equivalent: (i) T is weakly compact. (ii) T .X / Y . (iii) T is weakly compact. 2 L.X I c0 .NI Y //. Lemma 66. Let X, Y be Banach spaces and let T D .Tk /1 kD1 Then T is weakly compact if and only if the following two conditions hold: (i) Tk 2 LwK .XI Y / for each k 2 N and (ii) lim kTk .F /k D 0 for any F 2 X . k!1

, so the equivalence Proof. In our case T W X ! `1 .NI Y /, T D .Tk /1 kD1 is clear from Theorem 65. t u Theorem 67 (Raymond A. Ryan, [Ry1]). Let X be a normed linear space. The following statements are equivalent: (i) X has the Dunford-Pettis property. (ii) LwK .X I Y / LwsC .XI Y / for every Banach space Y . (iii) LwK . nX I Y / LwsC . nXI Y / for every Banach space Y and every n 2 N. (iv) PwK .X I Y / PwsC .XI Y / for every Banach space Y . Proof. (iii))(iv))(ii),(i) are clear. We prove (i))(iii) by induction. For n D 1 it is just the implication (i))(ii). Assume that (iii) holds for n 2 N and let M 2 LwK . nC1XI Y /. By Theorem 26 it j suffices to show that given weakly convergent sequences fxk g1 X, j D 1; : : : ; n, kD1 1 and a weakly null sequence fyk gkD1 X we have limk!1 M.xk1 ; : : : ; xkn ; yk / D 0. We will use the following notation: For any S 2 L. nC1X I Z/, where Z is a normed linear space, and for x 2 X we define SŒx 2 L. nXI Z/ by SŒx.´1 ; : : : ; ´n / D S.´1 ; : : : ; ´n ; x/. Clearly kSŒxk kSkkxk. We put Ak D M Œyk for k 2 N. For any ´1 ; : : : ; ´n 2 X the linear mapping x 7! M.´1 ; : : : ; ´n ; x/ is weakly compact, so limk!1 M.´1 ; : : : ; ´n ; yk / D 0 by the Dunford-Pettis property. Hence A D .Ak /1 2 L. nX I c0 .NI Y // with kAk kM k supk2N kyk k. We will show kD1 that A 2 LwK . nX I c0 .NI Y //, which enables us to finish the proof by induction. Indeed, by the inductive hypothesis we obtain A 2 LwsC . nXI c0 .NI Y // and so limk!1 A.xk1 ; : : : ; xkn / D .wj /j1D1 2 c0 .NI Y /. Since kM.xk1 ; : : : ; xkn ; yk /k D kAk .xk1 ; : : : ; xkn /k kAk .xk1 ; : : : ; xkn / sup kAj .xk1 ; : : : ; xkn / j 2N

wj k C kwk k

and this upper estimate tends to zero with k ! 1, we are done.

wk k C kwk k

Section 2. Weak continuity and spaces of polynomials

157

Fact 25 implies that A 2 LwK . nXI c0 .NI Y // is equivalent to the linearisation LA 2 L.˝n X I c0 .NI Y // being weakly compact. Clearly LA D .LAk /1 , where kD1 LAk 2 L.˝n XI Y / are the linearisations of Ak . By Fact 25, LAk 2 LwK .˝n XI Y /, ./k D 0 for every and hence by Lemma 66 it remains to show that limk!1 kLA k n n 2 .˝ X / D L. X/ (Theorem 7). To do this, fix 2 L. nX/ and consider the operator T W X ! L. nC1X/ defined by hT .x/; Si D h; SŒxi;

S 2 L. nC1X/:

It is easy to see that T is well-defined, linear, and bounded. So for every hLA ./; i D h; LA . /i D h; k k

D hT .yk /;

B LAk i D h;

B M i D hT .yk /;

2 Y

B Ak i

B LM i D hLM .T .yk //; i:

Therefore LA ./ D .LM B T /.yk / for all k 2 N. Fact 25 implies that the operator k LM is weakly compact and so LM 2 LwK ..˝nC1 X/ I Y / by Theorem 65. Thus LM B T 2 LwK .X I Y / Lwsc .XI Y / by the DPP. Since fyk g is weakly null, it follows that limk!1 LA ./ D limk!1 .LM B T /.yk / D 0. t u k

So if X is a normed linear space with the DPP and Y is any Banach space, then we can update our chains of inclusions as follows: Pwu .X I Y / D Pw .X I Y / PK .X I Y / PwK .XI Y / Pwsc .XI Y / D PwsC .X I Y / Lwu .XI Y / D Lw .XI Y / D LK .XI Y / LwK .X I Y / Lwsc .XI Y / D LwsC .X I Y / It follows that if X is a normed linear space with the Dunford-Pettis property, then P .X / D PK .X/ D PwsC .X/. Thus in this case any P 2 P .XI Y /, where Y is any Banach space, maps weakly Cauchy sequences to weakly Cauchy sequences (consider the compositions with linear functionals). This is a result of Aleksander Pełczy´nski, [Peł4]. By a similar argument to that of Theorem 67 we get a nice and useful variant below. Theorem 68 ([GGu1]). Let X be a normed linear space with the DPP and Y a Banach space such that L.X I Y / D LwK .X I Y /. Then P .X I Y / D PwsC .XI Y /. Proof. We prove that L. nXI Y / D LwsC . nX I Y / by induction on n. For n D 1 this follows from the assumption and the DPP. Assume that the statement holds for n 2 N and let M 2 L. nC1X I Y /. By Theorem 26 it suffices to show that given weakly j convergent sequences fxk g1 X, j D 1; : : : ; n C 1, where fxk1 g is weakly null, kD1 nC1 1 we have limk!1 M.xk ; : : : ; xk / D 0. Define Tk 2 L.XI Y / by Tk .x/ D M.xk1 ; : : : ; xkn ; x/ and T W X ! c0 .NI Y / by T .x/ D .Tk .x//1 . By Theorem 26 the operator T maps indeed into c0 .NI Y /. kD1 Clearly T is a linear operator and kT k kM k supk2N kxk1 k kxkn k. We claim that T is weakly compact, and consequently T 2 LwsC .XI c0 .NI Y // by the DPP. Then

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Chapter 3. Weak continuity of polynomials and estimates of coefficients

limk!1 T .xknC1 / D .wj /j1D1 2 c0 .NI Y /. Since kM.xk1 ; : : : ; xknC1 /k D kTk .xknC1 /k kTk .xknC1 / sup kTj .xknC1 / j 2N

wk k C kwk k

wj k C kwk k

and this upper estimate tends to zero with k ! 1, we are done. To prove the claim note that by the assumption Tk 2 LwK .XI Y / for each k 2 N, so by Lemma 66 it remains to show that limk!1 kTk .F /k D 0 for every F 2 X . To do this, fix F 2 X . By the assumption the linear operator ´ 7! M.x1 ; : : : ; xn ; ´/ is weakly compact and by Theorem 65 so is its second adjoint, which then maps into Y . Thus we may define a mapping QF W X n ! Y by QF .x1 ; : : : ; xn / D ´ 7! M.x1 ; : : : ; xn ; ´/ .F /: It is easy to see that QF 2 L. nXI Y / and Tk .F / D QF .xk1 ; : : : ; xkn /. Indeed, hTk .F /; i D hF; Tk . /i D hF; M.xk1 ; : : : ; xkn ; / . /i D hQF .xk1 ; : : : ; xkn /; i for any 2 Y . As QF 2 LwsC . nXI Y / by the inductive hypothesis, we obtain limk!1 Tk .F / D limk!1 QF .xk1 ; : : : ; xkn / D 0, since fxk1 g is weakly null. t u Corollary 69 ([Peł4]). Let K be a compact space and Y a Banach space with no subspace isomorphic to c0 . Then P ..C.K/I Y / D PwsC .C.K/I Y /. Proof. By Theorem 41, C.K/ spaces have the DPP. By the classical Theorem 47 we have L.C.K/I Y / D LwK .C.K/I Y /, so the result follows from Theorem 68. t u The next proposition shows that the compactness passes from the mapping to its derivatives. Proposition 70. Let X be a normed linear space, Y a Banach space, U X open, and let f W U ! Y be T k -smooth at a 2 U with the approximating polynomial P 2 P k .XI Y /. If P … PK .XI Y /, resp. P … PwK .XI Y /, then f .U / is not relatively compact, resp. relatively weakly compact. Proof. Without loss of generality we may assume that a D 0. First assume that P … PK .X I Y /. Since T k -smoothness implies T m -smoothness for m < k, we may without loss of generality assume that P D Q C Pk , where Q 2 PKk 1 .XI Y / and Pk 2 P . kXI Y / n PK . kX I Y /. There are " > 0 and fxn g BX such that fPk .xn /g is an "-separated set. By the definition of T k -smoothness there is ı > 0 such that kf .h/ P .h/k 4" khkk whenever h 2 X, khk ı. Since Q is compact, by passing to a subsequence we may assume that there is y 2 Y such that kQ.ıxn / yk < 8" ı k for all n 2 N. Therefore kf .ıxm /

f .ıxn /k kPk .ıxm /

Pk .ıxn /k

kQ.ıxm /

Q.ıxn /k

" P .ıxn /k ı k 4 whenever m ¤ n, which means that f .U / is not relatively compact. kf .ıxm /

P .ıxm /k

kf .ıxn /

Section 3. Weak continuity and `1

159

In case that P … PwK .XI Y / we may similarly assume that P D Q C Pk , where k 1 Q 2 PwK .X I Y / and Pk 2 P . kXI Y / n PwK . kXI Y /. There is a net fx˛ g BX such that fPk .x˛ /g has no weak cluster point. Let y 2 Y be a w -cluster point of fPk .x˛ /g in Y . Clearly y … Y . Let " D dist.y; Y / > 0. There is ı > 0 such that kf .h/ P .h/k 2" khkk whenever h 2 X, khk ı. Note that in particular it w means that f .ıBX / is bounded. Let fxˇ g be a subnet of fx˛ g such that Pk .xˇ / ! y. w

By the compactness there is a subnet fx g of fxˇ g such that Q.ıx / ! u for some w

u 2 Y and f .ıx / ! ´ for some ´ 2 Y . We have kf .ıx / w

P .ıx /k

" k 2ı

and f .ıx / P .ıx / ! ´ u ı k y, which implies k´ u ı k yk 2" ı k . Since dist.ı k y; Y / D "ı k , it follows that ´ u … Y and hence ´ … Y . This means that ff .ıx /g has no weak cluster point. t u

3. Weak continuity and `1 In this section we continue our study of weakly continuous mappings initiated in Section 2, focusing on the case when the domain does not contain `1 . All the spaces in this section are real. We start by recalling some important classical results concerning Banach spaces not containing `1 . Theorem 71 (Rosenthal’s `1 -theorem [Ros2], [FHHMZ, Theorem 5.37]). Every bounded sequence in a Banach space has a subsequence which is either weakly Cauchy or equivalent to the canonical basis of `1 . Recall that a function f on a topological space T is of Borel class 1 if f 1 .G/ is an F set for every open set G R. It is easy to see that a pointwise limit of a sequence of continuous functions on T is of Borel class 1. If T is metrisable, then also the converse statement holds, namely f is of Borel class 1 if and only if f is a pointwise limit of a sequence of continuous functions on T ([Kec, Theorem 24.10]). Theorem 72 (Edward Wilfred Odell and Haskell Paul Rosenthal, [OR]; [FHHMZ, Theorem 5.40]). Let X be a separable Banach space. The following statements are equivalent: (i) X does not contain `1 . (ii) Every element of BX is a w -limit of a sequence in BX . (iii) card X D card X D c. Definition 73. A subset A of a topological space T is called relatively countably compact in T if every sequence in A has a cluster point (in T ). A topological space T is said to have countable tightness if for every A T and x 2 A there is a countable set S A such that x 2 S. A topological space T is called angelic if for every relatively countably compact A T the following two conditions hold: (i) A is relatively compact, (ii) given x 2 A there is a sequence fxn g A such that xn ! x.

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Chapter 3. Weak continuity of polynomials and estimates of coefficients

It is not difficult to show that the above properties are hereditary (i.e. they pass to arbitrary subsets). Note that if X is a Banach space and .BX ; w / is angelic, then the points in the weak closure of a bounded set A X are w -limits of sequences from A. The following observation notes that the same holds for weak closures in X provided that the bidual is angelic. Fact 74. Let X be a normed linear space such that .BX ; w / is angelic. If A X w w is bounded and x 2 A , then there is a sequence fxn g A such that xn ! x. Proof. Without loss of generality we may assume that A BX . Then A is relatively w -compact in BX . We have x 2 A w

.X;w/

sequence fxn g A such that xn ! x in

A

X .

.X ;w /

BX and so there is a w

But this means that xn ! x in X.

t u

We will rely on the following fundamental result. Theorem 75 (Irving Kaplansky [FHHMZ, Theorem 3.54]). If X is a normed linear space, then the topological space .X; w/ has countable tightness. If a compact space K does not have countable tightness (e.g. K D Œ0; !1 ), then for X D C.K/ we have K .SX ; w /, and so .X ; w / does not have countable tightness either. Thus Kaplansky’s theorem in general does not hold for the weak topology. Nevertheless, for some Banach spaces .BX ; w / is angelic, which is stronger (Fact 74) than the countable tightness of .BX ; w/; see e.g. [EW]. An important example is provided by separable Banach spaces not containing `1 . Let us describe the situation in more detail. Recall that a Polish space is a separable and completely metrisable topological space, and a Souslin space is a topological space that is a continuous image of a Polish space. Theorem 76 (Jean Bourgain, David H. Fremlin, and Michel Talagrand, [BFT]; see [FHHMZ, Theorem 5.49]). If S is a Souslin space, then the space of functions on S of Borel class 1 with the topology of pointwise convergence is angelic. As a corollary we obtain the following result. Theorem 77. If X is a separable Banach space that does not contain `1 , then .X ;w / is angelic. Proof. As X is separable, .BX ; w / is a Polish space, [FHHMZ, Proposition 3.101]. Since the topological sum of a sequence of Polish spaces is La Polish space, it follows that .X ; w / is a Souslin space (as a continuous image of 1 nD1 .nBX ; w /). Consider X the canonical embedding .X ; w / R . Since X C .X ; w / , in view of Theorem 72 the elements of X are functions on .X ; w / of Borel class 1. Hence we are in a position to apply Theorem 76. t u Corollary 78. Let X be a Banach space that does not contain `1 . If A X is bounded w w and x 2 A , then there is a sequence fxn g A such that xn ! x.

Section 3. Weak continuity and `1

161

Proof. By Kaplansky’s theorem (Theorem 75) there exists a countable S A such w that x 2 S . Let Y D span.S [ fxg/. Then Y is separable and hence .Y ; w / is angelic by Theorem 77. Fact 74 finishes the proof. t u We now continue in developing our theory of weakly continuous mappings. Theorems 79 and 81 below are strengthenings of the observation mentioned after Definition 16: if X is a normed linear space with X separable, then the weak uniformity on bounded subsets of X is metrisable and hence Cwsc .U I Y / D Cw .U I Y / and CwsC .U I Y / D Cwu .U I Y / for U X convex and Y any Banach space. Theorem 79 ([CHL], [FGL], [Gut]). Let X , Y be Banach spaces such that X does not contain `1 and U X a convex set. Then Cwsc .U I Y / D Cw .U I Y /. Proof. We only need to prove that Cwsc .U I Y / Cw .U I Y /. By contradiction, let f 2 Cwsc .U I Y / n Cw .U I Y /. Then there are a CCB set V U , " > 0, A V , w and x 2 A such that dist.f .x/; f .A// > ". By Corollary 78 there is a sequence w fxn g A such that xn ! x. As f 2 Cwsc .U I Y /, we have lim f .xn / D f .x/, a contradiction. t u Combining Theorem 79 and Corollary 27 we obtain the next result. Corollary 80 ([AHV]). Let X, Y be Banach spaces such that X does not contain `1 . Then Pwsc .XI Y / D Pwu .XI Y /. Theorem 81 ([CHL]). Let X, Y be Banach spaces such that X does not contain `1 and let U X be a convex set. Then CwsC .U I Y / D Cwu .U I Y /. Proof. We only need to prove that CwsC .U I Y / Cwu .U I Y /. By contradiction, let f 2 CwsC .U I Y / and let V U be a CCB set such that f is not w–kk uniformly continuous on V . Then there is " > 0 such that for each finite ˚ BX and each n 2 N there are x˚;n ; y˚;n 2 V satisfying max2˚ j.x˚;n y˚;n /j < n1 and kf .x˚;n / f .y˚;n /k ". Put A D fx˚;n y˚;n I ˚ BX finite; n 2 Ng, which is w a bounded set. Clearly 0 2 A and so by Corollary 78 there is a weakly null sequence f´k g A satisfying ´k D xk yk , kf .xk / f .yk /k " for all k 2 N. Using Rosenthal’s `1 -theorem (Theorem 71), by passing to subsequences we may assume that fxk g and fyk g are weakly Cauchy. Since the sequence x1 ; y1 ; x2 ; y2 ; : : : is then also weakly Cauchy, we have reached a contradiction. t u The previous results do not hold in general: Since `1 has the Schur property, L.`1 I `1 / D Lwsc .`1 I `1 / D LwsC .`1 I `1 /, while Id … LK .`1 I `1 / D Lwu .`1 I `1 /. Let X be a Banach space that does not contain `1 , Y a Banach space, and U X a convex set. By combining the above results with the inclusions obtained earlier in this chapter we conclude the following relations: C .U I Y / CwK .U I Y / K Cwu .U I Y / D CwsC .U I Y / Cw .U I Y / D Cwsc .U I Y / Pwu .XI Y / D Pw .XI Y / D PwsC .X I Y / D Pwsc .X I Y / PK .X I Y / PwK .XI Y / Lwu .XI Y / D Lw .XI Y / D LwsC .X I Y / D Lwsc .X I Y / D LK .X I Y / LwK .XI Y /

162

Chapter 3. Weak continuity of polynomials and estimates of coefficients

Note also that if X does not contain `1 and has the DPP (which is equivalent to X being a Schur space, Theorem 41), then all the spaces of polynomials above are equal. By combining Theorem 81 with Corollary 59 and Theorem 63 we obtain the following improvement of the non-separable case: Corollary 82 ([Peł1], [ALRT],[GGu3], [GJ1]). Let be any set, 1 < p < 1, and n .` . // and P .c . // D P .c . //. n 2 N, n < p. Then P n .`p . // D Pwu p 0 wu 0 Further, if is any set, 1 q < 1, and n < pq , then P n `p . /I `q ./ D n ` . /I ` ./ and P c . /I ` ./ D P Pwu p q 0 q wu c0 . /I `q ./ . For the definition of type and some elementary properties see Section 4. Theorem 83 ([CHL]). Let X , Y be Banach spaces such that X does not contain `1 and Y is of type p > 1. Let p1 C q1 D 1 and suppose that P . ndX/ D Pwsc . ndX/ for n .X I Y /. some n; d 2 N, d > q. Then P n .X I Y / D Pwu Proof. By Corollary 80 it suffices to prove that P . mXI Y / D Pwsc . mX I Y / for all 1 m n. We proceed by contradiction. Let 1 m n be minimal such that P . mX I Y / ¤ Pwsc . mXI Y /. By Lemma 30 there exist P 2 P . mXI Y /, a normalised weakly null sequence fxj g X , and ı > 0 such that kP .xj /k > ı, j 2 N. By the assumption and Proposition 29, B P 2 Pwsc . mX/ for any 2 Y , so we conclude that fP .xj /g is weakly null in Y . By Corollary 4.43 there exist a subsequence fyk g of fP .xj /g and a polynomial Q 2 P . dY / such that Q.yk / D 1, k 2 N. Hence Q B P 2 P . mdX/ n Pwsc . mdX/, a contradiction with Proposition 29. t u To be able to lift weakly Cauchy sequences from quotients we need the following lemma. Lemma 84 (Robert Henry Lohman, [Loh]). Let X be a Banach space, Y X a subspace that does not contain `1 , and Q W X ! X=Y the canonical quotient mapping. If fyn g BX=Y is weakly Cauchy, then there is a weakly Cauchy sequence fxn g X such that limkxn k 1 and fQ.xn /g is a subsequence of fyn g. Proof. By contrary, assume that fyn g BX=Y is weakly Cauchy and there is no weakly Cauchy sequence fxn g X such that limkxn k 1 and fQ.xn /g is a subsequence of fyn g. For each n 2 N let xn 2 X be such that Q.xn / D yn and kxn k 1 C n1 . By passing to a subsequence we may assume that limkxn k 1. By the assumption fxn g has no weakly Cauchy subsequence and so by Rosenthal’s `1 -theorem we may assume that fxn g is a basic sequence equivalent to the canonical basis fen g of `1 . w Put ´n D y2n y2n 1 . Since ´n ! 0, by Mazur’s theorem ([FHHMZ, Corollary 3.46]) applied to the sets f´n I n pg, p 2 N there are an increasing sequence fpn g N and a sequence fvn g X=Y such that vn 2 convf´j I pn j < pnC1 g PpnC1 1 PpnC1 1 and vn ! 0. Let ˛j 0 be such that vn D j Dp ˛j ´j and j Dp ˛j D 1 for n n PpnC1 1 all n 2 N, and put un D j Dpn ˛j .x2j x2j 1 /. Then fun g is a block basis of fxn g

Section 3. Weak continuity and `1

163

and so it is a basic sequence equivalent to a (semi-normalised) block basis of fen g; thus it is equivalent to fen g ([FHHMZ, Proposition 4.45]). Since Q.un / D vn , it follows that dist.un ; Y / D kvn k ! 0. Thus by passing to a subsequence we may find vectors wn 2 Y sufficiently close to un so that fwn g is a basic sequence equivalent to fun g ([FHHMZ, Theorem 4.23]). In particular, Y contains `1 , which is a contradiction. u t Lemma 85. Let X be a Banach space, Y X a separable subspace that does not contain `1 , and Q W X ! X=Y the canonical quotient mapping. If fyn g BX=Y is weakly Cauchy, then there is a weakly Cauchy sequence fxn g X such that limkx2k 1 k 1, limkx2k k 3, fQ.x2k 1 /g is a subsequence of fy2k 1 g, and fQ.x2k /g is a subsequence of fy2k g. Proof. By Lemma 84 there are weakly Cauchy sequences fuk g X, fvk g X such that limkuk k 1, limkvk k 1, fQ.uk /g is a subsequence of fy2k 1 g, and fQ.vk /g is a subsequence of fy2k g. By the w -completeness of 2BX (which follows from the w -compactness) there exist limits u D w -lim uk 2 X and v D w -lim vk 2 X . Since Q is w –w continuous, we have Q .u v/ D w -lim Q .uk vk / D w-lim Q.uk vk / D 0. So u v 2 ker Q D Y (consider the canonical embedding Y X and note that F 2 X belongs to Y if and only if F .f / D 0 for every f 2 Y ? , and that Q is onto Y ? ). Clearly ku vk 2 and so by the Goldstine .Y ;w /

theorem u v 2 2BY D 2BY . By Theorem 77 the compact .2BY ; w / is angelic and so there is a sequence f´k g 2BY such that u v D w -lim ´k . Then w -lim vk C ´k D u, Q.vk C ´k / D Q.vk /, and by passing to a subsequence we may assume that limkvk C ´k k 3. So the sequence fxn g defined by x2k 1 D uk , x2k D vk C ´k satisfies the requirements. t u We will use the following criterion several times throughout the book. Fact 86. Let fxn g be a sequence in a metric space P . Then fxn g is not Cauchy if and only if it has a subsequence fyn g such that dist fy2k 1 I k 2 Ng; fy2k I k 2 Ng > 0. Proof. ( is clear. ) Without loss of generality we may assume that P is complete. If fxn I n 2 Ng is totally bounded, then the sequence fxn g has two distinct cluster points x; y 2 P and we choose the subsequence fyn g so that y2k ! x and y2kC1 ! y. Otherwise there are " > 0 and an infinite "-separated subset of fxn I n 2 Ng, so the statement follows. u t The following result goes in the same direction as Theorem 41(v). Proposition 87 ([CHL]). Let X be a Banach space and let Y X be a separable subspace that does not contain `1 . Let Z be a Banach space and suppose that n .X=Y I Z/. P . nXI Z/ D Pwsc . nXI Z/ for some n 2 N. Then P n .X=Y I Z/ D Pwsc Proof. Assuming the contrary, there is P 2 P . mX=Y I Z/ n PwsC . mX=Y I Z/ for some 1 m n (Corollary 27). Therefore by Fact 86 there is a weakly Cauchy sequence fyj g X=Y such that dist fP .y2k 1 /I k 2 Ng; fP .y2k /I k 2 Ng > 0. Denote by Q W X ! X=Y the canonical quotient mapping. By Lemma 85 there is a weakly Cauchy sequence fxj g X such that fQ.x2k 1 /g is a subsequence of fy2k 1 g

164

Chapter 3. Weak continuity of polynomials and estimates of coefficients

and fQ.x2k /g is a subsequence of fy2k g. However, since by Proposition 29 and the assumption P B Q 2 PwsC . mX I Z/, the sequence fP .Q.xj //g is Cauchy, which is a contradiction. t u We close this section by showing that the assumption that X does not contain `1 in the above results cannot be avoided. This fact is closely connected with the classical result of Aleksander Pełczy´nski. For the proof see [HMVZ, Theorem 7.29]. Theorem 88 (Aleksander Pełczy´nski [Peł5], James Hagler [Hag1]). A Banach space X has a subspace isomorphic to `1 if and only if X has a subspace isomorphic to L1 .Œ0; 1/. In particular, it follows that if X has a subspace isomorphic to `1 , then X has a subspace isomorphic to `2 ([FHHMZ, Theorem 4.53]). Since on `2 the weak and weak topologies coincide, we conclude by duality the following fact. Corollary 89. Let X be a Banach space that has a subspace isomorphic to `1 . Then X has a quotient isomorphic to `2 . We are going to prove a useful variant of this corollary. Lemma 90 ([D’H]). Let X be a Banach space that has a subspace isomorphic to `1 and p 2. Then there exist T 2 L.X I `p / and a normalised basic sequence fxj gj1D1 X equivalent to the canonical basis of `1 such that T .xj / D ej , where fej gj1D1 is the canonical basis of `p . Proof. It suffices to prove the case p D 2, as then we can compose T with the formal identity Id W `2 ! `p , which is a bounded linear operator. Let R W `2 ! L1 .Œ0; 1/ be an isomorphic embedding, [FHHMZ, Theorem 4.53]. By Theorem 88 there is an isomorphic embedding S W L1 .Œ0; 1/ ! X . Put fj D S BR.ej /, j 2 N, where fej g is the canonical basis of `2 . Then ffj g is a basic sequence in X equivalent to fej g and so it is weakly null. By Theorem 56 we may assume (passing to a subsequence) that there is a sequence fxj g X biorthogonal to ffj g. It is easy to see that f.R.ej /I S .xj //g is a biorthogonal system in L1 .Œ0; 1/ L1 .Œ0; 1/. Since L1 .Œ0; 1/ has the DPP and fR.ej /g is weakly null, by Proposition 40 the sequence fS .xj /g has no weakly Cauchy subsequence and hence the same holds for fxj g. By Rosenthal’s `1 -theorem there is a subsequence of fxj g equivalent to the canonical basis of `1 , which we still denote by fxj g. We finish the proof by setting T D .S B R/ . Since fT .xj /g is biorthogonal to fej g, it follows that T .xj / D ej . Finally we can P normalise the Psequence fxj g and compose T with the isomorphism U W `2 ! `2 , U aj ej D aj kxj kej . t u Proposition 91 ([D’H]). Let X be a Banach space that has a subspace isomorphic to `1 and let k 2 N, k 2. Then there exist a polynomial P 2 P . kX/ and a normalised basic sequence fxj gj1D1 X equivalent to the canonical basis of `1 such that ! 1 1 X X P aj xj D ajk j D1

j D1

for every .aj /j1D1 2 `1 . In particular, the restriction of P to bounded sets is not weakly continuous at the origin, so P . kX/ ¤ Pw . kX/.

Section 4. .p; q/-summing operators

165

Proof. Let T W X ! `k be (the restriction of) the linear operator and fxj g SX the P basic sequence from Lemma 90. Define Q 2 P . k`k / by Q.y/ D j1D1 yjk and put P D Q B T . To prove the last assertion let W D fx 2 XI jj .x/j < ı; j D 1; : : : ; ng be a weak neighbourhood of the origin given by ı > 0 and 1 ; : : : ; n 2 X . By the 1 boundedness there is a subsequence fxjs g1 sD1 such that fl .xjs /gsD1 converges for each l D 1; : : : ; n. Thus there are distinct k; l; m 2 N such that jj .xk / j .xl /j < ı and jj .xk / j .xm /j < ı, j D 1; : : : ; n. So for x D xk 12 xl 21 xm we have x 2 B.0; 2/ \ W but P .x/ 1 22k 12 . t u

4. .p; q/-summing operators In this section we survey some results concerning the .p; q/-summing operators, some of which will be needed in the proof of the main results in this chapter, i.e. the estimates of the coefficients of polynomials in P . nc0 I `p /. Most of these results with full proofs can be found in [DJT], or in some cases in [FHHMZ] or [Dies2]. The theory has classical roots, based on the study of unconditional convergence of series in Banach spaces. The next classical theorem has played an important role in the theory of summing operators. P Theorem 92 (Władysław Orlicz, [Dies2, p. 108]). P Let 1 nD1 xn be an unconditionally 1 convergent series in Lp .Œ0; 1/, 1 p 2. Then nD1 kxn k2 < C1. We introduce the basic concepts of the theory of summing operators. Definition 93. Let X be a Banach space and let 1 p < 1. A sequence fxj gj1D1 X P is called p-summable if j1D1 kxj kp < C1. We define its p-summable norm by 1 X

k.xj /j1D1 kp D

! p1 kxj kp

:

j D1

P A sequence fxj gj1D1 X is called weakly p-summable if j1D1 j.xj /jp < C1 for every 2 X . Analogously as P in Fact 44 we can show that fxj g is weakly p-summable if and only if sup2BX j1D1 j.xj /jp < C1. Thus we can define its weakly p-summable norm by k.xj /j1D1 kpw D sup

2BX

1 X

! p1 j.xj /jp

:

j D1

We denote by `p .NI X/ the space of all p-summable sequences in X equipped with the p-summable norm. Similarly, we denote by `pw .NI X/ the space of all weakly p-summable sequences in X equipped with the weakly p-summable norm.

166

Chapter 3. Weak continuity of polynomials and estimates of coefficients

To shorten the notation we will use `p .X/ D `p .NI X/ and `pw .X/ D `p .NI X/ in the rest of the chapter. This will hopefully not lead to a confusion with the usual notation `p . / D `p . I K/. The spaces `p .X/ and `pw .X/ are Banach spaces ([DJT, Chapter 2]). By Fact 44 we see that weakly 1-summable sequences coincide with wuC sequences. Hence the Orlicz theorem in combination with the Bessaga-Pełczy´nski theorem (Theorem 46) states that every weakly 1-summable sequence in Lp .Œ0; 1/, 1 p 2, is 2-summable. Proposition 94. Let X be a Banach space, 1 < p 1, and p1 C q1 D 1. A sequence fxj gj1D1 X has an upper p-estimate with constant C > 0 if and only if it is weakly q-summable with k.xj /kw q C. 1 P1 q q D K < C1. By the Hölder inProof. Suppose that sup2BX j D1 j.xj /j equality

n

! n n

X

X X

aj xj D sup aj xj D sup aj .xj / Kk.aj /jnD1 k`p :

2BX

2BX j D1

j D1

j D1

Hence each weakly q-summable sequence has an upper p-estimate. To see the converse, let C > 0 be the constant from the upper estimate and 2 BX . We use the classical duality of `p and `q to obtain ! ! q1 n n n X X X q D sup aj .xj / D sup aj xj C: j.xj /j k.a /k 1 k.a 1 /k ` ` j j p p t u j D1 j D1 j D1 Corollary 95. Let X be a Banach space, 1 < p 1, and p1 C q1 D 1. Then the space L.`p I X / (resp. L.c0 I X/ if p D 1) is isometrically isomorphic to `w q .X/ via 1 the canonical isomorphism T 7! .T .ej //j D1 . Corollary 96. Let 1 < p 1, let fej g be the canonical basis of `p (resp. c0 if p D 1), and p1 C q1 D 1. Then k.ej /kw q D 1. We denote by `pw0 .X/ `pw .X/ the closed subspace consisting of the elements .xk /1 such that limn!1 k.0; : : : ; 0; xn ; xnC1 ; : : : /kpw D 0. kD1 Proposition 97 ([DefFl, Section 8.2], [DFS, Corollary 1.1.12]). Let X be a Banach space and 1 p < 1. Then `p ˝" X is isometric to the space `pw0 .X/ via the natural mapping I W `p ˝" X ! `pw0 .X/ defined on a dense subspace by ! ! ! n m n n X X X X j j j al el ˝ xj D a1 xj ; : : : ; am xj ; 0; 0; : : : : I j D1

lD1

j D1

j D1

The vector valued sequence space `p .X/ can be identified, in the isometric and canonical fashion, with the tensor product space `p ˝p X, where the tensor norm p is defined so that the canonical embedding `p ˝p X ! `p .X/ which is given by .aj /j1D1 ˝ x 7! .aj x/j1D1 becomes an isometry, [DefFl, Chapter 7].

Section 4. .p; q/-summing operators

167

Definition 98 ([MiPe]). Let X , Y be Banach spaces. Given 1 p; q < 1 and an operator T 2 L.X I Y /, suppose that there is C > 0 such that ! p1 ! q1 n n X X kT .xj /kp C sup j.xj /jq 2BX

j D1

j D1

for every x1 ; : : : ; xn 2 X, n 2 N. Then we say that T is .p; q/-summing. The infimum of C for which the inequality holds is denoted by p;q .T /, and it is called the .p; q/-summing norm of T . By considering the sequences fj xg it is easy to see that if T is a non-zero operator that is .p; q/-summing, then q p. Clearly a .p; q/-summing operator maps weakly q-summable sequences to p-summable sequences and by using the closed graph theorem similarly as in Fact 44 we can see that this is actually a characterisation. It is easy to see that composing any .p; q/-summing operator T with a bounded operator S from the left or from the right preserves the summing property and yields p;q .S B T / kSkp;q .T /, resp. p;q .T B S/ kSkp;q .T /. If T 2 L.X I Y / and q p, then the formal identity Id W `q ! `p is bounded and by Proposition 12 we conclude that Id ˝ T W `q ˝" X ! `p ˝" Y is a bounded operator. If T is .p; q/-summing, then the range space `p ˝" Y may be equipped with the stronger p -norm still preserving the boundedness of Id ˝ T . In this case we are going to use the slightly inaccurate notation Id ˝ T W `q ˝" X ! `p .Y /, which is very convenient for inductive arguments involving multiple tensor products, since `pn ˝p `pm ˝p X D `pnm ˝p X D `pnm .X/. The following result gives a number of equivalent conditions for an operator to be .p; q/-summing. Theorem 99. Let X , Y be Banach spaces. Given 1 p; q < 1, C > 0, and an operator T 2 L.X I Y /, the following statements are equivalent: (i) T is .p; q/-summing and p;q .T / C . (ii) For every fxj gj1D1 X 1 X

! p1 kT .xj /kp

j D1

C sup 2BX

1 X

! q1 j.xj /jq

:

j D1

(iii) kId ˝ T W `nq ˝" X ! `pn .Y /k C for every n 2 N. (iv) kId ˝ T W `q ˝" X ! `p .Y /k C . (v) For every L 2 L.`q I X/, q1 C q1 D 1 (or L 2 L.c0 I X/ if q D 1) the operator T B L is .p; q/-summing and p;q .T B L/ C kLk. Sketch of proof. Statements (i) and (ii) are clearly equivalent. Statement (iv) follows from (ii) using the identification from Proposition 97 and the remarks above. On the other hand, (iv) implies (iii), which readily implies (i) by using Proposition 97. (i))(v) is clear and for the reverse implication fix fxj g `w q .X/ and let L 2 L.`q I X/ (resp. L.c0 I X /) be such that L.ej / D xj and kLk D k.xj /kw q (Corollary 95).

168

Chapter 3. Weak continuity of polynomials and estimates of coefficients

1 1 Pn Pn p p C kLkk.e /kw D C kLk D p p D Then j q j D1 kT B L.ej /k j D1 kT .xj /k C k.xj /kw , using also Corollary 96. t u q The next result is a generalisation of Orlicz’s theorem. Given 1 p q 1, p < 1 we define the number r.p; q/ D

2 1C2

1 p

˚ : max q1 ; 12

Theorem 100 ([Ben], [Carl]). If 1 p q 1, p < 1, and r D r.p; q/, then Id W `p ! `q is .r; 1/-summing. This fundamental result can be reformulated as follows. Theorem 101. Let 1 p q 1, p < 1, and r D r.p; q/. There is C > 0 such that for every L 2 L.c0 I `p / 1 X

! 1r kL.ej /krq

C kLk:

j D1

Proof. Let X D `p , Y D `q , and T D Id W X ! Y . Then T is .r; 1/-summing by Theorem 100, so T B L is also .r; 1/-summing and r;1 .T B L/ r;1 .T /kLk. The statement then follows using Corollary 96. t u In Section 5 we will show an extension of this result for multilinear mappings. The summing operators are intimately related to the notions of type and cotype, introduced by Jørgen Hoffmann-Jørgensen in [Ho] (type), and by Bernard Maurey and Gilles Pisier in [MaPi2] (cotype). Definition 102. We say that a normed linear space X is of type p, 1 p < 1, if there exists a C > 0 such that for every x1 ; : : : ; xn 2 X

n

p ! 1 ! p1 n

p X 1 X

X

"j xj C kxj kp : (5)

2n "j D˙1 j D1

j D1

The smallest C allowed here is called the type-p constant of X and is denoted by Tp .X /. We say that a normed linear space X is of cotype q, 1 q < 1, if there exists a C > 0 such that for every x1 ; : : : ; xn 2 X

n

q ! 1 ! q1 n

q X 1 X

X

q "j xj C n : (6) kxj k

2 j D1

"j D˙1 j D1

The smallest C allowed here is called the cotype-q constant of X and is denoted by Cq .X /.

Section 4. .p; q/-summing operators

169

Clearly, every normed linear space X is of type 1 with T1 .X/ D 1 and it is of cotype 1 (inPthis case the inequality in the definition naturally transforms to maxj kxj k C jnD1 kxj k) with C1 .X/ D 1. Further, using the Hölder inequality it is easily seen that if X is of type p, then it is also of type r for any 1 r p and Tr .X / Tp .X /, and if X is of cotype q, then it is also of cotype s for any s q and Cs .X / Cq .X/. By considering xj D x for all j , Khintchine’s inequality ([FHHMZ, Lemma 4.54]) implies than no non-trivial normed linear space can be of type p > 2 or cotype q < 2. Thus we say that a normed linear space is of a non-trivial type if it is of type p 2 .1; 2 and it is of a non-trivial cotype if it is of cotype q 2 Œ2; C1/. The generalised parallelogram law in a Hilbert space H implies that H is of type 2 and cotype 2 with T2 .H / D C2 .H / D 1. More generally, the infinite-dimensional space Lp ./, 1 p < 1 is of type minfp; 2g and cotype maxfp; 2g and both are the best possible ([AK, Theorem 6.2.14]). Moreover, Tp .Lp .// D Cq .Lq .// D 1 for 1 p 2 and q 2. By Kahane’s inequality (Theorem 2.76) we see that we can replace the exponent p on the left-hand side of (5) by any other exponent 1 r < 1 (of course the constant C changes to Kp;r C ) and similarly we can replace the exponent q on the right-hand side of (6) by any other exponent 1 r < 1. Of particular interest is r D 1 or r D 2. A simple consequence of Fact 42 is the following observation, which can be viewed as another generalisation of Orlicz’s theorem. Proposition 103. If a Banach space X is of cotype q, then IdX is .q; 1/-summing and q;1 .IdX / Cq .X/. Obviously both the type and cotype are isomorphic invariants, are inherited by subspaces, and pass to finite direct sums. If X is of type p, then X is of cotype q, where p1 C q1 D 1, and Cq .X / Tp .X/, see e.g. [AK, Proposition 6.2.12]. (We also remark that the converse does not hold: the space `1 is of cotype 2, while c0 is not of a non-trivial type, see the remarks after Theorem 4.2.) If X has an equivalent norm with modulus of smoothness of power type p, then it is of type p ([DGZ, Proposition IV.5.10]), and if X has an equivalent norm with modulus of convexity of power type q, then it is of cotype q ([DGZ, Proposition IV.5.12]). These results can be reversed provided that X is a super-reflexive Banach lattice ([LiTz3, Chapter 1]). Theorem 104 ([MaPi2], [DJT, Lemma 14.10]). Let X be a Banach space of cotype q. Then for any measure the space Lq .I X/ is of cotype q with the constant Cq .Lq .I X // D Cq .X/. For the convenience of the reader we also recall a version of the classical Minkowski inequality, see e.g. [HLP, Section 2.11]: Let aj;k 2 R, j D 1; : : : ; M , k D 1; : : : ; N , and 0 < r s < 1. Then ! rs ! 1s ! rs ! 1r N M M N X X X X r s jaj k j jaj k j : (7) j D1

kD1

kD1

j D1

170

Chapter 3. Weak continuity of polynomials and estimates of coefficients

5. Estimates of coefficients of multilinear mappings In the present section we obtain precise estimates on the values of coefficients of multilinear mappings and homogeneous polynomials in P . nc0 I `p /, leading to a quantitative (and polynomial) version of condition (iii) in Theorem 47. These results are based on a systematic use of the properties of .p; 1/-summing operators, whose relevance to the problem of estimating the coefficients can be seen from Proposition 1.24. Definition 105. Let X1 ; : : : ; Xn ; Y be Banach spaces and 1 p; q1 ; : : : ; qn < 1. We say that M 2 L.X1 ; : : : ; Xn I Y / is multiple .pI q1 ; : : : ; qn /-summing if there is a constant K > 0 such that for every choice of m1 ; : : : ; mn 2 N and xjk 2 Xk , 1 j mk , 1 k n,

˘

X

kM.xj11 ; : : : ; xjnn /kp

1 p

K

1jk mk 1kn

n Y

k k.xjk /jmD1 kw qk :

(8)

kD1

In this case we define the multiple .pI q1 ; : : : ; qn /-summing norm of M by pIq1 ;:::;qn .M / D inf fKI K satisfies (8)g: A multiple .pI q; : : : ; q/-summing multilinear mapping will be called for short multiple .pI q/-summing and the norm will be denoted by pIq . The next fact is straightforward. Fact 106. Let X1 ; : : : ; Xn , Y be Banach spaces and 1 p; q1 ; : : : ; qn < 1. Then M 2 L.X1 ; : : : ; Xn I Y / is multiple .pI q1 ; : : : ; qn /-summing if and only if there is a constant K > 0 such that for every choice of m2 ; : : : ; mn 2 N and xjk 2 Xk , k 1 j mk , 2 k n, with k.xjk /jmD1 kw qk 1 the associated linear operator m2 mn S W X1 ! `p .Y / given by S.x/ D M.x; xj22 ; : : : ; xjnn / 1jk mk (9) 2kn

is .p; q1 /-summing and satisfies p;q1 .S/ K:

(10)

In that case pIq1 ;:::;qn .M / D inf fKI K satisfies (10)g. Theorem 107 ([BPV]). Let X1 ; : : : ; Xn be Banach spaces and let Y be a Banach space of cotype q. Then every M 2 L.X1 ; : : : ; Xn I Y / is multiple .qI 1/-summing and qI1 .M / Cq .Y /n kM k. Proof. For n D 1 it suffices to notice that M D IdY B M and apply Proposition 103. We proceed by induction. Suppose the statement has been proved for n 1. Fix k m2 ; : : : ; mn 2 N and elements xjk 2 Xk , 1 j mk , with k.xjk /jmD1 kw 1 1, k D 2; : : : ; n. Using Fact 106 it suffices to show that q;1 .S/ Cq .Y /n kM k, where S is defined by (9). By the inductive hypothesis for every x 2 X1 the mapping

Section 5. Estimates of coefficients of multilinear mappings

171

Mx 2 L.X2 ; : : : ; Xn I Y / given by Mx .x2 ; : : : ; xn / D M.x; x2 ; : : : ; xn / is multiple .qI 1/-summing and qI1 .Mx / Cq .Y /n

1

kMx k Cq .Y /n

1

kM kkxk:

So for every x 2 X1 kS.x/k qI1 .Mx / Cq .Y /n

1

kM kkxk;

2 mn whence kSk Cq .Y /n 1 kM k. By Theorem 104, `m .Y / is of cotype q with the q m2 mn constant Cq .`q .Y // D Cq .Y /. Then from the linear case (i.e. n D 1) we obtain that S is .q; 1/-summing and

q;1 .S/ Cq .Y /Cq .Y /n

1

kM k D Cq .Y /n kM k:

t u

Using xjk D ej in (8), where fej g is the canonical basis of c0 , and combining the above theorem with Corollary 96 we conclude the following result. Corollary 108. Let Y be a Banach space of cotype q. Then every M 2 L. nc0 I Y / satisfies ! q1 X kM.ej1 ; : : : ; ejn /kq Cq .Y /n kM k; j1 ;:::;jn 2N

where

fej gj1D1

is the canonical basis of c0 .

Since Cq .`q / D 1 for q 2, we obtain the following corollary. Corollary 109 ([BPV]). If 2 q < 1, then every M 2 L. nc0 I `q / satisfies ! q1 X

kM.ej1 ; : : : ; ejn /kq

kM k;

j1 ;:::;jn 2N

where fej gj1D1 is the canonical basis of c0 . Corollary 110. Let Y be a Banach space of cotype q and P 2 P . nc0 I Y /. If P is given by the formal expression X P .x/ D x ˛ y˛ ˛2I.1;n/

(see Proposition 1.24), then ! q1 X ˛2I.1;n/

where C D maxˇ 2I.n;n/

n ˇ .

ky˛ kq

}k; C Cq .Y /n kP

172

Chapter 3. Weak continuity of polynomials and estimates of coefficients

Proof. By Proposition 1.24 each y˛ , ˛ D .˛1 ; ˛2 ; : : : / is uniquely determined by n } ˛ 1 ˛2 y˛ D P . e1 ; e2 ; : : : /; ˛ where fej gj1D1 is the canonical basis of c0 . Hence ! q1 X

ky˛ kq

! q1 X

C

}.ej1 ; : : : ; ejn /kq kP

}k: C Cq .Y /n kP

j1 ;:::;jn 2N

˛2I.1;n/

t u

In [DeSe] the above estimates were further improved for Y D `q , q < 2 (spaces of cotype 2). The method of proof relies on complex interpolation. Of course, the real case then follows by passing to the complexification, paying the price by getting worse constants in the estimates. Theorem 111 ([DeSe]). Let n 2 N and 1 p q 1, p < 1. There is C > 0 such that for every M 2 L. nc0 I `p / ! 1 X C kM k; kM.ej1 ; : : : ; ejn /kq j1 ;:::;jn 2N

where

fej gj1D1

is the canonical basis of c0 and

˚

D

nC2

p

1 p

2n ˚ max q1 ; 21

if p < 2, if p 2.

The proof of the theorem is based on two lemmata together with Theorem 100. Note that while Theorem 99 gives an estimate on the operator norm of Id ˝ T , the next lemma gives an estimate on the summing norm of this operator. Lemma 112 ([DeSe]). Let X , Y be Banach spaces and suppose that Y is of cotype 2. If T 2 L.X I Y / is .r; 1/-summing for some 1 r < 1, then for each m; n 2 N the mn operator Id ˝ T W ˝n" `m 1 ˝" X ! `2 .Y / is also .r; 1/-summing and p n 2C2 .Y / r;1 .T /: r;1 .Id ˝ T / consider first the case n D 1. Take any Proof. We use induction on n. Let us P m x1 ; : : : ; xN 2 `m ˝ X and let x D " k j D1 ej ˝ xk .j / where xk .j / 2 X for 1 k D 1; : : : ; N . We want to show that ! 1r N N X X p r sup j.xk /j: k.Id ˝ T /.xk /k`m .Y / 2C2 .Y /r;1 .T / 2

kD1

2B.`m ˝" X / kD1 1

Using p successively the cotype 2 of the space Y and Kahane’s inequality (with K1;2 2), the Minkowski inequality (7), the fact that T is .r; 1/-summing, and

Section 5. Estimates of coefficients of multilinear mappings

173

Fact 42 we get the estimate N X

! 1r k.Id ˝ T /.xk /kr`m .Y / 2

D

kD1 N X p 2C2 .Y / kD1

! 21 !r ! 1r

N X

m X

kD1

j D1

kT .xk .j //k2

m

!r ! 1 r

1 X

X

" T .x .j //

j k

2m "j D˙1 j D1

m

r ! 1 N X

r X

"j T .xk .j //

"j D˙1 kD1 j D1 ˇ !ˇ N ˇ m ˇ X X p 1 X ˇ ˇ 2C2 .Y /r;1 .T / m sup "j xk .j / ˇ ˇ ˇ ˇ 2 2B X kD1 j D1 "j D˙1

m

N

X X p

j ak xk .j / : 2C2 .Y /r;1 .T / sup sup

jak j1 jj j1 p

1 X 2C2 .Y / m 2

j D1

kD1

On the other hand, by Fact 42 and the definition of the injective tensor norm

N

N

X

X

sup j.xk /j D sup ak xk

m 2B.`m ˝" X/ kD1 jak j1 kD1 `1 ˝" X 1 ˇ N m ˇ ˇX X ˇ ˇ ˇ D sup sup ˇ ak j .xk .j //ˇ ˇ ˇ jak j1 2B m `1

2BX

kD1 j D1

ˇ !ˇ N X m ˇ ˇ X ˇ ˇ D sup sup ˇ ak j xk .j / ˇ ˇ ˇ 2B jak j1 X kD1 j D1

2B`m

1

N m

X X

ak j xk .j / : D sup

jak j1 2B`m

kD1 j D1

1

This finishes the proof for n D 1. Now let n > 1 and assume that the statement of the lemma holds for n 1. Put mn 1 U D ˝n" 1 `m .Y /, and S D Id ˝ T W U ! V . By Theorem 104, 1 ˝" X , V D ` 2 V is of cotype 2 with C2 .V / D C2 .Y /. Further, by the inductive hypothesis S is .r; 1/-summing. Thus by the case n D 1 and the inductive hypothesis p m 2C2 .V /r;1 .S/ r;1 Id ˝ S W `m 1 ˝" U ! `2 .V / p p n 1 2C2 .Y / 2C2 .Y / r;1 .T /; which completes the proof.

t u

174

Chapter 3. Weak continuity of polynomials and estimates of coefficients

In the proof of the next lemma we will make use of the following observation. Let 1 1 T 2 L.XI Y / for Banach spaces X, Y , and 1 p; q; 1, p C q D 1.mConsider n n m m ST 2 L L. `p I X/I ` .Y / defined by ST .M / D T B M.ej1 ; : : : ; ejn / j1 ;:::;jn D1 . Then using the identification in Proposition 10 it is not difficult to see that ST can be mn identified with an operator Id ˝ T 2 L .˝n" `m q / ˝" X I ` .Y / and both operators have the same norm. Lemma 113 ([DeSe]). Let X, Y be complex Banach spaces where Y is of cotype 2, M 2 L. nc0 I X/, and let T 2 L.X I Y / be .r; 1/-summing for some 1 r 2. Then ! 1 X p n 1 r;1 .T /kM k; 2C2 .Y / kT B M.ej1 ; : : : ; ejn /k j1 ;:::;jn 2N

where D

2n nC2. 1r

1 2

/

and fej gj1D1 is the canonical basis of c0 .

Proof. By the remark preceding the lemma it suffices to show that for each m 2 N p

n 1

Id ˝ T W .˝n `m / ˝" X ! `mn .Y / r;1 .T /: 2C2 .Y / " 1 We use induction on n. For n D 1 the statement is clear from the fact that T is .r; 1/-summing and Theorem 99. For the induction we use the notation D n (as r is fixed). Let us assume that the n m estimate holds for n 1. Denote L D Id ˝ T W .˝n" `m 1 / ˝" X ! .˝" `1 / ˝" Y . In what follows we will obtain estimates on the norm of L in various norms on the target space, which will allow us to use another estimate on the norm using interpolation. First, Lemma 112 implies p n 1 mn 1 2C2 .Y / r;1 .T /; .Y / r;1 Id ˝ T W .˝n" 1 `m 1 / ˝" X ! `2 hence by Theorem 99 p

n

L W .˝n `m / ˝" X ! `m .`mn 1 .Y // 2C2 .Y / " 1 r 2

1

r;1 .T /:

(11)

On the other hand, as we shall see, from the induction we are able to obtain estimates n 1 m for the target space `m n 1 .`2 .Y //. Our goal is to get an estimate of the norm of L in the n 1

m target space `m 2 .`n 1 .Y //, which will be amenable to an easy interpolation with (11). To this end we use the Minkowski inequality (7), which however transposes the order of summation. To counter this effect, we will first compose L with the transposition operator. m So, denoting S D Id ˝ T W p `m 1 ˝" X ! `2 .Y /, Lemma 112 implies that S is .r; 1/-summing and r;1 .S/ 2C2 .Y /r;1 .T /. Recall that `m 2 .Y / is of cotype 2 with C2 .`m .Y // D C .Y / by Theorem 104. Thus by the inductive assumption used 2 2 on S in place of T we obtain

Id ˝ S W .˝n 1 `m / ˝" .`m ˝" X/ ! `mn 1 .`m .Y // " 1 1 n 1 2 (12) p n 1 r;1 .T /: 2C2 .Y /

Section 5. Estimates of coefficients of multilinear mappings n 1

Next, let Q W `m n

1

n 1

m m .`m 2 .Y // ! `2 .`n

n 1 m . ..yj k /m / kD1 j D1

1

175 n 1

.Y // be defined by Q ..yj k /jmD1 /m kD1

D

Then kQk 1 by the Minkowski inequality (7), using also the fact n 1 `m / ! .˝n 1 `m / ˝ `m be defined that n 1 2. Further, let t W `m " 1 " 1 ˝" .˝" 1 1 by t .ej1 ˝ .ej2 ˝ ˝ ejn // D .ej2 ˝ ˝ ejn / ˝ ej1 . Clearly, ktk D 1, and so also kt ˝ IdX k D 1 (using the injective tensor product). Moreover, observe that L D Q B L B .t ˝ IdX / in the algebraic sense. Thus in combination with (12) we obtain p

n 1

L W .˝n `m / ˝" X ! `m .`mn 1 .Y // r;1 .T /: 2C2 .Y / " 1 2 n 1 Finally, we use the complex interpolation on the above estimate and (11) with D n1 . By [BerLö, Theorem 4.1.2] and [BerLö, Theorem 4.2.1] we get the estimate p

n 1

L W .˝n `m /˝" X ! `m .`mn 1.Y //; `m .`mn 1.Y // 1 r;1 .T /: 2C2 .Y / " 1 2 n 1 r 2 n

But by [BerLö, Theorem 5.1.2], m mn 1 mn 1 mn 1 mn 1 mn 1 .Y //; `2 .`n 1 .Y //; `m .Y / 1 D `m .Y // 1 D `m .` r .`2 Œ`n 1 .Y /; `2 n

n

where 1 1 n1 1 1 1 D C n and D 2 r n Since D D n , this completes the proof.

1 n 1

C

1 n

2

: t u

Proof of Theorem 111. The case 2 p q 1 follows from Corollary 109 together with the fact that kykq kykp for any y 2 `p . Case 1 p q 2: By Theorem 100 the identity Id W `p ! `q is .r; 1/-summing, where 1r D 21 C p1 q1 . Using Lemma 113 together with the fact that `q is of cotype 2 we get the desired estimate. Case 1 p < 2 < q 1: By Theorem 100 we know that Id W `p ! `2 is .p; 1/-summing. We then apply Lemma 113 combined with the fact that kykq kyk2 for any y 2 `2 . t u By taking p D 1 and q D 2 in Theorem 111 we immediately obtain the following special case. Corollary 114 (Henri Frédéric Bohnenblust and Einar Hille, [BH]). If M 2 L. nc0 I C/, then ! nC1 2n X p n 1 2n 2 kM k: jM.ej1 ; : : : ; ejn /j nC1 j1 ;:::;jn 2N

The constant in the above corollary is due to Alexander Munro Davie [Da] and Sten Kaijser [Kai]. For n D 2 this result is known as Littlewood’s 34 -formula. The polynomial case of Theorem 111 follows readily. The proof is the same as that of Corollary 110.

176

Chapter 3. Weak continuity of polynomials and estimates of coefficients

Corollary 115 ([DeSe]). Let n 2 N and 1 p q 1, p < 1. Then there is C > 0 such that if P 2 P . nc0 I `p / is given by the formal expression X P .x/ D x ˛ y˛ ; ˛2I.1;n/

then

! 1 X

ky˛ kq

C kP k;

˛2I.1;n/

where is defined in Theorem 111.

6. Bohr radius Let Y be a complex Banach space and U C n . According to [Dieu, Section 9.3] a mapping f W U ! Y is said to be analytic if for every a D .a1 ; : : : ; an / 2 U there are r1 ; : : : ; rn > 0 and y˛ 2 Y , ˛ D .˛1 ; : : : ; ˛n / 2 N0n , such that X f .´/ D .´1 a1 /˛1 .´n an /˛n y˛ ˛2N0n

for every ´ D .´1 ; : : : ; ´n / in some open polydisc U.a1 ; r1 / U.an ; rn / U and the power series is absolutely convergent there. Formally this definition is much stronger than our definition in Section 1.8, which only requires the convergence of the abstract power series, in which all the n-homogeneous terms are “pre-summed”. However, from the Cauchy formula it easily follows that the Taylor expansion of a holomorphic mapping on U C n converges also in this stronger sense on any polydisc contained in U . Indeed, from Theorem 1.163 we obtain for any ˛ 2 N0n 1 @j˛j f .a/ ˛1 Š ˛n Š @´˛1 1 : : : @´˛nn Z Z 1 f .a1 C ´1 ; : : : ; an C ´n / D d´n d´1 ; .2 i /n 1 ´˛1 1 C1 ´˛nn C1

n

y˛ D

(13)

where j W Œ0; 2 ! C, j .t/ D rj e it , j D 1; : : : ; n. Thus there is M > 0 such that ky˛ k ˛1M ˛n for every ˛ 2 N0n , from which the absolute convergence in r1 rn

U.a1 ; r1 / U.an ; rn / follows. See also [Dieu, Theorem 9.10.1]. In this section we are going to be concerned with the absolute convergence of the classical (“expanded”) power series in finite-dimensional complex Banach spaces. Since such power series are in fact generalised series, their convergence is always understood as an absolute convergence. Let X be a complex n-dimensional Banach space with a basis fej gjnD1 satisfying

n

n

X

X

j aj ej aj ej for all aj ; j 2 C, jj j 1, j D 1; : : : ; n, (14)

j D1

j D1

Section 6. Bohr radius

177

and denote by D D UX its open unit ball. All coordinates below are understood with respect to this basis. The Bohr radius of D, denoted by K.D/, isPdefined as a supremum of r 2 Œ0; 1 satisfying the following: for any power series ˛2N n a˛ ´˛ , ˇP ˇ 0 a˛ 2 C, absolutely convergent in D and satisfying sup´2D ˇ ˛2N n a˛ ´˛ ˇ 1 we 0 P have sup´2rD ˛2N n ja˛ ´˛ j 1. 0 P Let ˛2N n a˛ ´˛ , a˛ 2 C, be absolutely convergent in D. In what follows we 0 P need some estimates on the coefficients a˛ . Put f .´/ D ˛2N n a˛ ´˛ and assume 0 that f is bounded on D. Fix ˛ 2 N0n and 0 < r < 1, and let x 2 rD be such that jx ˛ j D maxrD j´˛ j. Since B.0; jx1 j/ B.0; jxn j/ D by (14), using (13) with

j .t / D jxj je i t we obtain ja˛ j

sup´2D jf .´/j jx1 j˛1 jxn j˛n

ja˛ j

and consequently

sup´2D jf .´/j : sup´2D j´˛ j

(15)

We will improve this estimate in the next lemma. Lemma 116 (Norbert Wiener, [AM], [BoKh]). Let D be an open unit ball of an n-dimensional Banach space with a basis fej gjnD1 satisfying the estimate (14) and let P f .´/ D ˛2N n a˛ ´˛ be a holomorphic function on D with jf .´/j < 1 for all ´ 2 D. 0 Then 1 ja0 j2 ja˛ j sup´2D j´˛ j for all ˛ 2 N0n , ˛ ¤ 0. Proof. Fix ˇ 2 N0n , ˇ ¤ 0. Let k D jˇj > 0 and let ! be a primitive kth root of unity. Then ( k 1 X jm 1 if m 0 .mod k/, ! D k 0 otherwise. j D1

Pk

Let g.´/ D k1 above we have

j D1 f .!

j ´/.

g.´/ D

Then jg.´/j < 1 for all ´ 2 D and by the property of !

X

a˛ ´˛ D a0 C

j˛j0 mod k

X

a˛ ´˛ C :

j˛jDk

´ a0 1 a0 ´ .

Next, we put '.´/ D Because ja0 j D jf .0/j < 1, the function ' is a holomorphic isomorphism of the open unit disc in C, [RudinW, Theorem 12.4]. Further, 1 '.´/ D 1 ja a0 / C o.´ a0 /; ´ ! a0 , and hence 2 .´ 0j X X a˛ h.´/ D ' B g.´/ D ´˛ C b˛ ´˛ : (16) 2 1 ja0 j j˛jDk

j˛j>k

It is still jh.´/j < 1 for all ´ 2 D. Applying the estimate (15) to the function h on D ja j we get 1 jaˇ j2 sup 1 j´ˇ j , which implies the desired estimate. t u 0

´2D

178

Chapter 3. Weak continuity of polynomials and estimates of coefficients

Theorem 117 ([Ai], [Boa], [BoKh]). Let 1 p 1 and D D U`pn . If 1 p < 2, then 1 1 p1 1 log n 1 p 1 K.D/ : < 3 1 n 3e 3 n If 2 p 1, then r 1 log n : p K.D/ < 2 n 3 n P Partial proof. Let f .´/ D ˛2N n a˛ ´˛ be a holomorphic function on D satisfying 0 jf .´/j < 1 for all ´ 2 D. Suppose first that 1 p < 2. A straightforward calculation p1 ˛˛ 1. By Lemma 116 shows that sup´2D j´˛ j D j˛j j˛j j˛j p1 j˛j ja0 j2 / .1 ˛˛

j˛jj˛j j˛jj˛j .1 ja0 j2 / ˛ ˛ ˛Š P for any ˛ 2 N0n , ˛ ¤ 0. Observe that k´kk`n D j˛jDk ˛k j´˛ j for any ´ 2 X, k 2 N 1 (Proposition 1.22). Thus it follows from the above estimate that for any ´ 2 X, k 2 N, ja˛ j .1

X

ja0 j2 /

X kk j´˛ j ˛Š j˛jDk kk X k j´˛ j D .1 ja0 j2 / kŠ ˛

ja˛ ´˛ j .1

ja0 j2 /

j˛jDk

D .1

ja0 j2 /

j˛jDk

By the Hölder inequality k´k`n1 n1 X

1 p

kk k´kk`n : 1 kŠ

k´k`pn . Hence if k´k`pn r D . n1 /1

ja˛ ´˛ j ja0 j C .1

ja0 j2 /

˛2N0n

1 p

q, then

1 X kk k q : kŠ

kD1

It is easy to verify that max t C 12 .1 t 2R

t 2 / 1:

(17)

P P 1 kk k Thus ˛2N n ja˛ ´˛ j 1 for all ´ 2 rB`pn if q is chosen so that 1 kD1 kŠ q D 2 . 0 1 1 (We refer to [Boa] for the precise computation q D 3 e 3 .) Now we consider the case p 2. Assume first that D D rU`n1 , r > 1. Let be the normalised Lebesgue measure on the torus T D f´ 2 C n I j´j j D 1g D. The monomials ´˛ are orthonormal in L2 ./, so using the Bessel inequality on (16) we ˇ a ˇ2 R 1 P P ˇ ˛ 2 ˇ jhj2 d 1 and hence ja˛ j2 2 1 ja0 j2 for get j˛jDk 1 ja0 j

j˛jDk

T

each k 2 N. The Cauchy-Schwartz inequality now implies that X ˛2N0n

ja˛ ´˛ j D ja0 j C

1 X X kD1 j˛jDk

ja˛ ´˛ j ja0 j C .1

ja0 j2 /

! 12

1 X

X

kD1

j˛jDk

j´˛ j2

:

Section 7. Notes and remarks

But

P

˛ 2 j˛jDk j´ j

X

Pn

j D1 j´j j

˛

179

2 k

ja˛ ´ j ja0 j C .1

˛2N0n

(Proposition 1.22), so if ´ 2 13 B`n2 , then

1 X 1 1 D ja0 j C .1 ja0 j / k 2 3 2

ja0 j2 / 1;

kD1

using (17) again. The lower estimate for K.U`pn / then follows from the fact that 1 1 1 n p B`n1 B`pn and n p 2 B`pn B`n2 . The proof of the upper estimates consists of constructions of particular forms and polynomials using probabilistic techniques, see [Boa] for details. t u

7. Notes and remarks Section 1. For the theory of tensor products we refer to the books [DefFl], [Ry3], or [DFS], and for the basics see also [FHHMZ, Chapter 16]. Symmetric tensor products were introduced by Raymond A. Ryan [Ry2], for more see e.g. [AnFl], [Fl]. Applications of this theory to complex analysis on infinite-dimensional spaces can be found in [Din], where various topological spaces of holomorphic functions and polynomials are studied. Section 2. The various weak continuity notions and results which occupy the beginning of the section were studied by Richard Martin Aron and his co-authors, e.g. in [AHV], [AP], [Aro1]; see [Din, Notes to Chapter 1] for references to a large amount of related literature. Our approach however is based on the systematic use of the theory of uniform spaces and the related results on uniformly continuous mappings. This simplifies some proofs and sometimes leads to stronger results. A good source of information on uniform spaces is e.g. [Eng]. The pioneering paper in this area seems to be Aleksander Pełczy´nski’s [Peł1], which was inspired by Stanisław Mazur, and complemented Jaroslav Kurzweil’s work in [Kur1]. It contains in particular the well-known Corollary 59 claiming that P .c0 / D Pwsc .c0 /, due independently to Witold M. Bogdanowicz [Bog]. It seems to have passed unnoticed that this result follows from much earlier and more precise coefficient estimates of Henri Frédéric Bohnenblust and Einar Hille, Corollary 114. Let us mention briefly some additional results on this property. In the papers [CCG] and [Far] several characterisations of the wsc property were obtained: Let X be a Banach space and n 2 N. The following statements are equivalent: (i) P . nX / D Pwsc . nX/. (ii) Each P 2 P . nX/ is weakly continuous on every weakly compact set A X. (iii) W X ! ˝n;s X , .x/ D ˝n x is w–w sequentially continuous. (iv) If fxk g is weakly null in X , then f˝n xk g1 is weakly null in ˝n;s X. kD1 Sets of weak continuity and wsc continuity of polynomials have been studied e.g. in [BR], [ArDim], [CGG]. In particular, if P 2 P 2 .X/ is weakly continuous at the origin, then it is wsc ([ArDim]). More generally, in [FGL] it is shown that Cwsc .XI Y /

180

Chapter 3. Weak continuity of polynomials and estimates of coefficients

consists precisely of the functions that are weakly continuous when restricted to every weakly compact set in X . Theorem 37 was motivated by a result due to [DL] which claims the following: Let X be a Banach space. Then PK . nXI `p / D P . nXI `p / for every n 2 N, 1 < p < 1, if and only if P . nX/ does not contain c0 for every n 2 N. Similar results were obtained in the linear case by Nigel J. Kalton in [Kal] and Kamil John [John], [EJ], motivated by some early work in [Th], [Ton], [TW] which concerns the complementation property of LK .XI Y / in L.X I Y /. The Dunford-Pettis property was introduced and studied by Alexander Grothendieck in [Gr] and Nelson Dunford and Billy James Pettis in [DuPe], who proved that C.K/ spaces and L1 ./ enjoy the DPP, see e.g. [AK], [Dies1]. The classical Theorem 47 combines several structural results concerning linear operators from C.K/ spaces due to A. Grothendieck [Gr] and A. Pełczy´nski [Peł2]. We refer to [DU, p. 180] for more historical remarks. The formulation of Theorem 56 appears to be new, although the ideas used for the proof have been around for a long time. Versions of Theorem 63 have been reproved by many authors. The oldest version seems to be that of A. Pełczy´nski ([Peł1], [ALRT], [GGu3], [GJ1]). For more results on polynomial images of various types of sequences see e.g. [GGu3], [GGu2], [GGu5], [GGu4]. A. Pełczy´nski [Peł4], [Peł3] initiated the study of the wsc property for polynomials on spaces with the DPP, proving in particular that P .X/ D Pwsc .X/ whenever X has the DPP. The main result in this direction is Theorem 67, due to R. Ryan. On the other hand, the DPP is far from being a necessary condition to conclude the wsc property for all scalar valued polynomials. In [CGG] examples of Banach spaces failing the DPP for which L. nX/ D Lwsc . nX/ have been studied in detail. Problem 118 ([CGG]). Let X be a Banach space such that P . nX/ D Pwsc . nX/. Is then L. nX / D Lwsc . nX/? By Theorem 67 and Theorem 7 we see that if X has the DPP and fxjk gj1D1 are weakly Cauchy sequences in X, 1 k n, then fxj1 ˝ ˝ xjn gj1D1 is weakly Cauchy in X ˝ ˝ X (the converse is not true in general, [Dies1]). In [BomVi] it is proved that the space C.K1 / ˝ ˝ C.Kn / has the DPP if and only if all Kj , 1 j n are scattered. This is closely related with Fact 25: Consider X D C.K/ for non-scattered K. Since X ˝ X lacks the DPP, there exist a Banach space Y and T 2 LwK .˝2 XI Y / n Lwsc .˝2 X I Y /. However, as X has the DPP (Theorem 41), by Theorem 67, T B ˝ 2 LwK . 2XI Y / Lwsc . 2XI Y /. This provides an example such that M 2 Lwsc . 2XI Y / but LM … Lwsc .˝2 X I Y /. On the other hand, observe that M 2 Lwu . nX I Y /

implies LM 2 Lwu .˝n X I Y /:

Indeed, by Lemma 18 and Fact 25 we have LM 2 LK .˝n X I Y / D Lwu .˝n XI Y /. The implication cannot be reversed in general, as LM is always weakly uniformly continuous for Y D R.

Section 7. Notes and remarks

181

According to [FaJ], a sequence fxk g X is said to be P -null provided that P .xk / ! 0 for every P 2 P .X/, P .0/ D 0. A Banach space X is said to have the polynomial DPP if k .xk / ! 0 holds for every pair of sequences fxk g that is P -null in X and fk g weakly null in X . It is shown there, and independently in [JP], that in super-reflexive spaces P -null sequences coincide with norm null ones. It is shown in [GGu5] that Schreier’s pace S has the polynomial DPP (even hereditary), although it does not have the DPP. In this paper P -convergent sequences in S are characterised. These sequences do not coincide with either weakly or norm convergent ones. A Banach space X is said to be polynomially Schur if every P -null sequence is norm null. (Polynomially Schur spaces has been introduced in [CCG] under a different name – the -property.) Note that by Corollary 4.43 if X is of a non-trivial type (which is equivalent to X being of a non-trivial type), then X is polynomially Schur. It is shown in [CCG], [FaJ], that if X has the DPP, then X is polynomially Schur if and only if X is Schur. The reflexivity of spaces of polynomials has been also extensively studied, see [Far] and the monograph [Din] for more results. R. Ryan obtained in [Ry2] the following characterisation: If X is a reflexive Banach space with the approximation property (Definition 7.18), then the following statements are equivalent: (i) P . nX / is reflexive. (ii) P . nX / D Pwsc . nX/. (iii) Each P 2 P . nX/ attains its norm on BX . More generally, [Far], [GJ1], if X and Y are reflexive Banach spaces with the approximation property, then P . nX I Y / D Pwsc . nXI Y / if and only if P . nXI Y / is reflexive. The next problem seems rather subtle. Problem 119 ([DiZa]). Is there a reflexive Banach space X such that P .X/ D Pwsc .X/ and simultaneously P .X / D Pwsc .X /? In [GGu4] the actions of polynomials on wuC sequences are studied. Recall that a Banach space X has the Grothendieck property if L.XI c0 / D LwK .XI c0 /. In [GGu2] it is shown that for every n 2 N, n > 1 the following statements are equivalent: (i) P . nX / is reflexive. (ii) P . nXI c0 / D PwK . nXI c0 /. (iii) ˝n;s X is a Grothendieck space. Similarly, for every n 2 N, n > 1 we have P . nX I c0 / D Pwsc . nXI c0 / if and only if L.X I c0 / D Lwsc .XI c0 /. Concerning other structural results for the spaces of polynomials, it is shown in [Bl] that P . kXI Y / is isomorphic to a complemented subspace of P . nXI Y / for every X, Y , and k < n. By [DíD], P . nX/ Š L. nX/, n 2 N holds for Banach spaces X isomorphic to their cartesian square, see also [AnFl]. The structure of L. n`p / has been studied in detail in [AriFa].

182

Chapter 3. Weak continuity of polynomials and estimates of coefficients

Section 3. The deep theory concerning Banach spaces not containing `1 , due to Haskell Paul Rosenthal, Edward Wilfred Odell, and Jean Bourgain, David H. Fremlin, Michel Talagrand, is presented in detail in [FHHMZ]. A long list of equivalent conditions to X not containing `1 is given in [Ros4]. In particular, this condition is equivalent to Lwsc .X I Y / D LK .XI Y / for every Banach space Y . The importance of this condition in our theory cannot be exaggerated. As a first example, consider Theorem 81 due to [CHL], which is one of the key tools in the theory of W -spaces, among other things. Theorem 83 streamlines results in [AAD], [AAF], [GJ1]. The opposite condition, i.e. containing `1 , is also very useful especially in dealing with stabilisation problems for polynomials as we will see in Chapter 4. Indeed, the well-known Corollary 89 to Theorem 88 of A. Pełczy´nski, [Peł5], states that X has a quotient isomorphic to `2 whenever X is a Banach space containing `1 . A variant of this result, Proposition 91, is instrumental for building diagonal polynomials on X , which are indispensable for proving the main results on infinite-dimensional polynomial algebras in Chapter 4. Section 4. For the theory of summing operators we refer to any of the following books: [DJT], [Pis4], [Tom2], [Pie], [JL]. Proposition 103 has a converse proved by M. Talagrand [Tal3], with some earlier results in [MaPi2]: If q > 2, then X is of cotype q if and only if IdX is .q; 1/-summing. For q D 2 the result is false, [Tal2]. Theorem 100 was proved independently by Grahame Bennett [Ben] and Bernd Carl [Carl]. An important special case when p D 1 was proved earlier by Stanisław Kwapie´n [Kw1]. Section 5. The whole section is based on [BPV] and [DPS]. The main result, Theorem 111, is optimal in the sense that the value of .n; p; q/, for given n 2 N, 1 p q 1, p < 1 cannot be improved (i.e. made smaller). The proof of the optimality requires a construction of a certain multilinear form by using random coefficients. The theory has many deep connections with other directions of research. We suggest the reader to check the recent works of Andreas Defant and his co-authors in this area, e.g. [DGM], [DeKal], [DFOOS], [DMP], [DefFr]. Section 6. The study of the radius of absolute convergence of complex power series started with a theorem of Harald Bohr stating that K.UC / D 31 . In the multidimensional case there are several alternative definitions of the notion of Bohr radius (cf. [DMP], [DePr], [BoKh], [Boa]). We adopted the one in [DGM], and we refer to [DT1] for historical comments and more references. Theorem 117 is due to Lev Aizenberg [Ai] for p D 1, and to [Boa] for 1 p 2, by using essentially the same argument. It is due to [BoKh] for p 2. The optimality of the asymptotic growth of the upper estimates can again be established using random techniques. For example, [Boa], if 1 p 1 and if n, d are integers larger than 1, then there exists a polynomial P 2 P . dC n / of the form (the signs are chosen using random techniques) X d P .´/ D ˙ ´˛ ˛ j˛jDd

Section 7. Notes and remarks

183

such that the supremum of jP .´/j on B`pn is at most (p 1 p if 1 p 2, n.d Š/1 p 32 log.6d / 1 C. 1 1 /d 1 n 2 2 p .d Š/ 2 if 2 p 1. Let X be a complex n-dimensional space with a basis fej g satisfying (14). In [DefFr] the following inequality has been obtained: 1 1 K.BX /; 4e d.X; `n1 / where d.X; Y / denotes the Banach-Mazur distance of X and Y . It is known that 1 d.`n1 ; `pn / n1 min.p;2/ , [Tom2], so this result contains, in an asymptotic sense, Theorem 117 as a special case. The problem of Bohr radius is closely connected with the problem of the existence of an unconditional basis in the space P . nX/, where X is an infinite-dimensional Banach space and n 2. This direction was researched e.g. in [DT1] and [DGM]. The problem was finally settled in [DeKal] showing that P . nX/, n 2 has an unconditional basis if and only if X is finite-dimensional. For bases in the multilinear case see e.g. [DiZa].

Chapter 4

Asymptotic properties of polynomials It this chapter we employ the concept of finite representability to the study of polynomials, their asymptotic behaviour, and the linear structure of the underlying spaces. We describe the ultrapower construction for a Banach space X , which leads to a much larger Banach space .X/U (possibly containing X ) that is finitely representable in X and that is well-suited for constructions of uniformly continuous mappings. We give, without proof, several deep results from the local theory of Banach spaces, that will be used later in our investigation of the structure of C k -smooth Banach spaces. Another important tool is the spreading model construction for X, which leads to spaces with a sub-symmetric basis that capture the asymptotic behaviour of infinite sequences in X. It is suited particularly well for the study of the upper and lower estimates of sequences, the Banach-Saks, and the p-Banach-Saks properties. These results are then applied to the weak sequential continuity properties of polynomials, relying also on the concept of P n -null sequences in Banach spaces. Some of the main results on Banach spaces with separating polynomials are given in Section 4. We show that if a Banach space has a separating d -homogeneous polynomial and a sub-symmetric basis, then it is isomorphic to some `p for p even and d D kp, k 2 N. Moreover, by a fundamental result of Robert Deville, every Banach space with a separating polynomial contains such `p . While spreading model techniques give us a very good idea of the limit behaviour of any uniformly continuous function at infinity when restricted to suitable subspaces, the results remain finite-dimensional in nature. In Section 5 we show that for polynomials on `p we are able to obtain a sub-symmetric behaviour for full tail vectors. These results can be applied to the case of polynomials on general Banach spaces. We show that for an arbitrary polynomial P on a Banach space X there is an infinitedimensional subspace Y of X such that the restriction of P to Y is either separating or asymptotically (and this cannot be strengthened) zero in a strong sense. The last Sections 6 and 7 are devoted to the study of algebras An .X/ of polynomials generated by polynomials of degree at most n on a Banach space X . Section 6 is devoted to the proof of our main technical tool, namely a finite-dimensional lemma P which claims that the symmetric polynomial snN .x/ D jND1 xjn on RN is not in the uniform closure of suitably defined sub-symmetric sub-algebra of An 1 .RN / provided that N is large enough. We proceed by applying this result in Section 7 to infinite-dimensional spaces using spreading model techniques. The main result implies in particular that A1 .`p / D D An 1 .`p / ¤ An .`p / ¤ AnC1 .`p / ¤ , where n D dpe.

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185

1. Finite representability and ultraproducts The important concept of finite representability has become one of the key notions in modern Banach space theory (see [FHHMZ, Section 6.1] for some basic information). The principle of local reflexivity gives a very precise description of the fact that X is always finitely representable in X . We describe a very general technique of building Banach spaces which are finitely representable in X and have additional properties. The ultrapower construction leads to a large Banach space .X/U (possibly containing also X ) that is well-suited for constructions of uniformly continuous mappings. The section ends with a list of several deep results from the local theory of Banach spaces that will be used later in our investigation of the structure of C k -smooth Banach spaces. Definition 1. Let X, Y be normed linear spaces. We say that Y is crudely finitely representable in X if there is K > 0 such that for every finite-dimensional subspace F of Y there is a linear isomorphism T W F ! T .F / X satisfying kT kkT 1 k K. We say that Y is finitely representable in X if for every " > 0 the space Y is crudely finitely representable in X with the constant K D 1 C ". Observe that finite representability (and also crude) is a transitive relation. (Crude) finite representability clearly preserves properties that depend only on finite-dimensional subspaces (even in a uniform way independent of dimension). For example it preserves type or cotype, and moreover if Y is finitely representable in X, then Tp .Y / Tp .X/ and Cq .Y / Cq .X/. One of the key results in this area is the following principle of local reflexivity ([LindeRo], [JRZ]): Theorem 2 (principle of local reflexivity, [FHHMZ, Theorem 6.3]). Let X be a Banach space, E X and F X finite-dimensional subspaces, and " > 0. Then there is a linear isomorphism T of E onto T .E/ X such that kT kkT 1 k 1 C ", f .T .// D .f / for any f 2 F and 2 E, and T is the identity on E \ X. In particular, X is finitely representable in X. According to this principle many properties of finite-dimensional subspaces of X pass to X unchanged. Thus the type or cotype of X passes to X , including the best constants. It should be noted that a Banach space X is of a non-trivial type if and only if X is of a non-trivial type, [DJT, Corollary 13.7, Theorem 13.10]. The same is not true for cotype: the space `1 is of cotype 2 but `1 is not of a non-trivial cotype. Also, type is stronger than cotype in the following sense: If a Banach space X is of a non-trivial type, then it is also of a non-trivial cotype. Indeed, by the above, X is then also of a non-trivial type and so X (and thus also X as its subspace) is of a non-trivial cotype. The converse does not hold (`1 is of cotype 2 but it is not of a non-trivial type). It is well-known that every normed linear space is finitely representable in c0 , [FHHMZ, Theorem 6.2]. The celebrated result of Aryeh Dvoretzky, [Dv], states that `2 is finitely representable in every infinite-dimensional normed linear space. For the proof see e.g. [AK, Theorem 11.3.13]. For a normed linear space X we define p.X / D sup fpI X is of type pg and q.X/ D inf fqI X is of cotype qg.

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Theorem 3. Let X be an infinite-dimensional Banach space. Then `p.X/ and `q.X/ are finitely representable in X. This theorem is due to [MaPi2] in case that X is of a non-trivial type, resp. cotype; for the proof see [MS, Theorem 13.2]. The case p.X/ D 1, i.e. when X is not of a non-trivial type, is due to [Pis1]. The case q.X/ D 1, i.e. when X is not of a non-trivial cotype, comes from [MaPi1]. For the proof see [AK, Theorem 11.1.14]. Next we review some basic facts on ultraproducts of Banach spaces. Let I be an arbitrary index set. Recall that an ultrafilter on I is called principal if it is of the form fA I I ˛ 2 Ag for some fixed element ˛ 2 I . A non-principal ultrafilter is also called a free ultrafilter. Further, an ultrafilter is called countably complete if it is closed under countable intersections. It is immediate that an ultrafilter U is countably incomplete if and only if there are An 2 U, n 2 N such that A1 A2 A3 T and 1 A nD1 n D ;. Every free ultrafilter on N is countably incomplete, as it contains the Fréchet filter (all subsets of N with a finite complement). Recall that if X is a topological space, I an index set, fx˛ g˛2I X, and U an ultrafilter on I , then limU x˛ D x 2 X (the limit with respect to the ultrafilter U) if f˛ 2 I I x˛ 2 V g 2 U for every neighbourhood V of x; this limit always exists if X is compact. Let X˛ , ˛ 2 I , be normed linear spaces for some index set I . Consider the normed linear space `1 .I I fX˛ g/ D f.x˛ /˛2I I x˛ 2 X˛ ; sup˛2I kx˛ k < C1g with the supremum norm. If U is an ultrafilter on I , then ˚.x˛ / 7! limU kx˛ k is a continuous semi norm on `1 .I I fX˛ g/ and so NU .fX˛ g/ D .x˛ / 2 `1 .I I fX˛ g/I limU kx˛ k D 0 is a closed subspace. Definition 4 ([DaKr]). The normed linear space `1 .I I fX˛ g/=NU .fX˛ g/ is called the ultraproduct of fX˛ g˛2I and it is denoted by .X˛ /U . In the important special case when X˛ D X for all ˛ 2 I we call this space the ultrapower of X and denote it by .X /U . The element of .X˛ /U represented by .x˛ / 2 `1 .I I fX˛ g/ will be denoted by .x˛ /U . It is easy to see that the canonical quotient norm is given by k.x˛ /U k D limU kx˛ k. The mapping x 7! .x˛ /U , where x˛ D x for all ˛ 2 I , clearly defines a canonical isometric embedding of X into .X/U . If X˛ are all Banach spaces, then .X˛ /U is obviously also a Banach space. The following proposition is not difficult to prove. Proposition 5 ([Ster], [DJT, Theorem 8.13]). Let X be a normed linear space and U an ultrafilter. Then .X/U is finitely representable in X. Theorem 6 ([Ster]). Suppose that a normed linear space Y is finitely representable in a normed linear space X . Then there exists an ultrafilter U such that Y is isometric to a subspace of .X/U . Proof. Let I D f.E; n/I E Y is a finite-dimensional subspace, n 2 Ng and pick T˛ 2 L.EI X / with kT˛ k D 1 and kT˛ 1 k < 1 C n1 for ˛ D .E; n/ 2 I . We put a partial ordering on I according to the relation .E; m/ .F; n/ if E F and m n. The order filter F on the directed set .I; / is a filter generated by all the sets of

Section 1. Finite representability and ultraproducts

187

the form fˇ 2 I I ˇ ˛g, ˛ 2 I . Finally, let U be an ultrafilter on I extending F . The embedding T W Y ! .X/U is now defined by T .y/ D .x˛ /U , where for each ˛ D .E; n/ 2 I ( T˛ .y/ if y 2 E, x˛ D 0 otherwise. It is easy to check that T is linear and an isometry.

t u

By combining the previous theorem with the fact that X is finitely representable in X we see that there is an ultrafilter U such that X is isometric to a subspace of .X /U . By utilising the full power of the principle of local reflexivity we obtain an isometric embedding with additional properties: Corollary 7. Let X be a Banach space. There are an ultrafilter U and an isometric embedding T W X ! .X/U such that T .x/ D .x/U for each x 2 X and w -limU x˛ D for every 2 X and every fx˛ g X such that T . / D .x˛ /U . Proof. Let I be the collection of all triples .E; F; n/ where n 2 N and E X , F X are finite-dimensional subspaces with E \ X ¤ f0g. By the principle of local reflexivity for each ˛ D .E; F; n/ 2 I there is an isomorphism T˛ W E ! T˛ .E/ X such that 1 kT˛ k 1 C n1 , 1 kT˛ 1 k 1 C n1 , f .T˛ .// D .f / for any f 2 F and 2 E, and T˛ is the identity on E \ X. We put a partial ordering on I according to the relation .E1 ; F1 ; n1 / .E2 ; F2 ; n2 / if E1 E2 , F1 F2 , and n1 n2 . The order filter F on the directed set .I; / is a filter generated by all the sets of the form fˇ 2 I I ˇ ˛g, ˛ 2 I . Finally, let U be an ultrafilter on I extending F . The embedding T W X ! .X/U is now defined by T .y/ D .x˛ /U , where for each ˛ D .E; F; n/ 2 I ( T˛ .y/ if y 2 E, x˛ D 0 otherwise. It is easy to check that T has the required properties.

t u

To be able to extend mappings from X to the ultrapowers, we first need to look at their domains. Let A be a subset of a normed linear space X and U an ultrafilter on I . By .A/U we denote the set f.x˛ /U 2 .X/U I x˛ 2 A for all ˛ 2 I g. Proposition 8. Let X be a normed linear space, A X, and U an ultrafilter on I . (i) If A is bounded, then so is .A/U . (ii) If A is convex, then so is .A/U . (iii) If U is countably incomplete or if A is closed, then .A/U is closed. (iv) If ´ 2 .X /U , then there is a representation ´ D .x˛ /U such that kx˛ k D k´k for all ˛ 2 I . In particular, .SX /U D S.X/U . Proof. (i) and (ii) are trivial. T To prove (iii), let Bn 2 U be such that B1 B2 B3 and 1 nD1 Bn D ; in case that U is countably incomplete; put Bn D I for all n 2 N otherwise. Let .x˛ /U 2 .A/U . There are .x˛n /U 2 .A/U such that x˛n 2 A for all n 2 N, ˛ 2 I and

188

Chapter 4. Asymptotic properties of polynomials

˚ T k.x˛n /U .x˛ /U k < n1 . Put Cn D ˛ 2 I I kx˛n x˛ k < n2 and Dn D Bn \ jnD1 Cj . T1 Then Dn 2 U for each n 2 N and D1 D2 D3 . Put D D nD1 Dn and note that D D ; in case that U is countably incomplete. Thus by the assumption x˛ 2 A for each ˛ 2 D. If ˛ 2 D1 n D, then there is a unique n 2 N such that ˛ 2 Dn n DnC1 and we put y˛ D x˛n . Set y˛ D x˛ for ˛ 2 D and y˛ D x˛1 for ˛ … D1 , and note that y˛ 2 A for all ˛ 2 I . Fix n 2 N. For each ˛ 2 Dn n D there is a unique m 2 N, 2 n2 . m n such that ˛ 2 Dm n DmC1 . Thus ky˛ x˛ k D kx˛m x˛ k < m ˚ 2 Hence Dn ˛ 2 I I ky˛ x˛ k < n . It follows that limU ky˛ x˛ k D 0 and so .x˛ /U D .y˛ /U 2 .A/U . ´˛ (iv) Let ´ D .´˛ /U and fix e 2 X with kek D k´k.ˇ Put x˛ D k´k ˇ k´˛ k if ´˛ ¤ 0, x˛ D e otherwise. Then limU kx˛ ´˛ k D limU ˇk´k k´˛ kˇ D 0 and hence .x˛ /U D .´˛ /U . t u Now if X is a normed linear space, Y a uniform space, A X, f W A ! Y a mapping that is uniformly continuous on bounded sets and maps bounded sets to relatively compact sets, and U an ultrafilter, then we can define .f /U W .A/U ! Y by .f /U .x˛ /U D lim f .x˛ /: U

It is easy tocheck that .f /U is well-defined by the uniform continuity. Moreover, .f /U .x/U D f .x/ for any x 2 X, so we may view .f /U as an extension of f . In view of Corollary 7 this extension can be in particular used to extend mappings with domain in X to mappings with domain in X . Such an extension depends of course on the ultrafilter U as well as on the particular embedding of X into .X/U (via the principle of local reflexivity). However, if X, Y are normed linear spaces and f W X ! Y is w–w uniformly continuous (in addition to being kk–kk uniformly continuous on bounded sets), then f has a unique w –w uniformly continuous extension g W X ! Y (see the remarks preceding Proposition 3.15). Let T be the embedding from Corollary 7 and consider the extension .f /U W .X/U ! .Y ; w /. Then given 2 X we have g. / D w -limU f .T . /˛ / D .f /U T . / and so g and .f /U coincide on X . Therefore the extension (on X ) is independent of U and T . In particular, if L 2 L.X I Y /, then .L/U X D L . The next result is also useful for preserving uniform properties with respect to finite representability. Theorem 9. Let X , Y be normed linear spaces and let be either the norm, weak, or weak star (in case that .Y; kk/ is a dual space) topology on Y . Further, let A X, and let f W A ! Y be a mapping that is uniformly continuous on bounded sets and maps bounded sets to relatively -compact sets. If U is an ultrafilter on I , then .f /U W .A/U ! .Y; / has the following properties: (i) If f is uniformly continuous with a modulus ! 2 M that is continuous, then so is .f /U . (ii) If f 2 P . nXI Y /, then .f /U 2 P . n.X/U I Y /, and if f 2 P n .X I Y /, then .f /U 2 P n ..X/U I Y /. Moreover, in both cases k.f /U k D kf k.

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189

(iii) If f is T k -smooth at x with the approximating polynomial P , then .f /U is T k -smooth at .x/U with the approximating polynomial .P /U . In particular, if f is Fréchet differentiable at x, then .f /U is Fréchet differentiable at .x/U with D.f /U ..x/U / D .Df .x//U . (iv) Suppose that A D X or that A is an open convex bounded set. If ˝ Ms is a convex cone and f 2 C k;˝ .AI Y /, then .f /U 2 C k;˝ .Int.A/U I Y /. Moreover, d j.f /U .x/Œh D w -limU d jf .x˛ /Œh˛ (in Y if ¤ w ) for any j D 1; : : : ; k, x D .x˛ /U 2 Int.A/U with x˛ 2 A, and h D .h˛ /U 2 .X/U . Proof. (i) Notice that kk is -lower semi-continuous. Let x D .x˛ /U 2 .A/U , y D .y˛ /U 2 .A/U with x˛ ; y˛ 2 A. Denote ı D kx yk and let " > 0 be arbitrary. Then V D f˛ 2 I I kx˛ y˛ k < ı C "g 2 U. Since kf .x˛ / f .y˛ /k !.ı C "/ for all ˛ 2 V , it follows that k.f /U .x/ .f /U .y/k limU kf .x˛ / f .y˛ /k !.ı C"/ for all " > 0. The continuity of ! now implies that k.f /U .x/ .f /U .y/k !.ı/. (ii) First we show the homogeneous case. Let us define M W .X/nU ! .Y; / by M .x˛1 /U ; : : : ; .x˛n /U D limU f}.x˛1 ; : : : ; x˛n /. It is clear that M is an n-linear sym , so .f /U 2 P . n.X/U I Y /. Obviously this also metric mapping and .f /U D M n implies that .f /U 2 P ..X/U I Y / in the non-homogeneous case. To show the boundedness (and the equality of the norms) let x 2 B.X/U . There are x˛ 2 BX such that x D .x˛ /U . Then k.f /U .x/k limU kf .x˛ /k kf k. (iii) Proposition 3.70 implies that P maps bounded sets to relatively -compact sets. Thus .P /U is well-defined by the -limit and it is a polynomial of degree at most k by (ii). Let " > 0. There is ı > 0 such that kf .x C h/ P .h/k "khkk for all h 2 X , khk ı. If .h˛ /U 2 .X/U is such that k.h˛ /U k ı, then by Proposition 8 we may assume that kh˛ k ı for all ˛ 2 I . Then kf .x C h˛ / P .h˛ /k "kh˛ kk and by passing to a limit we obtain

.f /U .x/U C .h˛ /U .P /U .h˛ /U limkf .x C h˛ / P .h˛ /k U

" limkh˛ kk D "k.h˛ /U kk : U

(iv) For x 2 A denote by Px the Taylor polynomial of the mapping f of order k at x. Let ! 2 ˝ be the modulus of continuity of d kf . Corollary 1.108 implies that 1 !.khk/khkk for all x 2 A, h 2 X with x C h 2 A. Put kf .x C h/ Px .h/k kŠ Z D Y if D w , Z D Y otherwise. Fix .x˛ /U 2 Int.A/U with x˛ 2 A and define P.x˛ /U W .X/U ! Z by the formula P.x˛ /U .h˛ /U D w -lim Px˛ .h˛ /: U

Using the uniform continuity of x 7! Px and the convexity of A it is not difficult to check that P.x˛ /U is well-defined and continuous. Assume that the spaces X and Y are real. Let .y˛ /U ; .h˛ /U 2 .X/U . By Theorem 2.50 kC1 X kC1 . 1/kC1 j Px˛ .y˛ C j h˛ / D 0 j j D0

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for all ˛ 2 I . Consequently, by passing to a w -limit with respect to the ultrafilter U, kC1 X kC1 . 1/kC1 j P.x˛ /U .y˛ /U C j.h˛ /U D 0 j j D0

and so by Theorem 2.50 again we may conclude that P.x˛ /U 2 P k .X/U I Z . Given .h˛ /U 2 .X /U such that .x˛ /U C.h˛ /U 2 Int.A/U , we may without loss of generality assume that x˛ C h˛ 2 A for all ˛ 2 I . Then 1 kf .x˛ C h˛ / Px˛ .h˛ /k !.kh˛ k/kh˛ kk kŠ for all ˛ 2 I . Since the topology on Y considered as a subspace of .Z; w / is weaker than or equal to , we have .f /U .x˛ /U C .h˛ /U D w -limU f .x˛ C h˛ /. Thus by passing to a limit in the inequality above, using also the w -lower semi-continuity of the norm on Z and the continuity of !, we obtain

1

.f /U .x˛ /U C .h˛ /U P.x˛ /U .h˛ /U !.k.h˛ /U k/k.h˛ /U kk : kŠ So we may apply Corollary 1.126 to obtain .f /U 2 C k;˝ .Int.A/U I Y /. The complex case can be shown similarly as in (ii). (Of course, the method of (ii) works in the real case too. The use of Theorem 2.50 is nevertheless formally somewhat shorter.) t u Often in various constructions we rely on a reasonably rich supply of functions of a given class that have certain separating properties. Definition 10. Let X be a set, A; B X, Y a metric space, and f W X ! Y . We say that f separates A from B if dist f .A/; f .B/ > 0. To simplify the notation, if A consists of a singleton fxg we say that f separates x from B. Let X be a normed linear space and f W X ! R. We say that f is a separating function if there is x 2 X and r > 0 such that f separates fxg from X n B.x; r/. Note that if a set S of functions on a normed linear space X is stable under affine transformations of the domain (in particular if S is any of the smoothness spaces from Section 1.5), then there is a separating function in S if and only if there is a function in S that separates 0 from X n BX . An example of a separating function is a bump function (or shortly a bump) – a real function on X with a bounded non-empty support. From Proposition 8 it immediately follows that if f W X ! R is uniformly continuous on bounded sets such that it separates 0 from X n BX , then .f /U separates 0 from .X /U n B.X /U . Combining this with Theorems 6 and 9 we obtain the next corollary. Corollary 11. Let ˝ Ms be a convex cone and let X be a normed linear space that admits a C k;˝ -smooth bump, resp. separating polynomial in P n .X/, resp. separating polynomial in P . nX/. If a normed linear space Y is finitely representable in X, then Y admits a C k;˝ -smooth bump, resp. separating polynomial in P n .Y /, resp. separating polynomial in P . nY /.

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191

We proceed by introducing the important classes of Lp -spaces, which will play an important role in the sequel (in particular in the case p D 1). We recall that the Banach-Mazur distance between the normed linear spaces X and Y is defined as d.X; Y / D inf fkT kkT 1 kI T 2 L.XI Y / is an isomorphismg. Definition 12 ([LP]). Let 1 p 1. A Banach space X is said to be a Lp; -space, 1, if for every finite-dimensional subspace E of X there is a finite-dimensional subspace F of X, E F , such that d.F; `pn / , n D dim F . We say that X is a Lp -space if X is a Lp; -space for some 1. Lp -spaces have been extensively studied in the literature. Although their definition seems to be purely finite-dimensional, these spaces have very strong structural properties. We recall some of the properties of this class of spaces for the future use. For more information and references we refer to [JL, Chapters 3 and 36]. Theorem 13 ([LP], [LindeRo]). Let 1 p 1. Then Lp ./ is a Lp; -space for every > 1. Moreover, every Lp -space is isomorphic to a subspace of Lp ./ for some measure ; if 1 < p < 1, then it is isomorphic to a complemented subspace of Lp ./. On the other hand, a complemented subspace of a Lp -space is either a Lp -space or it is isomorphic to a Hilbert space. If X is a Lp -space, 1 p 1, then .X/U is again a Lp -space for any ultrafilter U, [BenLi, Proposition F.3] or [BourgJ2, Proposition 1.22]. Moreover, if X D Lp ./, 1 p < 1, then .X/U is isometric to Lp ./ for some measure , [He, Theorem 3.3]. Since any separable subspace of Lp ./, 1 p < 1 is isometric to a subspace of some separable Lp .1 / ([Woj, Proposition III.A.2]), which is isometric to a subspace of Lp .Œ0; 1/ ([JL, pp. 14–15]), it follows that if a separable Banach space is finitely representable in Lp .Œ0; 1/, then it is isometric to a subspace of Lp .Œ0; 1/. We recall that a Banach space is a P -space if it is -complemented in every superspace. The following should be compared with Proposition 7.78. Proposition 14 ([FHHMZ, Proposition 5.13]). Let X be a Banach space and 1. The following statements are equivalent: (i) X is a P -space. (ii) For every Banach space Y , every T 2 L.Y I X/, and every Banach space Z Y there is an extension T 2 L.ZI X/ of T with kT k kT k. (iii) For every Banach spaces Z and Y X, and for every T 2 L.X I Z/ there is an extension T 2 L.Y I Z/ of T with kT k kT k. Theorem 15 ([LP], [LindeRo], [Li1]). Let K be a compact space. Then C.K/ is a L1; -space for every > 1. Every complemented subspace of C.K/ is a L1 -space. If X is a L1; -space, then X is a P -space. Theorem 16 ([LindeRo], [Steg1]). Let X be a Banach space. Then X is isomorphic to `1 if and only if X is a separable L1 -space that does not contain `1 . It turns out that L1 -spaces, and in particular isomorphic preduals of `1 , need not contain c0 , [BD], [BourgJ2]. In fact, L1 -spaces may even be saturated with spaces

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Chapter 4. Asymptotic properties of polynomials

isomorphic to `2 , [Hay4]. Every Banach space with a separable dual can be embedded into some isomorphic predual of `1 , [FOS].

2. Spreading models In this section we develop some basic facts concerning the spreading model construction for a Banach space X, which leads to a Banach space with a sub-symmetric basis which captures the asymptotic behaviour of infinite sequences in X . We then pass to the closely related concepts of the Banach-Saks and the p-Banach-Saks properties. These results will be later applied to the weak sequential continuity properties of polynomials. Definition 17. Let K 1. We say that a sequence fxn g1 nD1 in a normed linear space is K-spreading if

k

k

X

X

a x a x K

j nj j mj

j D1

j D1

whenever k 2 N, a1 ; : : : ; ak are arbitrary scalars, and mj ; nj 2 N are such that m1 < m2 < < mk , n1 < n2 < < nk . In particular, a sequence fxn g is 1-spreading if and only if

k

k

X

X

aj xnj D aj xj

j D1

j D1

whenever k 2 N, a1 ; : : : ; ak are any scalars, and nj 2 N satisfy n1 < n2 < < nk . Note that from Rosenthal’s `1 -theorem (Theorem 3.71) it follows that any K-spreading sequence in a Banach space is either equivalent to the canonical basis of `1 , or it is weakly Cauchy (use the fact that the linear operator T W spanfxnj g ! spanfxj g, T .xnj / D xj is bounded and hence w–w uniformly continuous). Proposition 18 ([BeaLa, Proposition I.4.2]). Let fen g be a K-spreading sequence in a Banach space X. Then fen g is a basic sequence if and only if it is not weakly convergent to a non-zero element of X. If moreover fen g is weakly null, then fen g is an unconditional basic sequence. Definition 19. A Schauder basis fen g1 nD1 of a Banach space is called symmetric 1 if fe.n/ g1 is equivalent to fe g n nD1 nD1 for any permutation of N. A Schauder of a Banach space is called sub-symmetric if it is unconditional and basis fen g1 nD1 1 fenk g1 is equivalent to fe g for every increasing sequence fnk g1 N. n nD1 kD1 kD1 We remark that a symmetric basis is automatically unconditional, and in fact subsymmetric, [Sin, Proposition II.22.2]. It can be shown using the Uniform boundedness principle that a sub-symmetric basis is K-spreading for some K 1 (the unconditionality here is substantial), and similarly if fen g is a symmetric basis of a Banach

Section 2. Spreading models

193

space X, then there is K 1 such that

1

1

1

X

X

X 1

an en an e.n/ K an en

K nD1

nD1

nD1

P1

for every nD1 an en 2 X and every permutation of N, [Sin, Theorems II.21.2 and II.22.1]. Further, it is easy to check that if fen g X is a sub-symmetric basis that is K-spreading, then the sequence ffn g X biorthogonal to fen g is a sub-symmetric basic sequence that is CK-spreading, where C is the unconditional basis constant of fen g. basis fen g1 Definition 20. Let X be a Banach space with nD1 . For any a Schauder P1 P1 a e D a e for those vectors mapping W N ! N denote T nD1 n n nD1 n .n/ P1 nD1 an en 2 X , for which the sum on the right converges. A subset U X is called symmetric (resp. sub-symmetric) if T .U / U for any permutation W N ! N (resp. for any increasing mapping W N ! N). Let U X be a symmetric (resp. sub-symmetric) set and A any set. A mapping f W U ! A is called symmetric (resp. sub-symmetric) if f T .x/ D f .x/ for all x 2 U and for any permutation W N ! N (resp. for any increasing mapping W N ! N). These notions will typically be applied to functions whose domain is a Banach space with a symmetric (resp. sub-symmetric) basis, or a subspace of a space with a Schauder basis consisting of finitely supported vectors. Definition 21. Let fxn g be a sequence in a Banach space X. We say that a sequence fen g in a Banach space Y is a spreading model of the sequence fxn g if for every " > 0 and k 2 N there is an N 2 N such that

k

k

k

X

X

X

.1 "/ aj ej aj xnj .1 C "/ aj ej

j D1

j D1

j D1

for all N n1 < n2 < < nk and all scalars a1 ; : : : ; ak . Note that a spreading model is a 1-spreading sequence. Moreover, if fxn g is a basic sequence with a spreading model fen g, then fen g is also a basic sequence with a basis constant bounded above by that of fxn g. It is easy to see that if a spreading model fen g of fxn g is a basic sequence, then it is uniquely determined in the sense that if fun g is a spreading model of fxn g, then fun g is a basic sequence and T 2 L.spanfen gI spanfun g/, T .en / D un is an isometry. Clearly if a sequence has a spreading model, then its subsequences have the same spreading model. For a given sequence fxn g a spreading model may not exist. However, we have the following fundamental result. Theorem 22 (Antoine Brunel and Louis Sucheston, [BrSu1]). Let X be a Banach space and suppose that fxn g X is a bounded sequence such that fxn I n 2 Ng is not relatively compact. Then fxn g has a subsequence with a spreading model.

194

Chapter 4. Asymptotic properties of polynomials

The proof is based on repeated use of Ramsey’s theorem and can be found e.g. in [FHHMZ, Theorem 6.6] (the main idea of the crucial step is exposed in the proof of a simpler Theorem 79). The following fact follows almost immediately from the definition. Fact 23. Let X , Y be Banach spaces and fxn g X a sequence with a spreading model fen g Y . Then spanfen g is finitely representable in X. The proof of the next proposition can be found in [FHHMZ, Theorem 6.7]. Proposition 24. Let X be a Banach space and fxn g X a weakly null sequence with a spreading model fen g. Then fen g is a sub-symmetric basic sequence with the unconditional basis constant at most 2. By passing to subsequences and diagonalising we obtain the following useful observation. Proposition 25. Let X, Y be a Banach spaces and fxn g X a sequence with a RC and fNk g1 N. There is a spreading model fen g Y . Let f"k g1 kD1 kD1 subsequence fyn g of fxn g such that

N

N

N k k k

X

X

X

aj ej aj ej aj ynj .1 C "k / .1 "k / (1)

j D1

j D1

j D1

for all k n1 < n2 < < nNk and all scalars a1 ; : : : ; aNk . Later on we will make use of the following additional result. We prefer to omit the proof, as it can be obtained by modifying the proof of Theorem 22, working simultaneously with the norm kk on X and P and keeping in mind that homogeneous polynomials form a closed set in the topology of uniform convergence on bounded sets (see also [GJ2, Theorem 1.8]). Theorem 26. Let X be a Banach space, fxn g X a semi-normalised basic sequence, P 2 P . dX /, f"k g1 RC , and fNk g1 N. There are a subsequence fyn g of kD1 kD1 fxn g with a spreading model fen g Y D spanfen g and a sub-symmetric polynomial Q 2 P . dY / such that (1) holds and ˇ ! !ˇˇ ˇ Nk Nk X X ˇ ˇ ˇQ aj ej P aj ynj ˇˇ "k ˇ ˇ ˇ j D1 j D1 for all k n1 < n2 < < nNk and all scalars a1 ; : : : ; aNk with

PNk

j D1 aj ynj

2 BX .

The behaviour of spreading models with respect to duality is described in the following theorem.

Section 2. Spreading models

195

Theorem 27 ([BeaLa, Corollaries III.1.5 and III.1.6]). Let X , Y be Banach spaces and fxn g X a weakly null basic sequence with a spreading model fen g Y . Further, let fxn g X be a basic sequence biorthogonal to fxn g and fen g Y a basic sequence biorthogonal to fen g. Then there exists an infinite set M N such that the sequence fxn gn2M spanfxn gn2M has a spreading model fen g1 nD1 . In particular, the sequence fen g1 is a spreading model of a sequence in a quotient of X . nD1 a normed linear space is called a Banach-Saks Definition 28. A sequence fxn g1 nD1 in ˚ 1 PN 1 sequence if the sequence N nD1 xn N D1 is convergent. We say that fxn g1 nD1 is a hereditarily Banach-Saks sequence provided that every subsequence of fxn g1 nD1 is a Banach-Saks sequence. Every convergent sequence is easily seen to be hereditarily Banach-Saks. The canonical basis of c0 is an example of a non-convergent hereditarily Banach-Saks sequence. Haskell Rosenthal proved that every weakly null sequence in a Banach space has a subsequence which is either hereditarily Banach-Saks, or its spreading model is equivalent to the canonical basis of `1 ([BeaLa, Proposition II.6.2]). Hence if X is a Banach space, then either X is a Schur space, or X contains a normalised weakly null hereditarily Banach-Saks sequence, or X contains a normalised weakly null sequence and every spreading model of every normalised weakly null sequence is equivalent to the canonical basis of `1 . Definition 29. Let X be a Banach space. We say that X has the Banach-Saks property if every bounded sequence in X has a Banach-Saks subsequence. We say that X has the weak Banach-Saks property if every weakly null sequence in X has a Banach-Saks subsequence. It follows readily from the James theorem [FHHMZ, Corollary 3.131] that every Banach space with the Banach-Saks property is reflexive. On the other hand, there exist reflexive spaces without the Banach-Saks property (the Baernstein space or the (Figiel-Johnson) Tsirelson space T , [CasaSh, p. 18]). Every Schur space (and in particular `1 ) enjoys the weak Banach-Saks property, but `1 clearly does not have the Banach-Saks property. Consequently, no Schur space has the Banach-Saks property. On the other hand, for reflexive spaces the weak Banach-Saks property clearly coincides with the Banach-Saks property. The spaces with the weak Banach-Saks property can be characterised, thanks to Rosenthal’s results mentioned above, as those spaces whose spreading models of weakly null sequences are not equivalent to the canonical basis of `1 ([BeaLa, Theorem II.3.8]). If X has the Banach-Saks property, then every bounded sequence in X has even a hereditarily Banach-Saks subsequence, and similarly if X has the weak Banach-Saks property, then every weakly null sequence in X has even a hereditarily Banach-Saks subsequence, [Rak], see also [Dies2, p. 253, pp. 192–197]. The weak Banach-Saks property can be conveniently refined using the following concept.

196

Chapter 4. Asymptotic properties of polynomials

Definition 30. Let 1 < p < 1. A sequence fxn g1 nD1 in a normed linear space is called a p-Banach-Saks sequence if there is C > 0 such that

N

X 1

xn CN p

nD1

fxn g1 nD1

for all N 2 N. We say that is a hereditarily p-Banach-Saks sequence if for some C > 0 every subsequence of fxn g1 nD1 is a p-Banach-Saks sequence with the constant C . We say that a Banach space X has the weak p-Banach-Saks property if there exists a C > 0 such that every normalised weakly null sequence in X has a p-Banach-Saks subsequence with the constant C . It is immediate that a p-Banach-Saks sequence is an r-Banach-Saks sequence for r < p and it is also a Banach-Saks sequence. Thus the weak p-Banach-Saks property implies the weak Banach-Saks property. It follows readily from the Gurarii-James theorem [FHHMZ, Theorem 9.25] that every super-reflexive Banach space has the weak p-Banach-Saks property for some p 2 .1; 1/, hence it has the Banach-Saks property. Fact 31. Let 1 < p < 1 and let fxn g be a hereditarily p-Banach-Saks sequence in a normed linear space X. Then there is a C > 0 such that for any subsequence fyn g1 nD1 1 of fxn g1 nD1 , any sequence fan gnD1 of scalars with jan j 1, n 2 N, and any N 2 N

N

X

1

an yn CN p :

nD1

fxn g1 nD1

Proof. Assume that is a hereditarily p-Banach-Saks sequence with a constant C . Let fyn g be a subsequence of fxn g and fix N 2 N. Assume first that X is real. By the Bauer maximum principle

N

X

M D max an yn

jan j1 nD1

is attained at an extremal point of B`N , i.e. for some fan gN nD1 f 1; 1g. Thus 1

!

X X

1

M max yn C yn 2CN p :

Af1;:::;N g n2A

n…A

For the complex case use the above estimate for the real and imaginary part.

t u

Clearly, if a sequence has an upper p-estimate, then it is hereditarily p-Banach-Saks. Conversely, we have the following: Lemma 32. Let 1 < p < 1 and let fxn g1 nD1 be a hereditarily p-Banach-Saks sequence in a Banach space. Then fxn g1 has an upper r-estimate for any 1 < r < p. nD1

Section 2. Spreading models

197

Proof. Let q be such that p1 C q1 D 1, let C be the constant from Fact 31, and P r denote D D 1 2nq. p 1/ 2 R. Choose N 2 N and any scalars a1 ; : : : ; aN with nD0 ˚ PN .nC1/ < ja j 2 n , n 2 N . Then, r j 0 j D1 jaj j 1. Define Bn D j 2 NI 2 using the Hölder inequality,

N

1 1 X 1

X

X X X r nr 1 1

n p 2n. p 1/ 2 p jBn j p aj xj aj xj C 2 jBn j D C

j D1

nD0 j 2Bn

CD

1 X

1 q

nD0

nD0

! p1 2

nr

jBn j

r p

D 2 CD

nD0 r p

2 CD

1 q

N X

1 q

1 X

! p1 2

.nC1/r

jBn j

nD0

! p1 jaj j

r

r

1

D 2 p CD q :

t u

j D1

Lemma 33. Let X be a Banach space, fxn g X a sequence with a spreading model fen g, and 1 < p < 1. Then fxn g has a hereditarily p-Banach-Saks subsequence if and only if fen g is p-Banach-Saks. Proof. ) is easy to see. ( Assume that fen g is a p-Banach-Saks sequence with a constant K. We may assume without loss of generality that fen g is normalised. Let fyn g be the subsequence of fxn g from Proposition 25 with "k D 12 and Nk D 2k . Note that then kyn k 2 for all n 2 N. For N 2 N set

N

2 4dlog2 N e

X C 1 ej : CN D 1 Np N p j D1 Note that lim supN !1 CN 2K and hence C D supN 2N CN 2 R. Let fnj g N be an increasing sequence and N 2 N. Putting m D dlog2 N e we have using (1)

N

m

N

N

1 1 1 2m 2

X

X

X

X

y y C y C e

n n n j j j j 1 1 1 1 1

N p j D1 N p j D1 N p j DmC1 Np N p j DmC1

N

m

N

2m 2 2 4m 2

X

X

X 1 C 1 ej C 1 ej 1 C 1 ej C:

Np

Np

Np Np Np j D1

j D1

j D1

t u Clearly, an upper p-estimate or a lower q-estimate passes from a sequence to its spreading model. Conversely, the following holds. Corollary 34 ([GJ1], [FaJ]). Let X be a Banach space, let fxn g X be a sequence with a spreading model fen g, and 1 < p; q < 1. If fen g is a p-Banach-Saks sequence, then fxn g has a subsequence with an upper r-estimate for all 1 < r < p. If fen g has a lower q-estimate and fxn g is weakly null, then fxn g has a subsequence with a lower r-estimate for all r > q.

198

Chapter 4. Asymptotic properties of polynomials

Proof. The first statement follows by combining Lemmata 33 and 32. The second statement follows by combining Theorem 27 and Fact 3.50 with the first statement. u t We say that a sequence fxn g has an upper p-estimate for non-negative coefficients if

P

PN p 1 an p for every N 2 N and there exists C > 0 such that N an xn C nD1

nD1

every a1 ; : : : ; aN 2 RC 0 ; similarly for a lower q-estimate. Since a spreading model of a weakly null sequence is unconditional (Proposition 24), we conclude the following corollary (compare this result also with Theorem 6.71). Corollary 35. Let 1 < p; q < 1 and let fxn g X be a semi-normalised weakly null sequence in a Banach space with an upper p-estimate (resp. lower q-estimate) for non-negative coefficients. Then fxn g has a subsequence with an upper r-estimate for all 1 < r < p (resp. lower r-estimate for all r > q). It is easy to see that if a Banach space X with an unconditional basis is of type p (resp. of cotype q), then it has property Sp (resp. Tq ). The converse is false even for spaces with a symmetric basis, as shown by Gilles Pisier ([LiTz3, Example 1.f.19]). This observation leads to the next result. Proposition 36. If a Banach space X is of type p > 1, then X has the weak p-BanachSaks property and it has property Sr for all 1 < r < p. If a Banach space X is of cotype q < 1, then X has property Tr for all r > q. Proof. Let fxn g X be a normalised weakly null sequence. By passing to a subsequence we may without loss of generality assume that fxn g has a spreading model fen g Y D spanfen g such that fen g is an unconditional basis of Y with the unconditional basis constant at most 2 (Proposition 24). Since Y is finitely representable in X (Fact 23), Y is of type p (resp. cotype q). Hence fen g has an upper p-estimate with the constant 2Tp .X/ (resp. lower q-estimate with the constant 2Cq .X/). The result for type now follows from Lemma 33 and Lemma 32 (notice that in the proof Lemma 33 the constant C in this case depends only on Tp .X/). The case of cotype follows from Corollary 34. t u

3. Polynomials and p-estimates In our study of polynomials and their relationship with the Banach space structure we are lead to make a fundamental distinction between weakly sequentially continuous ones (which are in some sense considered the easy ones) and the rest. Let X be a Banach space that does not contain `1 and 1 < p < 1. By Fact 3.49 if a seminormalised basic sequence ffj gj1D1 in X has an upper p-estimate, then the linear operator S W `p ! X , S.ej / D fj is bounded. By the reflexivity of `p there is T 2 L.X I `q /, p1 C q1 D 1, such that T D S. Since S is non-compact, so is T . By Proposition 3.33 there are L 2 L.XI `q / and a weakly null sequence fxn g X P dqe such that L.xn / D en . We define Q 2 P . dqe`q / by Q.u/ D j1D1 uj and put

Section 3. Polynomials and p-estimates

199

P D Q B L 2 P . dqeX/. Then P is a prototype of a non-wsc polynomial. Refinements of this basic approach are the subject of the present section. Unlike linear operators, continuous polynomials are not necessarily w–w continuous. 2 1 For example, P 2 P . 2 `2 I `1 /, P .xk /1 D .x / does not map weakly null kD1 k kD1 sequences to weakly null sequences. Fact 37. Let fen g be a semi-normalised unconditional basic sequence in a Banach space X. If fen g is not weakly null, then it has a subsequence with a lower 1-estimate. Consequently, the subsequence is equivalent to the canonical basis of `1 . Proof. We may assume without loss of generality that there is f 2 SX such that Re f .en / ˛ > 0 for all n 2 N. Denote by K the unconditional basis constant of fen g. Let fan gN nD1 be a finite sequence of scalars. Then

N

ˇ

N

!ˇ N

ˇ

X

X 1 ˇˇ 1

X

ˇ

jan jen jan jen ˇ an en

ˇf

2K ˇ ˇ

2K nD1 nD1 nD1 ˇ N ˇ N ˇ 1 ˇˇ X ˛ X ˇ D jan jf .en /ˇ jan j: ˇ ˇ 2K 2K ˇ nD1

t u

nD1

We will generalise this fact to the polynomial setting. Definition 38. A sequence fxk g1 in a normed linear space is called P n -null if kD1 n limk!1 P .xk / D 0 for every P 2 P .X/, P .0/ D 0. Lemma 39. Let X be a real Banach space, n 2 N, n 2, and let fxk g1 be a kD1 P n 1 -null sequence in X . If P .xk / ! 6 0 for some P 2 P . nX/, then fxk g has a subsequence with a lower n-estimate for non-negative coefficients, a lower r-estimate for every r > n, a lower n-estimate if n is even. Proof. ([Dev1]) Without loss of generality we may suppose that P .xk / 1 for all k 2 N. Put ık D 2k n1nŠ2k , k 2 N and observe that for a fixed m 2 N we have P P 1 1 1k1 kn Dm ım 2nŠ2m . Hence 1k1 kn ıkn 2nŠ . We will construct an }.xk ; : : : ; xk /j < ıl whenever increasing sequence of integers fnk g1 such that jP n 1 kD1 n fkj gj D1 fnk g, k1 kn , k1 < kn , and kn D nl . To this end, put n1 D 1 and continue by induction. Suppose that n1 ; : : : ; nm are already defined for some m 2 N. }.xk ; : : : ; xk ; n rx/ is a homogeneous For each k1 ; : : : ; kr 2 N the mapping x 7! P r 1 polynomial of degree less than n. Thus there exists nmC1 2 N, nmC1 > nm such that }.xk ; : : : ; xk ; n rxnmC1 /j < ımC1 whenever r < n, fk1 ; : : : ; kr g fn1 ; : : : ; nm g, jP r 1 and k1 kr .

200

Chapter 4. Asymptotic properties of polynomials

Denote yj D xnj , j 2 N. Let a1 ; : : : ; aN 2 R and moreover aj 0, j D 1; : : : ; N in case that n is odd. Then P

N X

! aj yj

j D1

N X j D1

N X

N X

ajn

jak1 akn jıkn

1k1 kn N k1 1, 1 1 1 p C q D 1, and let fxk gkD1 X be a semi-normalised basic sequence. Then for each s > q there is a subsequence fxnk g1 such that there exists a bounded linear kD1 operator T W X ! `s satisfying T .xnk / D ek , where fek g is the canonical basis of `s . Further, there is a subsequence fxnk g1 such that for each n 2 N, n > q there is kD1 P 2 P . nX / such that P .xnk / D 1 for all k 2 N. Proof. If X is of type p > 1, then X does not contain `1 , and by Proposition 36, X has property Sr for every 1 < r < p. Thus we may apply Proposition 42. For the second part we may use a fixed s 2 .q; Œq C 1/. t u Proposition 44 ([GJ2]). Let X be a real Banach space that admits a C k;˛ -smooth bump, k 2 N, ˛ 2 .0; 1. Then any normalised P k -null sequence in X has a subsequence with an upper .k C ˛/-estimate. Proof. Let f 2 C k;˛ .X/ be such that f .0/ D 0 and f .x/ 2 for kxk 1. For each x 2 X denote by P .x/ 2 P k .X/ the polynomial h 7! T .x/Œh f .x/, where T .x/ is the Taylor polynomial ofˇ order k of f at x. Put q D kˇ C ˛. By Corollary 1.108 there is K 1 such that ˇf .x C h/ f .x/ P .x/Œhˇ Kkhkq for all h 2 X. k Let fxn g1 nD1 X be a normalised P -null sequence in X . We will construct a 1 subsequence fxnj gj D1 by induction. Suppose that n1 < n2 < < nm 1 2 N are already defined for some m 2 N. The mapping x 7! P .x/ is continuous and so the set ( k

Qm D Q 2 P .X/I Q.h/ D P

m X1

!

)

aj xnj Œam h; jaj j 1; j D 1; : : : ; m

j D1

is a compact subset of P k .X/. Since fxn g is P k -null, it follows that there is nm 2 N, nm > nm 1 such that jQ.xnm /j 21m for all Q 2 Qm . 1 PN aj q q and b D Now if a1 ; : : : ; aN 2 R are not all zero, we set a D . j j D1 jaj j aK 1=q PN 1 q Then j D1 jbj j D K and jbj j 1, j D 1; : : : ; N . Hence using the definition of Ql

Section 4. Separating polynomials. Symmetric and sub-symmetric polynomials

we can estimate ˇ ! N N ˇ X X ˇ f bj xnj ˇf ˇ j D1

l X

l 1 X

!ˇ ˇ ˇ bj xnj f bj xnj ˇ ˇ j D1 j D1 lD1 ˇ ˇ ! N N l 1 ˇ X ˇ X X 1 ˇ ˇ q Kjbl jq C l < 2; Kjbl j C ˇP bj xnj Œbl xnl ˇ ˇ ˇ 2 j D1 lD1

!

203

lD1

PN

1 1 P N q q.

q j D1 bj xnj < 1 and so j D1 aj xnj < K j D1 jaj j t u

P which implies that N

4. Separating polynomials. Symmetric and sub-symmetric polynomials In the present section we combine the spreading model techniques developed in the previous sections with the averaging technique for polynomials to obtain some of the main results on Banach spaces with separating polynomials. We show that if X has a separating d -homogeneous polynomial and a sub-symmetric basis, then it is isomorphic to some `p for p even integer and d D kp; k 2 N. Moreover, by a fundamental result of R. Deville every Banach space with a separating polynomial contains such `p . We show that for an arbitrary polynomial P on X there is an infinitedimensional subspace Y of X such that the restriction of P to Y is either separating or asymptotically (and this cannot be strengthened) zero in a strong sense. We start with a few facts concerning separating polynomials. Fact 45 ([Kur1], [FPWZ]). Let X be a real normed linear space and let P 2 P n .X/, ´ 2 X, and r > 0 be such that P separates ´ from fx 2 X I kx ´k D rg. Then X admits a polynomial Q 2 P n .X/ such that Q.0/ D 0 and infSX Q > 0, a separating polynomial of degree at most 2n that separates 0 from X n B.0; s/ for every s > 0, a homogeneous separating polynomial that separates 0 from X n B.0; s/ for every s > 0. Proof. If dim X < 1, then the assertions are trivial, so assume that dim X D 1. Put q.x/ D P .rx C ´/ P .´/. Then q.0/ D 0 and since SX is connected we have either infSX q > 0, in which case we put Q D q, or supSX q < 0, in which case we put Q D q. Let D infSX Q and let Qj 2 P . jX/, j D 1; : : : ; n, be such P P P 2nŠ=j . From Fact 1.28 it that Q D jnD1 Qj . Put R D jnD1 Qj2 and S D jnD1 Qj 2n 2nŠ follows that R 2 P .X/ and S 2 P . X/. x Let s > 0 and fix x 2 X , kxk > s. There is j 2 f1; : : : ; ng such that Qj kxk n. 2 2 2 2n 2 2j 2nŠ= l Then R.x/ Qj .x/ kxk n minfs ; s g n . Similarly, as Ql .x/ 0

204

Chapter 4. Asymptotic properties of polynomials

for each l 2 f1; : : : ; ng, it follows that S.x/ Qj .x/2nŠ=j kxk2nŠ ˚ 2nŠ s 2nŠ min 1; n .

2nŠ=j n

t u

We note that the polynomial .x; y/ 7! 2x 2 x 4 C y 4 on X D `22 separates 0 from SX but 0 from X nB.0; r/ for any r > 0. Further, the polynomial Pdoes not separate P .x; y/ 7! j1D1 xj4 C j1D1 yj6 is a separating polynomial on `4 ˚ `6 of degree 6, but as it follows from Corollary 61 below, homogeneous separating polynomials on this space are of degree 12k, k 2 N. Notice that a separating polynomial is of degree at least 2 and that if a real Banach space X admits a separating polynomial of degree 2, then X is isomorphic to a Hilbert space. Indeed, by Fact 45 there is P 2 P 2 .X/ such that P .0/ D 0 and infSX P > 0. Letting Q.x/ D P .x/ C P . x/ we have Q 2 P . 2X/ and D infSX Q > 0. Hence } x/ kQkkxk2 for every x 2 X. It follows that Q } is an kxk2 Q.x/ D Q.x; inner product on X which gives rise to an equivalent norm on X. In connection with this notice also that every non-negative 2-homogeneous polynomial is convex (use Lemma 1.21). Suppose that a real normed linear p space X admits a separating k-homogeneous polynomial P that is convex. Then k P is an equivalent norm on X . Indeed, it is easily seen to be a quasi-convex positively homogeneous even function and hence it is a norm on X. Such a norm is of course a k-polynomial norm and it is obviously an analytic norm. Notice also that if a real normed linear space admits a separating polynomial, then by composition with a suitable smooth real function we can produce a C 1 -smooth bump with all derivatives bounded (in particular it is C k;1 -smooth for all k 2 N). The results in Chapter 3 immediately imply the next statement. We note though that later we show much stronger results. Corollary 46. Suppose that p 2 Œ1; C1/ is not an even integer. Then there is no separating polynomial on `p of degree at most p. Proof. Suppose that P 2 P n .`p / is a separating polynomial, where n p. By Fact 45 we may assume that P .0/ D 0 and infSX P > 0. Put Q.x/ D P .x/ C P . x/. Then Q is of even degree and hence deg Q < p. Further, Q.0/ D 0 and infSX Q > 0. If p D 1, then Q D 0, which is impossible. For p > 1 we have en ! 0 weakly but infn2N Q.en / > 0, a contradiction with Corollary 3.59. t u Fact 47. Let X be a real normed linear space and let fPj gjkD1 P n .X/, ´ 2 X, and ˚ r > 0 be such that inf maxj jPj .x/ Pj .´/jI kx ´k D r > 0. Then X admits a family of non-constant homogeneous polynomials fQj I 1 j 2k ng of degree at most n such that infX nB.0;s/ maxj Qj > 0 for every s > 0, a separating polynomial of degree at most 2n that separates 0 from X n B.0; s/ for every s > 0. Proof. We put qj .x/ D Pj .rx C ´/ Pj .´/, j D 1; : : : ; k. Then qj .0/ D 0 and P D infSX maxj jqj j > 0. Let qj D nlD1 Qj;l , where Qj;l 2 P . lX/, j D 1; : : : ; k.

Section 4. Separating polynomials. Symmetric and sub-symmetric polynomials

205

Then the family f˙Qj;l I 1 j k; 1 l ng satisfies the requirements. Indeed, ˇ ˇ ˇqj x ˇ . given s > 0 and x 2 X n B.0; s/ there is j 2 f1; : : : ; kg such that kxk ˇ ˇ x ˇ n and consequently It follows that there is l 2 f1; : : : ; ng such that ˇQj;l kxk jQj;l .x/j kxkl n minfs; s n g n . To see the second statement it suffices to take the P 2 polynomial j;l Qj;l . t u A finite family of polynomials that satisfies the assumptions of Fact 47 is called a separating family of polynomials. Fact 48. Let X be a normed linear space and Y a finite-codimensional subspace of X. If Y admits a separating family of polynomials of degree at most n (resp. separating n-homogeneous polynomial), then so does X . Proof. Clearly we may assume that dim X D 1 and so n > 1. Using induction on the codimension of Y it suffices to prove the statement for codim Y D 1. Let f 2 SX be such that Y D ker f and v 2 X such that f .v/ D 1. Suppose that fP1 ; : : : ; Pk g P n .Y / are polynomials such that Pj .0/ D 0, j D 1; : : : ; k,and infY nBY maxj Pj 1. Put Q0 .x/ D kvk2 f .x/2 and Qj .x/ D Pj x f .x/v for x 2 X, j D 1; : : : ; k. Then Qj .0/ D 0, j D 0; : : : ; k. Let x 2 X, kxk 2. If kx f .x/vk > 1, then there is j 2 f1; : : : ; kg such that Pj x f .x/v 1 and so Qj .x/ 1. Otherwise we have jf .x/jkvk 1 and so Q0 .x/ 1. In the case of a separating polynomial P 2 P . nY /, infSY P 1, we can take Q.x/ D P x f .x/v C kvkn f .x/n . t u As we have seen above, in general the existence of a separating polynomial of degree n does not imply the existence of a separating n-homogeneous polynomial. However, in some special cases this is true. Proposition 49. Let X be an infinite-dimensional Banach space with a separating k 1 .X/ (in particular, if X has polynomial of degree at most k 2 N. If P k 1 .X/ D Pwsc property Sk ), then X admits a k-homogeneous separating polynomial. Proof. By Theorem 5.44 we may assume that X is separable. Since X is an Asplund space (Theorem 5.2), X is separable. Let ffn g be a dense sequence in SX . P We may assume that P D jkD1 Pj , where Pj 2 P . jX/, and that infSX P > 0. Assume that there is no k-homogeneous separating polynomial on X. By Fact 48 finite-codimensional subspaces of X do not admit a k-homogeneous separating polyT nomial either. Therefore for each n 2 N there is xn 2 SX \ jnD1 ker fj such that jPk .xn /j < n1 . Then fxn g is weakly null and so Pj .xn / ! 0 for each j D 1; : : : ; k, which is a contradiction. If X has property Sk , then we appeal to Corollary 3.58. u t Fact 50. Let X be a vector space, ' W X ! R a convex function, x1 ; : : : ; xn 2 X , and m 2 N, m n. Then ! ! m n X X 1 X 1 X ' "j xj n ' "j xj : 2m 2 "j D˙1

j D1

"j D˙1

j D1

206

Chapter 4. Asymptotic properties of polynomials

Proof. We have '.y/ D ' 21 .y C x/ C 12 .y x/ 21 '.y C x/ C '.y any x; y 2 X . The statement now follows by a trivial induction on n.

x/ for t u

Theorem 51 ([Dev1]). Let X be a Banach space with a separating family of polynomials fPl gm . Then X is of cotype q D maxl deg Pl . lD1 Proof. By Fact 47 we may assume that Pl are homogeneous and maxl Pl .x/ 1 for x 2 SX . Let K D max1kq Cq;k , where Cq;k are the constants from Proposition 2.78. To show that X is of cotype q choose arbitrary x1 ; : : : ; xn 2 X n f0g. Since the inequality in the definition of cotype is homogeneous, we may assume without lossPof generality

Pn that x1 ; q: : : ; xn 2qBX and x1 2 SX . From Fact 50 it follows that 1

kx1 k D 1. "j D˙1 j D1 "j xj 2n x For each j D 1; : : : ; n there is l.j / 2 f1; : : : ; mg such that Pl.j / kxj k 1. Since j Pn Pm P q D q , it follows that there exists k 2 f1; : : : ; mg kx k kx k j j j D1 kD1 l.j /Dk P 1 Pn q 1 .k/ f1; : : : ; ng and let such that l.j /Dk kxj kq m j D1 kxj k . Denote A D l r D deg Pk q. Then, using Proposition 2.78 and Fact 50, n X

kxj kq m

j D1

X

kxj kq m

j 2A

mKkPk k

X

kxj kr m

j 2A

1 2jAj

X

Pk .xj /

j 2A

q ! r

q X

X

"j xj

"j D˙1 j 2A

n

q !r

n

q

q

1 X 1 X

X

X

mKkPk k n "j xj "j xj : mKkPk k n

2 2 "j D˙1 j D1

"j D˙1 j D1

t u It is of course easier to work with polynomials that have some regular structure. In what follows we will be dealing with symmetric and sub-symmetric polynomials. These are defined on (sub-)symmetric linear subspaces of spaces with Schauder basis. In particular, if X is a normed linear space with a Schauder basis fen g1 nD1 , then the linear subspace spanfen g is both symmetric and sub-symmetric. Of course, if fen g is K-spreading (resp. symmetric), then X itself is sub-symmetric (resp. symmetric). From Corollary 1.31 we immediately obtain the following observation. Fact 52. Let X be a sub-symmetric (resp. symmetric) linear subspace of a normed linear space with a Schauder basis, Y a vector space, and P 2 P .X I Y /. Then P is sub-symmetric (resp. symmetric) if and only if all of its homogeneous summands are sub-symmetric (resp. symmetric). For n; d 2 N we denote I C .n; d / D f˛ 2 I.n; d /I ˛j > 0; j D 1; : : : ; ng, Sd C n nD1 I .n; d /, and N .n/ D f D .1 ; : : : ; n / 2 N I 1 < 2 < < n g. Sd Clearly there is a natural bijection between I.1; d / and nD1 I C .n; d / N .n/ such that .˛; / 2 I C .n; d / N .n/ corresponds to a multi-index in I.1; d / whose j th I C .d / D

Section 4. Separating polynomials. Symmetric and sub-symmetric polynomials

207

coordinate equals ˛j , j D 1; : : : ; n. Thus if X is a normed linear space with a Schauder basis fej gj1D1 , Y a vector space, and P 2 P . dX I Y /, then by Proposition 1.24 there is a unique collection of vectors fy˛; I ˛ 2 I C .n; d /; 2 N .n/; n D 1; : : : ; d g Y such that the formula X P .x/ D x˛11 x˛nn y˛; (2) ˛2I C .n;d / 2N .n/ nD1;:::;d

P holds for every finitely supported vector x D xj ej 2 X , where d } ˛1 y˛; D P . e1 ; : : : ; ˛nen /: ˛

(3)

Definition 53. Let X be a normed linear space with a Schauder basis fej gj1D1 and denote X0 D spanfej gj1D1 . For ˛ 2 I C .n; d / we define the polynomial P˛ 2 P . dX0 / by X P˛ .x/ D x˛11 x˛nn 2N .n/

P

for all x D xj ej 2 X0 . Clearly, each P˛ is a sub-symmetric polynomial and it is called an elementary sub-symmetric polynomial. Further, we denote sd D P.d / , i.e. sd .x/ D

1 X

xjd

j D1

P

for all x D xj ej 2 X0 . Clearly, each sd is a symmetric polynomial and it is called a power sum symmetric polynomial. In general the elementary sub-symmetric (or power sum symmetric) polynomials may not be bounded. They are, however, in the presence of a lower estimate. Fact 54. Let X be a Banach space with a Schauder basis fej gj1D1 that has a lower q-estimate, 1 q < 1, with a constant C . Let ˛ 2 I C .n; d / be such that ˛l q, l D 1; : : : ; n. Then ! ! dq k n k X Y X 1 jxj jq jP˛ .x/j jxj j˛l d kxkd C j D1 j D1 lD1

P for every finitely supported x 2 X , x D jkD1 xj ej . In particular, P˛ can be uniquely extended to a continuous d -homogeneous polynomial on the whole of X , which is then sub-symmetric in case that fej g is K-spreading. Similarly, each sd , d q, can be uniquely extended to a continuous d -homogeneous polynomial on the whole of X , which is then symmetric in case that fej g is symmetric. Conversely, if fej g is unconditional and sd is bounded on spanfej g, then fej g has a lower d -estimate.

208

Chapter 4. Asymptotic properties of polynomials

P Pk q ˛l =q . The extension Proof. The estimates are clear using jkD1 jxj j˛l j D1 jxj j follows from Proposition 1.25. Considering the assumption of the unconditional basis in the last statement, check s1 on c0 with the summing basis. t u For a sub-symmetric homogeneous polynomial the coefficients y˛; in (2) do not depend on by (3) and the Polarisation formula. Thus we obtain the following. Fact 55. Let X be a normed linear space with a Schauder basis fej gj1D1 , Y a vector space, and X0 D spanfej g. If P 2 P . dX0 I Y / is sub-symmetric, then there is a unique collection of vectors fy˛ I ˛ 2 I C .d /g Y such that for every x 2 X0 X P .x/ D P˛ .x/y˛ : (4) ˛2I C .d /

The coefficients y˛ are given by y˛ D

d } ˛1 ˛n ˛ P . e1 ; : : : ; en /

when ˛ 2 I C .n; d /.

The usefulness of sub-symmetric polynomials stems from the fact that the space of sub-symmetric polynomials is finite-dimensional (the previous fact), while a general polynomial can be asymptotically approximated by a sub-symmetric one (Theorem 26). The following is a classical theorem, see e.g. [Sta, Corollary 7.7.2]. Theorem 56. Let X be a normed linear space with a Schauder basis fej gj1D1 , Y a vector space, and X0 D spanfej g. If P 2 P . dX0 I Y / is symmetric, then there is a unique polynomial r 2 P .Kd I Y / such that for every x 2 X0 P .x/ D r s1 .x/; : : : ; sd .x/ : (5) }.˛1el ; : : : ; ˛nel / D P }.˛1e1 ; : : : ; ˛nen / for every Proof. Since P is symmetric, P n 1 C ˛ 2 I .n; Pd / and every distinct l1 ; : : : ; ln 2 N (use the Polarisation formula). For any x D jmD1 aj ej 2 X0 we have X }.ek ; : : : ; ek / P .x/ D ak1 akd P 1 d 1kj m j D1;:::;d

D

d X

}.˛1el ; : : : ; ˛nel / c˛ al˛11 al˛nn P n 1

X

nD1 ˛2I C .n;d / 1l1 ;:::;ln m distinct

D

d X

X

}.˛1e1 ; : : : ; ˛nen / c˛ P

nD1 ˛2I C .n;d /

d c˛ D ˛1

1 d ˛1 1 d 1 ˛2 1

al˛11 al˛nn ;

1l1 ;:::;ln m distinct

where

X

˛1 ˛3

˛2 1

1

˛n ˛n

1 : 1

This can be seen as follows: for each D .k1 ; : : : ; kd / 2 f1; : : : ; mgd we enumerate the distinct values in in the order as they appear, denoting them l1 ; : : : ; ln , and we set

Section 4. Separating polynomials. Symmetric and sub-symmetric polynomials

209

˛j to the number of occurrences of lj in , j D 1; : : : ; n. The number of different s that lead to the same .l1 ; : : : ; ln / and ˛ 2 I C .n; d / is given by c˛ , as k1 D l1 and the other ˛1 1 occurrences of l1 can be distributed into d 1 places. Disregarding the already distributed values l1 , we now have d ˛1 places available for the rest and we proceed inductively. So, it is clearly sufficient to prove that each of the polynomials Q˛ , ˛ 2 I C .n; k/, n k, given by X Q˛ .x/ D al˛11 al˛nn 1l1 ;:::;ln m distinct

P for any x D jmD1 aj ej 2 X0 can be expressed in the form (5). This can be done by induction on n. For n D 1 there is nothing to prove. Let n 2 N, ˛ 2 I C .n; k/, and assume that the statement holds for each Qˇ , ˇ 2 I C .n 1; l/, for every l n 1. Then ! ! m m X X X Q˛ aj ej D al˛11 al˛22 al˛nn j D1

l1 D1

1l2 ;:::;ln m distinct X ˛1 C˛2 al2 al˛nn 1l2 ;:::;ln m distinct

X

al˛22 al˛n1 C˛n

1l2 ;:::;ln m distinct

and we may use the inductive hypothesis on each of the summands. This finishes the proof of the existence of the polynomial r satisfying (5). Considering the uniqueness it clearly suffices to show that r D 0 if P D 0. Let Z D spanfe1 ; : : : ; ed g and define F W Z ! Kd by F .x/ D .s1 .x/Z ; : : : ; sd .x/Z /. Then the Jacobian JF of F is a constant multiple of the Vandermonde determinant. Thus JF is non-zero on an open subset of Z. By the Inverse mapping theorem ([Dieu, Theorem 10.2.5]) there is an open U Z such that F .U / is open. It follows that r D 0 on F .U / and so r D 0 on the whole of Kd by Fact 1.39. t u The next theorem was proved in an unpublished version of [HaHá] for `p spaces. The averaging proof of the slightly more general formulation here is due to [Gon2]. Theorem 57. Let X be a Banach space with a sub-symmetric Schauder basis fej gj1D1 and let P 2 P . dX/ be a sub-symmetric polynomial. If P satisfies the formula (4) and q D minf˛j I ˛ 2 I C .d /; y˛ ¤ 0g, then fej g has a lower q-estimate. Proof. The proof is similar to the proof of Lemma 40. Since the basis is sub-symmetric, it is automatically semi-normalised and in particular it is bounded by M 1. It is easy to check that if X is real, then fej g is also a sub-symmetric basis of the complexification XQ and the extension PQ is sub-symmetric. Thus we may without loss of generality assume that X is complex. Further, we assume for simplicity that there is ˛ D .˛1 ; : : : ; ˛k / 2 I C .d / such that y˛ ¤ 0 and ˛1 D q. Put }.˛1e1 ; : : : ; ˛kek /. By Fact 55, y˛ D d b and hence b ¤ 0. Since P is subb DP ˛ }.˛1en1 ; : : : ; ˛kenk / D b for any n1 ; : : : ; nk 2 N, n1 < < nk . symmetric, P

210

Chapter 4. Asymptotic properties of polynomials

Now fix any x1 ; : : : ; xN 2 C and choose 1 ; : : : ; N 2 C, jj j D 1, such that q q q j xj b D jxj bj. Then using Lemma 2.69 we obtain jbj

N X

jxj jq D

j D1

D

N X j D1 N X

q q } ˛1 j xj P . ej ; ˛2eN C2 ; : : : ; ˛keN Ck /

} q.j xj ej /; ˛2eN C2 ; : : : ; ˛keN Ck P

j D1 N

q 1 X} P D N q lD1

q

N X

! q;N rj .l/j xj ej

! ˛2

˛k

; eN C2 ; : : : ; eN Ck

j D1

N

q

X

}k .2C / M kP xj ej ;

q

d

j D1

where C is the unconditional basis constant of fej g

t u

Corollary 58. Let P 2 P . d`p /, 1 p < 1, be sub-symmetric. If P satisfies (4), then ˛j p, j D 1; : : : ; n for every ˛ D .˛1 ; : : : ; ˛n / 2 I C .d / with y˛ ¤ 0. For symmetric polynomials we have the following version. Theorem 59 ([GGJ]). ˚ Let X be a Banach space with a symmetric Schauder basis fej gj1D1 and q D min k 2 NI fej g has a lower k-estimate . Suppose that P 2 P . dX/ is symmetric. If d < q, then P D 0. If d q, then there is a unique polynomial r 2 P .Kd qC1 / such that P .x/ D r sq .x/; : : : ; sd .x/ for every x 2 X . Proof. Let ˛ 2 I C .n; d / for some 1 n d . Suppose that ˛1 < q and put }.˛1x; ˛2e2 ; : : : ; ˛nen /. Then Q 2 P . ˛1X/ and Q.ej / D Q.e1 / for all Q.x/ D P j > n by the symmetry of P . Now if Q.e1 / ¤ 0, then fej gj1DnC1 has a lower ˛1 -estimate by Lemma 40. It is not difficult to check that then also fej gj1D1 has a lower ˛1 -estimate, which contradicts the definition of q. Similarly we can see that }.˛1e1 ; : : : ; ˛nen / D 0 whenever ˛j < q for some j 2 f1; : : : ; ng. The result now P immediately follows from the proof of Theorem 56. t u Proposition 60 ([GGJ]). Let X be a Banach space with a K-spreading (resp. symmetric) basis fej gj1D1 . If X admits a k-homogeneous separating polynomial, then there exists a k-homogeneous sub-symmetric (resp. symmetric) separating polynomial on X . Proof. If the basis is K-spreading, then the spreading model from Theorem 26 is equivalent to fej g and so we can transfer the sub-symmetric polynomial

P Q onto X.

In the symmetric case we may assume that the norm on X satisfies j1D1 aj e.j / D

P1

j D1 aj ej for every permutation of N ([LiTz2, p. 114]). Further, we may assume that P 2 P . kX/ satisfies P .x/ 1 for x 2 SX . We define Qn 2 P . kX/ by ! ! 1 n X X 1 X Qn aj ej D P a.j / ej ; nŠ j D1

2Sn

j D1

Section 4. Separating polynomials. Symmetric and sub-symmetric polynomials

211

where Sn is the set of all permutations of f1; : : : ; ng. It is easily seen that Qn is symmetric on spanfej I 1 j ng and Qn .x/ 1 for x 2 SX , max supp x n. Since fQn I n 2 Ng is equi-Lipschitz on bounded sets, by passing to subsequences and diagonalising we can find a subsequence fQnj g such that it converges pointwise on a countable set dense in X . Consequently, it converges for every x 2 X. Thus Q.x/ D limj !1 Qnj .x/ is a k-homogeneous symmetric separating polynomial (Theorem 1.29). t u Corollary 61 ([HaHá]). Suppose that `p admits a d -homogeneous separating polynomial. Then p is an even integer and d is an integer multiple of p. Proof. By Proposition 60 we may assume that there is a symmetric d -homogeneous polynomial P on `p with P .x/ 1 whenever kxk D 1. Let n D dpe. By Theorem 59 there is r 2 P .Kd nC1 / such P that P .x/ D r sn .x/; : : : ; sd .x/ . By Proposition 1.23, r.x1 ; : : : ; xd nC1 / D ˛2J.d nC1;D/ y˛ x ˛ for some D 2 N. Put P 1/j ej 2 `p , m 2 N. Then kum k D 1 and limm!1 sl .um / D 0 um D jmD1 .m1=p for all l > p. Since P .um / 1, it follows that p D n and y˛ ¤ 0 for some ˛ D .˛1 ; 0; : : : ; 0/. Since P is d -homogeneous, we necessarily have p˛1 D d . Also, 1 if p is odd, then jsp .um /j m t u ! 0, again a contradiction with P .um / 1. Corollary 62. If a Banach space X admits a separating polynomial, then p.X/ D 2 and q.X / is an even integer. Proof. By Fact 45 the space X has a homogenous separating polynomial. By Theorem 3, `p.X / and `q.X/ are finitely representable in X . Hence these spaces have a homogeneous separating polynomial thanks to Corollary 11. Corollary 61 finishes the proof. t u We are going to show in Chapter 5 that in fact X is of type 2 and cotype q.X/. Corollary 63 ([Kur1], [GJ2]). Let X be a Banach space with a sub-symmetric basis fej gj1D1 and a separating polynomial. Then fej g is equivalent to the canonical basis of some `p , p an even integer. Proof. By Fact 45 and Proposition 60 we may assume that there is a sub-symmetric P 2 P . dX / with P .x/ 1 whenever kxk D 1. Let P be given by (4). By Theorem 57 there is p 2 N such that fej g has a lower p-estimate with a constant C . By Fact 54 we have ! pd k X X X X 1 d kxkd jy˛ j kxkd P .x/ D P˛ .x/y˛ jy˛ j jxj jp C C C C j D1 ˛2I .d /

for any x D

Pk

j D1 xj ej

˛2I .d /

2 X . Finally, p must be even by Corollary 61.

˛2I .d /

t u

Theorem 64 ([GGo]). Let X be a Banach space with a separating d -homogeneous polynomial. Then every normalised weakly null sequence fxj gj1D1 in X has a subsequence equivalent to the canonical basis of `p , p even integer, such that p q.X/ and d is an integer multiple of p.

212

Chapter 4. Asymptotic properties of polynomials

Proof. Let n be the greatest integer such that fxj gj1D1 is P n 1 -null. Clearly n d . By Proposition 44 we can assume that fxj gj1D1 has an upper n-estimate. Passing to further subsequence using Theorem 22 we may assume that fxj gj1D1 has a spreading model fej gj1D1 that is sub-symmetric by Proposition 24. Let Y D spanfej g. By Corollary 11 and Fact 23, Y admits a d -homogeneous separating polynomial. By Corollaries 63 and 61, fej gj1D1 is equivalent to the canonical basis of `p , where p is an even integer and d is an integer multiple of p. The upper n-estimate clearly passes to fej gj1D1 , so we have n p. By Lemma 39, fxj gj1D1 has a subsequence with a lower r-estimate for every r > n. Hence fej gj1D1 has a lower r-estimate for every r > n, so n p. It follows that n D p, which is an even integer. Thus by Lemma 39 a subsequence of fxj gj1D1 has a lower p-estimate. Consequently, this subsequence is equivalent to the canonical basis of `p . The fact that q.`p / D p q.X/ follows automatically, as the cotype passes to subspaces. t u Corollary 65. Let X be a Banach space with a separating polynomial. Let r be the smallest number such that `r embeds into X. Then X has properties Sr and Tq.X/ , and for every p such that `p embeds into X we have r p q.X/. Our results obtained so far can be used to get an important structural step needed for the proof of Deville’s fundamental theorem (Theorem 5.67). But first we recall the classical Mazur’s technique for constructing basic sequences. Lemma 66 (Stanisław Mazur). Let X be a normed linear space, Y a finite-dimensional subspace of X, and " > 0. Then there is a finite-codimensional subspace Z of X such that kyk .1 C "/ky C ´k for every y 2 Y and ´ 2 Z. Proof. Assume additionally that " 1. Let fyj gjmD1 be an 2" -dense set in SY and T let fj 2 SX be such that fj .yj / D 1 for j D 1; : : : ; m. Put Z D jmD1 ker fj . y yj < 2" . Therefore Given y 2 Y , y ¤ 0 there is j 2 f1; : : : ; mg such that kyk

" 1 ky C´k > yj C ´ " fj yj C ´ D 1 " 1 for any ´ 2 Z. u t kyk

kyk

2

kyk

2

2

1C"

One of the main tools is the following diagonalisation lemma. Its proof is similar to that of Lemma 39. Lemma 67. Let X be an infinite-dimensional Banach space and let q 2 N be such that X does not have a separating family of polynomials of degree less than q. Let fPl gm P q .X/ be a finite family of non-constant homogeneous polynomials. Then lD1 P there is a normalised basic sequence fxn g X such that if x D jNDM aj xj 2 BX , then jPl .x/j 21M whenever deg Pl < q and ˇ ˇ N ˇ ˇ X 1 ˇ ˇ q aj Pl .xj /ˇ M ˇPl .x/ ˇ ˇ 2 j DM

whenever deg Pl D q. Moreover, if X is separable, then fxn g can be chosen to be weakly null, and if X has a Schauder basis, then fxn g can be chosen as its block basis.

Section 4. Separating polynomials. Symmetric and sub-symmetric polynomials

213

Proof. First note that by Fact 48 no finite-codimensional subspace of X has a separating family of polynomials of degree less than q. If X is separable, then we choose a sequence ffj gj1D1 dense in SX ; otherwise we just set fj D 0, j 2 N. 1 Let Nl D deg Pl and ık D 2kC1 qŠ4 q k q . We construct a normalised basic sequence 1 }l .xk ; : : : ; xk /j < ık fxn gnD1 X with the basis constant at most 2 such that jP 1 Nl Nl whenever 1 l m, 1 k1 kNl , and Nl < q or k1 < kNl . In other words, for every n 2 N jR.n;lIk1 ;:::;kr / .xn /j < ın whenever 1 l m, 0 r < Nl , Nl

r < q, k1 kr < n, (6)

}l .xk ; : : : ; xk ; Nl rx/, which is a homogeneous polynowhere R.n;lIk1 ;:::;kr / .x/ D P r 1 Q mial of degree less than q. To this end let f"n g RC be such that 1 nD1 .1 C "n / 2. We construct the sequence fxn g by induction so that for every n 2 N it moreover satisfies kyk .1 C "n /ky C xnC1 k

for all y 2 spanfx1 ; : : : ; xn g and 2 K,

fj .xn / D 0 for every j n. The last condition clearly ensures that fxn g is weakly null in case that X is separable. The first step is identical to the inductive step except for setting Z D X, so we describe just the inductive step from n 1 to n. Let Z be the finite-codimensional subspace of TX from Lemma 66 used on Y D spanfx1 ; : : : ; xn 1 g and " D "n 1 . Put W D Z\ jnD1 ker fj . Then W is finite-codimensional and so by the assumption there is xn 2 SW such that (6) holds, as the family of polynomials R.n;lIk1 ;:::;kr / considered

P

P

in P (6) is finite. Finally, jnD1 aj xj .1 C "n / .1 C "k 1 / jkD1 aj xj 2 jkD1 aj xj for all n < k and all a1 ; : : : ; ak 2 K. Thus fxn g is a basic sequence with the basis constant P at most 2. Choose any x D jNDM aj xj 2 BX . Since the basis constant of fxn g is at most 2, it follows that jaj j 4, j D M; : : : ; N . Thus if deg Pl < q, then X ˇ ˇ }l .xk ; : : : ; xk /ˇ Nl Šˇak1 akNl P jPl .x/j 1 Nl M k1 kNl N

X

qŠ4q ıkNl

M k1 kNl N

qŠ4q k Nl ık

kDM

while if deg Pl D q, then ˇ ˇ N ˇ ˇ X ˇ ˇ q aj Pl .xj /ˇ ˇPl .x/ ˇ ˇ

X

X

j DM

N X

1 ; 2M

ˇ ˇ }l .xk ; : : : ; xk /ˇ qŠˇak1 akq P q 1

M k1 kq N k1 1; n N

Therefore X

jP .x/j

ja˛; jjx1 j˛1 jxn j˛n

.˛;/2K.N;d /

ja.0/;.1/ j C

d X k X

X

X

ja˛; jjx1 j˛1 jxn j˛n

kD1 nD1 ˛2I C .n;k/ 2N .n/ n N

C CC

d X k X

! X

kD1 nD1 ˛2I C .n;k/

C CC

d X k X

X

kD1 nD1 ˛2I C .n;k/

D C C 2C

d X

2k

1

X

jx2 jp jxn jp C

2N .n/ 1 D1; n N

1 .n

D C C 2C.2d

1 C 1/Š nŠ

X

jx1 jp jxn jp

2N .n/ 1 >1; n N

d X k X k C C 2C n kD1 nD1

1 1

1/ 2d C1 C;

kD1

since jI C .n; k/j D nC.kn 1n/ 1 D kn 11 (indeed, from the combinatorial point of view it represents the number of distributions of k identical balls into n distinct boxes such that each box is non-empty). t u

Section 5. Stabilisation of polynomials

217

By Fact 55 for every sub-symmetric polynomial Q 2 P d .`p0 / there is a collection fa˛ I ˛ 2 I C .k/; 1 k d g R such that X Q.x/ D Q.0/ C a˛ P˛ .x/ (9) ˛2I C .k/ kD1;:::;d

for every x 2 `p0 . Lemma 71. Let P 2 P d .`p0 /, 1 p < 1 be given by (8), let M N be an increasing sequence, and let Q 2 P d .`p0 / be sub-symmetric and given by (9). Then (7) holds if and only if lim sup ja˛; a˛ j D 0: N !1 .˛;/2K.M;d / 1 N

Proof. We may assume without loss of generality that P .0/ D Q.0/ D 0. For any P P 2 N .k/ we define T W `pk ! `p0 by T jkD1 xj ej D jkD1 xj ej and put P D P B T . Then P 2 P d .`pk /. Note that (7) can be reformulated this way: for every k 2 N and every " > 0 there is N 2 N such that jQ.x/ P .x/j " for every x 2 B`pk and every 2 N .k/ satisfying M and 1 N . In other words, for every k 2 N lim sup kQ`pk P k D 0: N !1

We have P

k X

2N .k/ M; 1 N

! xj ej

X

D

j D1

a˛;.1 ;:::;n / x˛11 x˛nn :

.˛;/2K.N;d / n k

Since the space P d .`pk / is finite-dimensional, all norms on this space are equivalent. In particular, the canonical norm is equivalent to the norm that takes the maximum of absolute values of the corresponding coefficients. Hence there is Kk 1 such that K1k kQ`pk P k max.˛;/2K.N;d / ja˛; a˛ j Kk kQ`pk P k for every 2 N .k/. As sup ja˛; a˛ j D max sup max ja˛; a˛ j 1kd

.˛;/2K.M;d / 1 N

D sup k2N

2N .k/ .˛;/2K.N;d / M; 1 N

sup

max

ja˛;

2N .k/ .˛;/2K.N;d / M; 1 N

a˛ j; t u

the statement of the lemma follows. P d .`p /,

Lemma 72. Let P 2 1 p < 1 be a polynomial consisting only of large powers. There are a subsequence fenj g of the canonical basis fej g and a sub-symmetric polynomial Q 2 P d .`p / such that

lim .Q P /spanfe g1 D 0: N !1

nj j DN

218

Chapter 4. Asymptotic properties of polynomials

Proof. By Theorem 26 there are an increasing sequence M N and a sub-symmetric Q 2 P d .`p / such that (7) holds. By Corollary 58 the polynomial Q consists only of large powers and so clearly the same holds also for Q P . Let Q be given by (9) and M D fnj g. Lemma 70 gives the estimate

.Q P /spanfe g1 2d C1 ja˛; a˛ j: sup nj j DN .˛;/2K.M;d / 1 N

t u

Lemma 71 now finishes the proof.

Lemma 73. Let M N be an increasing sequence and suppose that P 2 P d .`p .M //, 1 p < 1 is given by (8). If P consists only of small powers and P .0/ D 0, then there is a subsequence M1 of M such that lim

sup

N !1 .˛;/2K.M1 ;d / 1 N

ja˛; j D 0:

Proof. By Theorem 26 there are a subsequence M1 of M and a sub-symmetric Q 2 P d .`p / such that (7) holds on M1 . Let Q be given by (9). For any ˛ 2 I C .n; k/ such that ˛j p for all j 2 f1; : : : ; ng we have a˛; D 0 for all 2 N .n/. From Lemma 71 it follows that a˛ D 0. Consequently Q D 0 by Corollary 58 and we appeal to Lemma 71 again. t u Proposition 74. Let M0 N be an increasing sequence and P 2 P d .`p .M0 //, 1 p < 1 a polynomial consisting only of small powers with P .0/ D 0. Then there is a subsequence M D fmj g of M0 such that limN !1 kP spanfemj gj1DN k D 0. Proof. To simplify the notation we denote the restriction of a polynomial Q to spanfej gj 2A by QA . Using induction we construct M together with auxiliary nested subsequences M0 M1 M2 , Mk D fmjk gj1D1 , and a sequence of polynomials fSk g P d .`p0 / with the following properties: (i) mjk D mj whenever j k, (ii) kSk k 21k , (iii) Sk fmk I j >kg D 0. j Pk P ˛1 ˛n (iv) Denote Rk D P j D1 Sj Mk . If Rk .x/ D .˛;/2K.Mk ;d / b˛; x1 xn for all x 2 `p0 .Mk /, then each .˛; / 2 K.Mk ; d / with b˛; ¤ 0 satisfies maxfj I ˛j < pg > mk . Let the polynomial P be given by the formula (8). By Lemma 73 there exists a subsequence M1 of M0 such that sup.˛;/2K.M1 ;d / ja˛; j 2d1C2 . We define m1 D m11 , A D f.˛; / 2 K.M1 ; d /I 1 D m1 ; ˛1 < p; ˛j p for j > 1g, and P S1 .x/ D .˛;/2A a˛; x˛11 x˛nn for x 2 `p0 . By Lemma 70 the polynomial S1 is continuous and kS1 k 12 . Since P consist only of small powers, the properties (iii) and (iv) are clearly satisfied. Now assume m1 ; : : : ; mk , Mk , and S1 ; : : : ; Sk are already defined for some k 2 N. y Fix y 2 `p with supp y fm1 ; : : : ; mk g and define a polynomial Rk 2 P d .`p0 /

Section 5. Stabilisation of polynomials

219

P y P y by Rk js D1 xj ej D Rk y C js Dmk C1 xj ej . By the inductive hypothesis Rk y consists only of small powers and Rk .0/ D 0. After re-grouping the coefficients we obtain X y Rk .x/ D cˇ; .y/xˇ11 xˇnn ; .ˇ;/2K.Mk ;d / 1 >mk

for every x 2 `p0 , where cˇ; .y/ D

dX jˇ j

s X

X

b. 1 ;:::; r ;ˇ1 ;:::;ˇn /;.1 ;:::;r ;1 ;:::;n / y 11 y rr

sD1 rD1 2I C .r;s/ 2N .r/ r mk

if ˇ D .ˇ1 ; : : : ; ˇn /. Since the set of the coefficients fb˛; g is bounded, it is not difficult to see that fy 7! cˇ; .y/I .ˇ; / 2 K.Mk ; d /; 1 > mk g is an equi-Lipschitz system of functions on B D fy 2 B`p I supp y fm1 ; : : : ; mk gg. Since B is compact, there 1 is a finite set F B such that each cˇ; .F / is 2d CkC3 -dense in cˇ; .B/. Therefore by y using Lemma 73 for each Rk , y 2 F , we can find a subsequence W D fmjkC1 gj1DkC1 1 of Mk n fm1 ; : : : ; mk g such that sup.ˇ;/2K.W;d / jcˇ; .y/j 2d CkC2 for each y 2 B. kC1 We put mkC1 D mkC1 and MkC1 D fm1 ; : : : ; mk g [ W . ˚ Next, we set A D .˛; / 2 K.MkC1 ; d /I maxfj I ˛j < pg D mkC1 and P SkC1 .x/ D .˛;/2A b˛; x˛11 x˛nn for x 2 `p0 . Fix any ´ 2 B`p0 .MkC1 /. Then ´ D y C x for some y 2 B and x 2 B`p0 with supp x > mk . Notice that X cˇ; .y/xˇ11 xˇnn : SkC1 .´/ D SkC1 .y C x/ D .ˇ;/2K.MkC1 ;d / 1 DmkC1 ; ˇ1

1

Because the polynomial x 7! SkC1 .y C x/ consists almost only of large powers 1 , and so (ii) holds. The propon fmjkC1 gj1DkC1 , Lemma 70 gives jSkC1 .´/j 2kC1 erty (iii) is clearly satisfied. Finally, to see (iv) note that RkC1 D .Rk SkC1 /MkC1 . P To finish the proof, let x D jkDN xmj emj with kxk 1. Then Rk .x/ D 0 P P by (iv) and hence P .x/ D jkD1 Sj .x/ D jkDN Sj .x/ by (iii). It follows that P t jP .x/j jkDN jSj .x/j 2N1 1 by (ii). u Proof ˚of Theorem 69. Suppose that the polynomial P is given by the formula (8). Put K D ..˛1 ; : : : ; ˛n /; / 2 K.N; d /I ˛j p; j D 1; : : : ; n and define R 2 P d .`p0 / P by R.x/ D .˛;/2K a˛; x˛11 x˛nn , i.e. R consists of all the large powers of P . Then R is continuous by Lemma 70 and by Proposition 1.25 it can be extended to a continuous polynomial on `p . Hence P R is a continuous polynomial on `p that consists only of small powers. The proof now follows easily by combining Lemma 72 with Proposition 74. t u We continue by applying these results in the setting of general Banach spaces.

220

Chapter 4. Asymptotic properties of polynomials

Definition 75. Let X be a real Banach space with a Schauder basis fej gj1D1 and let f W X ! R be a p-homogeneous function (i.e. f .x/ D p f .x/ for all > 0). We say that f stabilises at zero level if 1 X lim sup jf .x/jI x D aj ej ; x 2 BX D 0: n!1

j Dn

It would be more precise to say that f stabilises at zero level with respect to the basis fej g. Note that if f stabilises at zero level with respect to the basis fej g and fuj g is a block basis of fej g, then f spanfuj g stabilises at zero level with respect to the basis fuj g. Theorem 76 ([HaHá]). Let X be a real infinite-dimensional Banach space and let P 2 P . dX /. Then there exists a basic sequence fxj gj1D1 X such that P spanfxj g is either separating or stabilises at zero level. Proof. If X does not contain `p , p even, then X does not have a separating family of polynomials (Fact 47 and Theorem 64). Thus by Lemma 67 there is a normalised basic sequence fxj gj1D1 X such that P spanfxj g stabilises at zero level. For the rest of the proof we may assume that X D `p , p even, and thanks to Theorem 69 that P is sub-symmetric. P Cn 1 1 e , where ftn g N is a suitable sequence so that fun g We put un D jtnDt n n1=p j is a block basis of the canonical basis fen g. Then kun k`p D 1, while limn!1 kun k`q D 0 for any q > p. Denote J D f˛ D .˛1 ; : : : ; ˛k / 2 I C .d /I ˛j p; j D 1; : : : ; kg. If Q ˛ ˛ D .˛1 ; : : : ; ˛k / 2 J, then jP˛ .x/j jkD1 kxk`˛j for every x 2 X (see Fact 54). j

It follows that P˛ spanfun g stabilises at zero level whenever there is j 2 f1; : : : ; kg with ˛j > p. P By Fact 55 and Corollary 58 there are b˛ 2 R such that P D ˛2J b˛ P˛ . There are two possibilities. Either d D kp for some k 2 N and b.p;:::;p/ ¤ 0, or the opposite holds. In the latter case P stabilises at zero level on spanfun g by the previous paragraph. In the former case, consider the polynomial spk . This is a sub-symmetric polynomial P and so there are cˇ 2 R such that spk D ˇ 2IC .d / cˇ Pˇ . From the multinomial formula we get X kxkd D sp .x/k D kŠP.p;:::;p/ .x/ C cˇ Pˇ .x/: ˇ 2J ˇ ¤.p;:::;p/

Since all Pˇ in the sum on the right stabilise at zero level on spanfun g, it follows that P.p;:::;p/ is separating on spanfun gnn0 for a suitable n0 2 N. Consequently, P is separating on spanfun gnn1 for some n1 2 N. t u Theorem 76 implies that odd degree homogeneous polynomials stabilise at zero level on infinite-dimensional subspaces of X. A similar result in the finite-dimensional case is the quantitative Theorem 2.45. We show that these two results cannot be combined in the infinite-dimensional spaces, in a rather strong and quantitative way. In particular,

Section 5. Stabilisation of polynomials

221

it follows from the result below that on every separable real Banach space X there is a homogeneous polynomial P , of any prescribed odd degree greater than one, which has no infinite-dimensional null space (i.e. a subspace Y X such that P .Y / D f0g). Theorem 77 ([ArHá2]). Let X be a real Banach space with w -separable dual and n > 1 an odd integer. Then there exists P 2 P . nX/ without any infinite-dimensional null space. More precisely, given any x 2 X n f0g, P .x/ D 0, there exists an N 2 N such that every null space Y X containing x satisfies dim Y N . Proof. Suppose that we have already proved the statement of the theorem for X D c0 and n D 3. Let P 2 P . 3c0 / be the polynomial with the required property. Given any Banach space X with w -separable dual and n D 3 C 2l, l 2 N0 , we construct the desired Q 2 P . nX/ as follows: 1 Let T 2 L.XI c0 / be a one-to-one operator (take for example T .x/ D j1 fj .x/ j D1 , where ffj gj1D1 BX is w -dense and hence it P1 1 2l P B T .x/. Clearly, any separates the points of X), and put Q.x/ D j D1 2j fj .x/ subspace Y X where Q vanishes translates via T onto a subspace of c0 of the same dimension where P vanishes, which concludes the proof of the general case. It remains to produce P on c0 . We put X P .x/ D ˛j k xj xk2 1j 0, and ˛ 2 I C .d / ja˛; j < d˛ kP there is N D N˛ .n; "; K/ such that for any polynomial R 2 P . d`N 1 / of the form (11) with ja˛; j K for all 2 N .k; N / there are A f1; : : : ; N g, jAj D n, and c 2 R

224

Chapter 4. Asymptotic properties of polynomials

such that kRY may take

cP˛n k < ", where Y D spanfek I k 2 Ag. It is then clear that we

N.n; d; "/ D N˛v : : : N˛2 N˛1 .n; v" ; d d /; v" ; d d : : : ; v" ; d d ; where ˛ 1 ; : : : ; ˛ v is an enumeration of I C .d /. " So fix ˛ 2 I C .k; d /, n 2 N, " > 0, and K > 0. Let ı D 2nŠ and M D K . By ı Ramsey’s theorem there is N 2 N such that for every 2.M C 1/-colouring of k-subsets (i.e. subsets of cardinality k) of f1; : : : ; N g there is A f1; : : : ; N g, jAj D n, such that d N 2P all k-subsets of A have the same colour. Now given R . `1 / of the form (11) with a˛; ja˛; j K for all 2 N .k; N / we put m./ D ı 2 f M 1; M; : : : ; M g. Note that ja˛; ım./j < ı. Each 2 N .k; N / uniquely determines a k-subset of f1; : : : ; N g and vice versa and so the function m induces a 2.M C 1/-colouring of the k-subsets of f1; : : : ; N g. Let A f1; : : : ; N g, jAj D n, be such that there is m0 2 N satisfying m./ D m0 for all A. Then ˇ ! !ˇ ˇ ˇ X X X n ˇ ˇ n ˛1 ˛k xj ej xj ej ˇ ı jx1 xk j ı 0 such that if M N , then sup jp.x/

snC1 .x/j "

x2B`M 1

for every p from the algebra generated by the sub-symmetric polynomials on RM of degree at most n. Proof. Applying Lemma 85 to K D n we obtain N 2 N and u; v 2 B`N such 1 that P .u/ D P .v/ for every P 2 H n;n .RN / but snC1 .u/ ¤ snC1 .v/. We put " D 21 jsnC1 .u/ snC1 .v/j. Let M N . Since all sub-symmetric polynomials from P n .RN / are contained in H n;n .RN /, from the remark after Fact 80 it follows that in particular P .u/ D P .v/ for every sub-symmetric P 2 P n .RM /. We conclude that p.u/ D p.v/ for every p from the algebra generated by the sub-symmetric polynomials from P n .RM /. The statement now easily follows. t u

7. Polynomial algebras on Banach spaces All spaces in this section are real. Let X be a Banach space. Observe that if A is a subalgebra of C.X/ such that each member of A is bounded on CCB subsets of X, then b A is also a subalgebra of C.X/. For every Banach space X we denote by An .X/ the algebra generated by P n .X/. Generally speaking, with a single exception when P .X / D Pwu .X/, there are no results giving a characterisation of the uniform closure b P .X / in any infinite-dimensional Banach space. The refinement of the problem is finding the characterisation of An .X/, and this is wide open as well. The results

230

Chapter 4. Asymptotic properties of polynomials

in this section focus on the natural question when An .X/ D AnC1 .X/ (of course, the inclusion always holds). More precisely, we are going to use the theory of subsymmetric polynomials developed in the previous section together with the asymptotic approach to polynomial behaviour to obtain rather general results showing that the inclusion is almost never satisfied. We begin by formulating a positive result, which easily follows from results proved later on in Chapters 6 and 7. Proposition 87. Let X be a Banach space such that it does not contain `1 and P . nX / D Pwsc . nX/. Then A1 .X/ D A2 .X/ D D An .X/: Proof. It is clear that Pf .X/ A1 .X/ (see Definition 7.10). Therefore we have n .X/ C .X/ D P .X/b A .X/, where the first equality follows P n .X / D Pwu wu f 1 from Proposition 3.29 and Corollary 3.80, and the second equality is Theorem 7.17. Since A1 .X / is an algebra, An .X/ A1 .X/, from which the result follows. t u By Corollary 3.69 and the classical Theorem 5.125, the above proposition can be applied to all C.K/ spaces, K scattered compact, and all n 2 N. All our results going in the opposite direction will rely on the following criterion. Theorem 88. Let X be a Banach space and n 2 N such that for each " > 0 there is P 2 P . nX / with the following property: For every N 2 N there exists a linearly independent set fej gjND1 SX such that ˇ ˇ ! N N ˇ ˇ X X ˇ ˇ sup aj ej ajn ˇ ": ˇP PN ˇ ˇ j D1 jaj j1

Then P . nX / 6 An

j D1

j D1

1 .X/.

S n .`m 1/

the algebra generated by all sub-symmetric polynomials Proof. Denote by of degree at most n. By Corollary 86 there is m 2 N and " > 0 such that on `m 1 n kQ sn k 3" for all Q 2 S n 1 .`m 1 /. Let P 2 P . X/ be the polynomial from the assumptions of the theorem. We claim that P … An 1 .X/. By contradiction, suppose that there exist P1 ; : : : ; Pk 2 P n 1 .X/ and r 2 P .Rk / such that for R D r B.P1 ; : : : ; Pk / we have kP Rk < ". Put K D 1Cmaxj kPj k and let 0 < 1 be such that jr.u/ r.v/j < " whenever u; v 2 KB`k1 , ku vk`k1 < . Using Theorem 79 recursively k.n 1/ times we find N 2 N such that for any linearly independent fej gjND1 SX there exist A f1; : : : ; N g, jAj D m, and sub-symmetric polynomials Q1 ; : : : ; Qk 2 P n 1 .Y / such that kPj Y Qj k < , j D 1; : : : ; k, where Y D spanfej I j 2 Ag with `1 -norm. Let fej gjND1 be the linearly independent set from the assumptions of the theorem and A f1; : : : ; N g, Q1 ; : : : ; Qk 2 P n 1 .Y / as above. Note that as fej g is normalised, kRY P Y kY kRY P Y kX < ". Put Q D r B .Q1 ; : : : ; Qk /. Then Q 2 S n 1 .`m P Y k C kP Y sn k < 3", 1 / and kQ sn k kQ RY k C kRY a contradiction. t u

Section 7. Polynomial algebras on Banach spaces

231

Theorem 89. Let X be a Banach space that contains `1 . Then A1 .X/ ¤ A2 .X/ ¤ A3 .X/ ¤ t u

Proof. Combine Proposition 3.91 and Theorem 88. The following easy fact will be used later in the proofs.

Fact 90. Let X be a Banach space and fxn g X. If there is a w -null sequence fn g X (in particular if X has a Schauder basis f.en I fn /g satisfying infn ken k > 0 and fn g is a subsequence of ffn g) such that " D infn jn .xn /j > 0, then fxn I n 2 Ng is not relatively compact. Proof. We construct a subsequence fxnk g of fxn g by induction. We put n1 D 1 and if nk is already defined for some k 2 N, then we find nkC1 > nk such that jj .xnk /j < 2" whenever j nkC1 . By our assumption K D supn kn k < C1. The " set fxnk I k 2 Ngˇ is then an 2K -separated set. Indeed, let k; l 2 N, k < l. Then ˇ 1 " 1ˇ jnl .xnl /j jnl .xnk /j 2K . t u kxnl xnk k K nl .xnl xnk /ˇ K Now we prove another type of a diagonalisation result (see also Lemma 67 and Lemma 6.72). Lemma 91. Let X be a Banach space with a Schauder basis f.xn I fn /g, Y a Banach space with a bounded shrinking Schauder basis, and let m 2 N be such that P m 1 .X I Y / D PKm 1 .X I Y /. Let P 2 P . mX I Y / and denote by Pk the kth component of P , i.e. the composition of P with the kth coordinate functional on Y . Then for each " > 0 there is a subsequence fxnk g such that for every k 2 N ˇ ˇ ˇPn .x/ am Pn .xn /ˇ " k k k k 2k P1 whenever x D j D1 aj xnj 2 BX is such that minfj I aj ¤ 0g jsupp xj. Proof. Note that we may assume without loss of generality that fxn g is normalised. For any polynomial Q 2 P .XI Y / we will use the convention that Qk denotes the kth component of Q, i.e. the composition of Q with the kth coordinate functional on Y . For ˛ 2 J.l; m/, q 2 N0 satisfying q C j˛j m, and k 2 N we define polynomials ˛;q Rk 2 P . m j˛j qX/ by m ˛;q Rk .x/ D P}k .˛1x1 ; : : : ; ˛lxl ; qxk ; m j˛j qx/: ˛; q; m j˛j q Note that using Proposition 1.22 we obtain ! 1 1 X X X ˛k 1 q ˛;q ˛1 Pk aj xj D a1 ak 1 ak Rk aj xj : j D1

˛2J.k 1;m/ 0qm j˛j

j DkC1

The proof of the lemma is two-stage. In the first step we discard the coordinates before k, ˛;q in the second step we get rid of the coordinates after k (i.e. the polynomials Rk ).

232

Chapter 4. Asymptotic properties of polynomials ˛;q

Clearly fRk I k 2 N; ˛ 2 J.l; m/; 0 q m j˛j; l 2 Ng is bounded in We claim that for fixed ˛ 2 J.l; m/ and q 2 N0 with q C j˛j m and ˛;q j˛j > 0 we have limk!1 kRk k D 0. Indeed, if this is not the case, then there are ˛;q ˛;q a sequence fvj g BX and a subsequence fRk g such that infj jRk .vj /j > 0. By j j ˛;p passing to further subsequences we may also assume that bp D limj !1 Rk .vj / j exists finite for all 0 p m j˛j. Obviously bq ¤ 0 and so there is t 2 R such that Pm j˛j p pD0 bp t ¤ 0. }.˛1x1 ; : : : ; ˛lxl ; m j˛j x/. Then Q 2 P . m j˛jXI Y / and so by our Put Q.x/ D P }/k D P}k assumption the polynomial Q is compact. By the Polarisation formula .P 1 Pm j˛j p ˛;p m and so Qk .txk C x/ D ˛;m j˛j .x/ by Lemma 1.21. Hence pD0 t R 1 Pm j˛j k p m limj !1 Qkj .txkj C vj / D ˛;m j˛j ¤ 0. This however contrapD0 bp t dicts the compactness of Q by Fact 90. Now we construct a subsequence fxmk g so that ˇ ! !ˇ 1 m 1 ˇ ˇ X X X " ˇ ˇ q q aj xmj ak Rmk aj xmj ˇ (14) ˇPmk ˇ ˇ 3 2k qD0 j D1 j DkC1 P .0;:::;0/;q q for any j1D1 aj xmj 2 BX , where Rk D Rk for short. Let K 1 be the basis constant of fxn g. We set m1 D 1. If m1 ; : : : ; mk are already defined, then we find ˛;q " mkC1 > mk so that mjJ.k; m/j.2K/m kRmkC1 k < 32kC1 for all ˛ 2 J.mk ; m/ such that ˛j D 0 if j … fm1 ; : : : ; mk g and j˛j > 0, and all 0 q m j˛j. It is easily checked that the sequence fmk g satisfies (14). q In the second step we show that the polynomials Rmk , q < m are asymptotically zero on a suitable subspace. Using induction we construct a subsequence fnk g of fmk g along with auxiliary nested subsequences fmk g fm1k g fm2k g . Set n1 D m1 . By Theorem 26 there are a subsequence fm1k g of fmk g, a Banach space E with a Schauder basis fen g that is a spreading model of fxm1 g, and sub-symmetric k q polynomials Sn1 2 P . m qE/, q D 0; : : : ; m 1, satisfying the following for r D 1: ˇ ! !ˇ N N ˇ ˇ X " ˇ q X ˇ q aj ej Rnr aj xmrl ˇ (15) ˇ S nr ˇ ˇ 3m.2K/m 2r j j D1 j D1 P for all N l1 < l2 < < lN , all scalars a1 ; : : : ; aN with jND1 aj xmrl 2 BX , P m .X /.

j

and all 0 q < m. Now assume that fmrk 1 g is already defined for some r 2 N, r > 1. Then we put nr D mrr 1 and again by Theorem 26 there are a subsequence q fmrk g of fmrk 1 g and sub-symmetric polynomials Snr 2 P . m qE/, q D 0; : : : ; m 1, satisfying (15). (Recall that subsequences of a sequence with a spreading model have the same spreading model.) Note that for each k 2 N the sequence fnj gj1DkC1 is a subsequence of fmjk gj1D1 . It follows that ˇ ! !ˇ N N ˇ ˇ X " ˇ q X ˇ q aj ej Rnk aj xnlj ˇ ˇSnk ˇ ˇ 3m.2K/m 2k j D1 j D1

Section 7. Polynomial algebras on Banach spaces

233

for all maxfk C 1; N g l1 < l2 < < lN , all scalars a1 ; : : : ; aN 2 R satisfying PN q < m, and k 2 N. Combining these estimates with (14) j D1 aj xnlj 2 BX , all 0 P q m and the fact that Pk .xk / D m qD0 Rk .0/ D Rk .0/ we obtain ˇ !ˇ m N ˇ ˇ X1 X ˇ ˇ 2 " ˇ ˇ q q m ˇPn .x/ fn .x/ Pn .xn /ˇ a e C .2K/ S ˇ j j ˇ k k k k ˇ ˇ nk 3 2k qD0

j D1 lj kC1

m X1 2 " m kSnqk k C .4K/ 3 2k qD0

PN

whenever x D j D1 aj xnlj 2 BX with N l1 < < lN (assuming, as we may, that (1) in Proposition 25 holds with "k 21 ). q We claim that limk!1 kSnk k D 0 for each 0 q < m. This finishes the proof by passing to a suitable subsequence of fnk g in the estimate above. By contradiction, q assume that this does not hold for some 0 q < m. Since fRk I k 2 Ng is a bounded q set, it follows that fSnk g is a bounded sequence in a finite-dimensional space of subq symmetric .m q/-homogeneous polynomials on E. Thus fSnk g has a non-zero cluster Pr point S. In particular, there is a finitely supported w D j D1 wj ej 2 21 BE such that q S.w/ ¤ 0, which means Pr that infk jSpk .w/j D 2ı > 0 for some subsequence fpk g of fnk g. Put wl D j D1 wj xnlCj and note that wl 2 BX for l large enough. Then q lim infl!1 jRpk .wl /j ı for each k 2 N (dropping finitely many members of fpk g if necessary). q m q q } Now let W l .x/ D m wl /. Then W l 2 P . qX I Y / D .˝;s X/˝ Y q P . x; q (Corollary 3.14). Since fwl g is bounded, so is fW l g. Since .˝;s X/ ˝ Y is sepw

arable, there is a subsequence fW lj g such that W lj ! W 2 P . qXI Y /, and in particular w -limj !1 W lj .xk / D W .xk / for each k 2 N (Corollary 3.14). Hence l q jWpk .xpk /j D limj !1 jWpjk .xpk /j D limj !1 jRpk .wlj /j ı for each k 2 N. But according to our assumption the polynomial W is compact, which contradicts Fact 90. t u Theorem 92. Let X be a Banach space and let m 2 N be such that there is a noncompact P 2 P . mXI `1 /. Then P . nX/ 6 An 1 .X/ for every n m. Proof. If X has a subspace isomorphic to `1 , then the result follows from Theorem 89, so for the rest of the proof we assume that X does not contain `1 . We may assume that m is minimal such that there is a non-compact P 2 P . mXI `1 /. Denote by fek g the canonical basis of `1 and by Pk the kth component of P , i.e. the composition of P with the kth coordinate functional on `1 . We may assume that there is a basic sequence fxk g X such that P .xk / D ek (although in the proof it suffices to have lim supjPk .xk /j > 0). Indeed, by Lemma 3.31 there is a weakly null sequence fxk g SX such that fP .xk /I k 2 Ng is not relatively compact. By passing to a subsequence we may assume that fxk g is a normalised basic sequence such that fP .xk /g is not relatively compact. Then we may use Proposition 3.36.

234

Chapter 4. Asymptotic properties of polynomials

Denote by ffk g the functionals biorthogonal to fxk g. Let " > 0 and let fxnk g be the subsequence from Lemma 91. Given n m we put Q.x/ D

1 X

fnk .x/n

m

Pnk .x/:

kD1

Then Q 2 P . nX/ (Theorem 1.29) and it satisfies the condition laid out in Theorem 88. P Indeed, given N 2 N we put yj D xnN Cj , j D 1; : : : ; N . If x D jND1 bj yj is such P that jND1 jbj j 1, then x 2 BX . We put aN Cj D bj , j D 1; : : : ; N and ak D 0 for P k … ŒN C 1; N C N . Then x D 1 kD1 ak xnk and so ˇ ˇ ˇ ˇ N 1 1 ˇ ˇ ˇX ˇ X X ˇ ˇ ˇ nˇ n m n bj ˇ D ˇ ak Pnk .x/ ak Pnk .xnk /ˇ ˇQ.x/ ˇ ˇ ˇ ˇ j D1

kD1

kD1

1 X ˇ ˇPn .x/ k

ˇ akm Pnk .xnk /ˇ ":

t u

kD1

Corollary 93. Let X be a Banach space admitting a non-compact T 2 L.XI `p /, 1 p < 1. Put n D dpe. Then An

1 .X/

¤ An .X/ ¤ AnC1 .X/ ¤ :

(16)

Proof. By Proposition 3.33 we may assume that T .BX / contains the canonical basis of `p . It then suffices to compose T with the polynomial P 2 P . n`p I `1 / given by P ..xj /j1D1 / D .xjn /j1D1 to obtain a non-compact n-homogeneous polynomial from X into `1 . It remains to apply Theorem 92. t u Corollary 94. Let X D Lp .Œ0; 1/, 1 p 1, or X D `1 , or X D C.K/, K a non-scattered compact. Then A1 .X/ ¤ A2 .X/ ¤ A3 .X/ ¤ Proof. The spaces L1 .Œ0; 1/, `1 , L1 .Œ0; 1/, and C.K/, K non-scattered, contain `1 ([FHHMZ, Proposition 5.4], Theorem 5.125) and so Theorem 89 applies. The spaces Lp .Œ0; 1/, 1 < p < 1 contain a complemented subspace isomorphic to `2 ([FHHMZ, Theorem 4.53]) and we may use Corollary 93. t u Corollary 95. Let 1 p < 1 and n D dpe. Then A1 .`p / D D An

1 .`p /

¤ An .`p / ¤ AnC1 .`p / ¤ :

Proof. By Proposition 87 and Corollary 3.59 we obtain An rest follows readily from Corollary 93.

1 .`p /

D A1 .`p /. The t u

Corollary 96. Let X be a Banach space such that X is of type p > 1. Then An where n D Œq C 1 and

1 p

1 .X/

C

1 q

¤ An .X/ ¤ AnC1 .X/ ¤ ;

D 1.

Section 8. Notes and remarks

235

Proof. Let s 2 .q; n/. By Corollary 43 there are a normalised basic sequence fxk g1 kD1 in X and T 2 L.X I `s / such that T .xk / D ek , where fek g is the canonical basis of `s . Thus T is not compact and an appeal to Corollary 93 finishes the argument. t u Corollary 97 ([DiGo]). Let X be a Banach space with an unconditional basis that does not contain `1 , and suppose that n is the smallest integer such that there exists P 2 P . nX / which is not weakly sequentially continuous. Then A1 .X/ D D An

1 .X/

¤ An .X/ ¤ AnC1 .X/ ¤

Proof. The initial sequence of equalities follows from Proposition 87. On the other hand, by Lemma 3.30 we may assume that there are a > 0 and a normalised weakly null block sequence fxj gj1D1 of the given unconditional basis such that jP .xj /j a, j 2 N. By Proposition 2.79 the space P . nX/ has a subspace isomorphic to `1 . To finish, we combine Theorems 3.37 and 92. t u We remark that by only formal changes in the argument the result holds for any Banach space with an unconditional FDD (this is the formulation in [DiGo]).

8. Notes and remarks Section 1. The important concept of finite representability was introduced by Robert Clarke James [Ja1] and has become one of the key notions in modern Banach space theory. Local theory of Banach spaces is studied in a number of monographs, e.g. [Pis4], [Pis5], [DJT], [Tom2], [MS]. Ultraproduct techniques are nicely exposed in [He]. Section 2. The notion of a spreading model was introduced by Antoine Brunel and Louis Sucheston [BrSu1], [BrSu2], and thoroughly studied in [BeaLa] or [Gue]. Basic facts can also be found in [FHHMZ]. The Banach-Saks property was first investigated by Stefan Banach and Stanisław Saks [BaSa] where it is shown that Lp -spaces, 1 < p < 1 have this property (the super-reflexive case can be deduced from this using the James-Gurarii result on the lower estimates for weakly null sequences in super-reflexive spaces). A reflexive counterexample is provided by the Baernstein space [Bae], [CasaSh]. William Buhmann Johnson [Johns2] introduced the notion of the weak p-Banach-Saks property, see also [CastSá1], [CastSá2]. It follows from [JohnsZip1] that quotients of `p have the weak p-Banach-Saks property. By [Johns2], for quotients of Lp , 2 < p < 1, this condition is equivalent to being a quotient of `p . These properties were also investigated by Aleksander Pełczy´nski in [Peł1] under the name of s -convergence, and used to study continuity of multilinear mappings under this topology. His notion of a 1 -null sequence is equivalent to the hereditarily p p-Banach-Saks sequence. For more information see [Rak], [Bea], [BeaLa]. Versions of Lemma 32 can be found in many places, e.g. [Rak], [GR], [CastSá2], [FaJ]. We reproduced the proof given in the last reference.

236

Chapter 4. Asymptotic properties of polynomials

Section 3. Versions of the key Lemma 39 were proved independently in [Dev1], [GJ2], [Há1], and [FaJ]. The important concept of P n -null sequences appears naturally in the study of polynomial versions of the Schur and Dunford-Pettis properties [Ry2], [CCG], [FaJ], see also [GuJL] for a nice survey of this area. The important Proposition 44 comes from [GJ2] and provides an essential ingredient to the proof of Theorem 64 in the next section. Section 4. Robert Deville’s Theorem 51 is a generalisation of results by Victor Zakharovich Meshkov [Me] in case n D 2. This work, and its methods of proofs based on averaging and Kahane’s inequality, is based on previous results of Tadeusz Figiel, and of Stanisław Kwapie´n in [Kw2]. In the works [Gon2], [GJ3], and [GGo] the Rademacher averaging technique from [ALRT] has been systematically used to obtain new interesting results as well as to considerably simplify the proofs (and often generalise the statements) of the results of R. Deville [Dev1], Jaroslav Kurzweil [Kur1], and [HaHá]. Corollary 63 has inspired many similar characterisations within classes of Banach spaces equipped with some invariant conditions on the domain: Let X be a separable rearrangement invariant function space on Œ0; 1. Then X admits a separating polynomial if and only if up to an equivalent renorming X coincides with L2k .Œ0; 1/ for some k 2 N ([GJ3]). Similarly ([GGJ]), let X be a separable rearrangement invariant function space on Œ0; C1/. Then X admits a separating polynomial if and only if X is isomorphic to L2n .Œ0; C1// \ L2m .Œ0; C1//. The proof of Theorem 64 from [GGo] follows a similar pattern as R. Deville’s argument, but it relies explicitly on the spreading models techniques. Of course, the additional important point obtained by Manuel González and Raquel Gonzalo is that it locates the canonical basis of `p within any weakly null sequence. The same techniques imply, [GGo], that if X is a Banach lattice with an exact cotype p, then X contains a complemented subspace isomorphic to `p . If moreover X is `p saturated, then every normalised weakly null sequence contains a subsequence spanning a complemented subspace isomorphic to `p . For examples showing that a Banach space with a separating polynomial X need not have a homogeneous separating polynomial of degree q.X/ see the discussion after Fact 45. It seems to be open if Theorem 51 can be strengthened as follows (for Banach lattices the answer is positive by the general theory): Problem 98. If X is a Banach space with a separating polynomial, does it have a renorming with modulus of convexity of power type q.X/? In connection with the previous problem we remark that there is a super-reflexive Banach space X such that q.X/ < inffqI X has an equivalent norm with modulus of convexity of power type qg, [Pis2]. Problem 99. Assume that a Banach space X admits a separating polynomial. Does it admit a separating polynomial that is convex?

Section 8. Notes and remarks

237

Problem 100. Is there a separable universal space for separable spaces with a separating homogeneous polynomial of degree n? Similarly, we may ask if the analogue of Mordecay Zippin’s result for reflexive (or Asplund) spaces holds. Namely: Problem 101. If X is a separable Banach space with a separating polynomial, is X isomorphic to a subspace of a Banach space with a Schauder basis that admits a separating polynomial? Section 5. All the proofs in this section are simplifications of the results first shown in an unpublished preprint, whose main results were listed in the communication [HaHá]. Given an equivalent norm jj on an infinite-dimensional Banach space .X; kk/ we ˚ jxj can define a distortion of jj on X by .jj/ D infY sup jyj I x; y 2 S.Y;kk/ , where Y runs over all infinite-dimensional subspaces of X . A norm jj is called distorted if .jj/ > 1. Solving a longstanding open problem, it was shown in [OS1] that on `p , 1 < p < 1 we can find an equivalent distorted norm. For more information on the distortion property see e.g. [OS2]. The results on polynomials in our section can be combined with this theory to show that polynomials are non-distorted, unless the Banach space is saturated with distorted copies of `p . Linear sets of zeros of polynomials under real or complex scalars have been studied in [PZ], [AroRu], [AGZ], [Fern], [Ferr], and [AT]. The main positive result for real Banach spaces is Theorem 78, whose proof relies on the use of the Erd˝os-Rado infinite combinatorial lemma. Section 6. The main part of the theory presented in Sections 6 and 7 was originally built in [Há4], but the proof of the key finite dimensional ingredient (Corollary 86) in this paper contains a serious flaw. A different proof was given in [DHJ], which forms the content of the present Section 6. A precursor to this work was the seminal paper [NS], where the authors show that the chain of closures of the algebras An .`p / does not become eventually constant by using a similar, but much easier observation based on the dimensions of the algebras of symmetric polynomials on Rn . Section 7. The theory presented in this section comes from [D’H]. The natural problem of finding the closure of the space of polynomials in the uniform topology goes back at least to Georgiy Evgenievich Shilov [Shi]. Under the assumption P .X / D Pwu .X/ it was solved in [AP], [Aro1], and this is the only case when the solution is known (see also [FGL] for related results). There seems to be no hope for a topological approach to the polynomial approximation problem. In fact, the natural candidate, the topology generated by polynomials, introduced and studied in [ACL], [HL], [GaJL], [GL], does not seem to be appropriate. Sufficient conditions for the polynomial approximability were found by Arkadij Semenovich Nemirovskij and S. M. Semenov, [NS]. They introduce a regular topology on `p generated by semi-norms fkU.x/k1 I U 2 L.`p I `p /g. A mapping f W `p ! Y is said to be regular if it is uniformly continuous in the regular topology on every

238

Chapter 4. Asymptotic properties of polynomials

bounded set. Every regular mapping f W `p ! Y is shown to be uniformly approximable by polynomials on bounded sets. As a corollary, every symmetric C 2;C -smooth function on `2 is uniformly approximable by polynomials. On the other hand, Jon Vanderwerff, [Va1], showed that there exist (non-symmetric, of course) C 2;C -smooth renormings of `2 which cannot be uniformly approximated by polynomials. Problem 102. Give a description of P .X/ for a general separable Banach space X . This problem is open even for X D `2 . In view of Corollary 93 we remark the following: If X contains `1 , then X contains `2 (see Theorem 3.88 and the remark after). In particular, X contains a weakly null sequence with an upper 2-estimate, although P .`1 / D Pwsc .`1 /. On the other hand, if X does not contain `1 , then the existence of upper estimates in X immediately leads to a construction of non-wsc polynomials, as was indicated in Section 3. Also, non-wsc polynomials give rise to non-compact operators into `p from some subspaces of X. Problem 103. Suppose that X does not contain `1 and P .X/ ¤ Pwsc .X/. Is there a non-compact bounded linear operator from X into `p for some 1 p < 1? Equivalently, is there a weakly null sequence in X with an upper q-estimate for 1 1 p C q D 1? An important remaining open problem is the following. Problem 104. Let X be a separable Banach space not containing `1 and let n 2 N be the smallest integer such that P . nX/ ¤ Pwsc . nX/. Is then A1 .X/ D D An

1 .X/

¤ An .X/ ¤ AnC1 .X/ ¤ ?

By combining Theorems 3.37 and 92 the answer to this problem is positive provided the following problem has a positive answer (note that the opposite implication in the statement below is true using Theorem 3.37 and a simple construction). Problem 105. Let X be a separable Banach space not containing `1 and let n 2 be an integer such that P . nX/ ¤ Pwsc . nX/. Does then the space P . nX/ contain c0 ? In particular, if the dual X has a subspace isomorphic to c0 , then Theorem 92 implies that (16) holds for n D 2. If the dual X has a super-reflexive subspace, then we can conclude that (16) holds for some n. Indeed, by the James-Gurarii theorem [FHHMZ, Theorem 9.25], X admits a non-compact linear operator into some `p . This leaves us with two possibilities. If X fails (16) for every n 2 N, then either X is `1 -saturated, or it contains a Tsirelson-like subspace Y , i.e. Y does not contain `1 , c0 , or a super-reflexive space. Problem 106. Suppose that X is a Banach space such that X is `1 -saturated. Is then P .X / D Pwsc .X/?

Chapter 5

Smoothness and structure In this chapter we are going to study the impact of higher smoothness on the structure and geometry of real Banach spaces. It turns out that even very mild smoothness assumptions, such as the existence of a separating twice Gâteaux differentiable function, have serious structural consequences for the space in question. We begin Section 1 by recalling some of the equivalent conditions for a Banach space to be an Asplund space. In particular, a separable Banach space is an Asplund space if and only if it has a C 1 -smooth renorming. We proceed by studying some basic properties of Legendre-Fenchel duality related to smoothness. Unfortunately, since all of these results are deeply intertwined, we have split the section on impact of smoothness on the structure into two parts and inserted a section on variational principles in between. In Section 2 we pass to the basic study of Banach spaces which admit a locally uniformly smooth bump function. We prove a version of the compact variational principle, which in the absence of c0 subspaces allows one to pass to bumps with uniformly continuous derivatives. This is shown to imply one of the fundamental results of our theory: if a Banach space X admits a bump with a locally uniformly continuous derivative, then either X contains a subspace isomorphic to c0 or X is super-reflexive. Section 3 begins with the celebrated Ekeland variational principle. This is followed by the first order Deville-Godefroy-Zizler variational principle and Stegall’s variational principle for spaces with the RNP, which will be our principal tool. Generally speaking, variational principles claim the existence of a point where a given lower semi-continuous function attains its (strong) minimum. These principles have striking consequences for Banach spaces with uniformly rotund norms and admitting bumps with higher derivatives. For example, if a space with a norm with modulus of convexity of power type q has a T q -smooth bump, then it admits a separating polynomial. In Section 4 we show that the modulus of convexity of power type q in Theorem 62 can be relaxed to cotype q, provided we upgrade the smoothness assumption on q the bump to be U Tc -smooth. We prove the fundamental result of Robert Deville concerning the structure of the spaces with C 1 -smooth bumps. We show that X is an Asplund space provided that it admits a bump with some rather weak differentiability properties. Similarly, under very weak convexity and differentiability assumptions we can already construct a norm with modulus of smoothness of power type 2 (and hence the space is super-reflexive). Finally, improving on the fundamental work of Victor Zakharovich Meshkov we show that X is isomorphic to a Hilbert space provided that both X and X admit a bump with certain smoothness properties.

240

Chapter 5. Smoothness and structure

Sections 5–6 are devoted to separable polyhedral Banach spaces. These spaces are characterised by admitting an equivalent renorming which locally depends on finitely many coordinates (LFC). The space c0 is the main polyhedral space, and every polyhedral space is an Asplund space and contains c0 . These spaces admit many characterisations, e.g. they can be renormed by means of C 1 -smooth and simultaneously LFC norms. In the remaining, rather computational, Sections 7–9 we find the best order of smoothness for the classical Banach spaces Lp , C.K/, and the Orlicz spaces, respectively. In this chapter every vector space is real.

1. Convex functions This section deals with various topics concerning the smoothness of convex functions, including norms. We first review (without proofs) some fundamental results about Asplund, weak Asplund, and weak Asplund spaces, which will we used throughout, and which are covered in detail elsewhere, notably in [DGZ]. Further, we define the Moreau-Rockafellar subdifferential, and a directional/pointwise modulus of smoothness for a general mapping. We show how these notions relate to the differentiability of convex functions, and how the power type of the moduli relate to the T p -smoothness. Using a Baire category argument we show that for convex functions on Banach spaces a directional pointwise modulus of smoothness of power type implies a pointwise modulus of smoothness of power type. Next, we prove some basic facts about convex functions and norms with uniformly continuous derivatives (also of higher order). We also show that the Minkowski functionals of sub-level sets of various smooth convex functions share the same smoothness properties and thus give smooth norms. We proceed by introducing the powerful concept of Legendre-Fenchel duality and show some elementary properties of this notion. We define a (pointwise) modulus of exposedness, show some of its basic properties and prove that it is dual to the pointwise modulus of smoothness. Finally, we prove that a Lipschitz convex function which on a large set has a modulus of exposedness of power type gives rise to an equivalent norm with modulus of convexity of some power type. This allows us to prove via duality various smoothness results in later sections. We end this section by showing that subdifferentials of weakly sequentially continuous convex functions exhibit some compactness, provided that the domain space does not contain `1 . The following concept plays an important role both in the theory of differentiability and in an investigation of the structure of Banach spaces. Definition 1. A Banach space X is called an Asplund space if for every continuous convex function f defined on an open convex subset U X the set of points at which f is Fréchet differentiable contains a dense Gı subset of U . A Banach space X

Section 1. Convex functions

241

is called a weak Asplund space if for every continuous convex function f defined on an open convex subset U X the set of points at which f is Gâteaux differentiable contains a dense Gı subset of U . Let us remark that the set of points of Fréchet differentiability of a continuous convex function defined on an open convex set is always a Gı set (Fact 13), while this is no longer true for the set of points of Gâteaux differentiability – examples can be found in [Tal1] or [HŠZ]. The norm of a normed linear space X is said to be C k -smooth if it is a C k -smooth function on X n f0g. Theorem 2. Let X be a Banach space. The following statements are equivalent: (i) X is an Asplund space. (ii) Every continuous convex function on X is Fréchet differentiable on a dense Gı subset of X . (iii) X has the Radon-Nikodým property (RNP). (iv) Every equivalent norm on X is Fréchet differentiable at some point. (v) Every separable subspace of X is an Asplund space. (vi) Every separable subspace of X has separable dual. (vii) Every separable subspace of X admits an equivalent C 1 -smooth norm. (viii) Every separable subspace of X admits a Fréchet differentiable bump. For the proof see [DGZ] and [FHHMZ, Theorem 11.8, Theorem 11.15]. From the equivalence of (i) and (viii) we immediately obtain the following corollary. Corollary 3. If a Banach space admits a Fréchet differentiable bump, then it is an Asplund space. We will show much stronger result later on in Theorem 72. We note that it is not known whether the converse holds. Theorem 4 (Stanisław Mazur). Let X be a separable Banach space and let f W X ! R be a continuous convex function. Then the set of points of Gâteaux differentiability of f is a dense Gı subset of X . In particular, X is a weak Asplund space. For the proof see [FHHMZ, Theorem 8.2]. Fact 5. Let X be a Banach space, U X an open convex set, and f W U ! R a lower semi-continuous convex function. Then f is continuous. Proof. Let Fn DSfx 2 U I f .x/ ng, n 2 N. Then each Fn is relatively closed in U and U D n2N Fn . Since U is a Baire space, there is n 2 N such that Fn has a non-empty interior. Hence f is bounded above on some open set and so f is continuous on U . t u Fact 6. Let X be a normed linear space, U X open, and let f W U ! R be w -lower semi-continuous. If f is Fréchet differentiable at x 2 U , then Df .x/ 2 X .

242

Chapter 5. Smoothness and structure

Proof. Denote F D Df .x/ 2 X . The functions fn .h/ D n f .x C n1 h/ f .x/ , h 2 BX are w -lower semi-continuous for n sufficiently large and fn ! F uniformly on BX . Hence F is w -lower semi-continuous on BX . As F is linear, it is w -continuous on BX . Therefore F is w -continuous on X by the BanachDieudonné theorem, which implies that F 2 X . t u Definition 7. A dual Banach space X is called a weak Asplund space if for every w -lower semi-continuous convex function f defined on an open convex subset U X the set of points at which f is Fréchet differentiable contains a dense Gı subset of U . Note that by Fact 5 each w -lower semi-continuous convex function f defined on an open convex set is continuous and hence each dual Asplund space is a weak Asplund space. Theorem 8 ([Col]). Let X be a Banach space. Then X is a weak Asplund space if and only if X has the Radon-Nikodým property. For the proof see [BourgR, Theorem 5.7.4]. Let A be a set. For f W A ! R [ fC1g we denote by dom f the effective domain of f , i.e. the set f 1 .R/. The function f is called proper if dom f is non-empty. The (closed) epigraph of f W A ! R is defined by epi f D f.x; t/ 2 A RI t f .x/g. Definition 9. Let X be a normed linear space, ' W X ! R[fC1g, and let x 2 dom '. We define the (Moreau-Rockafellar) subdifferential of ' at x by @'.x/ D ff 2 X I f .h/ C '.x/ '.x C h/ for all h 2 X g: Note that the subdifferential @'.x/ is always a w -closed convex subset of X . If ' W X ! R [ fC1g is a convex function continuous on Int dom ' and x 2 Int dom ', then the subdifferential @'.x/ is non-empty and bounded. This follows from the HahnBanach theorem used on the zero functional on f0g and the dominating convex function h 7! '.x C h/ '.x/. The boundedness is a consequence of the fact that ' is locally Lipschitz. Note also that every convex function ' W C ! R defined on a convex subset C of a vector space can be extended to a convex function defined on the whole space by setting '.x/ D C1 for x … C . Definition 10. Let X, Y be normed linear spaces, U X open, and f W U ! Y . For x 2 U , h 2 X we define a directional pointwise modulus of smoothness of f at x in the direction h by f

x;h .t / D sup kf .x C sh/ C f .x sh/ 2f .x/k for t 0 such that x ˙ th 2 U . 0st

Further, we define a pointwise modulus of smoothness of f at x by f

xf .t/ D sup x;h .t/ h2SX

for t 0 such that B.x; t/ U .

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243

Let X be a normed linear space and f W X ! R [ fC1g. Then for x 2 dom f f f the moduli x;h and x are well-defined on Œ0; C1/ by the formulae above. If f is f moreover convex, then it is easy to check that x;h .t/ D f .x Cth/Cf .x th/ 2f .x/ f and x .t / D suph2SX f .x C th/ C f .x th/ 2f .x/ . Lemma 11. Let f W U ! R be a continuous convex function defined on an open convex subset U of a normed linear space X and x 2 U . Then the following statements are equivalent: (i) The function f is Gâteaux differentiable at x. .x/ exists for every h 2 X . (ii) The directional derivative @f @h (iii) The subdifferential @f .x/ is a singleton. f (iv) x;h .t / D o.t/; t ! 0C for every h 2 SX . f .x C th/ C f .x th/ 2f .x/ (v) lim D 0 for every h 2 X. t !0 t If f is Gâteaux differentiable at x, then @f .x/ D fıf .x/g. Proof. (i))(v) is obvious and (iv) is just a reformulation of (v). (v))(iii) We already know that the subdifferential @f .x/ is non-empty. Suppose that there are functionals g1 ; g2 2 @f .x/ and h 2 X such that g1 .h/ > g2 .h/. Then f .x C t h/ C f .x th/ 2f .x/ g1 .th/ C g2 . th/ D t.g1 .h/ g2 .h// for every sufficiently small t > 0, which contradicts (v). (iii))(ii) Extend f to the whole of X with the value C1 outside U . Fix h 2 X nf0g. From the convexity of f it follows that for all t 2 R n f0g f .x C th/ f .x/ t 7! is non-decreasing. (1) t @f @f 1 1 Thus @h C .x/ D lim t f .xCth/ f .x/ and @h .x/ D lim t f .xCth/ f .x/ t !0C

t!0

@f @f @f always exist (finite) and @h C .x/ @h .x/. Next, we set g1 .th/ D t @hC .x/ and @f g2 .t h/ D t @h .x/. Obviously gj .th/ f .x C th/ f .x/ for all t 2 R, j D 1; 2. By the Hahn-Banach theorem there is an extension of gj to the whole of X such that gj 2 X and gj .u/ f .x C u/ f .x/ for all u 2 X, j D 1; 2. By (iii) it follows @f @f @f that g1 D g2 and hence @h C .x/ D @h .x/, which means that @h .x/ exists. (ii))(i) Using the convexity of f we obtain for every h; k 2 X and sufficiently small t > 0 f x C t .h C k/ f .x/ f .x C 2th/ f .x/ f .x C 2tk/ f .x/ C ; t 2t 2t @f from which it follows that the function h 7! @h C .x/ is sub-additive. Similarly we @f .x/ is super-additive. Hence h 7! @f .x/ is linear obtain that the function h 7! @h @h (the homogeneity is obvious). Further, from (1) and the continuity of f we obtain that @f @f h 7! @h C .x/ is upper semi-continuous, while h 7! @h .x/ is lower semi-continuous. Hence h 7! @f .x/ is continuous. @h The last statement follows from (iii) and (1). t u

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Note that from (1) it follows that if f is convex, then both t 7! f t 7! 1t x .t / are non-decreasing.

1 f t x;h .t/

and

Lemma 12. Let f W U ! R be a continuous convex function defined on an open convex subset U of a normed linear space X and x 2 U . Then the following statements are equivalent: (i) The function f is Fréchet differentiable at x. f (ii) x .t / D o.t/; t ! 0C. f .x C h/ C f .x h/ 2f .x/ D 0 for every h 2 X . (iii) lim khk h!0 Proof. (i))(iii))(ii) is obvious. (ii))(i) By Lemma 11 we know that f is Gâteaux differentiable at x. So for small h 2 X we have 0 f .x C h/

f .x/

ıf .x/Œh D f .x C h/

f .x C h/ C f .x

h/

f .x/ C ıf .x/Œ h

2f .x/ xf .khk/ D o.khk/; h ! 0:

t u

f x

We note that if f is not convex, then .t/ D o.t/; t ! 0C does not imply 2y the smoothness of f at x: for the function f .x; y/ D x 2xCy 2 , f .0; 0/ D 0 we have f .0;0/ .t / D 0, but f is not even Gâteaux differentiable at the origin. Fact 13. Let f W U ! R be a continuous convex function defined on an open convex subset U of a normed linear space X. Then the set of points of Fréchet differentiability of f is a Gı set. By Lemma 12 the set of points of Fréchet differentiability of f is equal to Proof. T G o n2N n , where [n Gn D x 2 U I 1t xf .t/ < n1 for t 2 .0; ı : ı>0

To see that each Gn is open fix x 2 Gn and let ı > 0 be such that f is Lipschitz ˚ f f on B.x; 2ı/ and 1ı x .ı/ < n1 . The family of functions y 7! y;h .ı/ h2SX is equif Lipschitz on B.x; ı/, and hence y 7! y .ı/ is Lipschitz on B.x; ı/. Thus there is f a neighbourhood V of x such that 1ı y .ı/ < n1 for all y 2 V . Since for f convex f

f

t 7! 1t x .t / is non-decreasing, it follows that 1t y .t/ < which shows that V Gn .

1 n

for all t 2 .0; ı and y 2 V , t u

Definition 14. Let X , Y be normed linear spaces, U X an open set, and f W U ! Y . f f f We say that x;h (or x ) is of power type p 2 .1; C1/, if x;h .t/ D O.t p /; t ! 0C f (or x .t / D O.t p /; t ! 0C). f

f

Clearly, if the modulus x;h (or x ) is of power type p, then it is also of power f type q for every q 2 .1; p/. It is obvious that x;h is of power type p if and only if f kf .x C t h/ C f .x th/ 2f .x/k D O.t p /; t ! 0C. Similarly, x is of power

Section 1. Convex functions

245

type p if and only if kf .x C h/ C f .x h/ 2f .x/k D O.khkp /; h ! 0. Also, f f x;ch .t / D x;h .ct/ for c 0 and small t. f Observe that if f is T p -smooth at x for p 2 .1; 2, then x is of power type p. f Similarly, if f is directionally T p -smooth at x for p 2 .1; 2, then x;h is of power type p for each h 2 SX . Fact 15. Let X be a normed linear space, U X an open convex set, f W U ! R f a convex continuous function, x 2 U , and p 2 .1; 2. If x;h is of power type p for f each h 2 SX , then f is weakly T q -smooth at x for every q 2 .1; p/. If x is of power q type p, then f is T -smooth at x for every q 2 .1; p/. Proof. In both cases the function f is Gâteaux differentiable at x by Lemma 11. By the convexity ıf .x/Œh f .x C h/ f .x/ for every h 2 X satisfying x C h 2 U . The assertion now follows from the inequalities 0 f .x C h/ f .x/ ıf .x/Œh D f .x C h/ f .x/ C ıf .x/Œ h f .x C h/ C f .x h/ 2f .x/. t u Proposition 16. Let X be a Banach space, U X an open convex set, f W U ! R a convex continuous function, and x 2 U . If for each h 2 SX the directional pointwise f f modulus x;h is of some power type ph , then the pointwise modulus x is of some f power type. Moreover, if p D inf fph I h 2 SX g > 1, then x is of power type p. Proof. Let ı > 0 be such that B.x; ı/ U and f is bounded on B.x; ı/. We put ˚ 1 Fn D h 2 BX I f .x C th/ C f .x th/ 2f .x/ nt 1C n for 0 t ı : f

From the assumption on the modulus x;h and from the boundedness of f on the S ball B.x; ı/ it follows that BX D n2N Fn . Each of the sets Fn is closed because ˚ T 1 Fn D t 2Œ0;ı h 2 BX I f .x C th/ C f .x th/ 2f .x/ nt 1C n and f is continuous. Further, each Fn is absolutely convex. Indeed, let u; v 2 Fn and 2 .0; 1/. Then f x C t .u C .1 /v/ C f x t.u C .1 /v/ 2f .x/ f .x C t u/ C .1

/f .x C tv/ C f .x D f .x C t u/Cf .x tu/ 2f .x/ C .1

tu/ C .1

/f .x

/ f .x C tv/Cf .x

tv/ tv/

2f .x/ 2f .x/

1

nt 1C n for 0 t ı. Obviously, if h 2 Fn , then also h 2 Fn . Hence by the Baire category theorem there is n 2 N such that Fn has a nonempty interior. It follows from the absolute convexity that there is > 0 such that 1C n1 B.0; / Fn . Therefore 0 f .x C h/ C f .x h/ 2f .x/ n khk for every f 1 h 2 B.0; ı/, which means that x is of power type 1 C n . Finally, notice that if p > 1, then instead of the sets Fn as above we may consider the sets ˚ Fn D h 2 BX I f .x C th/ C f .x th/ 2f .x/ nt p for 0 t ı : t u

246

Chapter 5. Smoothness and structure

Combining Proposition 16 with Fact 15 we obtain the following corollary. Corollary 17. Let X be a Banach space, U X an open convex set, f W U ! R a convex continuous function, x 2 U , and p 2 .1; 2. If f is directionally T p -smooth at x, then f is T q -smooth at x for every q 2 .1; p/. Corollary 18. Let X be a Banach space that admits a separating continuous directionally T p -smooth convex function f for some p > 1. Then X is an Asplund space. Proof. By Corollary 17 f is Fréchet differentiable. Hence by composing f with a suitable smooth real function we obtain a Fréchet differentiable bump (Theorem 1.69). Thus X is an Asplund space by Corollary 3. t u Next, we turn our attention to convex functions with uniformly continuous derivatives. Lemma 19. Let X be a normed linear space, U X an open convex set, and let ˝ Ms be the cone generated by ! 2 Ms . (i) If Y is a Banach space and f 2 C 1;˝ .U I Y /, then there is C > 0 such that

f .x C h/ C f .x h/ 2f .x/ C !.khk/khk for every x; h 2 X satisfying x ˙ h 2 U . (ii) Suppose that f W U ! R is a continuous convex function. If there is C > 0 such that f .x C h/ C f .x

h/

2f .x/ C !.khk/khk

for every x; h 2 X satisfying x ˙ h 2 U , then f 2 C 1;˝ .V / for each non-empty bounded open V U satisfying dist.V; X n U / > 0. Proof. (i) Suppose that f 2 C 1 .U I Y / and Df is uniformly continuous on X with modulus K! for some K > 0. By Theorem 1.89 for any x; h 2 X satisfying x ˙h 2 U we get

Z

Z 1

1

f .x C h/ C f .x h/ 2f .x/ D Df .x C th/Œh dt C Df .x th/Œ h dt

0 0 Z 1

Df .x C th/ Df .x th/ khk dt 0

K!.2khk/khk 2K!.khk/khk: (ii) First, notice that f is Fréchet differentiable on U by Lemma 12. Let V U be a non-empty ˚ bounded open set for which dist.V; X n U / > 0. Put d D diam V and " D min d2 ; dist.V; X n U / . Fix any x; y 2 V . For an arbitrary h 2 X satisfying khk D d" kx yk (note that x Ch 2 U , y h 2 U ) we use the fact that Df .x/ 2 @f .x/

Section 1. Convex functions

247

and Df .y/ 2 @f .y/ to obtain Df .x/Œh

Df .y/Œh D Df .x/Œh C Df .y/Œ h

f .x C h/

f .x/ C f .y

D f .x C h/ C f .y

h/

2f

h/ f .y/ x C y 2 x C y

f .x/ C f .y/

f .x C h/ C f .y h/ 2f 2 x C y x C y x y x y Df C Ch Cf 2

2 2x y 2 x y

C h C h C! 2 2 1 " 1 " C C! C kx yk kx yk 2 d 2 d C !.kx yk/kx yk: Taking supremum over h 2 X, khk D " kx d

ykkDf .x/

" kx d

h

2f

2f

x C y 2

x C y 2

yk, we get

Df .y/k C !.kx

yk/kx

which shows that Df is uniformly continuous on V with modulus

yk; d " C !.

t u

We note that for !.t/ D t (the Lipschitz case) it can be shown that for continuous convex functions the condition f .x C h/ C f .x h/ 2f .x/ C khk2 for x; h 2 X , x ˙ h 2 U , is equivalent to f 2 C 1;1 .U /. (We restrict kx yk to be small and use the fact that mappings on normed linear spaces that are Lipschitz for small distances are globally Lipschitz with the same constant.) Further, for convex functions that are defined on the whole space X we can take khk D 12 kx yk in the previous proof, which gives the following version. Lemma 20. Let f W X ! R be a continuous convex function on a normed linear space X and let ˝ Ms be the cone generated by ! 2 Ms . Then f 2 C 1;˝ .X/ if and only if there is C > 0 such that for every x; h 2 X f .x C h/ C f .x

h/

2f .x/ C !.khk/khk:

Fact 21. Let .X; / be a normed linear space such that the norm is C k -smooth. Then d j.cx/ D c j1 1 d j.x/ for every x 2 X n f0g, c > 0, and every j 2 f1; : : : ; kg. Proof. Just differentiate the equality .cx/ D c.x/.

t u

Let .X; / be a normed linear space. The norm is said to be C k;˝ -smooth if it is on X n f0g and d k is uniformly continuous on SX with modulus ! for some ! 2 ˝. C k -smooth

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Chapter 5. Smoothness and structure

Fact 22. Let .X; / be a normed linear space and let ˝ Ms be a convex cone. If the norm is C 1;˝ -smooth, then for each ı > 0 there is ! 2 ˝ such that D is uniformly continuous on X n B.0; ı/ with modulus !. If the norm is C k;˝ -smooth, k > 1, then for each 0 < ı < 1 there is ! 2 ˝ such that d k is uniformly continuous on B.0; 1ı / n B.0; ı/ with modulus !. Proof. For convenience we will also denote kxk D .x/. Suppose k > 1. Let 2 ˝ be such that d k is uniformly continuous on SX with modulus . Fix x0 2 SX and denote C D kd k.x0 /kC .2/. Then kd k.x/k kd k.x0 /kCkd k.x/ d k.x0 /k kd k.x0 /k C .kx x0 k/ C for any x 2 SX . Using Fact 21 we can estimate for x; y 2 B.0; 1ı / n B.0; ı/

y

k

1 k

d .x/ d k.y/ D 1 d k x d

kxkk 1 kxk kyk kykk 1

ˇ

ˇ y

k x

k y ˇ 1 1 ˇˇ 1 k

ˇ d d C d kxk kyk kyk ˇ kxkk 1 kykk 1 ˇ kxkk 1 x ˇ ˇ 1 y C

ˇkxkk 1 kykk 1 ˇ k 1

C k 1 k 1 kxk kyk ı kxk kyk 2 C 1 kx yk C 2k 2 kxkk 2 C kxkk 3 kyk C C kykk 2 kx yk k 1 ı ı ı 2 C.k 1/ 1 .kx yk/ C 3k 4 kx yk: k 1 ı ı ı The assertion now follows from Fact 1.120. The proof for k D 1 is analogous. t u Lemma 23. Let X be a normed linear space, C X an open convex set, and let f W C ! R be a continuous convex function. Suppose that there are x0 2 C and a 2 R, a > f .x0 / such that the set fx 2 C I f .x/ ag is a closed bounded subset of X. (This in particular holds if f is separating.) Assume further that there is an open ˝ C such that f 2 C k .˝/, where k 2 N [ f1; !g, and that S D fx 2 C I f .x/ D ag ˝. Then X admits an equivalent C k -smooth norm ˇ. If we suppose further that d jf is bounded and uniformly continuous on S with modulus ! 2 Ms for each j 2 f1; : : : ; kg, then there is K > 0 such that d jˇ is bounded and uniformly continuous on S with modulus K! for each j 2 f1; : : : ; kg. Proof. Assume without loss of generality that x0 D 0. Put B D fx 2 C I f .x/ ag. Since by the assumption f .0/ < a, the set B is a closed convex bounded neighbourhood of 0 and hence the Minkowski functional of B, denoted by , is a continuous function and @B D fx 2 XI .x/ D 1g. On the other hand, using the facts that B is closed, f .0/ < a, and f is convex, we obtain @B D S . It follows that for every x 2 X n f0g and y 2 RC we have f yx D a if and only if y D .x/. Let H W X RC ! X be given by H.x; y/ D yx . The set G D H 1 .˝/ is open by the continuity of H . Define F W G ! R by F .x; y/ D f yx . Then F 2 C k .G/ (Corollary 1.84 and Theorem 1.83, or Proposition 1.173 and Theorem 1.172). Note that by the convexity of f and the fact that f .0/ < a there is ı > 0 such that Df .u/Œu ı

Section 1. Convex functions

249

x x for every u 2 S. It follows that D2 F x; .x/ D Df .x/ ¤ 0 for every .x/2 x 2 X n f0g. Therefore the Implicit function theorem (Theorem 1.87, Theorem 1.174) implies that 2 C k .X n f0g/. Moreover, x x i 1 1 x h Df D.x/ D Df .x/ .x/2 .x/ .x/ (2) x 1 D Df : x x .x/ Df .x/ .x/ We show the last statement of the theorem (with ˇ replaced by ) by induction 1 Df .x/ for x 2 S , it is easily on k. Suppose first that k D 1. Since D.x/ D Df .x/Œx checked that D is bounded and uniformly continuous with modulus K! on S for some K > 0 (recall that Df .u/Œu ı for u 2 S and use also Fact 1.120). Now suppose that the statement holds for k 1. Let " W X X ! R be the evaluation operator, i.e. ".g; x/ D g.x/. Then " 2 L.X ; XI R/. By Proposition 1.79 and Proposition 1.71 we can see that d j" is bounded and Lipschitz on bounded sets for 1 all j 2 N. Notice that (2) can be rewritten as D D p B q, where p D "B.Df;Id Df / x and q.x/ D .x/ . By the inductive hypothesis and Proposition 1.128 together with Proposition 1.129 there is L > 0 such that both d jp and d jq are bounded and uniformly continuous with modulus L! on S for all j 2 f1; : : : ; k 1g. So, using Proposition 1.128 once again, we conclude that there is K > 0 such that d j .D/ is bounded and uniformly continuous with modulus K! on S for all j 2 f1; : : : ; k 1g, which together with the case k D 1 gives that d j is bounded and uniformly continuous with modulus K! on S for all j 2 f1; : : : ; kg. Finally, ˇ.x/ D .x/ C . x/ is the desired equivalent norm. t u We note that a function which is not separating but satisfies the weaker assumptions of the previous lemma is used e.g. in the proof of Theorem 104. The following concept is very important in convex analysis. Definition 24. Let X be a normed linear space and ' W X ! R. We define the (Legendre-Fenchel) conjugate function ' W X ! R by ' .f / D sup ff .x/ Further, for

W X ! R we define

D

'.x/I x 2 X g: . X

Note that for a given f 2 X we have ' .f / D inf f'.x/

f .x/I x 2 Xg

D sup fc 2 RI f .x/ C c '.x/ for all x 2 Xg and if ' .f / 2 R, then the last supremum is attained. Further, ' is always a convex w -lower semi-continuous function.

250

Chapter 5. Smoothness and structure

Fact 25. Let X be a normed linear space and ' W X ! R. Then .' / ' and ˚ .' / .x/ D sup f .x/ C cI f 2 X ; c 2 R; f .y/ C c '.y/ for all y 2 X : (3) In particular, if ' is convex and lower semi-continuous, then .' / D '. Dually, if W X ! R, then . / and ˚ . / .f / D sup f .x/ C cI x 2 X; c 2 R; g.x/ C c .g/ for all g 2 X : If the function is convex and w -lower semi-continuous, then . particular, ..' / / D ' for any ' W X ! R.

/

D

. In

Proof. For any x 2 X we have ˚ '.x/ sup f .x/ C cI f 2 X ; c 2 R; f .y/ C c '.y/ for all y 2 X ˚ D sup f .x/ C cI f 2 X ; c 2 R; c inf f'.y/ f .y/I y 2 Xg ˚ D sup f .x/ C cI f 2 X ; c 2 R; c ' .f / D .' / .x/: Now suppose that ' is convex and lower semi-continuous and there is x 2 X such that '.x/ > .' / .x/. Let ˛ 2 R be such that '.x/ > ˛ > .' / .x/. By the separation theorem used on the closed convex set epi ' and a point .x; ˛/ there are f 2 X and c 2 R such that f .x/ C c ˛ and f .y/ C c < '.y/ for every y 2 X. (Some extra care is needed if x … dom '.) This contradicts the formula (3). The dual statement is proved in the same way. We only use the separation theorem in the locally convex space .X ; w / R. t u Lemma 26. Let X be a normed linear space, ' W X ! R [ fC1g, x 2 dom ', and f 2 X . The following statements are equivalent: (i) '.x/ C ' .f / D f .x/. (ii) f 2 @'.x/. (iii) x 2 @' .f / and .' / .x/ D '.x/. Proof. (i))(ii) For any y 2 X we get '.y/ '.x/ D '.y/ C ' .f / f .x/ '.y/ C f .y/ '.y/ f .x/ D f .y x/ and so f 2 @'.x/. (ii))(i) We have f .y/ '.y/ f .x/ '.x/ for every y 2 X . Thus we obtain ' .f / D sup ff .y/ '.y/I y 2 X g D f .x/ '.x/. (i))(iii) For any g 2 X we get ' .g/ ' .f / D ' .g/ C '.x/ f .x/ g.x/ '.x/ C '.x/ f .x/ D x.g f / and therefore x 2 @' .f /. Moreover, '.x/ .' / .x/ f .x/ ' .f / D '.x/. (iii))(i) From (ii))(i) used on ' we obtain ' .f / C .' / .x/ D f .x/ and so (i) follows. t u From the definition we immediately obtain that if '1 '2 on X, then '1 '2 on X . The next lemma shows that in some circumstances this dual behaviour holds even locally.

Section 1. Convex functions

251

Lemma 27. Let X be a normed linear space, '1 ; '2 W X ! R [ fC1g where '2 is convex, x 2 dom '2 , and f 2 @'2 .x/. If there is r > 0 such that '1 '2 on B.x; r/ and ˚ D inf '2 .y/ f .y/ '2 .x/ f .x/ I y 2 X; ky xk D r > 0; then '1 '2 on B.f; r /. Dually, if '2 is moreover lower semi-continuous and there is r > 0 such that '1 '2 on B.f; r/ and ˚ D inf '2 .g/ g.x/ '2 .f / f .x/ I g 2 X ; kg f k D r > 0; then '1 '2 on B.x; r /. Proof. Fix g 2 B.f; r /. Let ´ 2 X be such that k´ xk > r. Set y D x C r k´´ xxk . By the convexity '2 .y/ f .y/ k´ r xk '2 .´/ f .´/ C 1 k´ r xk '2 .x/ f .x/ . Therefore '2 .´/ g.´/ '2 .x/ g.x/ '2 .´/ f .´/ '2 .x/ f .x/ kg f kk´ xk k´ xk '2 .y/ f .y/ k´ xk 0; '2 .x/ f .x/ r r which implies that sup fg.´/ '2 .´/I ´ 2 Xg D sup fg.´/ '2 .´/I ´ 2 B.x; r/g. Thus '1 .g/ sup fg.´/ '1 .´/I ´ 2 B.x; r/g sup fg.´/ '2 .´/I ´ 2 B.x; r/g D '2 .g/. To prove the dual statement, note that x 2 @'2 .f / by Lemma 26. Thus by the first statement used on '1 and '2 we obtain .'1 / .'2 / on B.x; r /. Hence t u '1 .'1 / .'2 / D '2 on B.x; r / by Fact 25. The following notion turns out to be a dual notion to the pointwise modulus of smoothness. Definition 28. Let X be a normed linear space, ' W X ! R [ fC1g, x 2 dom ', and f 2 @'.x/. We define a modulus of exposedness of ' at x by f by ˚ ' x;f ."/ D inf '.y/ f .y/ C '.´/ f .´/ 2 '.x/ f .x/ I y; ´ 2 X; ky ´k " for " 0. '

'

Note that x;f is a non-decreasing non-negative function, x;f .0/ D 0, and ' ."/ D C1 for " > diam dom '. Further, '.x C h/ '.x/ C f .h/ C x;f .khk/ ' for all h 2 X. This in turn implies that if '.x C h/ < '.x/ C f .h/ C x;f ."/, then khk < ". On the other hand the following fact holds. ' x;f

Fact 29. Let X be a normed linear space, ' W X ! R [ fC1g, x 2 dom ', and let f 2 @'.x/. If there is a non-decreasing function W Œ0; C1/ ! R such that ' '.x C h/ '.x/ C f .h/ C .khk/ for all h 2 X , then x;f ."/ . 2" /. Proof. It suffices to notice that if y; ´ 2 X are such that ky kx yk 2" or kx ´k 2" .

´k ", then either t u

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The next fact justifies the name of the modulus. Fact 30. Let X be a normed linear space, ' W X ! R [ fC1g, x 2 dom ', and f 2 @'.x/. Then .x; '.x// is a strongly exposed point of the set epi ' exposed by ' .f; 1/ 2 X ˚ R if and only if x;f ."/ > 0 for all " > 0. Proof. Let F D .f; 1/ 2 .X ˚ R/ , i.e. F .y; t/ D f .y/ t. As f 2 @'.x/, we have F .y; t / f .y/ '.y/ f .x/ '.x/ D F .x; '.x// for any .y; t/ 2 epi '. ( Suppose that F .xn ; tn / ! F .x; '.x// forxn 2 dom ', tn '.xn /. Then ' x;f .kxn xk/ '.xn / f .xn / '.x/ f .x/ tn f .xn / '.x/ f .x/ D ' F .x; '.x// F .xn ; tn / ! 0. Since x;f is non-decreasing, this implies that xn ! x, which together with F .xn ; tn / ! F .x; '.x// gives .xn ; tn / ! .x; '.x//. ' ) By contrary, let " > 0 be such that x;f ."/ D 0. Then for each n 2 N there are yn ; ´n 2 X satisfying 0 '.yn / f .yn / C '.´n / f .´n / 2 '.x/ f .x/ n1 and kyn ´n k ". It follows that 0 '.yn / f .yn / '.x/ f .x/ n1 and also 0 '.´n / f .´n / '.x/ f .x/ n1 , which can be rewritten as 0 F .x; '.x// F .yn ; '.yn // n1 and 0 F .x; '.x// F .´n ; '.´n // n1 . The strong exposedness now gives yn ! x and ´n ! x, a contradiction. t u Lemma 31. Let X be a normed linear space, ' W X ! R [ fC1g, x 2 dom.' / , and let ' be lower semi-continuous at x. Suppose that there is f 2 @.' / .x/ such .' / that x;f ."/ > 0 for every " > 0. Then x 2 dom ' and .' / .x/ D '.x/. Proof. Assume that the statement does not hold. Let ı 2 R, ı > 0 be such that .' / .x/Cı '.x/. From the lower semi-continuity of ' at x there is " > 0 such that '.y/ > .' / .x/ C 2ı for each y 2 U.x; "/. Without loss of generality we may assume ˚ .' / that kf k" 4ı . Put D min x;f ."/; 4ı and note that > 0. Now for y 2 U.x; "/ we have f .y/C.' / .x/ f .x/C < kf k"C.' / .x/C .' / .x/C 2ı < '.y/, while for y 2 X with ky xk " we have f .y/ C .' / .x/ f .x/ C .' / f .y x/ C .' / .x/ C x;f .ky xk/ .' / .y/ '.y/. Thus f .y/ C c '.y/ for all y 2 X, where c > .' / .x/ f .x/, a contradiction with Fact 25. t u '

For a convex function ' the infimum in the definition of the modulus x;f can be taken only over ky ´k D ": Lemma 32. Let X be a normed linear space, ' W X ! R [ fC1g a convex function, x 2 dom ', and f 2 @'.x/. Then for " 0 we have ˚ ' x;f ."/ D inf '.y/ f .y/ C '.´/ f .´/ 2 '.x/ f .x/ I y; ´ 2 X; ky ´k D " : '.x/ f .x/ for u 2 X and note that is Proof. Set .u/ D '.u/ f .u/ ' convex. Also note that x;f ."/ D inf f .y/ C .´/I y; ´ 2 X; ky ´k "g. Let " > 0 and pick any y; ´ 2 X satisfying ky ´k ". Without loss of generality we may assume that .y/ .´/. Put v D y C " k´´ yyk . Then ky vk D " and .v/ k´ " yk .´/ C 1 k´ " yk .y/ .´/. Thus .y/ C .v/ .y/ C .´/, from which the statement of the lemma follows. t u

Section 1. Convex functions

253 '

'

For formal reasons we extend the moduli x and x;f to the whole R to be even ' ' ' ' functions, i.e. x .t/ D x . t/ for t < 0, x;f ."/ D x;f . "/ for " < 0. Now we come to our main duality tool. Proposition 33. Let X be a normed linear space, ' W X ! R [ fC1g a convex ' ' function, x 2 dom ', and f 2 @'.x/. Then f D .x;f / . If ' is moreover lower ' ' semi-continuous, then x D .f;x / . '

Proof. By Lemma 26, f 2 dom ' and so f is well-defined. For any t 0 we have ˚ ' f .t / D sup ' .f C tg/ C ' .f tg/ 2' .f /I g 2 SX ˚ D sup f .y/ C tg.y/ '.y/ Cf .´/ tg.´/ '.´/ 2' .f /I g 2 SX ; y; ´ 2 X ˚ D sup tky ´k C f .y/ '.y/ C f .´/ '.´/ 2' .f /I y; ´ 2 X ˚ D sup t " Cf .y/ '.y/ Cf .´/ '.´/ 2' .f /I y; ´ 2 X; ky ´k D "; " 0 ˚ D sup t " C sup ff .y/ '.y/ C f .´/ '.´/ 2.f .x/ '.x//I y; ´ 2 X; ky ´k D "gI " 0 ˚ ˚ ' ' ' D sup t " x;f ."/I " 0 D sup t " x;f ."/I " 2 R D .x;f / .t/; where we used Lemma 26 and Lemma 32. For t < 0 the formula follows from the obvious fact that if W R ! R [ fC1g is even, then so is . ' Further, by Lemma 26, x 2 @' .f / and so f;x is well-defined. If ' is moreover lower semi-continuous, then similarly as above and using moreover Fact 25 we obtain ˚ x' .t / D sup .' / .x C th/ C .' / .x th/ 2'.x/I h 2 SX ˚ D sup u.x/ C tu.h/ ' .u/ C v.x/ tv.h/ ' .v/ 2'.x/I h 2 SX ; u; v 2 X ˚ D sup tku vk C u.x/ ' .u/ C v.x/ ' .v/ 2'.x/I u; v 2 X ˚ D sup t " C u.x/ ' .u/ C v.x/ ' .v/ 2'.x/I u; v 2 X ; ku vk D "; " 0 ˚ ˚ ' ' ' D sup t " f;x ."/I " 0 D sup t " f;x ."/I " 2 R D .f;x / .t/: t u Corollary 34. Let X be a normed linear space and let ' W X ! R [ fC1g be lower semi-continuous. Suppose that ' is Fréchet differentiable at f 2 X and put x D D' .f /. Then x 2 dom ', .' / .x/ D '.x/, and f 2 @.' / .x/ \ @'.x/. Proof. Fact 6 implies x 2 X . Denote D .' / and note that D ' by Fact 25. By Lemma 26 we have x 2 dom and f 2 @ .x/, and so x;f is well-defined. Suppose that there is "0 > 0 such that x;f ."0 / D 0. Proposition 33 implies that ' f .t / D f .t/ "0 t for all t > 0, which contradicts Lemma 12. Thus x;f ."/ > 0 for every " > 0, which in turn implies that x 2 dom ' and .' / .x/ D '.x/ by Lemma 31. Since f .h/ C '.x/ D f .h/ C .x/ .x C h/ '.x C h/ for every h 2 X, we get f 2 @'.x/. t u

254

Chapter 5. Smoothness and structure

Definition 35. Let X be a normed linear space, ' W X ! R [ fC1g, x 2 dom ', and ' f 2 @'.x/. We say that x;f is of power type q 2 .1; C1/ if '

lim inf

x;f ."/ "q

"!0C

> 0:

'

Note that since x;f is non-decreasing, it is of power type q if and only if for each ' r > 0 there is K > 0 such that x;f ."/ K"q for all " 2 Œ0; r. The following fact is proved by an elementary analysis. Fact 36. Let p 2 .1; C1/ and '.t/ D where p1 C q1 D 1.

1 p p jtj ,

t 2 R. Then ' .u/ D

1 q q juj ,

u 2 R,

Corollary 37. Let X be a normed linear space, ' W X ! R[fC1g a convex function, ' ' x 2 dom ', and f 2 @'.x/. If f is of power type p, then x;f is of power type q, ' 1 1 where p C q D 1. Similarly, if ' is moreover lower semi-continuous and if x is of ' power type p, then f;x is of power type q. Proof. Combine Proposition 33, Lemma 27, and Fact 36 together with the formula t u .C / .u/ D C . Cu / for any C > 0. Lemma 38. Let X be a normed linear space such that its norm has modulus of convexity of power type q. Set '.x/ D kxk2 . Then for each x 2 ˚X there is a constant ' C > 0 such that for every f 2 @'.x/ we have x;f ."/ C min "q ; "2 for all " 0. Proof. Denote by ı the modulus of convexity of the norm of X and let K > 0 be such that ı."/ K"q for " 2 Œ0; 2. Note that q 2. By Fact 29 it suffices to show that for each x 2 X there is a constant ˚ C > 0 such that for every f 2 @'.x/ we have '.x C h/ '.x/ f .h/ C min khkq ; khk2 . For x D 0 we have @'.0/ D f0g and K the inequality obviously holds with C D 1. Now fix x 2 X, x ¤ 0. Put L D .9kxk/ q 2 ˚1 1 L and C D min 4 ; 2q 1 ; 2q 1 . Suppose that f 2 @'.x/. It is not difficult to check that 1 kf k D 2kxk and f .x/ D 2kxk2 . Put g D 2kxk f . Then g 2 SX and g.x/ D kxk. Hence jg.tx/j D ktxk for all t 2 R. Choose any h 2 X. If khk 8kxk, then kxChk2 kxk2 f .h/ khk kxk

2

1 kxk2 kf kkhk D khk2 4kxkkhk khk2 : 2

So now suppose that khk < 8kxk. Put h1 D g.h/ x and h2 D h h1 . Note that h1 is a kxk multiple of x and h2 2 ker g. We will use the fact that on spanfh1 g the square of the norm is just a quadratic function, and on ker g, which gives a supporting hyperplane x , we can use the modulus of convexity. So, we have to BX at kxk kxChk2 kxk2 f .h/ D kxCh1 Ch2 k2 kxCh1 k2 CkxCh1 k2 kxk2 f .h1 /: (4)

Section 1. Convex functions

255

First we deal with the increment in the direction h1 : g.h/ 2 kx C h1 k2 kxk2 f .h1 / D kxk2 1 C kxk 2

kxk2

g.h/ f .x/ kxk

(5)

2

D g.h1 / D kh1 k : Next, if x C h1 D 0, then from (4), (5), and kh1 k2 C kh2 k2 21 khk2 the required xCh1 xCh inequality immediately follows. Otherwise we put y1 D kxChk and y2 D kxChk . Notice that kx C h1 k D jg.x C h1 /j D jg.x C h/j kx C hk, and so ky1 k D 1 and kh2 k 2 . On the other hand, ky2 k 1. Therefore 1 k y1 Cy y2 k/ D ı kxChk 2 k ı.ky1

ˇ ˇ

y1 C y2 ˇ ˇ

ˇg y1 C y2 ˇ D jg.x C h1 /j D kx C h1 k :

ˇ ˇ 2 2 kx C hk kx C hk Putting the last two estimates together with the fact that ı is of power type q yields kh2 kq kh2 k kx C h1 k kx C h1 k2 K ı 1 1 ; kx C hkq kx C hk kx C hk kx C hk2 which gives Kkh2 kq kx C hkq .9kxk/q

2 2

kx C hk2

kx C h1 k2

kx C h1 C h2 k2

kx C h1 k2 :

If we now get back to (4) and use the last inequality together with (5), we obtain

Lkhkxkk

kx C hk2

2

C kh1 kq minfL; 1g 2q1 1 khkq Lkh2 k2 C kh1 k2 minfL; 1g 21 khk2 kh1 k2 > 14 khk2 2

f .h/ Lkh2 kq C kh1 k2

q

if kh1 k 1, if kh2 k 1, if kh1 k > 1 and kh2 k < 1.

To see the last inequality, note that in this case khk kh1 k C kh2 k < kh1 k C 1 < 2kh1 k. This finishes the proof. t u Theorem 39. Let X be a Banach space which admits a Lipschitz convex function ' W U ! R defined on a non-empty open convex set U X, with the following property: There is a set ˝ residual in U such that for each x 2 ˝ there is f 2 @'.x/ ' for which x;f is of some power type qx . Then X admits an equivalent norm with modulus of convexity of some power type. If q D supx2˝ qx < C1, then X admits an equivalent norm with modulus of convexity of power type q. Proof. Without loss of generality we may assume that ' is 1-Lipschitz on U . For a convex function W X ! R [ fC1g which is 1-Lipschitz on Int dom and for x 2 Int dom we define ˚ ıx ."/ D inf .y/C .´/ 2 .x/Ck2x .y C´/kI y; ´ 2 X; ky ´k " ; " 0: Note that ıx is non-decreasing and for any f 2 @ .x/ we have x;f ıx .

256

Chapter 5. Smoothness and structure

For each n 2 N set ˚ Fn D x 2 U I ıx' ."/ n1 "n for all " 2 Œ0; 1 : Each Fn is closed in U . Indeed, let fxk g Fn , xk ! x 2 U . Then for any " 2 Œ0; 1 and y; ´ 2 X, ky ´k " we have '.y/ C '.´/ 2'.xk / C k2xk .y C ´/k n1 "n and hence also '.y/ C '.´/ S2'.x/ C k2x .y C ´/k n1 "n . S By our assumption ˝ n2 Fn and since U is a Baire space, it follows that n2 Fn is of the second category in U . Therefore there are n 2 N, n 2, x 2 U , and 0 < r < 21 such that B.x; r/ Fn . Define .h/ D 21 '.x C h/ C 12 '.x h/ '.x/ for all h 2 X. Then is an even convex function, 1-Lipschitz on dom , with .0/ D 0 and 0 2 @ .0/. For every u 2 B.0; r/ and " 2 Œ0; 1 we have ˚ ıu ."/ D inf .y/ C .´/ 2 .u/ C k2u .y C ´/kI y; ´ 2 X; ky ´k " ˚ inf 12 '.x C y/ C 12 '.x C ´/ '.x C u/ C 12 k2u .y C ´/kI y; ´ 2 X; ky ´k " ˚ C inf 12 '.x y/ C 12 '.x ´/ '.x u/ C 21 k2u .y C ´/kI y; ´ 2 X; ky ´k " '

D 21 ıxCu ."/ C 21 ıx'

u ."/

n1 "n :

In particular, for any y 2 B.0; 21 / we can use ´ D y in the definition of ı0 to 1 n n obtain 2 .y/ ı0 .2kyk/ n1 2n kykn . Next, we put a D 2n 2 r . Then the set B D fx 2 XI .x/ ag B.0; r/ is an absolutely convex bounded neighbourhood of 0. Hence the Minkowski functional of B is an equivalent norm on X . We claim that the modulus of convexity of is of power type n. Let 0 < C 1 be such that kxk C .x/ for all x 2 X. Fix any " 2 Œ0; 1. Let y; ´ 2 X be such that .y/ D .´/ D 1and .y ´/ D ". Note that ky ´k C " and yC´ 12 . Put 2 yC´ yC´ x D 2 = 2 . Then .x/ D .y/ D .´/ D a. Further, 1 n n nC "

ıx .C "/ D ky C ´k

.y/ C 1

yC´ 2

.´/ 2 .x/ C k2x .y C ´/k D k2x .y C ´/k 2r yC´ 4r 1 : 1 yC´ 1 yC´ 2 2 2

This shows that the modulus of convexity of is of power type n. To see instead of the sets ˚ the last' statement, Fn as above we consider the sets 1 q Fn D x 2 U I ıx ."/ n " for all " 2 Œ0; 1 . t u At the end of this section we look at subdifferentials of weakly sequentially continuous convex functions on spaces which do not contain `1 . Fact 40. Let X be a Banach space which does not contain `1 , U X an open convex set, and let ' W U ! R be a weakly sequentially continuous convex function. Then @'.x/ is compact for each x 2 U .

Section 2. Smooth bumps and structure I

257

Proof. Assume the contrary. Let x 2 U be such that @'.x/ is not compact. Since @'.x/ is closed (even w ) and bounded, there is a bounded sequence ffn g @'.x/ which is not relatively compact. Let ı > 0 be such that B.x; ı/ U . By Theorem 3.56 there are a subsequence ffnk g, a weakly null sequence fhk g B.0; ı/, and " > 0 such that fnk .hk / " for each k 2 N. Then '.x C hk / '.x/ C fnk .hk / '.x/ C " for all k 2 N, which contradicts the fact that '.x C hk / ! '.x/. t u Proposition 41. Let X be a Banach space which does not contain `1 , U X an open convex set, andSlet ' W U ! R be a weakly sequentially Cauchy-continuous convex function. Then x2V @'.x/ is relatively compact for each bounded V U satisfying dist.V; X n U / > 0. S Proof. Assume the contrary. Let V U be bounded such that x2V @'.x/ is not relatively compact and ı D 21 dist.V; X n U / > 0. There are sequences fxn g V , ffn g X such that fn 2 @'.xn / and ffn g is not relatively compact. By Rosenthal’s `1 -theorem we may assume that fxn g is weakly Cauchy. There are a subsequence ffnk g, a weakly null sequence fhk g B.0; ı/, and " > 0 such that fnk .hk / " for each k 2 N (if ffn g is bounded, then we use Theorem 3.56; otherwise it is obvious). Then '.xk C hk / '.xk / C fnk .hk / '.xk / C " for all k 2 N. On the other hand, since the sequence fyk g, where y2k 1 D xk , y2k D xk C hk , is weakly Cauchy, it follows that lim '.xk C hk / D lim '.xk /, which is a contradiction. t u

2. Smooth bumps and structure I In this section we give an hors d’oeuvre of the results that show how the existence of bump functions (or norms) with certain kind of smoothness impacts the structure of the space. First we show a few results related to separating polynomials. Then we show how local uniform smoothness together with convexity combine to produce a global uniform smoothness. Next we introduce a lemma by Marián Fabian and Robert Deville that constructs a “strongly separating” function from a single bump on spaces that k;˝ do not contain c0 . This will be used to prove that on such spaces Cloc -smoothness k;˝ -smoothness. Further, in case of the first order smoothness leads to a global C we construct a C 1;˝ -smooth norm from a C 1;˝ -smooth bump. Combining these two 1;C results we obtain that if X admits a Cloc -smooth bump, then it is either super-reflexive, or it contains c0 . We start with rather simple observation about the smoothness of (powers of) norms at the origin. Note that for every normed linear space and every 1 q < p < C1 the function x 7! kxkp is T q -smooth at 0 with the polynomial P from the definition of T q -smoothness being zero. Proposition 42. Let X be a normed linear space such that the function x 7! kxkp is weakly T p -smooth at 0. Then p is an even integer and the norm of X is p-polynomial.

258

Chapter 5. Smoothness and structure

In particular, if the square of the norm of X is G 2 -smooth at 0, then X is a Hilbert space. Proof. The weak T p -smoothness implies that there is a polynomial Q 2 P Œp .X/ such that kt hkp Q.th/ D o.jtjp /; t ! 0 for each h 2 X. However, since kt hkp D o.jtjq /; t ! 0 for q < p, by the uniqueness of the polynomial expansion each k-homogeneous summand of Q is zero for k < p. It follows that p 2 N and p Q 2 P . pX /. So for a given h 2 X we have khkp Q.h/ D o.tt p / ; t ! 0C, whence khkp D Q.h/. From this we also get Q. h/ D Q.h/, thus p is even. t u Corollary 43. Let X be a normed linear space such that the norm of X ˚p R is weakly T p -smooth at Œ0; w for some w 2 R. Then p is an even integer and the norm of X is p-polynomial. Proof. By the assumption the function .x; y/ D kxkp C jyjp is weakly T p -smooth at Œ0; w (remark after Proposition 1.115). Thus also the restriction of to X C Œ0; w is weakly T p -smooth at Œ0; w, i.e. x 7! x p C jwjp is weakly T p -smooth at 0 and we may use Proposition 42. t u The following theorem shows that the existence of a separating polynomial (of a given degree) is separably determined. This will later allow us to use certain smoothing techniques that are inherently separable in a general setting. Theorem 44. Let X be a normed linear space and n 2 N. If every separable closed subspace of X admits a separating polynomial of degree at most n (resp. n-homogeneous), then X admits a separating polynomial of degree at most n (resp. n-homogeneous). ˚ Proof. Let us define cY D sup infSY P I P 2 P n .Y /; kP k 1; P .0/ D 0 for a subspace Y X. Notice that by the assumption cY > 0 for every separable closed subspace Y of X . Then c D inf fcY I Y is a separable closed subspace of Xg > 0: Indeed, otherwise there wouldS be a sequence of separable closed subspaces fYn g of X with cYn < n1 . Set Y D span 1 nD1 Yn . Then Y is a separable closed subspace of X and 0 < cY cYn < n1 for every n 2 N, a contradiction. Now for every separable closed subspace Y of X we find a polynomial P Y 2 P n .Y / such that kP Y k 1, P .0/ D 0, and infSY P Y 2c . Denote by PjY , j D 1; : : : ; n, the j -homogeneous terms of P Y and note that by Fact 1.42 there is K > 0 such that kPjY k K for every separable closed subspace Y of X and every j 2 f1; : : : ; ng. Define QjY W X ! R by QjY .x/ D PjY .x/ for x 2 Y , QjY .x/ D 0 otherwise. Let be a family of all separable closed subspaces of X ordered by inclusion. Then fQjY BX gY 2 , j D 1; : : : ; n, are nets in the compact space Œ K; KBX . Thus Y we can extract convergent subnets fQj BX g 2 , j D 1; : : : ; n. Since each of the Y functions QjY is j -homogeneous, it follows that the nets fQj g 2 converge pointwise Y on the whole of X. Define Qj W X ! R by Qj .x/ D lim Qj .x/. It is not difficult to check using the Polarisation formula that Qj 2 P . jX/.

Section 2. Smooth bumps and structure I

259

P Put Q D jnD1 Qj . Then Q 2 P n .X/ and Q.0/ D 0. It is easy to see that kQk 1 and since for every x 2 SX there is ˇ 2 such that x 2 Y for ˇ, we t u have infSX Q 2c . The homogeneous case is clear from the above. Corollary 45 ([Li2]). Assume that every separable subspace of a Banach space X is isomorphic to a Hilbert space. Then X is isomorphic to a Hilbert space. The next theorem shows that local uniform smoothness of the norm turns to global uniform smoothness under additional convexity assumptions. Theorem 46 ([FWZ]). Let ˝ Ms be a convex cone and let X be a Banach space k;˝ -smooth norm such that its unit ball has at least one strongly exposed with a Cloc point. Then X admits an equivalent C k;˝ -smooth norm. Proof. Let w 2 SX be a strongly exposed point of BX exposed by f 2 SX and let ı > 0 be such that kk is C k;˝ -smooth on U.w; 2ı/. Denote H D ker f . By the strong exposedness there is > 0 such that kw C hk 1 C whenever h 2 H , khk ı. Indeed, otherwise there would be a sequence fhn g H , khn k ı, such that kwChn k 1C n1 . This means that 1 D f .w/ D f .wChn / kwChn k 1C n1 ! 1 and hence hn ! 0, a contradiction. Thus h 7! kw C hk is separating on H \ U.0; 2ı/ k;˝ -smooth equivalent norm on H . Now it suffices to and by Lemma 23 there p is a C take the norm x 7! .x f .x/w/2 C f .x/2 . t u The following lemma is an important tool for constructing a “strongly separating” function from a single bump without any additional structure. Lemma 47. Let X be a Banach space which does not contain c0 . Let U be a bounded symmetric (i.e. U D U ) open neighbourhood of the origin and let f W U ! R be an even lower semi-continuous function such that f .0/ 0 and inf f .@U / > 0. Then there exist a symmetric compact set K U and a neighbourhood V of 0 such that K C V U and for each ı > 0 there are " > 0 and a finite subset F K such that for each x 2 V , kxk ı there is ´ 2 F satisfying f .´ C x/ > f .´/ C ". Proof. Let m D inf f .@U /. We will inductively construct a sequence fxj gj1D0 X such that for each n 2 N0 it satisfies ˚Pn (i) Kn U , where Kn D "j xj I ."j / 2 f 1; 1gnC1 , j D0 1 (ii) f .´/ m 2 1 2n for all ´ 2 Kn . Set x0 D 0. If x0 ; x1 ; : : : ; xn 1 have already been defined, we put m 1 En D x 2 X I ´ C x 2 U; f .´ C x/ 1 for all ´ 2 Kn 1 : 2 2n From the inductive hypothesis it follows that at least 0 2 En . We put ˛n D supx2En kxk and choose xn 2 En such that kxn k ˛n =2. Using the symmetry of Kn 1 and U , the fact that f is even, and the inductive hypothesis we check that the properties (i) and (ii) are satisfied. S Let K D 1 nD0 Kn . Obviously K is symmetric. Further, K U , hence K is bounded. From the Bessaga-Pełczy´nski theorem (Theorem 3.46) it follows that K

260

Chapter 5. Smoothness and structure

is compact and xn ! 0, hence also ˛n ! 0. Since f is lower semi-continuous, f .x/ m 2 for each x 2 K and thus K \ @U D ;. Hence there is a neighbourhood V of 0 satisfying K C V U . m Pick any ı > 0. Let n 2 N be such that ˛n < ı and put " D 2nC1 and F D Kn 1 . If x 2 V and kxk ı, then kxk > ˛ and hence x … E . Thus there is ´ 2 F satisfying n n m 1 1 f .´Cx/ > 2 1 2n . On the other hand from (ii) it follows that f .´/ m 2 1 2n 1 t u and so f .´ C x/ > f .´/ C ". Theorem 48. Let X be a Banach space which does not contain c0 . Let ˝ M be a k;˝ convex cone. If X admits a Cloc -smooth bump, then X admits C k;˝ -smooth bump. k;˝ .X/ satisfying f .0/ D 0 and Proof. By the assumption there is a function f 2 Cloc f .x/ D c > 0 for x outside some ball. Replacing f with x 7! f .x/ C f . x/ we may additionally assume that f is even. Let K X be the compact set and V the neighbourhood of 0 from Lemma 47 used on f . Let r > 0 be such that U.0; r/ V . There is a covering of K by open balls U.xj ; ıj /, j D 1; : : : ; n, such that f is C k;˝ -smooth on each U.xj ; 2ıj /. Put ı D 21 minfr; ı1 ; : : : ; ın g. For this ı we find " > 0 and F K finite from Lemma 47 such that for each x 2 X, ı kxk < r there is ´ 2 F satisfying f .´ C x/ > f .´/ C ". Define g W U.0; 2ı/ ! R by X 2 g.x/ D f .´ C x/ f .´/ : ´2F

Since for each ´ 2 F there is j 2 f1; : : : ; ng such that U.´; 2ı/ U.xj ; 2ıj /, it follows from Proposition 1.128 that g 2 C k;˝ U.0; 2ı/ . Further, g.0/ D 0 and g.x/ > "2 for each x 2 X satisfying ı kxk < 2ı. Let 2 C 1 .RI Œ0; 1/ be a function with all derivatives bounded and such that .0/ D 1 2 and .t / D 0 for t " . Put h.x/ D g.x/ for x 2 U.0; 2ı/ and h.x/ D 0 elsewhere. Then it is easy to check again using Proposition 1.128 that h 2 C k;˝ .X/. Finally, observe that h.0/ D 1 and h.x/ D 0 for x 2 X , kxk ı. t u As the next proposition shows, the assumption that X does not contain c0 is necessary 1;1 -smooth bump on c0 , see Theorem 94). in the previous theorem (note that there is a Cloc Proposition 49 ([We1]). There is no C 1;C -smooth bump on c0 . Proof. Assume such a bump f exists. Without loss of generality we may assume that f .0/ D 1 and f .x/ D 0 whenever kxk D 1. Let n 2 N be such that 1 1 for any x; y 2 c0 , kx yk . 2 n Let A be the set of all edges of a cube in the first 2n coordinates of c0 : kDf .x/

Df .y/k

(6)

n

AD

2 n [ j0 D1

x 2 c0 I jxj j D

1 n

o if j 2 f1; : : : ; 2n g n fj0 g, jxj0 j n1 , xj D 0 if j > 2n :

Section 2. Smooth bumps and structure I

261

Since A is symmetric and connected, and Df .0/ is even and continuous, there is h1 2 A with Df .0/Œh1 D 0. Moreover by changing the sign of h1 if necessary we may assume that at least 2n 1 coordinates of h1 is equal to n1 . Similarly there is h2 2 A with Df .h1 /Œh2 D 0 and such that at least 2n 2 coordinates of h1 C h2 equals n2 . By induction we construct hk 2 A for k D 3; : : : ; n such that Df .h1 C C hk 1 /Œhk D 0 and h1 C C hk has at least 2n k coordinates equal to kn . Then kh1 C C hn k D 1 and so ˇ 1 D ˇf .h1 C C hn /

n ˇ X ˇ ˇf .h1 C C hk / f .0/ˇ

f .h1 C C hk

ˇ ˇ

1/

kD1 n X ˇ ˇDf .h1 C C hk D

D

kD1 n X

ˇ ˇDf .h1 C C hk

1

ˇ C tk hk /Œhk ˇ

1

C tk hk /Œhk

for some 0 < tk < 1 Df .h1 C C hk

ˇ ˇ

1 /Œhk

kD1 n X 11 1 1 khk k n D ; 2 2n 2 kD1

which is a contradiction. (The second equality follows from the Mean value theorem t u and the second inequality is assured by (6), since ktk hk k n1 .) Theorem 50 ([FWZ]). Let ˝ Ms be a convex cone. Let X be a Banach space that admits a C 1;˝ -smooth bump. Then X admits an equivalent C 1;˝ -smooth norm. Proof. By shifting and composing with a suitable smooth function on R we construct from a C 1;˝ -smooth bump a C 1;˝ -smooth function ' W X ! Œ0; 6 such that '.0/ D 0 and '.x/ D 6 for kxk 1 (Proposition 1.128). Define a function W B.0; 6/ ! R by ( n X ˛j '.xj /I .x/ D inf j D1

xD

n X j D1

˛j xj ;

n X

) ˛j D 1; ˛j 0; xj 2 B.0; 6/; j D 1; : : : ; n; n 2 N :

j D1

It is easy to check that is convex. Clearly, .x/ '.x/ for any x 2 B.0; 6/. Hence is convex and bounded above, so it is continuous on U.0; 6/. Note also that kxk 6 .x/ kxk 6 ' kxk x C 1 6 '.0/ D kxk for all x 2 B.0; 6/. By Lemma 19(i) there is ! 2 ˝ such that '.xCh/C'.x h/ 2'.x/ !.khk/khk for x; h 2 X . We show that this implies .x C h/ C .x h/ 2 .x/ 2!.khk/khk for x; h 2 X, x ˙ h 2 U.0; 2/. So fix x; h 2 X, x ˙ h 2 U.0; 2/ and notice that khk < 2. Choose any " > 0 satisfying kxkPC " < 2. Let n 2 N, P xj 2 B.0; 6/, and ˛j 0, j 2 f1; : : : ; ng, be such that jnD1 ˛j D 1, x D jnD1 ˛j xj , and

262

Chapter 5. Smoothness and structure

Pn

.x/ C ". To estimate the values of .x ˙ h/ we need to disj D1 ˛j '.xj / < tribute the h only to those xj s that lie in B.0; 2/, otherwise xj C h may not lie in B.0; 6/. Without loss of generality we may assume that there is k 2 N0 such that x1 ; : : : ; xk 2 B.0; 2/, while xkC1 ; : : : ; xn … B.0; 2/. We have ! n n k X X X 2 > kxk C " .x/ C " > ˛j '.xj / 6 ˛j D 6 1 ˛j : j D1

P Put ˛ D jkD1 ˛j . The above inequality gives j 2 f1; : : : ; kg. So we can estimate .x C h/ C

0 there is x0 2 P such that '.x/ '.x0 / ".x; x0 / for all x 2 P . Moreover, the function x 7! '.x/ C 2".x; x0 / attains a strong minimum at x0 . Proof. Choose any y1 2 dom ' and define inductively ynC1 2 P so that '.ynC1 / C ".ynC1 ; yn / 1 min inf f'.x/ C ".x; yn /I x 2 P g C n ; '.yn / : 2

(7)

264

Chapter 5. Smoothness and structure

This is always possible as we may take ynC1 D yn if the infimum above is attained in yn , otherwise the infimum is strictly smaller than '.yn / and we can choose a suitable ynC1 by the definition of infimum. The sequence f'.yn /g is non-increasing and bounded below, thus it is converP 1 gent in R. Moreover, fyn g is Cauchy. Indeed, .ym ; yn / m kDn .ykC1 ; yk / Pm 1 '.ykC1 / =" D '.yn / '.ym / =" for m > n, and since f'.yn /g is kDn '.yk / Cauchy, so is fyn g. As P is complete, yn ! x0 2 P . Finally, (7) implies '.ynC1 / '.x/ C ".x; yn / C 21n ".ynC1 ; yn / for every x 2 P and n 2 N. From the lower semi-continuity of ' it follows that 1 '.x0 / lim '.yn / lim '.x/ C ".x; yn / C n ".ynC1 ; yn / n!1 n!1 2 D '.x/ C ".x; x0 /: To show the statement about the strong minimum, let fxn gn2N P be such that '.xn / C 2".xn ; x0 / ! '.x0 /. Then ".xn ; x0 / D ".xn ; x0 / C '.x0 / '.x0 / '.xn / C 2".xn ; x0 / '.x0 / ! 0. t u The following variational principle deals with the first order smoothness. See [DGZ, Theorem I.2.3]. Theorem 54. Let X be a Banach space that admits a Fréchet (resp. Gâteaux) differentiable Lipschitz bump. Let ' W X ! R [ fC1g be a bounded below proper lower semi-continuous function. Then for every " > 0 there exists a Lipschitz function g which is Fréchet (resp. Gâteaux) differentiable on X such that supx2X jg.x/j ", supx2X kıg.x/k ", and ' g attains a strong minimum on X . Theorem 55 (Stegall’s variational principle, [Steg2], [Steg3]). Let X be a Banach space with the RNP and let A be a non-empty closed bounded subset of X . Let ' W A ! R [ fC1g be a bounded below proper lower semi-continuous function. Then the set of f 2 X such that the function ' f attains a strong minimum on A is a dense Gı subset of X . Note that the fact that ' f attains a strong minimum at x0 can be reformulated as ' f 2 @'.x0 / and x0 ;f ."/ > 0 for all " > 0 (Fact 29). For the proof of the theorem see [BorVa, Corollary 6.6.17]. Let X be a normed linear space. We say that a function f W X ! R [ fC1g is > 0. Similarly we say that f is quadratically linearly coercive if lim infkxk!C1 '.x/ kxk coercive if lim infkxk!C1

'.x/ kxk2

> 0.

Corollary 56 ([Fab1]). Let X be a Banach space with the RNP and assume that ' W X ! R [ fC1g is a bounded below proper lower semi-continuous linearly coercive function. Put r D lim infkxk!C1 '.x/ . Then the set of f 2 X such that the kxk function ' f attains a strong minimum is residual in UX .0; r/.

Section 3. Smooth variational principles

265

Proof. Denote by M the set of f 2 X such that the function ' f attains a strong minimum. It suffices to show that M is residual in UX .0; r ı/ for every > r 2ı for kxk R1 . Put 0 < ı < r. So fix 0 < ı < r. Find R1 > 0 such that '.x/ kxk ˚ R D max R1 ; 2.'.0/C1/ . Denote by Mı the set of f 2 X such that .' f /B.0;R/ ı attains a strong minimum. By Theorem 55 the set Mı is residual in X . Fix f 2 Mı \ U.0; r ı/. Let x0 2 B.0; R/ be such that .' f /B.0;R/ attains a strong minimum at x0 . For every x 2 X n B.0; R/ we have '.x/

f .x/ '.x/

kf kkxk > r

ı kxk 2

.r

ı/kxk

(8) ı ı D kxk > R '.0/ C 1 D '.0/ f .0/ C 1: 2 2 It follows that the function ' f attains its minimum only at the point x0 . Further, if '.xn / f .xn / ! '.x0 / f .x0 /, then eventually '.xn / f .xn / < '.0/ C 1, and hence by (8) eventually xn 2 B.0; R/. Therefore ' f attains a strong minimum at x0 , which implies that f 2 M . Finally, since M \ U.0; r ı/ Mı \ U.0; r ı/, the set M is residual in U.0; r ı/. t u The following observation shows that the perturbed function can be chosen so that its strong minimum is arbitrarily close to the infimum of the original function. Corollary 57. Let X be a Banach space with the RNP and let ' W X ! R [ fC1g be a bounded below proper lower semi-continuous linearly coercive function. Then for every " > 0 there is f 2 X such that kf k < ", the function ' f attains a strong minimum at x0 2 X, and '.x0 / < infX ' C ". Proof. Without loss of generality we may assume that infX ' D 0. Let ı > 0 and R > 0 be such that '.x/ ıkxk whenever kxk > R. By Corollary 56 there is f 2 X " such that kf k < minf"; 2ı ; 4R g and ' f attains a strong minimum at x0 2 X . We need to show that '.x0 / < ". First observe that kx0 k R. Indeed, otherwise for every x 2 B.0; R/ we can estimate '.x/ D '.x/

f .x/ C f .x/ '.x0 /

f .x0 / C f .x/ ı ı ı kx0 xk kx0 k kxk kx0 k R > 0; > ıkx0 k 2 2 2 which together with the fact that '.x/ > ıR for kxk > R contradicts the fact that infX ' D 0. Suppose that '.x0 / ". Then for every x 2 B.0; R/ we can estimate " " " kx0 xk " 2R D ; '.x/ '.x0 / f .x0 / C f .x/ > " 4R 4R 2 which again contradicts the fact that infX ' D 0. t u For super-reflexive spaces we have the following quantitative version.

266

Chapter 5. Smoothness and structure

Theorem 58. Let X be a Banach space such that its norm has modulus of convexity of power type q. Let ' W X ! R [ fC1g be a bounded below proper lower semicontinuous quadratically coercive function. Then there are f 2 X , x0 2 X , and C > 0 such that for every x 2 X ˚ '.x/ f .x/ '.x0 / f .x0 / C C min kx x0 kq ; kx x0 k2 : '.x/ ˛kxk2 Proof. Let us put ˛ D 21 lim infkxk!C1 kxk 2 and notice that x 7! '.x/ is a bounded below proper lower semi-continuous quadratically coercive function. Therefore Corollary 56 implies that there exist f1 2 X and x0 2 X such that '.x0 / ˛kx0 k2 f1 .x0 / D infx2X '.x/ ˛kxk2 f1 .x/ . By Lemma ˚ 38 qthere are 2 2 f2 2 X and C > 0 such that ˛kx0 C hk ˛kx0 k f2 .h/ C min khk ; khk2 for all h 2 X. Put f D f1 C f2 . Then for every x 2 X '.x/ f .x/

D '.x/ ˛kxk2 f1 .x/ C ˛kxk2 ˛kx0 k2 f2 .x x0 / C ˛kx0 k2 f2 .x0 / ˚ '.x0 / ˛kx0 k2 f1 .x0 / C C min kx x0 kq ; kx x0 k2 C ˛kx0 k2 f2 .x0 / ˚ D '.x0 / f .x0 / C C min kx x0 kq ; kx x0 k2 : t u Let be a class of functions. We say that a Banach space X admits an -variational principle provided that for every bounded below proper lower semi-continuous function ' W X ! R [ fC1g there is f 2 defined on X such that the function ' f attains its minimum. We say that a Banach space X admits a strong -variational principle if for every bounded below proper lower semi-continuous function ' W X ! R [ fC1g and for every " > 0 there is f 2 defined on X such that the function ' f attains a strong minimum at x0 2 X and '.x0 / < infX ' C ". Corollary 59. Let X be a Banach space with the RNP. Let S C.X/ be such that g C f 2 S , g 2 S , and x 7! g.x C a/ 2 S for every g 2 S , f 2 X , and a 2 X. If S contains a linearly coercive bounded below function, then X admits a strong S-variational principle. Proof. Let g 2 S be linearly coercive and bounded below. Let ' W X ! R [ fC1g be a bounded below proper lower semi-continuous function and " > 0. Find a; b 2 X such that '.a/ < infX ' C 4" and g.b/ < infX g C 4" . Put h.x/ D g.x a C b/. The function ' C h is a bounded below proper lower semi-continuous function which is linearly coercive. By Corollary 57 there is f 2 X such that the function ' C h f attains a strong minimum at x0 2 X and '.x0 / C h.x0 / < infX .' C h/ C 2" . Then hCf 2 S and ' . hCf / attains a strong minimum at x0 . Finally, '.x0 /Ch.x0 / < infX .' C h/ C 2" '.a/ C h.a/ C 2" D '.a/ C g.b/ C 2" < infX ' C infX g C ", and so '.x0 / < infX ' C infX g C " h.x0 / D infX ' C infX h h.x0 / C " infX ' C ". u t Lemma 60. Let X be a normed linear space and let S C.X/ be a set of functions that is closed under shifting and scaling of the domain and under compositions with functions from C 1 .R/ with all derivatives bounded, and closed under sums that are finite on bounded sets. If X admits a bump function from S , then X admits a linearly coercive function from S.

Section 3. Smooth variational principles

267

Proof. By the assumptions there is f1 2 S such that supp f1 2 BX and f1 .0/ ¤ 0. By composing with a suitable C 1 -smooth function with all derivatives bounded we produce a non-negative function f 2 S for which f .x/ P D 1 for kxk 1 and there 1 is r > 0 such that f .x/ D 0 for x 2 B.0; r/. Put g.x/ D 1 nD1 f . n x/. On bounded sets this is a finite sum and hence g 2 S. Since g.x/ kxk > kxk 1, it follows that lim infkxk!C1 g.x/ 1. t u kxk Notice that if X admits a (strong) -variational principle and is closed under taking restrictions to subspaces of X, then also every closed subspace of X admits a (strong) -variational principle (for Y X it suffices to extend the function ' from Y by putting '.x/ D C1 for x 2 X n Y .) As the next theorem shows, in case of the higher order smoothness the assumption of the Radon-Nikodým property does not lead to a loss of generality. Note that it also covers the C k -smooth case (when ˛ D 0) and the C 1 -smooth case (just consider the class C 1 in all the statements). Theorem 61. Let X be a Banach space, k 2 N [ f1g, and ˛ 2 Œ0; 1 such that k C ˛ > 1. The following statements are equivalent: (i) X admits a strong C k;˛ -variational principle. k;˛ (ii) X admits a Cloc -variational principle. (iii) X is super-reflexive, admits an equivalent norm with modulus of smoothness of power type minfk C ˛; 2g, and admits a C k;˛ bump. (iv) X has the RNP and admits a C k;˛ bump. k;˛ (v) X does not contain c0 and admits a Cloc bump. Proof. (i))(ii) and (iii))(iv))(v) are clear. k;˛ (ii))(v) First we show that X admits a Cloc bump. Let ' W X ! R [ fC1g be k;˛ 1 defined as '.x/ D kxk for x ¤ 0 and '.0/ D C1. As X admits a Cloc -variational k;˛ principle, there is f 2 Cloc .X/ such that f .x0 / D '.x0 / and f .x/ '.x/ for every 1 x 2 X. Thus x0 ¤ 0 and f .x/ kxk < 2kx10 k D 21 f .x0 / whenever kxk > 2kx0 k. Since f .x0 / > 0, by composing f with a suitable smooth function we obtain a k;˛ Cloc -smooth bump (see the end of the proof of Theorem 48). 1;C Next, we show by contradiction that the space c0 does not admit a Cloc -variational principle. Let kk be a LUR norm on c0 and let ' W c0 ! R [ fC1g be defined as 1;C 1 '.x/ D kxk for x ¤ 0 and '.0/ D C1. Suppose that c0 admits a Cloc -variational 1;C principle. Then there are g 2 Cloc .c0 / and x0 2 c0 such that g.x0 / D '.x0 / and g.x/ '.x/ for every x 2 c0 . It follows that x0 ¤ 0 and there is f 2 c0 , kf k D 1, such that f .x0 / D kx0 k. Put H D ker f . First we show that kk attains a strong minimum at x0 on the hyperplane x0 C H . For any h 2 H we have kx0 k D f .x0 / D f .x0 C h/ kx0 C hk. Suppose that vectors hn 2 H are such that kx0 C hn k ! kx0 k. Then we have 2kx0 k D f .2x0 C hn / k2x0 C hn k kx0 k C kx0 C hn k ! 2kx0 k and so

268

Chapter 5. Smoothness and structure

2kx0 k2 C 2kx0 C hn k2 kx0 C x0 C hn k2 ! 0. Since the norm kk is LUR, it follows that hn ! 0. Now let ı > 0 be such that Dg is uniformly continuous on U.x0 ; 6ı/. From the above it follows that there is " > 0 such that kx0 C hk kx0 k C " for h 2 H , khk ı. Hence 1 1 g.x0 C h/ for all h 2 H , khk ı. (9) kx0 C hk kx0 k C " Let P 2 L.c0 I H / be the projection of c0 onto H defined by P .x/ D x Notice that kP k 2. Define a function G W c0 ! R by G.x/ D g x0 C P .x/

1 kx0 k

f .x/ kxx00 k .

f .x/2 :

It is easy to check that DG is uniformly continuous on U.0; 3ı/. We show that the function G is separating on U.0; 3ı/. Let x 2 c0 , kxk 2ı. If kP .x/k ı, then G.x/ kx01kC" kx10 k by (9). Otherwise ı > kP .x/k kxk jf .x/j 2ı jf .x/j

1 1 and hence G.x/ kx0 CP f .x/2 f .x/2 < ı 2 . Since G.0/ D 0, kx0 k .x/k by composing G with a suitable smooth function on R we obtain a C 1;C -smooth bump on c0 (see the end of the proof of Theorem 48). This however contradicts Proposition 49. (iv))(i) We check that the construction works also for P C k;˛ bumps. P1 1 in1 Lemma 60 1 1 k k ˛ ˛ Indeed, kd g.x/ d g.y/k nD1 nk C n˛ kx yk D C kx yk nD1 nkC˛ . Thus we can use Corollary 59. (v))(iii) By Theorem 48 the space X admits a C k;˛ bump (note that it works also in the case ˛ D 0 by considering the C k 1;1 -smoothness). The rest follows from Corollary 51. t u

We note that the same proof gives the statement of the previous theorem with C k;˛ replaced by C k;˝ with a general convex cone ˝ M, see also the formulation of Theorem 48. We only need to take extra care when k D 1 – when checking that the construction in P Lemma 60 works we need to employ some careful estimates of the 1 partial sums of n. The next theorem is a further example of an interplay between the geometry and smoothness of the space. Theorem 62. Let X be a Banach space such that its norm has modulus of convexity of power type q. Then the following statements are equivalent: (i) X admits a T q -smooth bump. (ii) X admits a separating polynomial. (iii) X admits a separating polynomial of degree at most q. Proof. (iii))(ii) is obvious. (ii))(i) follows from the fact that a polynomial is clearly T p -smooth for every p 2 Œ0; C1/, so a composition with a suitable smooth real function gives the desired bump (Proposition 1.115).

Section 3. Smooth variational principles

269

(i))(iii) Let g be a T q -smooth bump on X. Define a function ' W X ! R [ fC1g by '.x/ D g.x/ 2 for x 2 suppo g and '.x/ D C1 otherwise. By Proposition 1.115 the function ' is T q -smooth on dom '. Obviously ' is a bounded below proper lower semi-continuous quadratically coercive function. Thus by Theorem 58 there are f 2 X , x 2 dom ', and C > 0 such that '.xCh/ f .xCh/ '.x/ f .x/CC khkq for every h 2 BX . Since ' is T q -smooth at x, there is P 2 P Œq .X/ satisfying P .0/ D '.x/ and '.x C h/ P .h/ D o.khkq /; h ! 0. Now put Q D P f '.x/. Notice that Q 2 P Œq .X/ because q 2. Then Q.0/ D 0 and Q.h/ D P .h/

f .h/ '.x/ C o.khkq / f .x/ C o.khkq /

'.x/ D '.x C h/

f .h/

D '.x C h/

f .x C h/ '.x/ C C khkq C o.khkq / khkq 2 for all h from a suitably small neighbourhood of 0. Therefore Q is a separating polynomial. t u Theorem 63. Let X be a Banach space such that its norm has modulus of convexity of power type 2. If X admits a weakly T 2 -smooth bump, then X is isomorphic to a Hilbert space. Proof. Similarly as in the proof of Theorem 62 we obtain a function ', x 2 X , a functional f 2 X , and a constant C > 0 such that ' is weakly T 2 -smooth at x and '.x C h/ f .x C h/ '.x/ f .x/ C C khk2 for every h 2 BX . Let P 2 P 2 .X/ satisfy P .0/ D '.x/ and '.x C th/ P .th/ D o.t 2 /; t ! 0 for each h 2 X . Now put Q.y/ D P .y/ C P . y/ 2'.x/. Notice that Q 2 P . 2X/. Fix h 2 SX . Then t 2 Q.h/ D Q.th/ D P .th/ C P . th/ D '.x C th/

f .x C th/

C '.x

th/

f .x

2'.x/ '.x/ th/

f .x/

'.x/

f .x/ C o.t 2 /

2C t 2 C o.t 2 / for all suitably small t. It follows that Q.h/ 2C and therefore Q is a 2-homogeneous separating polynomial. t u Theorem 64. Let X be a Banach space. The following statements are equivalent: (i) X admits a strong polynomial variational principle. (ii) X admits a strong C 1 -variational principle. (iii) X admits C k -variational principle for every k 2 N. (iv) X admits a separating polynomial. (v) X is super-reflexive, admits an equivalent norm with modulus of smoothness of power type 2, and admits a C 1 bump. (vi) X does not contain c0 and admits a C k bump for every k 2 N. Proof. (i))(ii))(iii) and (v))(vi) are obvious. (ii),(v) and (iii),(vi) follow from Theorem 61.

270

Chapter 5. Smoothness and structure

(iv))(i) follows from Corollary 59 and Fact 4.45. (vi))(iv) By Corollary 51 the space X is super-reflexive. Thus X has an equivalent norm with modulus of convexity of some power type, see [DGZ, Theorem IV.4.8]. Then X admits a separating polynomial by Theorem 62. t u

4. Smooth bumps and structure II We continue with our quest of investigating how the existence of smooth bumps (or norms) affects the structure of the underlying space. We show that we can replace the modulus of convexity in Theorem 62 by cotype q, but we need to assume a q U Tc -smooth bump. Then we deal with the structure of spaces that admit C 1 -smooth bumps. Next, we prove some basic arithmetic rules for the pointwise modulus of smoothness and we show that X is an Asplund space provided that it admits a bump with some rather weak differentiability properties. The main result here is Theorem 73, which says that under very weak convexity and differentiability assumptions we can already construct a norm with modulus of smoothness of power type (and hence the space is super-reflexive). We present some corollaries stating that X is isomorphic to a Hilbert space provided that both X and X admit a bump with certain smoothness properties. These vastly generalise for example the classical result of Victor Zakharovich Meshkov, which requires C 2 -smoothness in both X and X . We finish with a theorem stating that in some cases a weakly T p -smooth bump suffices to give a polynomial with a certain kind of separation property. This will be used later in Section 7 when considering the smoothness of Lp spaces. We start with an auxiliary observation. Fact 65. Let X be a normed linear space that does not admit a separating family of polynomials of degree at most n 2 N. Let F P n .X/ be a finite family satisfying P .0/ D 0 for each P 2 F . Then for every r > 0 and " > 0 there is x 2 SX such that supjjr jP .x/j < " for each P 2 F . Proof. Assume moreover that r 1. For a polynomial P 2 P .X/ denote by Pj ˚ its j -homogeneous summand. Since the family Pj I P 2 F ; j 2 f1; : : : ; ng is not separating, there is x 2 SX such that jPj .x/j < nr1n " for each P 2 F , j 2 f1; : : : ; ng. P P Fix P 2 F . Then jP .x/j jnD1 jjj jPj .x/j < jnD1 r j nr1n " " for each 2 Œ r; r. t u Theorem 62 has a counterpart in which we weaken the convexity assumption, but strengthen the smoothness assumption, and also the conclusion is slightly weaker. Theorem 66. Let X be a Banach space saturated with spaces of cotype q. If X admits q a U Tc -smooth bump, then X admits a separating family of polynomials of degree at most q.

Section 4. Smooth bumps and structure II

271

Proof. By contradiction, suppose there is no separating family of polynomials of degree at most q on X . By Fact 4.48 no finite-codimensional subspace of X admits a separating family of polynomials of degree at most q. By the assumption there is q a U Tc -smooth function f W X ! R satisfying f .0/ D 0 and f .x/ 2q C 1 for x 2 X, kxk 1. For each x 2 X let us denote by Px 2 P Œq .X/ the polynomial satisfying Px .0/ D 0 and f .x C h/ f .x/ Px .h/ D o.khkq /; h ! 0. We construct a normalised basic sequence fxn g1 nD1 in X such that jPx .h/j ın (10) sup x2spanfx1 ;:::;xn g h2spanfxj I j >ng x;h2BX

P1 C for each n 2 N, where fın g1 nD1 ın 1. nD1 R satisfies Q To this end put ın D 21n and let fn g RC be such that 1 nD1 .1 C n / 2. Put Q1 n D j Dn .1 C j /. We construct the sequence fxn g by induction so that for every n 2 N it satisfies n X 1 for each 1 j < n, and (11) sup jPx .h/j 2k x2spanfx1 ;:::;xj g h2spanfxj C1 ;:::;xn g x;h2n BX

kyk .1 C n

1 /ky

C xn k

kDj C1

for all y 2 spanfx1 ; : : : ; xn

1g

and 2 R.

(12)

This clearly implies (10). Choose x1 2 SX arbitrarily. Let n 2 N and assume that x1 ; : : : ; xn are already defined such that (11) and (12) hold for this n. We find xnC1 such that (11) and (12) hold for n C 1. Consider the family of polynomials n A D P I P .h/ D Px .y C h/ Px .y/; o x 2 spanfx1 ; : : : ; xj g; y 2 spanfxj C1 ; : : : ; xn g; x; y 2 n BX ; j D 1; : : : ; n : An inspection of the proof of Theorem 1.110 reveals that y in the statement of that theorem can be restricted to a subspace. Thus from the proof we obtain that the q U Tc -smoothness of f implies continuity of the mapping x 7! Px on the subspace spanfx1 ; : : : ; xn g. Therefore the mapping .x; y/ 7! Px .y C / Px .y/ is continuous on spanfx1 ; : : : ; xn g X. It follows that the set A is compact in P Œq .X/ and so it contains a finite subset F such that for each P 2 A there is Q 2 F satisfying 1 . suph24BX jP .h/ Q.h/j < 2nC2 Let Z be the finite-codimensional subspace of X from Lemma 4.66 used on Y D spanfx1 ; : : : ; xn g and " D n . By Fact 65 there is xnC1 2 SZ such that 1 1 supjj4 jP .xnC1 /j < 2nC2 for each P 2 F and hence supjj4 jP .xnC1 /j < 2nC1 for each P 2 A. The property (12) is clearly satisfied and so kyk .1Cn /nC1 D n and jj ky C xnC1 k C kyk nC1 C n < 4 whenever y 2 spanfx1 ; : : : ; xn g and ky C xnC1 k nC1 . Using this together with the definition of A and the inductive hypothesis we obtain (11). This concludes the construction g. Moreover,

P

P

of fx nP

by (12), jnD1 aj xj .1 C n / .1 C m 1 / jmD1 aj xj 2 jmD1 aj xj for all n < m and all a1 ; : : : ; am 2 R. Thus fxn g is a basic sequence in X.

272

Chapter 5. Smoothness and structure

By the assumption there is an infinite-dimensional subspace of spanfxn g which is of cotype q. So there is a normalised block basic sequence fyn g of fxn g such that Y D spanfyn g is of cotype q with the cotype constant C ([FHHMZ, Theorem 4.26]). Observe that (10) remains true when passing to a block sequence of fxn g, but we also have to pass to a subsequence of fın g. Next, we will inductively construct a block ˚P n nC1 . Set u D 0. sequence fun g of fyn g and sets Kn D " u I ." / 2 f 1; 1g j 0 j D0 j j If u0 ; u1 ; : : : ; un 1 have already been defined, we put n En D h 2 spanfyj I j > max supp un 1 g \ BX I o ˇ ˇ ˇf .x ˙ h/ f .x/ Px .˙h/ˇ 1q khkq for all x 2 Kn 1 : C It is clear that En contains some non-zero vectors. We put ˛n D suph2En khk and choose un 2 En such that kun k > ˛n =2. S There are two possibilities. Either K D 1 nD1 Kn BX , or Kn n BX ¤ ; for some n 2 N. In the first case, since Y is of a non-trivial cotype, it does not contain c0 (check the canonical basis of c0 ). From the Bessaga-Pełczy´nski theorem (Theorem 3.46) it follows that K is compact and un ! 0, hence also ˛n ! 0. However, since f is q U Tc -smooth, there is ı > 0 such that jf .x C h/ f .x/ Px .h/j C1q khkq for all h 2 X , khk ı, and all x 2 K. This contradicts the fact that ˛n ! 0. In the latter

P

case, let n 2 N be the smallest such that Kn n BX ¤ ;. Then

n j uj > 1 for some choice of signs .j / 2 f 1; 1gn . On the other hand,

PjnD1

Pn 1

n

j D1 "j uj j D1 "j uj C kun k 2 for every ."j / 2 f 1; 1g . Hence using the definition of Ek and (10) we obtain ! ! ! n n k k X X X X1 f j uj D f j uj f j uj j D1

kD1 n X

j D1

j D1

1 n X 1 X 1 q ık C q kuk kq .k uk / C q kuk k < 1 j D1 j uj C C kD1 kD1 kD1

n

q

X X 1

1C n "j uj 1 C 2q ;

2 n

PPk

."j /2f 1;1g

which is a contradiction.

j D1

t u

The next result contains the main structural information concerning C 1 -smooth Banach spaces. Theorem 67 ([Dev1]). Let X be an infinite-dimensional Banach space which does not contain c0 and admits a C k -smooth bump for every k 2 N. Then X is of type 2, of exact cotype q D q.X/ which is an even integer, and admits a separating polynomial of degree d , where q d 2q but not less. Further, every homogeneous separating polynomial on X is of degree mq for some m 2 N and X has a subspace isomorphic to `q .

Section 4. Smooth bumps and structure II

273

Proof. The space X has a separating polynomial and is of type 2 according to Theorem 64. We know from Corollary 4.62 that q D q.X/ is an even integer. By Theorem 66 the space X has a separating family of polynomials of degree at most q. By Theorem 4.51 we conclude that X is of cotype q and has no separating polynomial of degree less that q. On the other hand by Fact 4.47 it has a separating polynomial of degree at most 2q. By Lemma 4.68 there exists a subspace of X isomorphic to `q . The statement about the degrees of homogeneous separating polynomials then follows from Corollary 4.61. t u Corollary 68. Let X be an infinite-dimensional Banach space that admits a C k -smooth bump for every k 2 N. Then X is saturated by spaces from f`p I p eveng [ fc0 g. By combining Theorem 66, Theorem 64, and Theorem 4.51 we obtain another corollary. Corollary 69 ([Dev1]). Let X be a Banach space that admits a C k -smooth bump for some k 2 N, k > 1, and suppose that X is saturated with spaces of cotype k. Then X admits a separating polynomial and is of type 2 and cotype k. The following lemma shows that the range of the derivative of a linearly coercive function is dense in a certain ball around origin. Lemma 70. Let X be a Banach space and let ' W X ! R [ fC1g be a proper linearly coercive lower semi-continuous bounded below function that is Gâteaux differentiable on dom '. Then ı'.dom '/ \ BX .0; r/ is dense in BX .0; r/, where . r D lim inf '.x/ kxk kxk!C1

Proof. Fix a functional f 2 UX .0; r/ and 0 < " r kf k. Since we have lim infkxk!C1 '.x/kxkf .x/ r kf k > 0, the function ' f is lower semi-continuous and bounded below. So by the Ekeland variational principle (Theorem 53) there is x 2 X such that '.x C h/ f .x C h/ '.x/ f .x/ "khk for all h 2 X. It follows that x 2 dom ' and '.x C h/ '.x/ f .h/ "khk for all h 2 X . This implies that ıg.x/Œh D lim t !0C 1t '.x C th/ '.x/ f .h/ "khk for every h 2 X . Thus jıg.x/Œh f .h/j "khk for every h 2 X and consequently kıg.x/ f k ". u t Let X, Y be normed linear spaces, A X, and f W A ! Y . We say that f is ˛-Hölder at x 2 A if there are ı > 0 and L 0 such that kf .y/ f .x/k Lky xk˛ for all y 2 U.x; ı/ \ A. We say that f is pointwise ˛-Hölder on A if it is ˛-Hölder at every x 2 A. We say that f is ˛-Hölder at x 2 A in the direction h 2 X if f xCspanfhg is ˛-Hölder at x. We say that f is directionally ˛-Hölder at x if for each h 2 SX it is ˛-Hölder at x in the direction h. We say that f is directionally pointwise ˛-Hölder on A if it is directionally ˛-Hölder at every x 2 A. Note that it follows directly from the definitions that if f is Fréchet differentiable at x, then it is Lipschitz at x, and if f has directional derivatives in all directions at x (in particular if f is Gâteaux differentiable at x), then it is directionally Lipschitz at x.

274

Chapter 5. Smoothness and structure

Fact 71. Let X , Y be normed linear spaces, U X open, f W U ! R, g W U ! Y , x 2 U , and h 2 X . If f , g are ˛-Hölder at x in the direction h, resp. ˛-Hölder at x, then fg

f

g x;h .t / O.1/x;h .t/ C O.1/x;h .t/ C O.t 2˛ /; t ! 0C;

resp.

xfg .t / O.1/xf .t/ C O.1/xg .t/ C O.t 2˛ /; t ! 0 C : If f is ˛-Hölder at x in the direction h, resp. ˛-Hölder at x, and f .x/ ¤ 0, then 1=f

f

x;h .t/ O.1/x;h .t/ C O.t 2˛ /; t ! 0C;

resp.

x1=f .t/ O.1/xf .t/ C O.t 2˛ /; t ! 0 C : Proof. Everything follows from the following expansions: 2 2f .x/ f .x C th/ f .x th/ D f .x/ f .x C th/f .x/ f .x C th/ f .x th/ f .x/ f .x th/ C ; f .x C th/f .x th/f .x/ 2 fg.x C t h/ C fg.x th/ 2fg.x/ D f .x C th/ C f .x th/ 2f .x/ g.x C th/ C g.x/ C g.x C th/ C g.x th/ 2g.x/ f .x C th/ C f .x/ C g.x th/ g.x/ f .x th/ f .x C th/ C f .x th/ f .x/ g.x th/ g.x C th/ : t u 1 1 C f .x C th/ f .x th/

Theorem 72 ([DGZ2], [Va2]). Let X be a Banach space that admits a continuous bump ' with one of the following properties: (i) For each x 2 X there exists ˛ > 21 such that the bump ' is ˛-Hölder at x and ' x .t / D o.t/; t ! 0C. (ii) ' is Gâteaux differentiable and for each x 2 X and h 2 SX there is p > 1 such that '.x C th/ '.x/ ı'.x/Œth D o.jtjp /; t ! 0. Then dens Y D dens Y for every subspace Y X. In particular X is an Asplund space. Note that the condition (ii) is satisfied in particular if for each x 2 X there is p > 1 such that ' is weakly T p -smooth at x. Proof. Since both properties pass to subspaces, we may assume that Y D X. Let Q be a dense subset of X with jQj D dens X. Define a function W X ! R [ fC1g by 1 .x/ D '.x/ .x/ D C1 otherwise. 2 for x 2 suppo ' and Assume first that (i) holds. From Fact 71 we obtain x .t/ D o.t/; t ! 0C for each x 2 dom . Fix f 2 X and " > 0. By the Ekeland variational principle (Theorem 53) " used on khk for f there is x 2 X such that .xCh/ f .xCh/ .x/ f .x/ 10 " each h 2 X . There is 0 < r 1 such that j .x Ch/C .x h/ 2 .x/j 10 khk for

Section 4. Smooth bumps and structure II

275

" h 2 B.0; 2r/. So 10 .x/ f .h/ D .x Ch/ .x/Cf . h/ khk .x Ch/ " khk 5" khk for all h 2 B.0; 2r/. .x C h/ .x/ C .x h/ .x/ C 10 Let ´ 2 Q \ B.x; r/, ´ ¤ x, and find n 2 N such that 12 k´ xk n1 k´ xk. Then ´ C n1 h 2 B.x; 2r/ for any h 2 BX . Hence

ˇ ˇ .´ C 1 h/ n

1 ˇ n f .h/

ˇ

.´/

ˇ ˇ D ˇ .´ C n1 h/ .´/ C .x/ C f .´ x/ˇ .x/ f .´ C n1 h x/ ˇ ˇ ˇ .´ C n1 h/ .x/ f .´ x/j .x/ f .´ C n1 h x/ˇ C j .´/ " " " k´ x C n1 hk C k´ xk 2k´ xk C n1 5 5 5 for every h 2 BX . It follows that ˇ ˇn

.´ C n1 h/

.´/

ˇ " f .h/ˇ .2nk´ 5

xk C 1/ ":

Now consider the function g´;n W BX ! R, g´;n .h/ D n .´ C n1 h/ .´/ . We have just shown that g´;n 2 `1 .BX / and kg´;n f k " when considering f as a member of `1 .BX /. So setting M D fg´;n I ´ 2 Q; n 2 N; g´;n 2 `1 .BX /g we have X M in `1 .BX / and hence dens X jM j jQj D dens X. Now assume that (ii) holds. The function is Gâteaux differentiable on dom by Theorem 1.70. Proposition 1.115 implies that for each x 2 dom and h 2 SX there is 1 < p < 2 such that .x C th/ .x/ ı .x/Œth D o.jtjp /; t ! 0. .x/ D C1, by Lemma 70 there is Fix f 2 X and " > 0. Since limkxk!C1 kxk " x 2 dom such that kf ı .x/k < 2 . Let be such that is bounded on B.x; /. Set ˚ Fn D h 2 BX I j .x C th/

.x/

1 ı .x/Œthj nkthk1C n ; 0 t :

By the continuity ofS each Fn is closed. From the properties of the function it follows that BX D n2N Fn . By the Baire category theorem there is K 2 N such that FK has a non-empty interior in BX . Thus there are y 2 BX and r > 0 such that j .x C t h/ .x/ ı .x/Œthj Kkthk1C˛ for all h 2 B.y; r/ and 0 t , 1 where ˛ D K . In other words, j .x C u/ .x/ ı .x/Œuj Kkuk1C˛ for all u 2 D D ft hI h 2 B.y; r/; 0 t g. 2 ˚ " r ;1 ˛ ; 2kykCr Put D0 D fx C thI h 2 B.y; 2r /; 0 t g. Let D min 20K and choose ´ 2 Q \ D0 \ B.x; /, ´ ¤ x. Further, let n 2 N be such that ˛ ˛ 1 xk1C 2 n1 k´ xk1C 2 k´ xk. Note that ´ x 2 D and also 2 k´ ´ x C n1 h 2 D for each h 2 BX . Indeed, ´ x D tv for some v 2 B.y; 2r / and k´ xk 0 < t . We have ´ x C n1 h D t.v C 1t n1 h/, and since t kykC r , we get kv C

11 t nh

yk

r 2

C

kykC 2r k´ k´ xk

˛

xk1C 2

r 2

˛

C .kyk C 2r / 2 r.

2

276

Chapter 5. Smoothness and structure

Hence ˇ ˇ ˇ .´ C 1 h/ .´/ ı .x/Œ n1 hˇ n ˇ ˇ .´ C n1 h/ .x/ ı .x/Œ´ Kk´

x C n1 hk1C˛ C Kk´

ˇ ˇ x C n1 hˇ C ˇ .´/

.x/

xk1C˛ K.21C˛ C 1/k´

for every h 2 BX . It follows that ˇ ˇ ˇ ˇn .´ C 1 h/ .´/ f .h/ˇ ˇn n

.´ C n1 h/

.´/

xk1C˛ C

ˇ xˇ

xk1C˛

ˇ " ı .x/Œhˇ C 2

" 10Kk´ 2 The proof can now be finished in the same way as in the case (i). 5Knk´

ı .x/Œ´

˛

xk 2 C

" ": 2 t u

Theorem 73 ([DGZ2], [MPVZ]). Let X be a Banach space such that X is a weak Asplund space. If X admits a continuous directionally pointwise Lipschitz bump ˇ ˇ with x;h of some power type for each h 2 SX , x 2 X , then X is super-reflexive. If ˇ moreover there is p 2 .1; 2 such that each x;h is of power type p, then X admits an equivalent norm with modulus of smoothness of power type p. We remark that some kind of convexity condition on X (such as the RNP, see 1;1 Theorem 8) is necessary in the previous theorem: the space c0 admits a Cloc -smooth norm (Theorem 94), but it is not super-reflexive. 1 Proof. For x 2 X put '.x/ D ˇ.x/ 2 if ˇ.x/ ¤ 0 and '.x/ D C1 otherwise. Note that ' is continuous on X and dom ' is an open set. By Fact 71 for each x 2 dom ', ˇ ' h 2 SX if x;h is of power type px;h , then x;h is of power type min fpx;h ; 2g. Since '.x/ D C1 for x outside some ball and ' 0, it follows that dom ' D X and ' is Lipschitz, as it is a supremum of a family of Lipschitz functions with their Lipschitz constant bounded. Since X is a weak Asplund space, there is a dense Gı subset ˝ of X such that ' is Fréchet differentiable at every point of ˝. Denote D .' / and note that ' and D ' by Fact 25. Fix f 2 ˝ and denote x D D' .f /. By Corollary 34 we have x 2 dom ', .x/ D '.x/, and f 2 @ .x/. Thus x;h .t/ D .x C th/ C .x th/ 2 .x/ ' '.x C t h/ C '.x th/ 2'.x/ x;h .t/ for all t 0, h 2 X . Hence x;h is of ' the same power type as x;h . The function is continuous on Int dom dom ', since is dominated by a continuous function '. Therefore x is of some power type (resp. of power type p) by Proposition 16. This in turn implies that f;x is of some power type (resp. of power type q, where p1 C q1 D 1) by Corollary 37. Now Theorem 39 and the standard duality [DGZ, Proposition IV.1.12] finishes the proof. u t

Note that the above theorem applies for example if X has the RNP and admits a continuous directionally T p -smooth bump (and in particular G 2 -smooth bump, Proposition 1.112).

Section 4. Smooth bumps and structure II

277

Corollary 74. Let X be a Banach space such that either X or X is a weak Asplund space, or equivalently such that either X or X has the RNP. If both X and X admit a continuous directionally pointwise Lipschitz bump with directional pointwise modulus of smoothness of power type 2, then X is isomorphic to a Hilbert space. Proof. The equivalence of the assumption of the RNP follows from Theorem 8. First assume that X is a weak Asplund space. Then X admits an equivalent norm with modulus of smoothness of power type 2 by Theorem 73. In particular, X is reflexive and so X is an Asplund space. Thus also X admits an equivalent norm with modulus of smoothness of power type 2 by Theorem 73. Hence X admits an equivalent norm with modulus of convexity of power type 2 ([DGZ, Proposition IV.1.12]). If Y is a separable subspace of X , then Y admits a G 2 -smooth bump by Corollary 7.72 (together with a version of Lemma 20 for norms), and hence Y is isomorphic to a Hilbert space by Theorem 63. So the space X is isomorphic to a Hilbert space by Corollary 45. If we assume that X is a weak Asplund space, then X admits an equivalent norm with modulus of smoothness of power type 2 by Theorem 73. In particular, X is an Asplund space, and so X is isomorphic to a Hilbert space by the first part of the proof. t u Corollary 75. Let X be a Banach space such that X admits a continuous directionally pointwise Lipschitz bump with directional pointwise modulus of smoothness of power type 2 and X admits a continuous bump ' with one of the following properties: (i) ' is pointwise Lipschitz with pointwise modulus of smoothness of power type 2, (ii) ' is Gâteaux differentiable and '.x C th/ '.x/ ı'.x/Œth D O.jt j2 /; t ! 0 for each h 2 SX , x 2 X . Then X is isomorphic to a Hilbert space. The same holds if we switch the roles of X and X . Note that condition (ii) is in particular satisfied if ' is weakly T 2 -smooth. Proof. Since X is an Asplund space by Theorem 72, we may use Corollary 74. If we switch the roles of X and X , then X is an Asplund space by Theorem 72, hence X has the RNP by Theorem 2, and we may again use Corollary 74. t u The following theorem says that in the non-separable setting a weakly T q -smooth bump together with the modulus of convexity of power type q is sufficient for the existence of a polynomial with certain separating properties. Compare this with Theorem 62. Theorem 76 ([MV]). Let X be a non-separable Banach space such that its norm has modulus of convexity of power type q, where q is not an even integer. If X admits a continuous weakly T q -smooth bump, then there is a polynomial on X of degree less than q which is not weakly sequentially continuous.

278

Chapter 5. Smoothness and structure

Proof. Let g W X ! R be the continuous weakly T q -smooth bump. We define a 1 function ' W X ! R [ fC1g by '.x/ D g.x/ 2 if g.x/ ¤ 0 and g.x/ D C1 otherwise. Then ' is a bounded below proper continuous quadratically coercive function and hence by Theorem 58 there are f 2 X , x 2 dom ', and C > 0 such that '.x C h/ '.x/ f .h/ C khkq for every h 2 BX . As ' is weakly T q -smooth at x (Proposition 1.115), there is P 2 P Œq .X/ satisfying P .0/ D 0 and such that '.x C th/ '.x/ P .th/ D o.jtjq /; t ! 0 for each h 2 X. Put Q.h/ D P .h/ C P . h/. Since q is not an even integer, deg Q < q. As X is a reflexive non-separable space, X has an uncountable total biorthogonal system f.x˛ I f˛ /g˛2 with fx˛ g BX (see e.g. [HMVZ, Corollary 4.11]). Assume that Q is weakly sequentially continuous. We claim that there is 2 such that Q.tx / D 0 for all t 2 Œ 1; 1. If this is not the case, then (since is uncountable) there are a sequence f˛n g of distinct elements of and a sequence ftn g Œ 1; 1 such that jQ.tn x˛n /j > ı for some ı > 0 and all n 2 N. By the weak sequential compactness of BX there is a subsequence ftnk x˛nk g of ftn x˛n g which is weakly convergent; and in fact weakly null by the totality of fx˛ I f˛ g. This contradicts the fact that Q.0/ D 0. Put h D x . Then 2C kt hkq '.x C th/

'.x/

f .th/ C '.x q

th/

'.x/

f . th/

q

D P .th/ C P . th/ C o.jtj / D Q.th/ C o.jt j / D o.jtjq /; t ! 0; which is a contradiction. It means that Q is not weakly sequentially continuous.

t u

Proposition 77. Let X be a Banach space with an analytic norm such that there is a non-empty open U X on which the norm is weakly sequentially Cauchy-continuous. Then X is separable. Proof. First notice that X is an Asplund space (Corollary 3) and so X does not contain `1 . Let V be a non-empty open ball in U with dist.V; X n U / > 0. Then Dkk.V / is relatively compact by Proposition 41. Further, since Dkk 2 C ! .X n f0gI X / and span Dkk.X n f0g/ D X by the Bishop-Phelps theorem, it follows from Corolt u lary 1.159 that span Dkk.V / D X and so X is separable.

5. Local dependence on finitely many coordinates We start by introducing the concept of local dependence on finitely many coordinates and show that the existence of an LFC bump has a strong impact on structure, namely the space is then a c0 -saturated Asplund space. We show how the LFC property is used to construct a C 1 -smooth norm on c0 . /. We prove that on a separable Asplund space a continuous LFC bump (or just a locally wsc bump) implies already the existence of a C 1 -smooth (and LFC) bump. Finally, we show how to construct a smooth approximation of a norm using a smooth generalised James boundary and the techniques of this section.

Section 5. Local dependence on finitely many coordinates

279

Definition 78. Let X be a topological vector space, ˝ X an open subset, E an arbitrary set, M X , and g W ˝ ! E. We say that g depends only on M on a set U ˝ if g.x/ D g.y/ whenever x; y 2 U are such that f .x/ D f .y/ for all f 2 M . We say that g depends locally on finitely many coordinates from M (g is LFC-M for short) if for each x 2 ˝ there are a neighbourhood U ˝ of x and a finite subset F M such that g depends only on F on U . We say that g depends locally on finitely many coordinates (g is LFC for short) if it is LFC-X . A canonical example of a non-trivial LFC function is the supremum norm on c0 , which is LFC-fej g away from the origin. Indeed, take any x D .xj / 2 c0 , x ¤ 0. Let n 2 N be such that jxj j < 12 kxk1 for j > n. Then kk1 depends only on fe1 ; : : : ; en g on U.x; 41 kxk1 /. Fact 79. Let X be a topological vector space, ˝ X an open subset, E an arbitrary set, M X , and g W ˝ ! E. The mapping g is LFC-M if and only if for each x 2 ˝ there are an open neighbourhood U ˝ of x, n 2 N, a biorthogonal system f.ej I fj /gjnD1 X M , an open set V Rn , and a map ping G W V ! E, such that g.y/ D G f1 .y/; : : : ; fn .y/ for all y 2 U , where Pn G.w/ D g x C j D1 wj fj .x/ ej for each w 2 V . Proof. Let U0 be an open neighbourhood of x and n 2 N such that g depends only on ff1 ; : : : ; fn g M on U0 . Without loss of generality we may assume that the functionals f1 ; : : : ; fn are linearly independent. Hence there are vectors e1 ; : : : ; en 2 X such that f.ej I fj /gjnD1 is a biorthogonal system (since \j ¤k ker fj 6 ker fk for each P 1 k n). Let ˚ W Rn ! X be defined as ˚.w/ D x C jnD1 wj fj .x/ ej . This is a continuous mapping and so the set V D ˚ 1 .U0 / is an open subset of Rn . Notice that G.w/ D g.˚.w// for each w 2 V . Let W X ! Rn be defined as .y/ D .f1 .y/; : : : ; fn .y//. This is a continuous mapping and so the set U D 1 .V / \ U0 is open. Moreover, ˚. .x// D x, hence U is an open neighbourhood of x. Now choose any y 2 U . Since .y/ 2 V , the composition G. .y// is well-defined. Further, using the facts that y 2 U0 , ˚. only on ff1 ; : : : ; fn g P .y// 2 U0 , g depends on U0 , and fk .˚. .y/// D fk .x/ C jnD1 fj .y/ fj .x/ fk .ej / D fk .y/ for each 1 k n, we may conclude that G. .y// D g ˚. .y// D g.y/. The other implication is obvious. t u From the previous fact it follows that the mappings G through which g is “locally factorised” share some properties of g – for example they are continuous (or smooth) if and only if g is continuous (or smooth). Notice further that if g W ˝ ! E is LFC and h W E ! F is any mapping, then the mapping h B g is also LFC. The following fact is also immediate. Fact 80. Let X, Y be normed linear spaces, E an arbitrary set, M Y , ˝ Y an open set, g W ˝ ! E an LFC-M mapping, and T 2 L.X I Y /. Then the mapping g B T W T 1 .˝/ ! E is LFC-T .M /.

280

Chapter 5. Smoothness and structure

A typical example of the use of the LFC notion is the following. Lemma 81. Let X, Y , Z be normed linear spaces, U X , V Y open sets, M Y , k 2 N [ f1g, ˚ W U ! V a continuous mapping such that f B ˚ is C k -smooth for each f 2 M , and let g 2 C k .V I Z/ be a LFC-M mapping. Then g B ˚ 2 C k .U I Z/. Proof. Fix x 2 U . By Fact 79 there are f1 ; : : : ; fn 2 M , an open neighbourhood W of ˚.x/, an open ˝ Rn , and G 2 C k .˝I Z/ such that g.y/ D G f1 .y/; : : : ; fn .y/ for all y 2 W . Then g.˚.´// D G f1 B ˚.´/; : : : ; fn B ˚.´/ for each ´ 2 ˚ 1 .W /, which is an open neighbourhood of x, and so the assertion follows. t u Let .X; / be a topological vector space, .Y; U/ a uniform space, ˝ X open, and f W ˝ ! Y . We say that f is locally weakly uniformly continuous if for each x 2 ˝ there is a -neighbourhood U of x such that f U is weakly uniformly continuous (or more precisely w–U uniformly continuous). Fact 82. Let X be a topological vector space, .Y; U/ a uniform space, ˝ X open, and let g W ˝ ! Y be a continuous LFC mapping. Then g is locally weakly uniformly continuous. Proof. We use the notation from the proof of Fact 79. Let K V be a compact neighbourhood .x/. Since G is clearly continuous, it is uniformly continuous on K. The mapping is obviously w–kk uniformly continuous and hence g D G B is w–U uniformly continuous on 1 .K/ \ U0 . t u The next lemma serves as an addition to the Implicit function theorem. Lemma 83. Let X, Y , Z be normed linear spaces, U X and V Y open sets, and let f W U V ! Z depend only on ff1 ; : : : ; fn g .X ˚ Y / on U V . Further, assume that there is a unique mapping u W U ! V satisfying f x; u.x/ D 0 for all x 2 U . Then u depends only on ff1 X ; : : : ; fn X g X on U . Proof. Let x; y 2 U be such that fj X .x/ D fj X .y/ for j D 1; : : : ; n. Then .x; u.x// 2 U V , .y; u.x// 2 U V , and fj .x; u.x// Dfj X .x/ Cfj .0; u.x// Dfj X .y/ Cfj .0; u.x// Dfj .y; u.x// for j D 1; : : : ; n. It follows that 0 D f x; u.x/ D f y; u.x/ , which by the uniqueness of u gives u.x/ D u.y/. t u The word “coordinate” in the term LFC originates of course from spaces with bases, where LFC was first defined using the coordinate functionals. In order to apply the LFC techniques to spaces without a Schauder basis, the notion was generalised using arbitrary functionals from the dual. However, as we will show, the generalisation does not substantially increase the supply of LFC mappings on Banach spaces with a Schauder basis. Interestingly, up to a small perturbation we can always in addition assume that the given LFC mapping in fact depends on the coordinate functionals. We begin with a simple related result for Markushevich bases.

Section 5. Local dependence on finitely many coordinates

281

Proposition 84. Let X be a separable Banach space, E a set, and g W X ! E an LFC mapping. Then there is a Markushevich basis f.ej I fj /gj1D1 X X such that g is LFC-ffj g. Proof. By the Lindelöf property of X we can choose a countable fhj g X such that it separates the points of X and g is LFC-fhj g. Then we can use the Markushevich theorem ([FHHMZ, Theorem 4.59]) to construct a Markushevich basis f.ej I fj /g such that spanffj g D spanfhj g. Now let x 2 X and U X be a neighbourhood of x such that g depends only on M D fh1 ; : : : ; hn g on U . Let M spanff1 ; : : : ; fm g. Then for any y; ´ 2 U satisfying fk .y/ D fk .´/ for k D 1; : : : ; m we also have hj .y/ D hj .´/ for all j D 1; : : : ; n and hence g.y/ D g.´/. Thus g depends only on ff1 ; : : : ; fm g on U . t u We would like to establish a similar result for Schauder bases. In this context shrinking Schauder bases emerge quite naturally, taking into account Theorem 91. Fact 85. Let X and Y be Banach spaces with equivalent Schauder bases fxj g and fyj g respectively. Then fxj g is shrinking if and only if fyj g is shrinking. Proof. Let fxj g be a shrinking basis and T W Y ! X be an isomorphism of Y onto X such that T yj D xj . Then T W X ! Y is an isomorphism of X onto Y such that T xj D yj and thus Y D T .X / D T .spanfxj g/ T .spanfxj g/ D span T .fxj g/ D spanfyj g: t u Lemma 86. Let X be a Banach space with a shrinking Schauder basis fej g. Let f 2 X , " > 0, and n 2 N. Then there are a (shrinking) Schauder basis fxj g of X and N 2 N, N > n, such that xj D ej for 1 j < N , fxj g is .1 C "/-equivalent to fej g, spanfxj gjmDk D spanfej gjmDk for all 1 k n and m k, and spanfxj I j N g ker f . Proof. If f .ej / D 0 for all j > n, then we may put N D n C 1 and xj D ej for all j 2 N. Otherwise by passing to a multiple of f we may without loss of generality assume that there is K n such that f .eK / D 1. Let us denote fk D f Pk 1 .f /, k 2 N, k > 1, where Pj are the projections associated with the basis fej g. As fej g is shrinking, kfk k ! 0 and hence we can find N 2 N such that N > K and " kfN k .2C"/ke . Put xj D ej for 1 j < N and xj D ej f .ej /eK for j N . Kk For any m1 ; m2 2 N, m1 m2 , and any sequence faj g of scalars we have

m

m

m2 2 2

X

X

X

aj xj D aj ej eK aj f .ej /

j Dm1 j Dm1 j Dmaxfm1 ;N g

m

! m2 2

X

X

aj ej C eK fN aj ej

j Dm1 j Dm1

m

m2 2

X

"

X

1 C keK kkfN k aj ej 1 C aj ej

2C" j Dm1

j Dm1

282

Chapter 5. Smoothness and structure

and

m

m 2 2

X

X

aj ej aj xj

j Dm1

j Dm1

eK fN

m2 X j Dm1

!

aj ej 1

" 2C"

m2

X

aj ej :

j Dm1

This implies that fxj g is a basic sequence .1 C "/-equivalent to fej g. Since xK D eK and n K < N , we have spanfxj gjmDk D spanfej gjmDk for all 1 k n and m k, and therefore spanfxj g D spanfej g, which implies that fxj g is a basis of X. Clearly, f .xj / D 0 for j N . t u It is certainly worth noticing that the method used in the previous lemma (and the next theorem) does not rely on the classical argument of perturbation by the norm-summable sequence. In fact our new basis is “far” away from the original one. Theorem 87. Let X be a Banach space with a shrinking Schauder basis fej g, let ffj g X be a countable set, and " > 0. Then there is a (shrinking) Schauder basis fxj g of X such that it is .1 C "/-equivalent to fej g, spanfxj gjmD1 D spanfej gjmD1 for all m 2 N, and spanffj g spanfxj g. Q Proof. Choose a sequence of "j > 0 such that j .1 C "j / 1 C " and put N0 D 1. We apply Lemma 86 to fej g, f1 , "1 , and n D 1. We obtain a basis fxj1 g which is .1 C "1 /-equivalent to fej g and N1 2 N such that spanfxj1 I j N1 g ker f1 . Moreover, xj1 D ej for j < N1 and spanfxj1 gjmD1 D spanfej gjmD1 for all m 2 N. We proceed by induction. Suppose the basis fxjk g and Nk 2 N have already been Q defined in such a way that the basis fxjk g is i k .1 C "i /-equivalent to the basis fej g, xjk D xjk 1 for j < Nk , spanfxjk gjmD1 D spanfej gjmD1 for all m 2 N, and finally spanfxjk I j Ni g ker fi for each 1 i k. We apply Lemma 86 to fxjk g, fkC1 , Q "kC1 , and n D Nk in order to obtain a basis fxjkC1 g which is i kC1 .1 C "i /-equivalent to the basis fej g, and a number NkC1 2 N satisfying NkC1 > Nk , such that spanfxjkC1 I j NkC1 g ker fkC1 . Moreover, xjkC1 D xjk for j < NkC1 and spanfxjkC1 gjmD1 D spanfxjk gjmD1 D spanfej gjmD1 for all m 2 N. Since also spanfxjkC1 gjmDNi D spanfxjk gjmDNi for all 1 i k and m Ni , we have

spanfxjkC1 I j Ni g ker fi for 1 i k C 1. Clearly, there is a sequence fxj g such that limk!1 xjk D xj for all j 2 N. (This is because the sequence fNk g is increasing and thus fxjk g1 is eventually constant.) kD1 m m It is straightforward to check that spanfxj gj D1 D spanfej gj D1 for all m 2 N, fxj g is a basis of X which is .1 C "/-equivalent to fej g, and spanfxj I j Ni g ker fi (which means that fi 2 spanfxj I j < Ni g) for any i 2 N. t u If a Banach space X has a shrinking Schauder basis, then using the Lindelöf property of X and Theorem 87 we obtain the following corollary, which usually allows us to work only with LFC-fej g mappings.

Section 5. Local dependence on finitely many coordinates

283

Corollary 88. Let X be a Banach space with a shrinking Schauder basis fej g, E any set, g W X ! E an LFC mapping, and " > 0. Then there is a (shrinking) Schauder basis fxj g of X , .1 C "/-equivalent to fej g, such that g is LFC-fxj g. As we shall see, the existence of an LFC bump function on a Banach space has a strong impact on its structure. The next lemma shows that if a space admits an arbitrary (even non-measurable) LFC bump, then it admits also an upper semi-continuous LFC bump. Lemma 89. Let X be a topological vector space, M X , and let g W X ! R be an LFC-M function. Then supp g is an upper semi-continuous LFC-M function. Proof. Since supp g is closed, supp g is upper semi-continuous. Fix x 2 X. There is an open neighbourhood U of X such that g depends only on ff1 ; : : : ; fn g M on U . We claim that supp g also depends only on ff1 ; : : : ; fn g on TU . Suppose that this is not the case. Then there are y; ´ 2 U such that ´ y 2 jnD1 ker fj , y 2 supp g, and ´ … supp g. There is a net fy g 2 X such that g.y / ¤ 0 for all 2 and y ! y. Since y C ´ y ! ´ and U is open, we way suppose that both fy g 2 U and fy C ´ yg 2 U . Hence g.y C ´ y/ D g.y / ¤ 0 for all 2 and so ´ 2 supp g, a contradiction. t u Theorem 90 ([PWZ]). An infinite-dimensional Banach space X that admits an (arbitrary) LFC bump contains c0 . Proof. By Lemma 89 there is a non-empty bounded closed set A X such that A is LFC. Without loss of generality we may assume that A contains the origin. Put g.x/ D 1 A .x/ C 1 A . x/. Then g is an even lower semi-continuous LFC function with g.0/ D 0 and g.x/ D 2 for all x outside some ball. Assume that X does not contain c0 . We apply Lemma 47 on the function g. Let K be the compact set and V the neighbourhood of 0 from Lemma 47. Let r > 0 be such that U.0; r/ V . Since g is LFC, there are a covering of K by open balls U.xj ; ıj /, j D 1; : : : ; n, and a finite set M X such that g depends T only on M on each U.xj ; 2ıj /. Put ı D minf 2r ; ı1 ; : : : ; ın g and pick any x 2 f 2M ker f with kxk D ı. Since x 2 V , by Lemma 47 there is ´ 2 K such that g.´ C x/ > g.´/. Let k 2 N be such that ´ 2 U.xk ; ık /. Since ´ C x 2 U.xk ; 2ık / and g depends only on M on U.xk ; 2ık /, we have g.´ C x/ D g.´/, a contradiction. t u Theorem 91 ([FZ1]). Let X be a Banach space, M X , and suppose that X admits an (arbitrary) LFC-M bump. Then span M D X . Proof. By Lemma 89 there is a non-empty bounded closed set A X such that A is LFC-M . Pick any f 2 X and " > 0 and notice that f is bounded on A. Let IA W X ! R [ fC1g be the indicator function of the set A, i.e. IA .x/ D 0 for x 2 A and IA D C1 for x 2 X n A. Put ' D IA f . Then ' is a lower semi-continuous bounded below function and so by the Ekeland variational principle (Theorem 53) there is x0 2 X such that '.x/ '.x0 / "kx x0 k for every x 2 X. Obviously x0 2 A and for every x 2 A we have f .x/ f .x0 / "kx x0 k from which it

284

Chapter 5. Smoothness and structure

follows that f .x

x0 / "kx

x0 k

for every x 2 A.

(13)

Let ı > 0 and f1 ;T : : : ; fn 2 M be such that A depends only on ff1 ; : : : ; fn g on U.x0 ; ı/. Put Z D jnD1 ker fj . For any ´ 2 Z, k´k < ı we have x0 C ´ 2 A and hence f .´/ "k´k by (13). This means that kf Z k ". By the Hahn-Banach theorem we can find a functional g 2 X such that g D f on Z and kgk ". Clearly, f g 2 span M and kf .f g/k ". t u Corollary 92. An infinite-dimensional Banach space X which admits an LFC bump is a c0 -saturated Asplund space. Proof. Since the existence of an LFC bump passes to subspaces, X is c0 -saturated by Theorem 90 and it is enough to show that X is separable provided that X is separable. Let g be an LFC bump on X. By the Lindelöf property of X there exists a countable set M X such that g is LFC-M . Then X D span M by Theorem 91, hence X is separable. t u We say that a norm on X is LFC-M it is LFC-M on the set X n f0g. Lemma 93. If in Lemma 23 we assume additionally that f is LFC-M on ˝ for some M X , then the norm ˇ is also LFC-M . Proof. We use the same notation as in the proof of Lemma 23. It is easy to check that if the function f is LFC-M on ˝, then the function F is LFC-N on G, where N D fg B P1 I g 2 M g [ fP2 g .X ˚ R/ , and P1 , P2 denote the canonical projections of X ˚ R onto X, R. Thus in this case is LFC-M by Lemma 83. t u The space c0 plays a prominent role in this theory both as a canonical example of a space with LFC structure and through Corollary 92. The LFC property usually allows for certain smooth constructions. A prime example is the next theorem. Theorem 94. For any set LFC-fe g norm.

the space c0 . / admits an equivalent C 1 -smooth

Proof. Let ' 2 C 1 .R/ be a convex non-negative function such that '.t/ D 0 for t 2 Œ 12 ; 21 and '.1/ D 1. Define a function ˚ W c0 . / ! R by X ˚.x/ D ' x. / :

2

Since the sum is locally finite, the function ˚ is C 1 -smooth and LFC-fe g. Obviously ˚ is convex and separating, since ˚.0/ D 0, while ˚.x/ 1 whenever kxk 1. Now we apply Lemma 93. t u Of course by composing the norm from Theorem 94 with a suitable C 1 -smooth function we can produce a C 1 -smooth LFC bump on c0 . /. Next, we show that on a separable Banach space the existence of a continuous LFC bump already implies the existence of a smooth bump. But first we need a few technical results.

Section 5. Local dependence on finitely many coordinates

285

Fact 95. Let f W R ! R be an even function that is non-decreasing on Œ0; C1/ and let ' W R ! R be an even function with bounded support that is non-increasing on R Œ0; C1/. Then the convolution f '.x/ D R f .x t/'.t/ dt is an even function that is non-decreasing on Œ0; C1/. Proof. Note that f ' is well-defined as the functions f and 'R are clearly Borel measurable and bounded on Rbounded sets. Further, f '. x/ D R f . x t/'.t/ dt D R t/'.t/ dt D f '.x/, using first the fact that f is R f .x C t /'.t/ dt D R f .x even and then the fact that ' is even. ' x 2y t Now pick any 0 x < y < C1. The function .t/ D ' y 2 x t is an odd function (this is obvious), such that .t/ 0 for t 0. Indeed, either we have 0 y 2 x t x 2 y t , or 0 < y 2 x t t x 2 y and in both cases we use the properties of '. Similarly we obtain that the function t 7! f xCy f xCy t 2 Ct 2 is non-negative for t 0. Therefore Z Z f xCy f .t/ '.y t/ '.x t/ dt D f '.y/ f '.x/ D .t/ dt 2 Ct R R Z Z C t .t/ .t/ dt D f xCy dt C f xCy 2 2 Ct . 1;0/ .0;C1/ Z Z dt C f xCy D f xCy t .t/ .t/ dt 2 2 Ct .0;C1/ .0;C1/ Z xCy D f xCy C t f t .t/ dt 0: 2 2 .0;C1/

t u Let X be a Banach lattice. We say that a function f W X ! R is a strongly lattice function if f .x/ f .y/ whenever jxj jyj. Recall that a BanachPspace X with an unconditional basis fej g has a natural lattice structure defined by j1D1 aj ej 0 if and only if aj 0 for all j 2 N, and similarly for `1 . Lemma 96. Let " > 0 and a sequence D fın g1 nD1 , ın > 0, be given. Denote ˚ V D x 2 `1 I jx.n0 /j ın0 > sup jx.n/j C ın0 for some n0 2 N ; n>n0

which is an open subset of `1 .ˇ There is a convex strongly lattice 1-Lipschitz function ˇ ˇ ˇ F W `1 ! Œ0; C1/ such that F .x/ kxk " for any x 2 `1 and F is LFC-fej g and C 1 -smooth on V , where ej are the coordinate functionals on `1 . Moreover, for each x 2 V satisfying kxk 4" there is a neighbourhood Vx of x such that F depends only on a finite subset of fej I j 2 supp xg on Vx . Proof. Let fej g `1 be the canonical basic sequence biorthogonal to fej g and Pn P the projection Pn .x/ D jnD1 ej .x/ej . Set "1 D minfı1 ; "g and "n D minfın ; "n 1 ; n1 g 1 for n > 1. Pick a sequence f'n g1 W R ! Œ0; C1/ nD1 of C -smooth even functions 'n R such that supp 'n Œ "n ; "n , 'n is non-increasing on Œ0; C1/, and R 'n .t/ dt D 1.

286

Chapter 5. Smoothness and structure

Define a sequence fFn g1 nD1 of functions Fn W `1 ! Œ0; 1/ by

n Z n X

Y

Fn .x/ D x t e 'j .tj / dn : j j

Rn

j D1

j D1

ˇ ˇ It is easily checked that each Fn is convex, 1-Lipschitz and ˇFn .x/ kxkˇ " for any x 2 `1 . We show that Fn is strongly lattice by induction. Put F0 D kk and note that F0 is strongly lattice. Assume that Fn 1 is strongly lattice for some n 2 N. Pick any x; y 2 `1 , x D .xj /, y D .yj /, satisfying jxj jyj. Define g W R ! R by g.u/ D Fn 1 .x C .u xn /en /. Then by the inductive hypothesis g is an even function non-decreasing on Œ0; C1/. Using the Fubini theorem we obtain Z Fn .y/ Fn .x/ D Fn 1 .y ten / Fn 1 .x ten / 'n .t/ dt ZR D Fn 1 .y ten / Fn 1 x C .yn xn t/en 'n .t/ dt R Z C Fn 1 x C .yn xn t/en Fn 1 .x ten / 'n .t/ dt Z R Fn 1 .y ten / Fn 1 x C .yn xn t/en 'n .t/ dt D R

C g 'n .yn /

g 'n .xn / 0;

because Fn 1 .y ten / Fn 1 .x C.yn xn t/en /, which follows from the inductive hypothesis (notice that jy ten j jx C .yn xn t/en j in the lattice sense), and also using Fact 95. R Using the Fubini theorem and the fact that R 'j D 1 for any j 2 N we obtain for m > n and any x 2 `1 ˇZ ˇ

! m

m n ˇ ˇ X X Y

ˇ ˇ

x

x

jFm .x/ Fn .x/j D ˇ t e t e ' .t / d ˇ j j j j j j m

ˇ Rm ˇ j D1

Z

j D1

j D1

m

m

X

Y

t e 'j .tj / dm "nC1 : j j

m

R

j DnC1

j D1

It follows that the sequence fFn g converges uniformly on `1 to some function F ˇ W `1 ! Œ0; ˇ C1/. Consequently F is convex, strongly lattice, 1-Lipschitz, and ˇF .x/ kxkˇ " for any x 2 `1 . Fix any x 2 V and let n0 2 N be such that jx.n0 /j ın0 > supn>n0 jx.n/j C ın0 . Note that the set Ux D fy 2 `1 I jy.n0 /j ın0 > supn>n0 jy.n/j C ın0 g V is an open neighbourhood of x. Pick any y 2 Ux and k > n0 . Then ky

tn0 en0 k; R as long as jtj j ın0 for n0 j k. Since "n ın0 for n n0 and R 'n D 1, it follows that Fk .y/ D Fn0 .y/ D Fn0 .Pn0 .y//. This means that F .y/ D Fn0 .Pn0 .y// t 1 e1

tk ek k D ky

t1 e1

tn0 en0 k D kPn0 .y/ t1 e1

Section 5. Local dependence on finitely many coordinates

287

0 and so F depends only on fej gjnD1 on Ux . Using substitution and Corollary 1.91 (similarly as in the proof of Lemma 7.69) it is easy to see that F is C 1 -smooth. Finally, let x 2 V be such that kxk 4" and let Ux be as above. Further, put Vx D Ux \ U.x; "/ and PM D fj 2 supp xI j n0 g. Pick any y 2 Ux and k > n0 , and denote PM .y/ D j 2M ej .y/ej . Then

2" < ky

t1 e1

tk ek k D kPM .y/

t1 e1

tn0 en0 k

whenever jtj j "j for each 1 j k. This follows from the discussion above combined with the fact that jy.j / tj j < 2" for 1 j n0 , j … supp x. Therefore F .y/ D Fn0 .PM .y// and so F depends only on fej I j 2 M g on Vx . t u Let X be a normed linear space, ˝ X open, and f W ˝ ! R. We say that f is locally weakly sequentially continuous (locally wsc for short) if for each x 2 ˝ there is a (norm) neighbourhood U of x such that f U is weakly sequentially continuous. Theorem 97. Let X be a separable Banach space. The following statements are equivalent: (i) X admits a continuous LFC bump. (ii) X is separable and X admits a bump that is locally weakly sequentially continuous. (iii) X admits a C 1 -smooth LFC bump. Proof. (iii))(i) is clear and (i))(ii) follows from Corollary 92 and Fact 82. (ii))(iii) Let ffn gn2N be a dense subset of SX and define a bounded linear operator T W X ! c0 by T .x/ D 2 n fn .x/ n2N . Denote Y D T .X/, which is a linear subspace of c0 . Notice that T is one-to-one and the linear operator T 1 W Y ! X has the following property: T 1 T .M / is kk–w continuous whenever M X is bounded. It follows that if A X is a closed convex bounded set, then so is T .A/ as a subset of a normed linear space Y . Indeed, if fyn g T .A/ is such that yn ! y 2 Y , w then T 1 .fyn g/ [ fT 1 .y/g is bounded and hence T 1 .yn / ! T 1 .y/. As A is closed and convex, it is weakly closed and hence T 1 .y/ 2 A, which in turn implies y 2 T .A/. As X admits a locally wsc bump, by shifting, scaling, and a composition with a suitable continuous function we produce a locally wsc function g W X ! Œ1; 2 such that g.0/ D 1 and g.x/ D 2 whenever kxk 1. By the Lindelöf property of X there exists a countable covering of X by open balls fUn gn2N such that gUn is wsc each Sfor n n 2 N. Without loss of generality we may assume that 0 2 U1 . Put An D j D1 Uj , n 2 N. Since each Uj , j 2 N, is weakly closed, it is easily checked that gAn is wsc for each n 2 N. Now for each n 2 N let us define Gn .y/ D g T 1 .y/ for y 2 T .An /. From the discussion above it follows that T .An / is closed in Y and Gn is continuous on T .An /, hence by Tietze’s theorem G n can be extended to a continuous function Gn W Y ! R. Note that g.x/ D Gn T .x/ for each x 2 An .

288

Chapter 5. Smoothness and structure

Choose a sequence of real numbers fn g decreasing to 1 such that 1 1 C 14 and a decreasing sequence D fın g such that 0 < ın < 14 .n nC1 / and ı1 18 . By Theorem 94 the space Y admits a C 1 -smooth LFC bump. The class of C 1 -smooth LFC functions is an approximation class and hence by Corollary 7.50 and Theorem 7.52 for each n 2 N there is Hn 2 C 1 .Y / which is LFC and satisfies supy2Y jHn .y/ n Gn .y/j < ın . Set hn D Hn BT . Then hn 2 C 1 .X/, hn is LFC by Fact 80, and supx2An jhn .x/ n g.x/j < ın . It follows that hn ! g locally uniformly. Further, let us define ˚ W X ! `1 by ˚.x/ D hn .x/ n . First we show that ˚ is continuous. Pick any x 2 X and " > 0. Let k 2 N be such that x 2 Uk and ık < 4" . There is a neighbourhood U Uk of x such that jg.x/ g.y/j < 4" whenever y 2 U . Then jhn .x/ hn .y/j jhn .x/ n g.x/j C n jg.x/ g.y/j C jn g.y/ hn .y/j " < 2ın C n < " 4 whenever y 2 U and n k. This, together with the fact that h1 ; : : : ; hk 1 are continuous, implies that ˚ is continuous at x. Next, consider the set V from Lemma 96. We claim that ˚.X/ V . Indeed, let x 2 X and n0 2 N be such that x 2 Un0 . Then for n > n0 we have hn0 .x/

ın0 > n0 g.x/

2ın0 > n0 C1 g.x/ C 2ın0

> n g.x/ C ın C ın0 > hn .x/ C ın0 ; where the second inequality follows from the definition of ın0 . We now apply Lemma 96 to the sequence and " D 81 in order to obtain the corresponding function F , and we set f D F B ˚ . The properties of F and hn together with the continuity of ˚ and Fact 79 imply that f is a C 1 -smooth LFC function (see also Lemma 81). Finally, since 0 2 U1 , 1 1 f .0/ D F ˚.0/ k˚.0/k C D sup hn .0/ C 8 8 n 1 1 3 sup n g.0/ C ın C D 1 C ı1 C : 8 8 2 n On the other hand, if kxk 1, then we find n0 such that x 2 Un0 and we get 1 1 1 D sup hn .x/ hn0 .x/ 8 8 8 n 1 7 1 > 2 ı1 : > n0 g.x/ ın0 8 8 4 Therefore f is a separating function on X and we obtain the desired bump by composing f with a suitable C 1 -smooth real function. t u f .x/ k˚.x/k

Let X be a normed linear space. Recall that a set B BX is called a James boundary of X if for each x 2 X there is f 2 B such that kxk D f .x/.

Section 5. Local dependence on finitely many coordinates

289

Definition 98. Let X be a normed linear space and let B be a set of functions on X. We say that B is a generalised James boundary of X if kxk D maxf 2B f .x/ for each x 2 X. The following approximation result is another corollary of Lemma 96. Theorem 99. Let .X; kk/ be a normed linear space and k 2 N [ f1g. If X has a countable James boundary, then the norm kk can be approximated by a C 1 -smooth LFC norm uniformly on bounded sets. If X has a countable equi-continuous generalised James boundary fgn gn2N such that each gn is a convex function C k -smooth on suppo gn , then the norm kk can be approximated by a C k -smooth norm uniformly on bounded sets. Proof. Let ffn I n 2 Ng SX be a countable James boundary of .X; kk/. Choose " > 0, a sequence of real numbers fn g decreasing to 1 such that 1 < 1 C ", and a sequence D fıng such that 0 < ın < 21 .n nC1 /. Let us define T 2 L.X I `1 / by T .x/ D n fn .x/ n . Consider the set V from Lemma 96. We claim that T .˝/ V , where ˝ D fx 2 XI kxk > 1g. Indeed, let x 2 ˝ and n0 2 N be such that fn0 .x/ D kxk. Then for n > n0 we have jn0 fn0 .x/j

ın0 D n0 kxk

ın0 > n0 C1 kxk C ın0

n kxk C ın0 jn fn .x/j C ın0 ; where the first inequality follows from the definition of ın0 . We now apply Lemma 96 to the sequence and " in order to obtain the corresponding function F , and we set f D F B T . Then f is a continuous convex function that is C 1 -smooth and LFC on ˝ (Fact 80). We have f .0/ D F .0/ ". Notice that kxk < kT .x/k 1 kxk < .1 C "/kxk for every x 2 X n f0g (the first inequality follows from the fact that ffn g is a James boundary). Set A D fx 2 XI f .x/ 1 C 2"g. Then for every x 2 A we have kxk kT .x/k F .T .x// C " 1 C 3", hence A is a CCB set. Moreover, for x 2 X satisfying f .x/ D 1 C 2" we have .1 C "/kxk > kT .x/k F .T .x// " D 1 C ", in particular x 2 ˝. Therefore by Lemma 93 there is an equivalent C 1 -smooth and LFC norm on X. Notice also that through the choice of " this norm can be made arbitrarily close to the norm kk. To prove the case of the generalised boundary we choose any 0 < " 14 , put fn D maxfgn ; 0g, and notice that each fn is a non-negative convex function C k -smooth on suppo fn and ffn gn2N is still an equi-continuous generalised James boundary of X. The proof is then almost identical as above with the following changes: T is not a bounded linear operator, but it is a continuous mapping (this follows from the equi-continuity of ffn g). The convexity of f follows easily from the fact that F is convex and strongly lattice, and each fn is convex and non-negative. Finally, for any x 2 ˝ there are a neighbourhood V of T .x/ and a finite M supp T .x/ such that F is C 1 -smooth and depends only on fen I n 2 M g on V (Lemma 96). There is a neighbourhood U of x in ˝ such that T .U / V and supp T .y/ \ M D M for each y 2 U , i.e. U suppo fn for each n 2 M . The C k -smoothness of f on U then follows from Fact 79. We finish by using Lemma 23. t u

290

Chapter 5. Smoothness and structure

6. Isomorphically polyhedral spaces This section deals with spaces isomorphic to polyhedral spaces. The main structural theorem gives several characterisations of separable isomorphically polyhedral Banach spaces, for example the existence of a renorming with a countable James boundary or the existence of a (smooth) LFC renorming. We also show that every separable polyhedral Banach space admits an equivalent analytic norm and conversely if a Banach space with an analytic norm has all polynomials weakly sequentially continuous, then it is a separable isomorphically polyhedral space. A closed convex bounded subset of a finite-dimensional normed linear space is called a (convex) polytope if it has only finitely many extreme points, or equivalently if it is a convex hull of finitely many points, or equivalently if it is a bounded intersection of finitely many closed half-spaces (see e.g. [Zie, Theorem 1.1]). We say that a normed linear space X is polyhedral, if the unit ball of every finitedimensional subspace of X is a polytope. Proposition 100 (Victor Klee). Let X be a normed linear space. The following statements are equivalent: (i) X is a finite-dimensional polyhedral space. (ii) X is a finite-dimensional polyhedral space. (iii) Ext BX is finite. (iv) X has a finite James boundary. Proof. (ii))(iii))(iv) are obvious. (iv))(i) Let T B be a finite James boundary of the space T X. It is easy to see that BX D Bı D f 2B fx 2 XI f .x/ 1g. In particular, f 2B ker f D f0g, hence X must be finite-dimensional. Since BX is an intersection of finitely many half-spaces, it is a polytope. (i))(ii) Using the reflexivity of X we obtain that X is a finite-dimensional polyhedral space. Now we can use the already proved implication (ii))(i). t u Lemma 101 ([GlMc]). Let X be a normed linear space. If there is a 1-norming set B BX such that no w -accumulation point of B with norm 1 attains its norm, then B is a James boundary of X and X is polyhedral. Proof. To see that B is a James boundary of X, choose x 2 SX . There is ffn g B such that fn .x/ ! 1. By the w -compactness of BX the sequence ffn g has a w -cluster point f 2 BX . Then f .x/ D 1 and consequently kf k D 1. It follows that f is not a w -accumulation point of B, hence f 2 B. Suppose further that X is not polyhedral. Then there is a finite-dimensional subspace Y X that is not polyhedral. Since M D ff Y I f 2 Bg \ SY is a James boundary of Y , the set M is infinite by Proposition 100. By the compactness of BY there is a sequence ffn g of distinct elements of M such that fn ! g 2 Y . Hence g 2 SY and g attains its norm (by the compactness of BY ). For each n 2 N there is hn 2 B such that fn D hn Y . By the w -compactness of BX the sequence fhn g has a w -cluster point h 2 BX , which is also a w -accumulation point of B, as all hn s are

Section 6. Isomorphically polyhedral spaces

291

distinct. Clearly hY D g. Therefore khk D 1 and h attains its norm, a contradiction with the assumptions. t u Let X be a normed linear space. We say that F SX is a facet of BX if there is a closed hyperplane H X supporting BX such that F D BX \ H and F has a non-empty relative interior in H . Lemma 102 ([Ve]). Let X be a polyhedral Banach space and x 2 SX . The following statements are equivalent: (i) The point x is an interior point of a facet of BX relative to the hyperplane defining the facet. (ii) There is f 2 SX such that kyk D f .y/ for y from a neighbourhood of x. (iii) The norm is Gâteaux differentiable at x. Proof. (i))(ii))(iii) are easy to see. (iii))(i) Put f D ıkk.x/ and Z D ker f . Then H D x C Z is a hyperplane supporting BX . We show that x is an interior point of F D BX \ H in H . Put C D F x, which is a closed convex subset of Z. Fix h 2 SZ . The space Y D spanfx; hg is a two-dimensional polyhedral space and so BY is a polygon. Since kkY is Gâteaux differentiable at x, SY contains a line segment L such that x is its interior point (Lemma 11). It follows that L F and so L x C , in particular t h 2 C for all t sufficiently small. This shows that C is absorbing in Z and so it has a non-empty interior in Z by Lemma 1.134. Now suppose that 0 is not an interior point of C in Z. Then by the separation theorem there is g 2 Z such that g.y/ 0 for each y 2 C . This however contradicts the fact that C is absorbing. t u Note that if x 2 X and f 2 BX , then f .x/ D kxk if and only if f 2 @kk.x/. Therefore B BX isS a James boundary of X if and only if @kk.x/ \ B ¤ ; for each x 2 SX . In particular, x2SX @kk.x/ is a James boundary of X. Let X be a normed linear space, ˝ X open, and f W ˝ ! R. We say that f is locally weakly sequentially Cauchy-continuous (locally wsC for short) if for each x 2 ˝ there is a (norm) neighbourhood U of x such that f U is weakly sequentially Cauchy-continuous. Theorem 103. Let X be a Banach space. The following statements are equivalent: (i) X is a separable isomorphically polyhedral space. (ii) X admits an equivalent norm with a countable James boundary. (iii) X admits an equivalent norm with a K James boundary. (iv) X is separable and admits an equivalent LFC norm. (v) X is separable and admits an equivalent norm which is C 1 -smooth and LFC. (vi) X is separable, does not contain `1 , and admits an equivalent norm that is locally weakly sequentially Cauchy-continuous on X n f0g. Proof. (v))(iv) is obvious. (iv))(iii) Let kk be an equivalent LFC norm on X. By the Lindelöf property of SX there exist a countable family of finite sets fMn gn2N , Mn X , and a countable

292

Chapter 5. Smoothness and structure

covering fUn gn2N of SX by open balls such that kk depends only on Mn on Un . For each n 2 N set Kn D .span Mn / \ SX , which is clearly a compact set. Now choose any x 2 SX . Let T g 2 SX be such that g.x/ D 1 and let n 2 N be such that x 2 Un . For any h 2 f 2Mn ker f sufficiently small so that x ˙ h 2 Un we have kx ˙ hk D kxk D 1. Hence T ˙g.h/ D g.x ˙ h/ g.x/ kx ˙ hk 1 D 0. It follows that h 2 ker g. Thus f 2Mn ker f ker g, which in turn implies that g 2 Kn . S Therefore 1 nD1 Kn is a K James boundary of .X; kk/. (iii))(ii) Let kk be an equivalent S norm on X with a K James boundary and let Kn SX be compact sets such that 1 nD1 Kn is a James boundary of .X; kk/. Take a sequence of real numbers f"S g 1 such that "n ! 0. Let n .0; Mn be a finite 1 ˚ "n of K . We put B D ˙.1 C " /f I f 2 M -dense subset n n n and define nD1 4 jjjxjjj D sup f .x/

(14)

f 2B

for x 2 X. We claim that jjjjjj is an equivalent norm on X such that B is a James boundary of .X; jjjjjj/. Choose any x 2 SX . Clearly, jjjxjjj 2. On the other hand, there are j 2 N " and g 2 Kj such that g.x/ D 1. Let h 2 Mj be such that kg hk < 4j . Then "j h.x/ > 1 4 and consequently "j "j 1C : .1 C "j /h.x/ > .1 C "j / 1 4 2 "

It follows that jjjxjjj > 1 C 2j and so jjjjjj is an equivalent norm on X . Furthermore, " since "n ! 0, there is n0 2 N such that .1 C "n /f .x/ 1 C 2j for n n0 and any f 2 SX . Thus the supremum in (14) is actually over a finite set and hence it is attained. This proves our claim. (ii))(i) Let ffn I n 2 Ng SX be a countable James boundary of .X; kk/ and put Q D f˙fn I n 2 Ng. By Rodé’s theorem (see [FHHMZ, Theorem 3.122]) BX D conv Q, hence both X and X are separable. Choose a sequence of real numbers fn g decreasing to 1 and put gn D n fn . Let B D f˙gn I n 2 Ng. Define a new norm on X by the formula jjjxjjj D supn2N jgn .x/j. Then kxk jjjxjjj 1 kxk for each x 2 X, so jjjjjj is an equivalent norm on X. Clearly B is a 1-norming set for .X; jjjjjj/. Suppose that g 2 X is a w -accumulation point of B. We claim that g does not attain its norm jjjgjjj. It is easy to see that g is also a w -accumulation point of Q and therefore kgk 1. Choose any x 2 X with jjjxjjj D 1. There is k 2 N such that fk .x/ D kxk. Thus g.x/ kgkkxk kxk D fk .x/ < gk .x/ jjjxjjj D 1: So the space .X; jjjjjj/ is polyhedral by Lemma 101. (i))(ii) Suppose that .X; kk/ is a polyhedral space. We show that SX can be covered by facets of BX . The set B D ff 2 SX I f 1 .1/ \ BX is a facet of BX g is then a (minimal) James boundary of X. From the separability of SX it follows that it can be covered only by countably many facets, hence B is countable.

Section 6. Isomorphically polyhedral spaces

293

x Let S be the union of all the facets of BX and Q D fx 2 X n f0gI kxk … S g. Obviously Q is a cone, i.e. ty 2 Q whenever y 2 Q and t > 0. We claim that for any u u 2 Q the set Q u is absorbing. So fix u 2 Q and put v D kuk . Choose any ´ 2 SX , ´ ¤ ˙v, and put Y D spanfv; ´g. Since Y is two-dimensional and X is polyhedral, BY is a polygon. As v 2 SY , it follows that there is w 2 SY , w ¤ v, such that the line segment Œv; w lies in SY , and w and ´ lie in the same half-plane in Y relative to spanfvg. Now it is not difficult to see that the line segment Œv; w/ 2 Q. Indeed if some y 2 .v; w/ belongs to a facet of BX , then the hyperplane defining this facet must contain the line segment Œv; w, a contradiction with v 2 Q. From this and the fact that Q is a cone we easily obtain that there is ı > 0 such that u C t´ 2 Q for t 2 Œ0; ı. This shows that Q u is absorbing. The norm kk is not Gâteaux differentiable at any point of Q (Lemma 102). So by Theorem 4 the set Q is contained in an F set with an empty interior, which means according to Lemma 1.134 that there is no u 2 Q such that Q u is absorbing. Therefore Q is empty. (ii))(v) The separability of X was already discussed above. The rest follows from Theorem 99. (iv))(vi) follows from Corollary 92 and Fact 82. S (vi))(iii) Proposition 41 combined with the Lindelöf property of SX implies that x2SX @kk.x/ is a countable union of relatively compact sets. Thus .X; kk/ has a K James boundary. t u

Theorem 104 ([DFH2]). Every separable isomorphically polyhedral Banach space X admits an equivalent analytic norm. Proof. By Theorem 103 we can consider X with an equivalent norm for which there is ffn I n 2 Ng SX a countable James boundary of X. Define a function ˚ W U.0; 1/ ! R by 1 X ˚.x/ D fn .x/2n : nD1

Obviously this is a convex function that is analytic (Theorem 1.168, see also Example 1.137). Let a 2 .0; 1/ and set A D fx 2 U.0; 1/I ˚.x/ ag. The set A is closed. Indeed, let fxn g A be such that xn ! x 2 X. Then obviously x 2 BX . Suppose that x 2 SX . Let pn 2 N be such that fn .x/ D 1. There is a neighbourhood V of x such that fn .y/ > 2n a for y 2 V . Hence ˚.y/ fn .y/2n > a for y 2 V \ U.0; 1/, which contradicts the fact that ˚.xn / a for all n 2 N. It follows that x 2 U.0; 1/ and so x 2 A by the continuity of ˚. Now we apply Lemma 23. t u The following is a corollary of Theorem 103. Corollary 105 ([HáTr]). Let X be a Banach space with an analytic norm. If all polynomials on X are weakly sequentially continuous, then X is separable and isomorphically polyhedral. For every x 2 X , x ¤ 0 there exists a neighbourhood U of x such that Proof. P kyk D 1 x/ for all y 2 U , where Pn 2 P . nX/, and the sum converges nD0 Pn .y

294

Chapter 5. Smoothness and structure

uniformly on U . Since each Pn is wsC (Corollary 3.27), it follows that kk is locally wsC on X n f0g. Thus X is separable by Proposition 77 and we may apply Theorem 103. t u

7. Lp spaces In this section we study in detail the smoothness of Lp spaces. We begin with a classical positive result of Robert Bonic and John Frampton on the smoothness of Lp ./ spaces. Then we prove some negative results on smoothness of `p spaces (which of course translate immediately to general Lp spaces). These show that the results of Bonic and Frampton are sharp in a strong sense. Then we proceed with the study of the smoothness of norms and bumps in the Lebesgue-Bochner spaces Lp .I X/, proving some basic results on the differentiability of homogeneous mappings along the way. Finally we show some sharper results on smoothness in Lp I Lq ./ spaces. Theorem 106 ([BF]). Let be a measure and 1 < p < C1. Let ˚ D kkp , where kk is the canonical norm of X D Lp ./. (i) If p is an even integer, then ˚ is a p-homogeneous polynomial. (ii) If p is an odd integer, then ˚ 2 C p 1;1 .X/. (iii) If p is not an integer, then ˚ 2 C Œp;p Œp .X/. R Proof. (i) For any f1 ; : : : ; fp 2 X we put M.f1 ; : : : ; fp / D f1 fp d. By repeated use of the Hölder inequality we can check that the integral exists. Obviously . Hence ˚ 2 P . pX/. M 2 L. pX / and ˚ D M (ii), (iii) Define ' W R ! R by '.t/ D jtjp . Let k D p 1 in the case (ii) and k D Œp in the case (iii). The function ' is C k;p k -smooth on R, i.e. there is C > 0 such that ˇ .k/ ˇ ˇ' .x/ ' .k/ .y/ˇ C jx yjp k for all x; y 2 R. (15) ˇ ˛ ˇ ˇ ˛ ˇ ˛ ˛ ˛ This follows from the fact that ˇjuj jvj ˇ ju vj and ˇjuj sgn u jvj sgn v ˇ 2ju vj˛ for all u; v 2 R and ˛ 2 .0; 1, which can be proved by standard analysis. By Corollary 1.108 ˇ ˇ k X ˇ ˇ 1 1 .j / jˇ p ˇ'.x C h/ ' .x/h for all x; h 2 R. (16) ˇ kŠ C jhj ˇ jŠ j D0

Z

For x; h 2 X and j 2 f0; : : : ; kg we have by the Hölder inequality Z pp j Z pj ˇ .j / ˇ ˇ .j / ˇ pp j jˇ p ˇ' ˇ ˇ jh.t/j d.t/ ' x.t/ x.t / h.t/ d.t/ d.t /

j C 1/kxkp j khkj : R Thus for any x 2 X and j 2 f0; : : : ; kg we may set Pjx .h/ D ' .j / x.t/ h.t/j d.t/ for h 2 X. Similarly as in (i) and together with the above estimate we can see that D p.p

1/ .p

Section 7. Lp spaces

295

Pjx 2 P . jX/. Using (16) we obtain ˇ ˇ ˇ˚.x C h/ ˇ

ˇ Z k X 1 x ˇˇ P .h/ˇ jŠ j

j D0

ˇ ˇ ˇ' x.t/ C h.t/ ˇ

1 C kŠ

Z

ˇ k X ˇ 1 .j / jˇ ' x.t/ h.t/ ˇ d.t/ jŠ

j D0

jh.t/jp d.t/ D

Now Theorem 1.125 implies that ˚ 2 C k;p

1 C khkp : kŠ

k .X/.

t u

Theorem 107. (i) If 1 p < C1 and p is not an even integer, then `p does not admit a T p -smooth bump. (ii) If 1 p < 2, then `p does not admit a continuous bump ˇ with the property that for each x 2 `p there is q > p such that ˇ is directionally T q -smooth at x. (iii) If 1 p < C1, p is not an even integer, and is an uncountable set, then `p . / does not admit a continuous weakly T p -smooth bump. Note that it means in particular that `p , 1 p < 2, does not admit a continuous G 2 -smooth bump, and `p . /, uncountable, 1 p < C1, p not an even integer, does not admit a continuous G dpe -smooth bump. Proof. The first part of the proof is the same for all three statements (i), (ii), and (iii). Denote by X the space `p in the case (i) or (ii), or `p . / in the case (iii). By S-smoothness we denote the smoothness of the type given in (i), (ii), or (iii) respectively. Let 1 ' W X ! R[fC1g be defined as '.x/ D kxk p for x ¤ 0 and '.0/ D C1. Assuming that X admits an S-smooth bump, by Corollary 59 (note that `1 . / has the RNP by Theorem 2) and Lemma 60 the space X admits also an S-variational principle. Thus there are an S-smooth function f W X ! R and x0 2 X such that f .x0 / D '.x0 / and f .x/ '.x/ for every x 2 X. It follows that x0 ¤ 0 and f .x0 / > 0, and so there is a neighbourhood U of x0 such that f > 0 on U . Hence g D f1 is S-smooth on U (Proposition 1.115), g.x0 / D kx0 kp and g.x/ kxkp for every x 2 U . (i) By the T p -smoothness there are P 2 P Œp .X/ satisfying P .0/ D 0, and ı > 0 such that g.x0 Ch/ g.x0 / P .h/ 21 khkp for khk ı. Put Q.h/ D P .h/CP . h/. Since p is not an even integer, deg Q < p. Further, Q.h/ g.x0 C h/ C g.x0 kx0 C hkp C kx0

h/ hkp

2g.x0 / 2kx0 kp

khkp khkp

for khk ı. The polynomial Q is weakly sequentially continuous (Corollary 3.59), and so limn!1 Q.ıen / D 0. (Note also that in case that p < 2 simply Q D 0). By the properties of the `p -norm we have limn!1 kx0 ˙ ıen kp D kx0 kp C ı p . Putting everything together we obtain 0 D lim Q.ıen / lim kx0 C ıen kp C kx0 ıen kp 2kx0 kp ı p D ı p ; n!1

n!1

which is a contradiction.

296

Chapter 5. Smoothness and structure

(ii) Let q > p be such that g is directionally T q -smooth at x0 . Without loss of generality we may assume that q < 2. For a given h 2 X there is a 2 R such that g.x0 C t h/ D g.x0 / C at C o.jtjq /; t ! 0. Since kx0 C t hkp C kx0

thkp

2kx0 kp g.x0 C th/ C g.x0

th/

2g.x0 /

q

D o.jtj /; t ! 0; it follows that there is > 0 such that 1 kx0 C thkp C kx0 tq

thkp

2kx0 kp 1

for t 2 .0; /. Since the expression on the left-hand side is obviously bounded for t 2 Œ; 1, we can conclude that 1 kx0 C thkp C kx0 thkp 2kx0 kp < C1 q t 2.0;1 t S for each h 2 X. Therefore X D 1 .h/ ng. Since is lower seminD1 fh 2 XI continuous, each of the sets fh 2 XI .h/ ng is closed and so by the Baire category theorem there are C > 0 and an open set V X such that .h/ C for h 2 V . Since is even, .h/ C also for h 2 V . The convexity of implies that there is ı > 0 such that .h/ C for h 2 B.0; ı/. Now similarly as in the proof of (i) we obtain 2t p ı p D lim kx0 C tıen kp C kx0 tıen kp 2kx0 kp C t q .h/ D sup

n!1

for all t 2 .0; 1, which is a contradiction. (iii) By the weak T p -smoothness there is a polynomial P 2 P Œp .X/ satisfying P .0/ D 0 and g.x0 C th/ g.x0 / P .th/ D o.jtjp /; t ! 0 for each h 2 X. Put Q.h/ D P .h/ C P . h/. Since p is not an even integer, deg Q < p, and hence each homogeneous summand of Q is countably supported (Corollary 3.62). Since also supp x0 is countable, there is 2 n supp x0 such that Q.te / D 0 for all t 2 R. Therefore, 2jt jp D kx0 C te kp C kx0 g.x0 C te / C g.x0 p

te kp te /

2kx0 kp 2g.x0 /

p

D Q.te / C o.jtj / D o.jtj /; t ! 0; which is a contradiction.

t u

We note that the statement (i) for p > 2 follows also from Theorem 62 and Corollary 4.46, and the statement (iii) for p > 2 follows also from Theorem 76 and Corollary 3.59. Notice that for p 2 N odd there is an equivalent norm on `p that is C p 1;1 -smooth and G p -smooth (variants of Corollary 7.72, Corollary 7.70), while there is no continuous G p -smooth bump on `p . / for uncountable (Theorem 107(iii)).

Section 7. Lp spaces

297

Theorem 108 ([LeSu]). Let .˝; / be a measure space with an infinite family of pairwise disjoint measurable subsets of ˝ of finite positive measure. Further, let X be a Banach space, 1 < p < C1, and k 2 N, k < p. The following statements are equivalent: (i) The norm of X is C k -smooth and its kth differential is bounded on SX . (ii) The canonical norm of the space Lp .I X/ is C k -smooth. (iii) The canonical norm of the space Lp .I X/ is C k -smooth with d jkk bounded on Lp .I X / n B.0; r/ for each r > 0, j 2 f1; : : : ; kg. We note that the assumption on the measure space is equivalent to Lp ./ (the scalar Lebesgue space) being infinite-dimensional and also equivalent to the fact that Lp .I X / contains a subspace isometric to `p .NI X/. Before proving the theorem we gather a few facts about the differentiability of p-homogeneous mappings, i.e. the mappings f satisfying f .x/ D p f .x/ for all > 0. Fact 109. Let X , Y be normed linear spaces, p 2 R, and let f W X n f0g ! Y be a p-homogeneous mapping that is k-times Fréchet differentiable at every x 2 SX . Then f is k-times Fréchet differentiable on X n f0g and d jf .x/ D p j d jf .x/ for each x 2 X n f0g, > 0, and j 2 f1; : : : ; kg. Proof. Given x 2 X , x ¤ 0, we have f .x C h/ f .x/ p 1 Df .x/Œh D p f .x C 1 h/ f .x/ Df .x/Œ 1 h D o.khk/; h ! 0. The rest follows by induction. t u Fact 110. Let X, Y be normed linear spaces, p 2 .1; C1/, let f W X ! Y be a p-homogeneous mapping with f .0/ D 0, and k 2 N, k < p. (i) For each 1 q < p the mapping f is weakly T q -smooth at 0 with the approximating polynomial being zero. (ii) For each 1 q < p the mapping f is T q -smooth at 0 if and only if f is bounded on SX . (iii) If f is G k 1 -smooth on X n f0g, then f is G k -smooth at 0 with ı jf .0/ D 0, j D 1; : : : ; k. Proof. (i) and (ii) follow directly from the definitions. (iii) follows by induction: lim t !0 1t ı j 1f .th1 /Œh2 ; : : : ; hj D lim t !0 t p j ı j 1f .h1 /Œh2 ; : : : ; hj D 0, using an analogue of Fact 109 for Gâteaux derivatives. t u Fact 111. Let X, Y be normed linear spaces, p 2 .1; C1/, f W X ! Y a p-homogeneous mapping, and k 2 N, k < p. The following statements are equivalent: (i) f 2 C k .X I Y /. (ii) f 2 C k .X I Y / and d jf is bounded on bounded sets for all j 2 f1; : : : ; kg. (iii) The mapping f is k-times Fréchet differentiable at the points of SX , d kf is continuous and bounded on SX , and f .0/ D 0.

298

Chapter 5. Smoothness and structure

Proof. (i))(ii) Let j 2 f1; : : : ; kg. Since d jf is continuous at 0, it is bounded on a neighbourhood of 0. By Fact 109 it is bounded on any bounded set. Note also that from Fact 109 it follows that d jf .0/ D 0. (ii))(iii) Since f is continuous, the p-homogeneity implies f .0/ D 0. The rest is obvious. (iii))(i) From Fact 109 it follows that f 2 C k .X n f0gI Y / and d kf is bounded on BX n f0g. Using repeatedly Proposition 1.72 we conclude that the mappings d k 1f; : : : ; df; f areLipschitz (and therefore bounded) on BX n f0g. It follows that h f .h/ D khkp f khk D o.khk/; h ! 0, which means that df .0/ D 0. Since by Fact 109 the mapping df is .p 1/-homogeneous, the rest follows by induction. u t Lemma 112. Let X be a normed linear space, dim X 2, and x; y 2 X, kxk kyk. There are points u; v 2 X such that the line segments Œx; u, Œu; v, and Œv; y lie in X n U.0; 13 kxk/ and ku xk C kv uk C ky vk 3ky xk. Proof. If Œx; y \ U.0; 13 kxk/ D ;, then the assertion is trivial, so assume the contrary. Let Z X be a two-dimensional subspace containing x and y. Let v 2 Œx; y be such that kvk D kxk and v ¤ x. From the convexity and the choice of v we get k´k kxk for all ´ 2 Œv; y. Let f 2 SZ be such that f .v x/ D 0 and f .x/ 0. By the compactness of BZ .0; kxk/ there is u 2 Z with f .u/ D kuk D kxk. For any t 2 Œ0; 31 we have kt uC.1 t/xk .1 t/kxk t kuk D .1 2t/kxk 31 kxk, while for t 2 Œ 31 ; 1 we have kt u C .1 t/xk f .tu C .1 t/x/ D t kxk C .1 t/f .x/ tkxk 31 kxk. Since f .v/ D f .x/ 0, we can show analogously that Œu; v \ U.0; 13 kxk/ D ;. Because Œx; v \ U.0; 13 kxk/ ¤ ;, it follows that kv xk 43 kxk. Therefore ku xkCkv ukCky vk 4kxkCky vk 3kv xkCky vk 3ky xk. u t Fact 113. Let X be a normed linear space, p 2 .1; C1/, '.x/ D kxkp , and k 2 N, k < p. Then ' 2 C k .X/ if and only if the norm is C k -smooth and its kth differential is bounded on SX . Proof. ) Follows from Fact 111 and Corollary 1.117. ( Using repeatedly Fact 109 and Proposition 1.71 together with Lemma 112 we conclude that d k 1kk; : : : ; d kk are Lipschitz (and therefore bounded) on X n B.0; 12 /. So ' 2 C k .X nf0g/ and d k' is bounded on SX by Corollary 1.117. Hence ' 2 C k .X/ by Fact 111. t u The proof of Theorem 108 is based on the following observation. Lemma 114. Let .˝; / be a finite measure space, X a normed linear space, and let f W ˝ ! X be a measurable mapping. Then for each " > 0 there is a measurable set Q ˝ such that f .Q/ is totally bounded and .˝ n Q/ < ". Proof. Let 'n W ˝ ! X be simple measurable mappings such that 'n ! f almost everywhere. Let " > 0 be given. By Egorov’s theorem there is a measurable set Q ˝ such that 'n ! f uniformly on Q and .˝ n Q/ < ". Therefore for every ı > 0 there is n 2 N such that the set 'n .Q/ is a finite ı-dense set for f .Q/. t u

Section 7. Lp spaces

299

Proof of Theorem 108. (i))(iii) We will show that the function ˚.x/ D kxkp is C k -smooth on Lp .I X/ using Theorem 1.110. Let ' W X ! R be defined as '.u/ D kukp . Then ' 2 C k .X/ and d j' are bounded by M > 0 on SX for all j 2 f1; : : : ; kg by Fact 113 and Fact 111. Hence d j'.u/ M kukp j for each u 2 X , j 2 f1; : : : ; kg by Fact 109. For x; h 2 Lp .I X/ and j 2 f1; : : : ; kg we have by the Hölder inequality Z pp j Z pj Z

j

j p p

d ' x.t / Œh.t/ d.t/

d ' x.t/ p j d.t / kh.t/k d.t/ Z

M kx.t /kp

j

pp j

pp j khkj D M kxkp d.t/

j

khkj :

It followsR that for agiven x 2 Lp .I X/ and j 2 f0; : : : ; kg we may define Pjx .h/ D d j' x.t/ Œh.t/ d.t/ for every h 2 Lp .I X/. Similarly as in the proof of Theorem 106 and together with the above estimate it is easily seen that Pjx 2 P . jLp .I X//. To estimate the remainder of the approximation by the polynomials as required in Theorem 1.110 we first use Corollary 1.108 and the Hölder inequality to obtain ˇ ˇ k X ˇ 1 y ˇˇ ˇ P .h/ˇ jR.y; h/j D ˇ˚.y C h/ jŠ j j D0

ˇ k X ˇ 1 j (17) d ' y.t/ Œh.t/ˇˇ d.t/ jŠ j D0 Z

1 sup d k' y.t/ C h.t/ d k' y.t/ kh.t/kk d.t/ kŠ 2Œ0;1 !pp k p Z

k p k 1 k khkk d ' y.t/ sup d ' y.t/ C h.t/ d.t/ kŠ 2Œ0;1 Z ˇ ˇ ˇˇ' y.t/ C h.t/

for any y; h 2 Lp .I X/. Now fix x 2 Lp .I X/ (or more precisely fix a concrete representative of the equivalence class x, which will be denoted by x again) and choose an arbitrary p " p k "R > 0. Set "1 D 621 p 2M . There is a measurable set A1 ˝ such that p ˝ D ˝ n A1 . (Take a simple A1 kx.t /k d.t/ < "1 and .˝1 / < C1, where R R 1 d and non-negative function .t/ kx.t/kp for which kx.t/kp d "1 < 1 .0/.) set A1 D R Further, there is > 0 such that A kx.t/kp d < "1R whenever A ˝ is measurable with .A/ < . This follows from the fact that A 7! A kx.t/kp d is a finite measure absolutely continuous with respect to . By Lemma 114 there is a measurable set Q ˝1 such R that K D x.Q/ is compact and .A2 / < , where A2 D ˝1 n Q. This means that A2 kx.t/kp d.t/ < "1 . By the compactness of K there is > 0 such

300

Chapter 5. Smoothness and structure

pp k " whenever u; v 2 X , dist.u; K/ < 2, that kd k'.u/ d k'.v/k < 2.Q/ dist.v; K/ < 2, and ku vk < . (Here we consider 10 D C1.) Let ı > 0 be such 1 ı p / < and 2ı < .3"1 / p . that 2. Now let y; h 2 Lp .I X/ be such that ky xk < ı and khk < ı (here we again choose concrete representatives of the equivalence classes for y and h). Set ˚ Q1 D t 2 QI ky.t/ x.t/k < ; kh.t/k < and A3 D Q n Q1 . It is easy to see that .A3 / .ft 2 QI ky.t/ x.t/k g/ C .ft 2 QI kh.t/k g/ R p ky xkp ı p p C khk p p < 2. / < . Therefore A3 kx.t/k d.t/ < "1 . We can finally estimate the remainder (17). Note that for t 2 Q1 and 2 Œ0; 1 we have dist.y.t /; K/ ky.t/ x.t/k < and dist.y.t/ C h.t/; K/ < 2. Hence p Z

k p k k d.t / sup d ' y.t/ C h.t/ d ' y.t/ Q1 2Œ0;1 (18) p .Q1 / 1 p "p k : "p k 2.Q/ 2 Next, for any t 2 ˝ and 2 Œ0; 1 we have by Fact 109

k

d ' y.t / C h.t/ d k' y.t/ d k' y.t/ C h.t/ C d k' y.t/ p M ky.t/ C h.t/kp k C ky.t/kp k 2M ky.t/k C kh.t/k

k

:

So, R settingpB D A1 [ A2 [ A3 and using the Minkowski inequality and the fact that Ai kx.t /k d.t/ < "1 , i D 1; 2; 3, we obtain Z B

p p k

d ' y.t/ d.t /

sup d k' y.t/ C h.t/

k

2Œ0;1

.2M /

p p k

.2M / p < .2M / p

Z B

ky.t /k C kh.t/k

Z

p k

B p k

p

kx.t /kp d.t/

d.t/ !p

p1 C ky

xk C khk

p p 1 1 p .3"1 / p C 2ı < .2M / p k 2p 3"1 D " p k : 2

1 This, together with (18) and (17) gives jR.y; h/j < kŠ "khkk for any y; h 2 Lp .I X/ satisfying ky xk < ı and khk < ı. Therefore by Theorem 1.110 the function ˚ is C k -smooth on Lp .I X/ and d j˚, j D 1; : : : ; k, are bounded on bounded sets by Fact 111. Hence by Corollary 1.117 the canonical norm on Lp .I X/ is C k -smooth with d jkk bounded on SLp .IX/ , j D 1; : : : ; k. Finally, d jkk are bounded on Lp .I X / n B.0; r/ for each r > 0, j 2 f1; : : : ; kg by Fact 109. (iii))(ii) is obvious. (ii))(i) By the assumption the spaces X and `p .NI X/ are isometric to certain subspaces of Lp .I X/, hence their norms are C k -smooth. From now on we will work

Section 7. Lp spaces

301

in the space `p .NI X/. Let ' W X ! R be defined as '.x/ D kxkp and similarly P k ˚ W `p .NI X / ! R as ˚.x/ D kxkp D 1 nD1 '.xn /. Note that ' 2 C .X n f0g/, ˚ 2 C k .`p .NI X/ n f0g/, and both ' and ˚ are G k -smooth and T k -smooth at 0 (Fact 110). Fix x; h 2 `p .NI X/ such that x ¤ 0 and supp h f1; : : : ; N g for some N 2 N. Then ˚.x C th/ D

N X

'.xn C thn / C

nD1

D

D

1 X

'.xn /

nDN C1

N X

k X tj

nD1

j D0

jŠ

ı2 '.x /Œh C o.t j

n

n

! k

/ C

1 X

'.xn /

nDN C1

2

N k 1 X X tj X j ı '.xn /Œhn C '.xn / C o.t k /; t ! 0: jŠ

j D0

nD1

nDN C1

Note that the Gâteaux derivatives are there to cater for the case that some xn D 0. P j Since ˚.x C t h/ D jkD0 tj Š d j˚.x/Œh C o.t k /; t ! 0, from the uniqueness of the P j approximating polynomials we obtain d j˚.x/Œh D N nD1 ı '.xn /Œhn , j D 1; : : : ; k. j The continuity of d ˚.x/ and the density of finitely supported vectors in `p .NI X/ P j imply that d j˚.x/Œh D 1 nD1 ı '.xn /Œhn for every x; h 2 `p .NI X/, x ¤ 0, and j 2 f1; : : : ; kg. Now assume that d k' is not bounded on SX . Then there are vectors xn ; hn 2 SX n n Q p 1 such that d k'.xn /Œhn 2n . Let xQ D .xn =2 p /1 nD1 and h D .hn =2 /nD1 . Then Q D 1, and x; Q hQ 2 `p .NI X /, kxk Q D khk 1 X xn hn k k Q d ˚.x/Œ Q h D d ' n n p 2 2p nD1 1 1 X X 1 n 1 p k 1 k k d '.x /Œh D 2 D C1; n n n n n 2 p p 2 2 nD1 nD1

2

2

a contradiction. Thus d k' is bounded on SX and by Fact 111 and Corollary 1.117 also d kkkX is bounded on SX . t u Note that if X is the space c0 with a C 1 -smooth norm, then by Theorem 108 and Proposition 49 the canonical norm of `2 .NI X/ is not even C 2 -smooth. Proposition 115. Let be a measure with at least two disjoint sets of finite nonzero measure, X a Banach space, and 1 p < C1. The following statements are equivalent: (i) The canonical norm of Lp .I X/ is weakly T p -smooth. (ii) p is an even integer and the norm of X is p-polynomial. (iii) p is an even integer and the canonical norm of Lp .I X/ is p-polynomial.

302

Chapter 5. Smoothness and structure

Proof. (i))(ii) By the assumption on the space Lp .I X/ contains a subspace isometric to X ˚p R and so Corollary 43 applies. (ii))(iii) For y 2 X denote Q.y/ D kykp . By the Polarisation formula we have !ˇ Z Z ˇˇ p ˇ X X ˇ ˇ 1 ˇ ˇ ˇQ } x1 .t /; : : : ; xp .t/ ˇ d.t/ D ˇ " " Q " x .t/ ˇ d.t/ 1 p j j ˇ 2p pŠ ˇ j D1 "j D˙1 ˇ

p

p !ˇ p

ˇ X Z ˇˇ X X 1 1

X

ˇ Q p " x .t/ " x d.t/ D ˇ

ˇ j j j j ˇ

ˇ 2 pŠ 2p pŠ "j D˙1

j D1

"j D˙1 j D1

} x1 .t/; : : : ; xp .t/ d.t/ for x1 ; : : : ; xp 2 Lp .I X/. Thus M.x1 ; : : : ; xp / D Q exists and obviously M 2 L. pLp .I X//. It follows that the canonical norm of Lp .I X / is p-polynomial. (iii))(i) is clear. t u R

Corollary 116. Let X be a Banach space. If the canonical norm of `2 .NI X/ is weakly T 2 -smooth, then X is a Hilbert space. Almost the same proof as that of Theorem 106 gives the following proposition. Proposition 117 ([DGJ]). Let be a measure, X; Y Banach spaces, 1 < p < C1, and let f 2 C k .XI Y / be a p-homogeneous mapping such that d kf is ˛-Hölder on SX and k C ˛ p. Then the mapping F W Lp .I X/ ! Y defined by Z F .x/ D f x.t/ d.t/ is a C k -smooth p-homogeneous mapping with d kF being ˛-Hölder on bounded sets. Corollary 118 ([DGJ]). Let be a measure with an infinite family of pairwise disjoint measurable sets of finite positive measure. Let X be a Banach space, 1 < p < C1, k 2 N, and ˛ 2 .0; 1 such that k C ˛ p. If the norm of X is C k -smooth with its kth differential ˛-Hölder on SX , then the canonical norm of Lp .I X/ has the same property. Further, the following statements are equivalent: (i) X admits a C k;˛ -smooth bump. (ii) Lp .I X / admits a C k;˛ -smooth bump. k;˛ (iii) Lp .I X / admits a Cloc -smooth bump. Proof. The statement about norms follows directly from Proposition 117. For bumps, (i))(ii) follows from Proposition 117 after constructing a suitable p-homogeneous function from the bump using a variant of [DGZ, Proposition II.5.1]. (ii))(iii) is obvious. k;˛ (iii))(ii) Since X is isometric to a subspace of Lp .I X/, it admits a Cloc -smooth L1 n ` bump. Assume first that X contains c0 . Then Lp .I X/ contains nD1 1 p , which is a reflexive space that is not super-reflexive, as every Banach space is finitely representable in it. This contradicts Corollary 51. Thus X does not contain c0 , and so it admits a C k;˛ -smooth bump by Theorem 48. t u

Section 7. Lp spaces

303

Corollary 119 ([DGJ]). Let ; be measures, 1 < p; q < C1, and X D Lp I Lq ./ . (i) If q is an even integer and p is a multiple of q, then the norm of X is p-polynomial. (ii) If q is an even integer, then the norm of X is C k;p k -smooth, where k D dpe 1. (iii) If q is not an even integer, then the norm of the space X is C k;r k -smooth, where r D minfp; qg and k D dre 1. Proof. (i) follows from Proposition 115. (ii), (iii) We use Proposition 117 on the function f W Lq ./ ! R, f .x/ D kxkp , which is p-homogeneous. If q is an even integer, then the norm of Lq ./ is polynomial, and so f is C k -smooth with d kf being .p k/-Hölder on SLq ./ (Proposition 1.128). If q is not an even integer, then f is C k -smooth with d kf being .r k/-Hölder on SLq ./ (Theorem 106, the fact that r q, and Proposition 1.128). t u L1 n Proposition 120 ([DGJ]). If the space X D nD1 `q p admits a separating polynomial, then p and q are even integers and p is an integer multiple of q. Proof. As `p is isometric to a subspace of X, p is an even integer by Corollary 4.61. Further, by the sliding hump argument, every normalised weakly null sequence in X has a subsequence equivalent to a block basis such that no two supports of different blocks share the same `nq , which means that the subsequence is equivalent to the canonical basis of `p . Therefore X is saturated by `p and has property Sp . By Corollary 69 the space X is of cotype p. Since `q is finitely representable in X , it follows that p q. By [Fi, Corollary 22] the norm of X has modulus of convexity of power type p. Thus X admits a p-homogeneous separating polynomial (Theorem 62 and Proposition 4.49). By Corollary 4.11, `q also admits a p-homogeneous separating polynomial, so finally by Corollary 4.61 we get the conclusion. t u L1 n p Theorem 121 ([DGJ]). Let 1 < p; q < C1. If X D nD1 `q p admits a T -smooth bump, then p and q are even integers and p is a multiple of q (and hence the norm of X is p-polynomial). Proof. We show that if X admits a T p -smooth bump, then it admits a separating polynomial. The rest is Proposition 120 and Corollary 119. Exactly as in the proof of Theorem 107 we find x0 2 X , x0 ¤ 0, P 2 P Œp .X/, and ı > 0 such that 1 P .h/ kx0 C hkp kx0 kp khkp (19) 2 for all h 2 X, khk ı. Since kkp is Lipschitz on bounded sets, there is > 0 such that ky C hkp < kx0 C hkp C 18 ı p and kykp > kx0 kp 81 ı p whenever y 2 X , ky x0 k < , and h 2 B.0; ı/. Denote by Tm W X ! X the projection Tm .´1 ; ´2 ; : : : / D .´1 ; : : : ; ´m ; 0; 0; : : : /, where ´n 2 `nq . Find N 2 N such that kx0 TN .x0 /k < and put H D .Id TN /.X/. Note that if u 2 TN .X/ and v 2 H , then ku C vkp D kukp C kvkp . Therefore for any h 2 H , khk D ı the estimate (19) implies 1 1 p 1 1 p 1 P .h/ > kTN .x0 / C hkp kTN .x0 /kp khkp ı D khkp ı D ıp : 2 4 2 4 4

304

Chapter 5. Smoothness and structure

It follows that H admits a non-negative homogeneous separating polynomial Q1 (Fact 4.45). As TN .X/ is finite-dimensional, it admits a non-negative homogeneous separating polynomial Q2 . The polynomial Q1 B .Id TN / C Q2 B TN is then a separating polynomial on X . t u Corollary 122 ([DGJ]). Let ; be measures with infinite families of pairwise disjoint measurable sets of finite positive measure, 1 < p; q < C1, and X D Lp I Lq ./ . (i) If q is an even integer and X admits a T p -smooth bump, then p is a multiple of q (and hence the norm of X is p-polynomial). (ii) If q is not an even integer, then X does not admit a T r -smooth bump, where r D minfp; qg. L1 n Proof. (i) follows from Theorem 121, as nD1 `q p is isometric to a subspace of X. (ii) The case p < q follows also from Theorem 121. The case q p follows from Theorem 107, as `q is isometric to a subspace of X. t u Note that `3 admits an equivalent norm that is C 2;1 -smooth and G 3 -smooth, but does not admit a T 3 -smooth bump (Theorem 106, variants of Corollary 7.72 and Corollary 7.70, Theorem 107). The space `2 .NI `4 / admits an equivalent norm that is C 1;1 -smooth and G 2 -smooth, but does not admit a T 2 -smooth bump (Corollary 119, variants of Corollary 7.72 and Corollary 7.70, Corollary 122).

8. C.K / spaces In the present section we collect, mostly without proof, some results on higher smoothness of C.K/ spaces. Fact 123. Let T be a topological space and X a Banach space of all bounded Borel functions on T with the supremum norm. Consider the space Œ 1; 1N with the product topology. Let ˚ W T ! Œ 1; 1N be a Borel mapping and L D ˚.T / (this is a metrisable compact space). Then the linear operator T W C.L/ ! X given by T .f / D f B ˚ is an isometry into. The mapping ˚ is Borel if and only if each n B ˚, n 2 N is Borel, where n is the projection onto the nth coordinate. Proof. T is an isometry because ˚.T / is dense in L. The last statement follows from the fact that the topology of Œ 1; 1 has a countable basis. The last statement follows from the fact that the topology of Œ 1; 1 has a countable basis of the form I1 In Œ 1; 1N , Ij intervals. t u Proposition 124. Let K be a compact space and X C.K/ a separable subspace. Then there are a metrisable compact L which is a continuous image of K and a subspace Y C.K/ such that Y is isometric to C.L/ and X Y . In particular, if K is scattered, then L is countable, and so Y is isometric to C.Œ0; ˛/ for some countable ordinal ˛.

Section 8. C.K/ spaces

305

Proof. Let ffn g be a dense subset of BX and let us define ˚ W K ! Œ 1; 1N by ˚.x/.n/ D fn .x/. Then ˚ is continuous and hence the operator T from Fact 123 is a linear isometry onto some Y C.K/. Note that fn D n B ˚ and hence T .n L / D fn . Thus fn 2 Y and consequently X Y . If K is scattered, then by [FHHMZ, Lemma 14.20] and [FHHMZ, Lemma 14.21] the space L D ˚.K/ is countable. So C.L/ is isometric to C.Œ0; ˛/ for some countable ordinal ˛ by the Mazurkiewicz-Sierpi´nski theorem ([HMVZ, Theorem 2.56]). t u Theorem 125. Let K be a compact space. The following statements are equivalent: (i) K is scattered. (ii) C.K/ is an Asplund space. (iii) C.K/ is isometric to `1 . / for some (in fact, we can choose D K). (iv) C.K/ does not contain `1 . (v) C.K/ is c0 -saturated. (i))(iii) is [FHHMZ, Theorem 14.24]. (i))(ii) follows from Theorem 2, Proposition 124, and the implication (i))(iii). (ii))(i) can be found in [FHHMZ, Theorem 14.25]. (iii))(ii) follows from Theorem 2 and the fact that `1 . / has the RNP. (ii))(iv) follows from Theorem 2. (i),(iv),(v) are results of Aleksander Pełczy´nski and Zbigniew Semadeni, [PS]. Since every Banach space admitting a C 1 -smooth bump function is an Asplund space (Corollary 3), we see that smooth renormings of C.K/ spaces may exist only for scattered K. Theorem 126. For any set the space c0 . / admits an equivalent norm which is simultaneously LFC-fe g, C 1 -smooth, and uniformly Gâteaux differentiable. We postpone the proof until Chapter 7, page 437. The following result improves an earlier result of Michel Talagrand which considers only C 1 -smoothness. Theorem 127 ([Hay1]). For every ordinal the space C.Œ0; / admits an equivalent C 1 -smooth LFC norm. b be a one-point compactification of a locally compact Theorem 128 ([Hay3]). Let T tree T . b/ admits a C 1 -smooth LFC bump. (i) The space C.T b/ admits an equivalent C 1 -smooth LFC norm if and only if it (ii) The space C.T admits an equivalent C 1 -smooth norm. Theorem 129 ([HáHa]). Let K be a -discrete compact space. Then C.K/ admits an equivalent C 1 -smooth LFC-K norm. S Proof. Let K D n2N Dn , where the sets Dn are pairwise disjoint and relatively discrete. First note that K is scattered. Indeed, by the Baire category theorem there is k 2 N such that Dk has a non-empty interior. Thus there are an open U Dk and x 2 Dk \ U . Since Dk is relatively discrete, there is an open neighbourhood

306

Chapter 5. Smoothness and structure

V of x such that V \ Dk D fxg. Thus also V \ Dk D fxg, and consequently V \ U D V \ U \ Dk D fxg, i.e. x is an isolated point of K. Since the -discreteness is hereditary, the scatteredness follows. Therefore K is sequentially compact ([Fab3, Lemma 2.1.1]). We define the new norm as a Musielak-Orlicz norm on `1 .K/ and show that it is smooth on its subspace C.K/. To this end we first define two functions ˛; ˇ W K ! .0; 1 that will provide endpoints of certain intervals determining the Orlicz Q functions t 7! '.x; t /, x 2 K. So fix rn 2 .0; 1/, n 2 N such that a D n2N rn > 0 and for x 2 K set Y Y rn ; ˇ.x/ D rn : ˛.x/ D 0 fnI x2Dn g

fnI x2Dn g

Notice that ˛.x/ D rk ˇ.x/, where k 2 N is the unique index such that x 2 Dk . This follows from the fact that Dk \ Dk0 D ; and Dn are pairwise disjoint. Therefore 0 < a ˛.x/ < ˇ.x/ 1. Further, we claim that the functions ˛; ˇ have the following property: ˇ.x/ lim inf ˛.xn / n!1

whenever xn ! x 2 K and xn 2 K are distinct.

(20)

To prove this, let fyn g be a subsequence of fxn g such that lim ˛.yn / D lim inf ˛.xn /. Next, by passing to further subsequences and then diagonalising we may assume that for each ˚ k 2 N the set fyn I n kg is either a subset of Dk or disjoint with Dk . Let M D k 2 NI fyn I n kg Dk . Then Y Y Y rk rk ˛.yn / rk ; kn k2M

k>n

kn k2M

Q and therefore limn!1 ˛.yn / D k2M rk . On the other hand, if k 2QM , then lim yn D x 2 Dk0 , as yn are distinct and fyn I n kg Dk . Hence ˇ.x/ k2M rk , which proves the claim. Let ' W K R ! Œ0; C1/ be such that for each x 2 K the function t 7! '.x; t/ is a C 1 -smooth even convex function non-decreasing on Œ0; C1/ and satisfying '.x; t / D 0 for jtj ˛.x/ and '.x; t/ > 1 for jt j ˇ.x/. Now define a function ˚ W `1 .K/ ! Œ0; C1 by X ˚.f / D ' x; f .x/ : x2K

Note that the function ˚ is convex, even, lower semi-continuous, ˚ D 0 on the ball B`1 .K/ .0; a/, and ˚.f / > 1 whenever kf k1 > 1. Since B.0; a/ B1 B.0; 1/, where B1 D ff 2 `1 .K/I ˚.f / 1g, the Minkowski functional of B1 is an equivalent norm on `1 .K/. Put B D ff 2 C.K/I ˚.f / 1g. We show that ˚ is LFC-K and C 1 -smooth on a neighbourhood of B in C.K/. Let f 2 B. We claim that there are ı > 0 and F K finite such that '.x; g.x// D 0 whenever kf gk1 < ı and x 2 K n F . Indeed, if this is not the case, then there are sequences ffn g C.K/ and fxn g K such that

Section 9. Orlicz spaces

307

xn are distinct, fn ! f uniformly on K, and '.xn ; fn .xn // > 0 for each n 2 N. It follows that jfn .xn /j > ˛.xn /. By passing to a subsequence we may assume that xn ! x 2 K. Since fn ! f uniformly on K and f is continuous, fn .xn / ! f .x/. Thus jf .x/j lim inf ˛.xn / ˇ.x/ by (20). This implies that '.x; f .x// > 1, a contradiction with ˚.f / 1. Clearly, ˚ depends only on F on U.f; ı/ \ C.K/ and is C 1 -smooth there as a finite sum of C 1 -smooth functions. Finally, set C D Intff 2 C.K/I ˚.f / 2g. Then C is a non-empty open convex set and B C , since ˚ is continuous on a neighbourhood of B. Further, B is closed in C.K/, as ˚ is lower semi-continuous. So, Lemma 23 together with Lemma 93 finishes the proof. t u Corollary 130 ([Há8]). Let K be a compact space with K .!1 / D ;. Then C.K/ admits an equivalent C 1 -smooth LFC-K norm. Recall that in general a dual LUR norm on X implies C 1 -smoothness of the norm on X, and this is often used to produce a C 1 -smooth renorming. The following corollary shows that in case of C.K/ spaces, this approach already gives a C 1 -smooth renorming. Corollary 131 ([HáHa]). Let K be a compact space. If C.K/ admits an equivalent dual LUR norm, then C.K/ admits an equivalent C 1 -smooth LFC-K norm. Proof. This follows from Theorem 129 combined with the result of Matías Raja [Raj] that C.K/ admits an equivalent dual LUR norm if and only if K is -discrete. t u Theorem 132 ([Há3]). Let K be a compact space. Then C.K/ has an equivalent analytic norm if and only if K is countable. Proof. ( follows from Theorem 103 and Theorem 104. ) Since C.K/ has an equivalent analytic norm kk, it is an Asplund space (Corollary 3) and so C.K/ is isometric to `1 .K/ (Theorem 125). Since C.K/ spaces enjoy the DPP, by Theorem 3.67 we get P .C.K// D PwsC .C.K//. For each x 2 C.K/ n f0g P there is a neighbourhood U of x such that kyk D 1 P x/ for all y 2 U , nD0 n .y where Pn 2 P . nX/, and the sum converges uniformly on U . It follows that kk is locally wsC on C.K/ n f0g. Thus C.K/ is separable by Proposition 77, which in turn implies that K is countable. t u

9. Orlicz spaces We briefly review some properties of Orlicz sequence spaces and then proceed to study how their smoothness depends on the behaviour of the Orlicz function. We then turn to polyhedrality of Orlicz sequence spaces which leads to an example of a separable c0 -saturated Asplund space that is not isomorphically polyhedral. This serves also as an example of a separable space that has C 1 -smooth norm but does not admit an equivalent analytic norm.

308

Chapter 5. Smoothness and structure

A function M W R ! Œ0; C1/ is called an Orlicz function if it is even, convex, non-decreasing on Œ0; C1/, M.0/ P D 0, and if M is not constant. We define ˚M W `1 ! Œ0; C1 by ˚M .x/ D 1 nD1 M.xn /, which is an even convex function. An Orlicz sequence space `M is a linear subspace of `1 consisting of x 2 `1 satisfying ˚M .x=/ < C1 for some > 0, equipped with the norm given by the Minkowski functional of fx 2 `M I ˚M .x/ 1g. With this norm the space `M is a Banach space. It is easily seen that the canonical coordinate projections in `M are continuous and so ˚M is lower semi-continuous on `M and it is continuous on U`M .0; 1/. From the convexity we obtain ˚M .x/ kxk for x 2 B`M , while ˚M .x/ kxk for x 2 `M , kxk > 1. An important subspace of `M is a space hM consisting of x 2 `M satisfying ˚M .x=/ < C1 for all > 0. The space hM is a closed subspace of `M and the canonical basis vectors fen g1 nD1 form a symmetric basis of hM . Further, ˚M is continuous on hM and for x 2 hM we have ˚M .x/ D 1 if and only if kxk D 1. If M is such that M.t/ D 0 for some t > 0, then it is called a degenerate Orlicz function. The associated space `M is then isomorphic to `1 and hM is isomorphic to c0 . For Orlicz functions M1 , M2 the spaces `M1 , `M2 are isomorphic if and only if the canonical bases of hM1 , hM2 are equivalent, if and onlyif the functions M1 , M2 are equivalent at 0, i.e. there is C 1 such that C1 M1 C1 t M2 .t/ CM1 .C t/ for each t 2 Œ0; 1. A non-degenerate Orlicz function M is said to satisfy the 2 -condition at 0 (at C1), if there is K > 0 such that M.2t/ KM.t/ for t 2 Œ0; 1 (t 2 Œ1; C1/). An Orlicz function M satisfies the 2 -condition at 0 if and only if hM D `M . For an introduction to Orlicz sequence spaces we refer the reader to [LiTz2]. To a non-degenerate Orlicz function M we associate the following numbers, called the Boyd indices: M.t/ ˛M D sup q 2 .0; C1/I sup < C1 ; q ;t 2.0;1 M.t / M.t/ ˇM D inf q 2 .0; C1/I inf >0 : ;t2.0;1 q M.t / It is not difficult to check that 1 ˛M ˇM C1 and that ˇM < C1 if and only if M satisfies the 2 -condition at 0. Theorem 133 ([LiTz2, Theorem 4.a.9]). Let M be a non-degenerate Orlicz function. The space `p , or c0 if p D C1, is isomorphic to a subspace of an Orlicz space hM if and only if p 2 Œ˛M ; ˇM . Corollary 134. Let M be a non-degenerate Orlicz function. The Orlicz space hM is c0 -saturated if and only if ˛M D C1. Proof. ) follows directly from Theorem 133. ( If the space hM is not c0 -saturated, then there is an infinite-dimensional subspace X hM that does not contain c0 . By [LiTz2, Proposition 4.a.7] there is a subspace

Section 9. Orlicz spaces

309

Y of X isomorphic to hN for some Orlicz function N . Since hN does not contain c0 , from Theorem 133 it follows that ˇN < C1 and hN contains `ˇN . Thus also hM contains `ˇN , a contradiction with ˛M D C1. t u Proposition 135 ([Kn]). Let M be a non-degenerate Orlicz function and 1 < p < ˛M . Then every weakly null sequence in hM has a subsequence with an upper p-estimate. In particular, every polynomial on hM with degree smaller than ˛M is weakly sequentially continuous. Proof. Let fxn g hM be a weakly null sequence. Without loss of generality we may assume that M.1/ 1 and fxn g BhM . We can find a subsequence fxnk g and a Pl 1 block basic sequence yk D jkC1 cj ej 2 BhM such that kxnk yk k 21k . Note Dlk PlkC1 1 that j Dl M.cj / 1, which also implies that jcj j 1 for all j 2 N. By the k definition of ˛M there is C 1 such that M.t/ C p p M.t/ for all ; t 2 Œ0; 1. 1 PN p p . Then Let a1 ; : : : ; aN be any scalars and denote D kD1 jak j N lkC1 X1 X kD1 j Dlk

M

ak cj C

N lkC1 X1 X kD1 j Dlk

Cp

jak j C

p M.cj /

lkC1 1 N X 1 X p jak j M.cj / 1; D p kD1

j Dlk

P which means that N

kD1 ak yk C . Hence

N

N

N

X

X

X 1

ak xnk ak yk C jak j k .C C 1/:

2 kD1

kD1

kD1

The last statement follows from Corollary 3.58.

t u

Somewhat weaker statement is obtained for polynomials of degree k ˛M when M satisfies certain condition. Note that an example of such function (where k > ˛M ) is M.t / D t pCsin logjlog tj , see also [LiTz2, Example 4.c.2]. Proposition 136. Let M be a non-degenerate Orlicz function such that it satisfies lim inf t !0C M.t/ D 0 for some k 2 N. Then fen g does not have a lower k-estimate. tk k Consequently, for each finitely supported x 2 X the sequence fS n .x/g1 nD1 is P -null, where S 2 L.hM I hM / is the right shift operator. Proof. For n 2 N denote n D ke1 C C en k. Suppose that fen g has a lower 1 k-estimate. Then there is C > 0 such that C n k n for each n 2 N. Hence e C C e 1 1 1 n D nM nM 1D˚ 1 n n C nk

310

Chapter 5. Smoothness and structure

for each n 2 N. Let ftn g .0; C1 be such that limn!1 1 k

1 k

M.tn / tnk

D 0. For each n 2 N

< tn C1 jn . Then find jn 2 N such that C1 .jn C 1/ jn C 1 M.tn / 2 M.tn / 1 < .jn C 1/M.tn / k 1 .jn C 1/M k k k ; 1 C jn tn C tn C.jn C 1/ k t u which is a contradiction. Thus we may apply Corollary 4.41. Theorem 137 ([MaT2]). Let M be a non-degenerate Orlicz function and the Orlicz space hM is not isomorphic to `p , p an even integer. If ˛M < C1, then hM does not admit a T ˛M -smooth bump. Proof. If ˛M is not an even integer, then the result follows from Theorem 133 and Theorem 107. So suppose that ˛M D k is an even integer and hM admits a T k -smooth bump. Without loss of generality we may assume that M.1/ D 1. Denote ˚ D ˚M . Analogously as in the proof of Theorem 107 (this time using the function ' D ˚1 ) we obtain x 2 hM and P 2 P k .hM / satisfying P .0/ D 0 such that for each " > 0 there is ı > 0 such that ˚.x C h/

˚.x/

P .h/ "khkk

for all khk ı.

(21)

Since hM is not isomorphic to `k , we have either lim sup t !0C M.t/ D C1, or tk M.t/ k lim inf t !0C t k D 0. In the first case let P D Q C Pk , where Q 2 P 1 .hM / and Pk 2 P . khM /. Let ı > 0 be such that (21) holds with " D 1 and let 0 < ı be such that M./ > kPk k C 1. Then k Q.en / ˚.x C en / M en .x/ C

Pk .en / k M en .x/ kPk kk

˚.x/

k :

Note that by the definition of ˛M we have lim t !0C M.t/ t q D 0 for all q < ˛M D k and thus limn!1 Q.en / D 0 by Proposition 136. So by passing to a limit as n ! 1 we obtain 0 M./ k kPk k C 1 > 0, a contradiction. Now we deal with the second case. By Theorem 133 there is a basic sequence fyn g hM which is equivalent to the canonical basis of `k . Since k > 1, fyn g is weakly null and hence there is a subsequence of fyn g equivalent to a block basic sequence of fen g (Proposition 3.32). Therefore there is a normalised block basic sequence fun g of fen g which is equivalent to the canonical basis of `k . For n 2 N denote n D ku1 C C un k. Since fun g is equivalent to the canonical basis of `k , 1 there is a > 0 such that an k n for each n 2 N. Let ı > 0 be such that (21) holds m with " D 12 ak and find m 2 N such that 1m ı. As ˚ u1 CCu D 1, we obtain m m a k u C C u X u m 1 j m 1D˚ D ˚ : m m m j D1 k u It follows that there is j 2 f1; : : : ; mg such that ˚ jm am . Set v D note that kvk D 1m ı.

uj m

and

Section 9. Orlicz spaces

311

Let S 2 L.hM I hM / be the right shift operator and let us denote vn D S n .v/. Then ˚.vn / D ˚.v/ and kvn k D kvk for each n 2 N. Proposition 136 implies that P .vn / ! 0, while obviously ˚.x C vn / ! ˚.x/ C ˚.v/. Therefore from k P .vn / ˚.xCvn / ˚.x/ 12 ak kvn kk we get 0 ˚.v/ 12 ak kvkk 21 am > 0, a contradiction. t u Similarly as the Orlicz sequence spaces we may also define Orlicz spaces using an arbitrary measure . Let M be the vector space of all of -measurable functions factorised R by equality almost everywhere. We define ˚M W M ! Œ0; C1 by ˚M .x/ D M.x.t// d.t/, which is an even convex function. An Orlicz space LM ./ is a linear subspace of M consisting of x 2 M satisfying ˚M .x=/ < C1 for some > 0, equipped with the norm given by the Minkowski functional of fx 2 LM ./I ˚M .x/ 1g. With this norm the space LM ./ is a Banach space. Obviously ˚M is continuous on ULM ./ .0; 1/ and using Fatou’s lemma it can be shown that ˚M is lower semi-continuous on LM ./. From the convexity we obtain ˚M .x/ kxk for x 2 BLM ./ , while ˚M .x/ kxk for x 2 LM ./, kxk > 1. The space HM ./ consisting of x 2 LM ./ satisfying ˚M .x=/ < C1 for all > 0 is a closed subspace of LM ./. Further, ˚M is continuous on HM ./ and for x 2 HM ./ we have ˚M .x/ D 1 if and only if kxk D 1. Recall that for a function ' W Œ0; C1/ ! R its non-decreasing envelope is given by '.t N / D sups2Œ0;t '.s/. Theorem 138 ([Mal1]). Let k 2 N and let M 2 C k .R/ be a non-degenerate Orlicz function which is C kC1 -smooth on RC and has the following property: There is p > k and C 2 such that M.t/ C p M.t/ for all 2 Œ0; 1, t 2 RC , and ˇ ˇ t j ˇM .j / .t/ˇ CM.C t/ for all t 2 RC , j D 1; : : : ; k C 1. Let !0 .t/ D t sup 2Œt;1 s2RC

M.s/ kC1 M.s/

(22) (23)

for t 2 .0; 1,

!0 .0/ D 0, and !0 .t/ D t for t 2 .1; C1/. Then !M D !N 0 2 M. Further, let ˝ M be the cone generated by !M . Then for any measure the function ˚M is k;˝ 1 1 C k -smooth on ULM ./ .0; 2C -smooth /, C k;˝ -smooth on ULM ./ .0; 4C /, and Cloc k;˝ on HM ./. In particular, the norm of HM ./ is Cloc -smooth and if ˚M is separating 1 / (e.g. if M satisfies the 2 -condition at 0 and C1), then there is on ULM ./ .0; 4C k;˝ an equivalent C -smooth norm on LM ./. We remark that the assumptions of the theorem can be somewhat relaxed to not require the C k -smoothness on R and to require (23) only for j D k C 1. One can then painfully prove using several inductions back and forth that our assumptions are actually satisfied. Since it turns out that this formal generalisation is not useful for us, we chose to trade it for slightly shorter proof.

312

Chapter 5. Smoothness and structure

The proof of this theorem is analogous to the proof of Theorem 106, but instead of the inequality (15) we use the following lemma. Lemma 139. Let M be an Orlicz function as in Theorem 138 and ! D !M . Then there is K > 0 such that ˇ .k/ ˇ ˇM .x C t´/ M .k/ .x/ˇv k K M.2C x/ C M.´/ C M.v/ !.t/ 1 for all x; ´ 2 R, v 2 Œ0; C1/, t 2 Œ0; 2C .

The proof of the lemma is based mainly on the estimate (iii) in the following fact. It seems that it would be natural to define ! as the non-decreasing envelope of 7! sups2RC M.s/ k M.s/ . This does not work however, and the particular formula for ! is chosen exactly so that the estimate (iii) below holds. Fact 140. Let M be a non-degenerate Orlicz function and let ! D !M as defined in Theorem 138. Then (i) !.t / t for each t 2 Œ0; C1/, (ii) M.t / !./k M.t/ for each 2 Œ0; 1, t 2 Œ0; C1/, (iii) kC1 M.a/ M. t a/ C M.a/ !.t/t k for each 2 Œ0; 1, a 2 Œ0; C1/, and t 2 RC . Proof. (i) and (ii) are clear from the definition of !. (iii) If t , then kC1 M.a/ t kC1 M.a/ M.a/!.t/t k by (i). If > t, then t t < 1 and hence !.t/ t

M. t t a/ . t /kC1 M. t a/

; t u

from which the inequality in (iii) follows.

Proof of Lemma 139. First we prove that there is C1 > 0 such that for each x; y 2 R not both zero ˇ .k/ ˇ M C maxfjxj; jyjg .k/ ˇM .y/ M .x/ˇ C1 jy xj: (24) maxfjxj; jyjgkC1 Without loss of generality we may assume that jxj jyj. If jy 0 < x y, or y x < 0, and so using (23) we obtain ˇ .k/ ˇM .y/

M

.k/

xj 12 jyj, then either

Z jyj ˇ M.C t/ ˇM .kC1/ .t/ˇ dt C dt t kC1 jxj jxj M C jyj M C jyj jy xj 2kC1 C jy C kC1 jxj jyjkC1

ˇ .x/ˇ

Z

jyj ˇ

xj:

Section 9. Orlicz spaces

313

xj > 12 jyj and x ¤ 0, then (23) and (22) imply ˇ ˇ .k/ ˇ ˇ .k/ ˇ M C jyj M C jxj .k/ ˇ ˇ ˇ ˇ ˇ CC M .x/ M .y/ C M .x/ C jyjk jxjk jxj M jyj C jyj M C jyj M C jyj 2 M C jyj DC CC C CC jyjk jxjk jyjk jyjk M C jyj jy xj < 2.C C C 2 / jyjkC1

On the other hand, if jy ˇ .k/ ˇM .y/

If x D 0, then it suffices to notice that (22) implies that M is T k -smooth at 0 with the approximating polynomial being zero and hence M .k/ .0/ D 0. Then we repeat the estimate above. 1 Now fix x; ´ 2 R, v 2 ˚Œ0; C1/, and t 2 Œ0; 2C . Clearly we may also assume that t ´ ¤ 0. Set D max jxj; jx C t´j . Obviously, t j´j 2. Also, since M is non-decreasing on Œ0; C1/, ˚ M.C / max M.2C jxj/; M.2C tj´j/ M.2C x/ C M.´/: (25) The estimate (24) implies ˇ .k/ ˇM .x C t´/ If v

C t ,

ˇ M.C / M .k/ .x/ˇv k C1 tj´j kC1 v k :

then by Fact 140(iii) and (25) ! C kC1 t j´j kC1 M.C / k tv kC1 t j´j kC1 v C M.C / C C tk C kC1 M.j´j/ C M.v/ C 2M.C / !.t/ C kC1 3M.´/ C M.v/ C 2M.2C x/ !.t/:

Finally, in case that v > Ct we use the inequality t j´j 2, (22), and Fact 140(ii) to obtain M.C / vk C k vk C tv 2C k M.tv/ 2C kC1 M.v/!.t/: tj´j kC1 v k 2 k M tv tv t u Proof of Theorem 138. Let us denote X D LM ./, H D HM ./, ˚ D ˚M , and ! D !M . From (22) it easily follows that !.t/ C maxft p k ; tg, t 2 Œ0; 1, and hence ! 2 M. By Corollary 1.108 and Lemma 139 ˇ ˇ k X ˇ ˇ 1 1 .j / j j ˇM.x C t h/ M .x/t h ˇˇ K M.2C x/ C 2M.h/ !.t/t k (26) ˇ jŠ kŠ j D0

1 . for all x; h 2 R, t 2 Œ0; 2C

314

Chapter 5. Smoothness and structure

Let x 2 UX .0; C1 / [ H , h 2 UX .0; C1 /, and j 2 f1; : : : ; kg. Fix concrete repon. Put resentatives for x; h and let .S; / be the˚measure space we are working T D fs 2 S I x.s/h.s/ ¤ 0g and A D s 2 T I jx.s/j jh.s/j . Then by (23) and (22) ˇ ˇ Z Z ˇ .j / ˇ ˇ h.s/ ˇj jˇ ˇ d.s/ ˇM ˇ x.s/ h.s/ d.s/ C M C x.s/ ˇ x.s/ ˇ T ˇ ˇ ˇ ˇ Z Z ˇ h.s/ ˇj ˇ h.s/ ˇj ˇ d.s/ C C ˇ ˇ DC M C x.s/ ˇˇ M C x.s/ ˇ x.s/ ˇ d.s/ x.s/ ˇ A T nA ˇ Z ˇ ˇ x.s/ ˇp j 2 2 ˇ ˇ C ˇ h.s/ ˇ M C h.s/ d.s/ C C˚.C x/ C ˚.C h/ C C˚.C x/: A Therefore for any given x 2 UX .0; C1 / [ H and j 2 f0; : : : ; kg we may define R Pjx .h/ D M .j / x.s/ h.s/j d.s/ for h 2 X. For a fixed -measurable function f 2 M the mapping .h1 ; : : : ; hj / 7!f h1h2 hj clearly belongs to L. j MI M/. Thus theR estimate above gives that h 7! .M .j / B x/ hj belongs to P . jXI L1 .//. Since g 7! g d is a bounded linear functional on L1 ./, it follows that Pjx 2 P . jX/. Using (26) we obtain ˇ ˇ ˇ˚.x C h/ ˇ

ˇ k X ˇ 1 .j / jˇ x.s/ h.s/ ˇ d.s/ M jŠ j D0 Z h.s/ 1 k d.s/ M 2C x.s/ C 2M K! khk khk kŠ khk h 1 ! khk khkk D K ˚.2C x/ C 2˚ kŠ khk 1 K ˚.2C x/ C 2 ! khk khkk kŠ

ˇ Z k X 1 x ˇˇ P .h/ˇ jŠ j

j D0

ˇ ˇ ˇM x.s/ C h.s/ ˇ

1 for any x; h 2 X, khk 2C . Now Theorem 1.110 together with the continuity of ˚ 1 k / and also C k -smooth on H . Finally, using implies that ˚ is C -smooth on UX .0; 2C Lemma 139 again, we get

k

d ˚.x/ d k˚.y/ D kP x P y k D sup jP x .h/ P y .h/j k k k k khk1

Z sup

ˇ .k/ ˇM x.s/

ˇ M .k/ y.s/ ˇjh.s/jk d.s/

khk1

K! kx

yk

Z sup

M 2C x.s/ C M

khk1

K! kx

yk ˚.2C x/ C 2

x.s/ kx

y.s/ C M h.s/ d.s/ yk

Section 9. Orlicz spaces

315

1 1 whenever x; y 2 UX .0; 2C . In particular, it follows that / [ H and kx yk 2C k;˝ 1 k;˝ ˚ 2C UX .0; 4C / and ˚ 2 Cloc .H /. The statements about the norms follow from Lemma 23, noting that the last part of the proof of Lemma 23 works also locally and that the minimal modulus of continuity of d k˚ is sub-additive due to the convexity of the domains. t u

Corollary 141 ([Mal1]). Let M be a non-degenerate Orlicz function such that hM is not isomorphic to any `p . If 1 < ˛M < C1, then set k D d˛M e 1. In this case hM k;q k admits an equivalent norm that is Cloc -smooth for any q 2 .k; ˛M /, but does not ˛ M admit a T -smooth bump. If moreover ˇM < C1, then hM admits an equivalent norm that is C k;q k -smooth for any q 2 .k; ˛M /. If ˛M D C1, then hM admits an equivalent C 1 -smooth norm. Proof. We construct an Orlicz function N which is equivalent to M at 0 and satisfies the assumptions of Theorem 138. Without loss of generality we may assume that M.1/ D e 1. We put M1 .t/ D M.t/ for t 2 Œ0; 1 and M1 .t/ D exp t 1 for t 2 .1; C1/. Further, set '.t; s/ D exp s s t for t > s 0 and '.t; s/ D 0 for 0 < t s. We define Z jt j M1 .s/ ' jt j; s ds N.t/ D s 0 for t 2 R. By the convexity, M1 .s/ .e 1/s for s 2 Œ0; 1 and hence N is real-valued. Since s 7! M1s.s/ is non-decreasing on .0; C1/ (this follows from the convexity of M and t 7! exp t 1), Z 3t t Z 3t 1 4 M1 .s/ 4 1 3 M1 . 2 / '.t; s/ ds e M t N.t / 1 ds D 1 t t t s 2e 3 2 2 2 2 Rt for t 2 RC . On the other hand, N.t/ M1t .t/ 0 1 ds D M1 .t/ for t 2 RC . Therefore M and N are equivalent at 0. It is easy to show by induction that there are constants cn;j 2 N, 0 j < n, n 2 N, such that n 1 s X cn;j s n 1 j @n ' .t; s/ D s exp @t n s t .s t/2n j j D0

for all t > s 0, n 2 N. Put f .t; s/ D M1s.s/ '.t; s/ for s; t 2 RC , f .t; 0/ D 0. n Then the mappings .t; s/ 7! @@t fn .t; s/ are continuous on RC RC 0 and hence by 1 C Corollary 1.91 the function N is C -smooth on R and Z t Z n 1 M1 .s/ @n ' 1 1 u X cn;j un 1 j .n/ .t; M .ut/ exp s/ ds D du N .t / D 1 s @t n tn 0 u 1 .u 1/2n j 0 j D0

for t 2 RC and n 2 N. It follows that N 0 is positive and non-decreasing on .0; C1/ and thus N is an Orlicz function.

316

Chapter 5. Smoothness and structure

Next, Z 1 n .n/ t jN .t /j M1 .t/ exp 0

u u

ˇ ˇn 1 ˇ X cn;j un 1 ˇ 1 ˇˇ .u 1/2n j D0

ˇ ˇ ˇ ˇ du D Cn M1 .t/ 2e 3 Cn N.2t/ jˇ ˇ

j

for each t 2 RC , n 2 N, where Cn are appropriate constants. This means that N satisfies (23) in Theorem 138 for any k 2 N. Since for each q < ˛M there is a constant A such that M.t/ At q for t 2 Œ0; 1, it follows from the above inequality that lim t !0C N .n/ .t/ D 0 whenever n < ˛M . Using the continuity at 0 it follows inductively that N 2 C n .R/ for all n < ˛M . It remains to show that for any 1 < p < ˛M the function N satisfies (22) in Theorem 138. First, we show this statement for M1 . Let K1 D sup;t 2.0;1 M.t/ p M.t / < C1, e < C1, and put K0 D maxfK1 ; K2 ; 1g. For ; t 2 Œ0; 1 K2 D sup2.0;1 p 1 .exp

1/

we have M1 .t/ D M.t/ K1 p M.t/ K0 p M1 .t/. If 2 .0; 1 and t 1, then M1 .t / D exp.t/ 1 < exp.t/ K2 p .exp t 1/ K0 p M1 .t/. And finally, if t > 1 and t < 1, then we use both of the previous estimates to obtain M1 .t / K0 .t/p M1 .1/ D K0 p t p M1 . 1t t/ K0 p t p K0 t1p M1 .t/ D K02 p M1 .t/. Now for 2 Œ0; 21 and t 2 RC we obtain N.t/ D N.2 2t / M1 .2 2t / 2 p p K0 2 M1 . 2t / 2pC1 e 3 K02 p N.t/, while of course N.t/ N.t/ 2p p N.t/ for 2 Œ 12 ; 1, t 2 RC . To sum up what we have proved so far: There is an Orlicz function N equivalent to M at 0 which is C n -smooth on R for each n < ˛M , it is C 1 -smooth on RC , satisfies (22) for any 1 < p < ˛M , and satisfies (23) for any k 2 N. Thus in case ˛M D C1 Theorem 138 implies that there is an equivalent C 1 -smooth norm on hM . If ˛M < C1, we put k D d˛M e 1 and choose any p 2 .k; ˛M /. It is easily checked, that for each q 2 .k; ˛M / there is L > 0 such that !N .t/ Lt q k for t 2 Œ0; 1. k;q k Hence the equivalent norm on hM given by Theorem 138 is Cloc -smooth, or even k;q k C -smooth in case ˇM < C1 (in this case N satisfies the 2 -condition at 0 and so ˚N is separating in any ball around 0). The negative statement is Theorem 137. t u We remark that the Orlicz spaces for a general measure can be treated in a similar way. Next we turn to study of polyhedrality in Orlicz sequence spaces. Theorem 142 ([Leu3]). The following statements are equivalent for every non-degenerate Orlicz function M : M.Kt/ t !0C M.t /

(i) There exists a constant K > 0 such that lim

D C1.

(ii) The Orlicz space hM is isomorphic to a subspace of C.Œ0; ! ! /. (iii) The Orlicz space hM is isomorphic to a subspace of C.K/ for some scattered compact K. All spaces satisfying the condition (ii) are isomorphically polyhedral (Theorem 103), and D. H. Leung in [Leu1] conjectured that conversely all polyhedral Orlicz sequence

Section 9. Orlicz spaces

317

spaces fall under this description. There is a strong evidence supporting this idea. First, it is shown in [Leu1] and [HJ3] that the naturally defined LFC renormings exist precisely for those spaces. Second, negating the condition in (i) we obtain the following statement M.Ktn / 8K > 0 9ftn g1 < C1: nD1 ; tn ! 0C W sup n2N M.tn / Reversing the order of the quantifiers leads to the following stronger (less general) condition 9ftn g1 nD1 ; tn ! 0 C

8K > 0 W sup n2N

M.Ktn / < C1: M.tn /

By Theorem 144 below, the Orlicz sequence spaces satisfying the last condition are not isomorphically polyhedral (although they may be c0 -saturated). Thus Leung’s theorem above is a near characterisation of polyhedrality for Orlicz sequence spaces, the gap lying in the exchange of quantifiers. The following theorem follows of course from Theorem 142, nevertheless we give a short direct proof. Theorem 143 ([Leu1]). Let M be a non-degenerate Orlicz function such that there is a K > 1 for which M.Kt/ D C1: lim t !0C M.t / Then the Orlicz space hM is isomorphically polyhedral. Proof. For k 2 N let bk D inf0 1 for all k 2 N and M.t /

1. 1 / k

Also notice that by the continuity of t 7! so we may set k D b bk 1 , k 2 N. Note that k > 1 and k ! 1. Define a new norm k on hM by jjjxjjj D sup k kPk .x /k;

(27)

k2N

where kk is the canonical norm on hM , Pk are the projections associated with the canonical basis, and if x D .xn / 2 hM , then x D .xn / is the non-increasing reordering of .jxn j/. It is easy to check that jjjjjj is an equivalent norm on hM and the canonical basis is monotone with respect to this norm. We prove that for each x 2 hM there is m 2 N such that jjjxjjj D jjjPm xjjj:

(28)

First we show that if x D .xn / is such that the sequence .xn / is non-negative and non-increasing, then there is k 2 N such that kxk k kPk xk. Without loss of 1 . generality we may assume that kxk D 1. Let k 2 N be such that kx Pk xk < K P1 Then nDkC1 M.Kxn / < 1. Suppose that k kPk xk < kxk D 1. Since .xn / is Pk non-increasing, from 1 > ˚M .k Pk x/ nD1 M.k xk / kM.xk / it follows

318

Chapter 5. Smoothness and structure

that M.xk / < k1 . For all n k we have M.xn / M.xk / < M.Kxn / bk M.xn /. Thus 1 D ˚M .x/ D ˚M .Pk x/ C

1 X

1 k

and therefore

M.xn /

nDkC1

1 1 ˚M .k Pk x/ C k bk

1 X

M.Kxn /

0. M.tn /

Then the Orlicz space hM does not admit any (even non-continuous) LFC bump function. Proof. Suppose that hM admits some LFC bump g. Without loss of generality we may assume that g D A for some set 0 2 A BX (by shifting, scaling, and composing with a suitable function) and that g is LFC-fei g. (Since hM is c0 -saturated by Theorem 90, it does not contain `1 . As fei g is unconditional, it is shrinking by James’s theorem. Now consider g B T , where T W hM ! hM is an equivalence isomorphism of the bases fei g and fxi g from Corollary 88.) Note that by our assumption the vectors with coordinates

Pk in the set ftn I n 2 Ng[f0g have the property of “bounded completeness”: If i D1 tmi ei 1 for all k 2 N, P where mi 2 N0 are not necessarily distinct (we put t0 D 0), then 1 i D1 tmi ei converges

Section 9. Orlicz spaces

319

in hM . Indeed, it follows that k X

Pk

i D1 M.tmi /

1 for all k 2 N. Further, k

M.Ktn / X M.Ktn / M.tmi / sup n2N M.tn / i D1 n2N M.tn / i D1 P1 for all K > 0 and all P k 2 N. Consequently, i D1 M.Ktmi / < C1 for all K > 0 and hence the sum 1 i D1 tmi ei converges in hM . We construct a sequence fxk g1 A by induction. Put x0 D 0 2 A and define kD1 natural numbers m0 D n0 D 1. If mk 1 2 N, nk 1 2 N, and xk 1 2 A are already defined, we put ˚ Mk D .m; n/ 2 N 2 I m mk 1 ; n > nk 1 ; and xk 1 C tm en 2 A : M.Ktmi / sup

As g depends only on some finite subset of fei g on a neighbourhood of xk 1 , and tm ! 0, we can see that Mk ¤ ;. Let .mk ; nk / D min Mk in the lexicographic ordering of N 2 and put xk D xk 1 C tmk enk . P As fxk g A BX and xk D kiD1 tmi eni , by the above argument xk ! x 2 hM . We can find ı > 0 and N 2 N such that g depends only on fei gi N . Then xj C tmj enj C1 2 A and therefore .mj ; nj C1/ 2 Mj C1 . But .mj ; nj C1/ < .mj C1 ; nj C1 /, which is a contradiction. u t Theorem 145 ([Leu1]). There is an Orlicz function M such that ˛M D C1, but the Orlicz space hM does not admit an LFC bump. In particular, hM is a separable c0 -saturated Asplund space that is not isomorphically polyhedral, and hM admits an equivalent C 1 -smooth norm, but not an LFC bump. In the proof we will use the following lemma. Lemma 146. Let fbn g1 nD0 .0; C1/ be a non-increasing sequence such that bnCm m K < C1 sup m;n2N0 bn for every K > 0. Let M be the even, continuous, piecewise affine function such that M.0/ D 0 and M 0 .t/ D bn for t 2 .2 n 1 ; 2 n /, n 2 N0 , M 0 .t/ D b0 for t 1. Then M is a non-degenerate Orlicz function with ˛M D C1. Proof. Clearly M is a non-degenerate Orlicz function. Fix any q 2 .0; C1/. Since 2 n 1 bn M.2 n / 2 n bn for each n 2 N0 , C D

sup m;n2N0

bnCm q M.2 .nCm/ / mq .2 2 2 sup M.2 n / m;n2N0 bn

1 m

/ < C1:

Given any ; t 2 .0; 1 we can find m; n 2 N0 such that 2 .2 t 2 .2 n 1 ; 2 n . Then t 2 .nCm/ and hence

m 1; 2 m

.nC1Cm 1/ / M.2 .nCm/ / M.t / q M.2 < D 4 2.m 1/q 4q C q M.t/ 2 .mC1/q M.2 n 1 / M.2 .nC1/ / whenever m 1. In case that m D 0 we have M.t/ M.t/ < 2q q M.t/.

and

t u

320

Chapter 5. Smoothness and structure

Proof of Theorem 145. We construct a sequence fbn g1 nD0 .0; C1/ which satisfies the assumptions of Lemma 146 and the resulting Orlicz function M satisfies the assumptions of Theorem 144. The rest follows from Corollary 134, Theorem 103, and Corollary 141. 1 Set ˛n D nŠ and note the following properties of the sequence f˛n g1 nD0 : it is nonincreasing, ˛0 D 1,P˛nCm ˛n ˛m for all m; n 2 N0 , and ˛n K n ! 0 for each K > 0. Further, put sn D jnD0 j for n 2 N0 , and define a sequence fcn g1 nD0 by c0 D 1 and cn cnC1 D ˛nC1 ˛snC2 cn . Finally, let bsn Ck D ˛n k for n; k 2 N0 , k n. The sequence fbn g1 nD1 is non-increasing. Indeed, let n; k; l 2 N0 with k < l n. c Then bsn Ck D ˛ncn k ˛cnn l D bsn Cl and bsnC1 D ˛nC1 D ˛snC2 cn cn D bsn Cn . nC1 Next, we show that bnCm ˛m bn for all m; n 2 N0 . Fix m; n 2 N0 and let i; j; k; l 2 N0 be such that n D si C k, k i, and n C m D sj C l, l j . If i D j , c c then m D l k and hence bnCm D bsj Cl D ˛jj l ˛m ˛j j k D ˛m bn . Otherwise cj i < j and m < sj C1 , and therefore bnCm bsj D ˛j D ˛sj C1 cj 1 ˛m ci D ˛m ˛0 bsi Ci ˛m bn . It follows that for any K > 0 sup m;n2N0

bnCm m K sup ˛m K m < C1: bn m2N0

Thus the requirements of Lemma 146 are satisfied. Now let M be the Orlicz function defined in Lemma 146. Set tn D 2 .sn Cn/ . We claim that the function M and the sequence ftn g1 nD1 satisfies the assumptions of Theorem 144. This sequence is clearly decreasing to 0. Fix m 2 N. Then for each n 2 N, n m, bs Cn m cn M.2m tn / D M.2 sn nCm / sn Cn m D s Cn m 2n 2n ˛m mC1 mC1 ˛0 bsn Cn 2 2mC1 2 sn n M.2 M.tn /; / D D ˛m 2sn CnC1 ˛m ˛m which means that supn2N

M.2m tn / M.tn /

< C1.

t u

Theorem 147 ([HáTr]). Let M be an Orlicz function. Then hM admits an equivalent analytic norm if and only if either hM is isomorphic to `2n , n 2 N, or hM is isomorphically polyhedral. In particular, (i) if there is K > 0 such that lim t !0C M.Kt/ D C1, then hM has an equivalent M.t / analytic norm; (ii) if ˛M D C1 and there exists a sequence ftn g1 nD1 decreasing to 0 such that M.Ktn / supn2N M.tn / < C1 for all K > 0, then hM does not admit an equivalent analytic norm. Proof. ( follows from Theorem 106 and Theorem 104. ) If M is degenerate, then hM is isomorphic to c0 and so it is isomorphically polyhedral. Otherwise, if hM admits an analytic norm, then either hM is isomorphic to some `2n , or ˛M D C1 (Theorem 137). In the latter case, by Proposition 135

Section 10. Notes and remarks

321

every polynomial on hM is weakly sequentially continuous. Thus by Corollary 105 the space hM is isomorphically polyhedral. (i) follows from Theorem 143, (ii) follows from Theorem 133, Theorem 144, and Theorem 103. t u From Theorem 145 and Theorem 147 we immediately obtain the following corollary. Corollary 148. There is a separable c0 -saturated Asplund space with a C 1 -smooth norm that does not admit an equivalent analytic norm.

10. Notes and remarks Our approach to parts of Sections 3, 4, and 7 was inspired by [FZ2]. Section 1. The theory of convex functions with respect to Legendre-Fenchel duality and differentiability is thoroughly studied in [BorVa], where many additional results can be found. Proposition 16 is essentially contained in [MPVZ] and is based on the idea in [BN]. The definition of the modulus of exposedness and the duality results are motivated by [BGV]. Theorem 39 is essentially implicitly contained in [DGZ2]. For the theory of moduli of smoothness and convexity of Banach spaces we refer to [DGZ], [LiTz3], or [God]. The fundamental result in this area is the Enflo-Pisier renorming theorem [Enf], [Pis3], which states that a Banach space has an equivalent norm with a modulus of convexity (resp. smoothness) of power type if and only if the space is super-reflexive. There is a vast number of papers studying these moduli for concrete classes of Banach spaces e.g. [Fi], [MaT1], [IT]. In connection with finite-dimensional convex analysis, recall that Alexandrov’s theorem states that a convex function on Rn is T 2 -smooth at almost every point in Rn . We refer to [BN] for some, mostly negative, results concerning possible generalisations to infinite-dimensional spaces. However, the following seems to be still open. Problem 149. Suppose that f is a convex continuous function on `2 . Is f weakly T 2 -smooth at some point? A very similar problem is open for convex functions on c0 . Problem 150 (M. Fabian). Let f be a convex continuous function on c0 . Are there x0 2 c0 and K; ı > 0 such that f .x0 C h/ C f .x0 h/ 2f .x0 / Kkhk2 for all h 2 U.0; ı/? Section 2. The idea of Theorem 44 comes from [Li2], where Corollary 45 is proved. The idea of Lemma 47 comes from [Fab2] and it is explicitly formulated in [DevFa]. Theorem 48 is essentially from [Fab2] (where only first order smoothness is considered), an earlier version with additional assumptions is in [FWZ].

322

Chapter 5. Smoothness and structure

Section 3. Variational principles have been studied and applied extensively in various parts of functional analysis. Their systematic use in problems on higher smoothness in Banach spaces was spurred by the paper of Jonathan M. Borwein and David Preiss [BorPr]. Theorem 61 for ˛ D 0 is proved in [FHV]. Theorem 62 appears in [DGJ] and some earlier versions in [FPWZ]. Theorem 64 (without variational principles) is proved in [FPWZ]. For a sample of other applications of variational principles see e.g. [DGZ3], [DeGh]. Section 4. Variants of Theorem 66 appear in [Dev1] and [Dev2]. The main structural Theorem 67, due to Robert Deville (except for the degrees of the homogeneous polynomials), is a culmination of the work of many mathematicians. The proof relies on ideas and techniques which originated in the papers of Jaroslav Kurzweil, Victor Zakharovich Meshkov, Boris Mikhailovich Makarov, Stanisław Kwapie´n, Tadeusz Figiel, and others. Regarding the assumptions of this theorem we pose the following well-known problem. Problem 151. Let X be a Banach space which admits a C k -smooth bump (resp. norm) for every k 2 N. Does X admit a C 1 -smooth bump (norm)? Regarding the lowest degree for a single homogeneous separating polynomial we pose the next problem. Problem 152. Let X be a Banach space admitting a separating polynomial. Consider the set S D fqI there is a subspace Y X such that q.Y / D qg. Let m be the least common integer multiple of the elements in S . Does there exist an m-homogeneous separating polynomial on X ? By our results we know that m is the lowest value for which a separating polynomial might exist. If S is a singleton, then every normalised weakly null sequence in X contains a subsequence equivalent to the canonical basis of `q . An example of such space is any subspace of `p . In [JO] it is shown that if a subspace X of Lp ./, 2 < p < 1, does not contain `2 , then X is isomorphic to a subspace of `p . /. Hence there are spaces with separating polynomials which are not isomorphic to a subspace of Lp ./ for some p even. For example `r ˚ `q for distinct even integers r; q > 2 has a separating polynomial, but does not embed into any Lp ./, p even. Indeed, according to the mentioned result and the well-known structural properties of `p -spaces ([FHHMZ, p. 209]), both spaces `r and `q would have to be isomorphic to a subspace of `p , which is a contradiction. Corollary 69 together with S. Kwapie´n’s characterisation implies the following result: If X is a Banach space that admits a C 2 -smooth bump and is saturated with spaces of cotype 2, then X is isomorphic to a Hilbert space. By Corollary 68 Tsirelson-like spaces (i.e. spaces not containing a subspace isomorphic to c0 or `p ) cannot have a C 1 -smooth bump. The next problem represents a natural generalisation of this result.

Section 10. Notes and remarks

323

Problem 153. If X has a C 2;C -smooth norm, does there exist a subspace of X isomorphic to `p for some p 2 .1; 1/? In a similar vein, it seems to be unknown if the 2-convexified Tsirelson space T .2/ constructed in [FiJ] has a C 2 -smooth bump (or norm). This space is of type 2 and weak cotype 2 (which implies that it is of cotype q for every q > 2), [Johns1]. On the other hand, [DeZi], if T .2/ had a C 2;˛ -smooth bump, then by Theorem 66 and Fact 4.47 it would have a separating polynomial, and hence it would have cotype 2 D q.T .2/ /. By S. Kwapie´n’s result again it would have to be isomorphic to `2 , a contradiction. We refer to [Pis5] for the theory of weak cotype 2 spaces, which is strongly related to this subject. Problem 154. If a separable Banach space X admits a C k -smooth bump, does it admit a C k -smooth norm? In particular, Problem 155. If a separable Banach space X admits a separating polynomial, does it admit a C 1 -smooth norm? According to the remark following the proof of Theorem 73, if X has the RNP and a G 2 -smooth bump, then X is super-reflexive. The same conclusion holds, due to 1;C Corollary 51, provided that X does not contain c0 and admits a Cloc -smooth bump. We do not know if these two sets of assumptions can be meshed together into a single and stronger one. Namely: Problem 156. If a Banach space X has a G 2 -smooth norm and does not contain c0 , is then X super-reflexive? Corollary 74 and Corollary 75 have a long history, starting with results of [Me] and subsequently gradually improved by a number of authors, e.g. [Fab2], [DGZ2], [Va2], and [MPVZ]. Theorem 76 has an analogue in [GJT]. Let X be a Banach space, 1 < p < 1, and suppose that X admits a U T p -smooth bump and does not contain `k , k even integer, k < p. Then every polynomial on X of degree less than p is weakly sequentially continuous. Section 5. The concept of local dependence on finitely many coordinates goes back to Nicolaas Hendrik Kuiper. The main structural results on Banach spaces admitting a bump function which depends locally on finitely many coordinates are Theorem 90 and Theorem 91 due to Jaroslav Pechanec, John H. M. Whitfield and Václav Zizler, [PWZ], resp. Marián Fabian and V. Zizler, [FZ1]. Independently, similar results were obtained by Vladimir P. Fonf, [Fo2] under the somewhat weaker assumption of having a LFC norm rather than just a bump. Proposition 84 and Theorem 87 are from [HJ4]. Theorem 94 is due to N. Kuiper and appears (for countable) in [BF]. Lemma 96 is essentially implicitly contained in [Há2]. Equivalence of (i) and (iii) in Theorem 97 is proved in [JS] based on some earlier results from [HJ4].

324

Chapter 5. Smoothness and structure

Section 6. Polyhedral spaces were introduced by Victor Klee, as a purely isometric concept of finite-dimensional character. They were thoroughly investigated by V. P. Fonf who studied among other things their isomorphic and topological aspects as well as renorming properties, e.g. [Fo1], [Fo3], [Fo4]. Parts of Theorem 103 appear in [Fo2] and [Há2]. The proof of (i))(ii) that we present comes from [Ve]. A version of Theorem 104 for the space c0 appeared in [FPWZ]. It is clear that polyhedral spaces, as well as spaces admitting an LFC bump are closed with respect to taking subspaces, and polyhedral spaces admit a LFC bump. Problem 157. If a separable Banach space X admits a LFC bump, is then X isomorphic to a polyhedral space? This problem has been addressed in [HJ3], where a tentative counterexample has been constructed, but the proof of non-polyhedrality of the candidate space is still missing. Polyhedral spaces, or more generally spaces admitting a LFC bump are c0 -saturated Asplund spaces. The converse fails, as in [CKT] there is a first example of a c0 -saturated space (in fact an Orlicz space) which is not isomorphically polyhedral and its quotient is isomorphic to `2 . Denny Ho-Hon Leung [Leu2] constructed first such example with an unconditional basis. Ioannis Gasparis improved this result in [GaspaI1], [GaspaI2] showing that every `p , 1 < p < 1, is isomorphic to a quotient of a polyhedral Banach space with an unconditional basis. Spiros A. Argyros, Vaggelis Felouzis, and Theocharis Raikoftsalis [ArgFe], [ArgRa] showed that every reflexive Banach space is a quotient of a c0 -saturated Banach space. Considering the relationship between polyhedral spaces and L1 -preduals, by the results in [JohnsZip2] if X is separable and X is isometric to L1 ./, then X is isometric to a quotient of C./. The isomorphic version is false, as there exist separable isomorphic preduals of `1 not containing a subspace isomorphic to c0 . First such spaces were constructed in [BD]. Problem 158. Is every separable polyhedral Banach space isomorphic to a subspace of an isometric predual of `1 ? Problem 159. Is every quotient of C.K/, K scattered, c0 -saturated? It is known ([Al]) that a quotient of C.Œ0; ! ! /, which is an isometric predual of `1 , need not be isomorphic to a subspace of any C.K/, K scattered. Section 7. The Banach spaces of the type Lp form the first class of spaces whose higher smoothness properties were investigated in quantitative detail. The upper estimate of the best smoothness of the classical Lp spaces summarised in Theorem 107 has its origins in the work of J. Kurzweil [Kur1], and also [BF]. The results are essentially from [DGJ], [FZ2], [MV], and [HeTr]. The closely related spaces Lp .I X/ were treated by Isaac Edward Leonard and Kondagunta Sundaresan in [LeSu], and in the special case X D Lq also in [DGJ].

Section 10. Notes and remarks

325

The following result from [DGJ] has a proof similar to those in Section 7: Let L1 n ` 1 < p; q < C1 and q is not an even integer. Then the space does not nD1 q p k;˛ admit a Cloc -smooth bump for k C ˛ > q. Among the classes of Banach spaces that are close to Lp -spaces and for which the best smoothness renormings have been found there are also the trace-class spaces cp .H / (the non-commutative `p -spaces). The results are similar to their classical `p -counterparts (this is not very surprising, since by [Fri] every subspace of cp .H / which is not isomorphic to `2 contains `p ). In particular, in [Tom1] and [GGo] the following is shown: Let H be a real infinite-dimensional Hilbert space and 1 < p < C1. (i) If p is an even integer, then the norm of cp .H / is p-polynomial. (ii) The norm of cp .H / is C dpe 1 -smooth. (iii) If p is not an even integer, then cp .H / does not admit a T p -smooth bump. The space c4 .`2 / is an example of a space with a separating 4-homogeneous polynomial which is not isomorphic to a subspace of L4 ./. Indeed, Charles McCarthy in [McC] proved that cp .H / does not embed into Lp ./, 2 < p < C1. To complete the picture, there is an equivalent C 1 -smooth norm on LK .H I H /, the space of compact operators on H [Tom1]. However, the following is open. Problem 160 (V. Zizler). Does the space of compact operators LK .`2 I `2 / admit a real analytic norm? Section 8. By the results in Chapter 6, every C 2 -smooth function on c0 . / depends locally on countably many coordinates. In particular we conclude the next result. Theorem 161 ([Há5]). If is uncountable, then there is no C 2 -smooth function on c0 . / which attains its minimum at one point. In particular, there is no equivalent rotund C 2 -smooth norm on c0 . /. There is a large number of papers devoted to renorming theory and smoothness of C.K/ spaces, apart from the results presented in this section. To begin with, by Theorem 125 we may restrict our attention to scattered compact sets K only. A testing ground for this theory is provided by a special class of scattered compacts – the one point compactifications of trees considered by Richard Haydon in his seminal paper [Hay3]. R. Haydon proved that C0 .T / has a C 1 -smooth bump for every tree T . He also characterised those trees for which X D C0 .T / admits a C 1 -smooth renorming by the existence of the so called Talagrand operator, i.e. a bounded linear operator T W X ! c0 .SX / such that for every x 2 X there exists .f; m/ 2 SX with f .x/ D kxk and T .x/.f; m/ ¤ 0 (this formulation in fact comes from the subsequent paper [FPST], and appears to be more general). In both cases mentioned above, the C 1 -smooth bumps and norms are LFC. By intricate analysis of the branching properties of trees, R. Haydon constructed examples of trees such that C0 .T / does not have even Gâteaux differentiable equivalent renorming. By contrast to Theorem 126 it is shown in [MoT] that C.K/ admits no equivalent uniformly Gâteaux differentiable renorming provided K is a separable and uncountable

326

Chapter 5. Smoothness and structure

scattered compact. In this context we point out that UG renormability for any Banach space was characterised in [FGZ]. In light of the above results and the results in this section, we pose the following well-known problem. Problem 162. If K is a scattered compact, does C.K/ admit a C 1 -smooth bump? In [FPST] an analogous theory has been built for polyhedral renormings. Namely, it is shown there that C.K/ admits a polyhedral renorming if and only if C.K/ admits a Talagrand operator. Apart from the trees such that C0 .T / has a C 1 -smooth renorming, this condition is also satisfied for all -discrete compacts K. It should be noted that in these cases one can even construct C 1 -smooth and LUR norms which are uniform limits of C 1 -smooth norms, [HP]. For more information on the subject of smooth renormings of non-separable Banach spaces we refer especially to [DGZ] and the survey paper [ST]. On the other hand, the following remains to be one of the main open problems in renorming theory. Problem 163. Does every Asplund space admit a Fréchet differentiable bump? Section 9. The best smoothness of Orlicz-type Banach spaces has been initially studied in particular by Rumen Petrov Maleev and Stanimir L. Troyanski in several papers [MaT3], [Mal1], [Mal2]. The polyhedral case received particular attention in the works of D. H. Leung, [Leu1], [Leu3]. Theorem 144 is from [Leu1] and [HJ3]. In [MZ] (and partially in [GJT]) the best smoothness of Orlicz-Musielak spaces has been studied. In connection with Theorem 143 there is a result of S. L. Troyanski: Let fpn g Œ1; C1/ and pn ! C1. Then the Nakano space hfpn g is isomorphically polyhedral. Results concerning the best Gâteaux smoothness for separable Banach spaces are quite rare. It is proved in [Tr] that Lp ./ for a -finite admits an equivalent G p -smooth norm for p an odd integer. (For separable Lp -spaces this result can be gleaned out by combining Theorem 106 and Corollary 7.72.) There are somewhat strengthened results in [Mal2]. The assumption on the measure space cannot be relaxed (see Theorem 107). This leaves us with the following natural problem. Problem 164. Let p > 2, p not an even integer. Does `p admit a G ŒpC1 -smooth bump or an equivalent norm?

Section 10. Notes and remarks

327

Table 1. Smoothness in various spaces space Lp ./

p-polynomial norm

p not even

`p . /,

k

norm, k D dpe

1

X has C k;˛ bump, kC˛ p

C k;˛ bump

Cor. 118

q not even

p-polynomial norm C k;p

k

norm, k D dpe k;r

1

T p bump

1

T r bump, r D minfp; qg

k

C norm, r D minfp; qg, k D dre

T p bump

q not even or p ¤ kq c0

analytic norm;

separable polyhedral

C 1 LFC norm

c0 . /,

C 1 LFC and UG norm

uncountable

C.K/

Th. 127 Th. 128

C

1

LFC norm

C k;q k norm, k D d˛M e 1, q 2 .k; ˛M /

˛M D C1

C 1 norm

M.Kt / t !0C M.t /

D C1

M.Ktn / M.tn /

< C1

M.Ktn / M.tn /

< C1,

lim

Th. 129 analytic norm

1 < ˛M , ˇM < C1

n2N

Th. 104 T. 126; C. 3.59, C. 105

C 1 LFC bump

1 < ˛M < C1

sup

Th. 94, Pr. 49,

K Db T

k;q k norm, Cloc k D d˛M e 1, q 2 .k; ˛M /

n2N

analytic norm

Th. 121

C 1 LFC norm

K uncountable

sup

C 1;C bump

Cor. 119, Cor. 122

K D Œ0;

K is -discrete

hM © `p

Th. 107 Th. 108

q even, p ¤ kq

p

Th. 106, Th. 107

continuous weakly T p bump C k norm

q even, p D kq

T p bump

X has C k norm with d kkk bounded, k < p

Lp I Lq ./

n nD1 `q

C k;p

remark Th. 106

uncountable, p not even

Lp .I X /

L1

does not admit

admits p even

T ˛M bump

polyhedral

Th. 132

Cor. 141

Th. 143 LFC bump

Th. 144

analytic norm

Th. 147

˛M D C1 LK .H I H / p not even

cp .H /

p even

`fpn g © `p

C dpe

p D lim inf pn < C1 pn ! C1

norm 1

norm

T p bump

p-polynomial norm C d˛' e

h' hfpn g

C

1

C dpe

1 1

[GGo]

norm

[MZ]

norm

polyhedral

Troyanski, unpubl.

pn ! q ¤ 2k

T q bump

pn ! 2k, pn 2k

C 2k bump

pn ! 2k, pn 2k

[Tom1]

T

2k

bump

[MZ] [GJT]

Chapter 6

Structural behaviour of smooth mappings In the present chapter we develop the theory of smooth mappings between Banach spaces with emphasis laid on the preservation of the structure of the initial space. In the first section we study general properties of weakly continuous mappings with regards to higher smoothness. The important Theorem 8 bootstraps the weak uniform continuity into the higher differentials provided that d kf is uniformly continuous. If X does not contain `1 , then these results combine into a very efficient tool for handling uniformly smooth mappings. In Section 2 we study the bidual extensions of mappings f 2 C k;C .BX I Y / into the bidual denoted by f 2 C k;C .BX I Y /. The construction requires several ingredients, namely the Converse Taylor theorem and the powerful ultrapower construction based on the principle of local reflexivity. The extension will be a key tool later on for generalising some results concerning C.K/ for K scattered to the general compact spaces K. In Section 3 we start developing the theory of W -spaces, i.e. all those X such that 1;C C .BX / CwsC .BX / holds for some 2 .0; 1. The main goal of this theory is to generalise Theorem 3.47 into the setting of uniformly smooth mappings. This objective is achieved in the main Theorem 57 after having completed several important steps. The first step is the quantitative finite-dimensional Lemma 27, which shows the rigidity of `n1 with respect to uniformly smooth functions. Using this essential observation, the second step consists of showing that C.K/ for K scattered are W1 -spaces. The third step is completed in Theorem 46. In Section 4 uniformly smooth non-compact mappings from C 1;C .Bc0 I X/ are studied in detail. They are characterised by the fact that their bidual extension has a point where the derivative is non-compact, and hence fixes a copy of c0 . This implies that X contains c0 . Moreover, non-compact uniformly smooth mappings from C.K/, K scattered compact space, can always be reduced to a suitable subspace isomorphic to c0 where the restriction remains noncompact and this wraps up the scattered case. The case of a general C.K/ space is taken up in Section 5. The new ingredient, Theorem 56, states that weakly Cauchy sequences in C.K/ spaces are uniformly close to weakly Cauchy sequences in some C.Œ1; ˛/ after passing to the bidual. In Section 6 we show that for non-compact polynomials from C.K/, K scattered, fixing a copy of c0 remains true, but we give an example of a uniformly smooth mapping from Bc0 where the fixing property fails.

Section 1. Weak uniform continuity and higher smoothness

329

In Section 7 we give some rather general results on the ranges of smooth mappings (and derivatives of smooth functions) which illustrate that the theory of the class W works under nearly optimal assumptions. Indeed, according to Theorem 69 no structure is preserved by surjective C 1 -smooth mappings from spaces with property B. In order to obtain positive results we introduce the notion of separating mappings. In the rest of the chapter our attention moves to smooth separating mappings from `p -spaces, proving a strong structural result in this case. In the whole chapter the scalars are real unless specified otherwise.

1. Weak uniform continuity and higher smoothness We continue with the study of higher smoothness in connection with the weak continuity. Definition 1. Let X , Y be normed linear spaces, U X a convex set with non-empty interior, and k 2 N0 (resp. k D 1). By C k .U I Y / we denote the locally convex space of C k -smooth mappings f W Int U ! Y such that every d jf , j D 0; : : : ; k (resp. j 2 N0 ), has a continuous extension to the whole U , and such that all d jf are bounded on CCB subsets of U . We will use the expression d jf .x/ for all x 2 U in the sense of these extensions. The space C k .U I Y / is endowed with the locally convex topology bk of uniform convergence of d jf , j D 0; : : : ; k (resp. j 2 N0 ) on CCB subsets of U . It is easy to see using Theorem 1.85 that if U D X (resp. if U is a CCB set) and Y is a Banach space, then .C k .U I Y /; bk / is a Fréchet locally convex space (resp. a Banach space, provided that k < 1). Let X , Y be normed linear spaces, U X a convex set with non-empty interior, k 2 N (resp. k D 1), and f 2 C k .U I Y /. If V U is a CCB set, then there is a CCB set W U satisfying V W D W \ Int US . Indeed, assuming without loss of generality that 0 2 Int U , we may take W D t2Œ0;1 tV . Thus applying Proposition 1.71 to W \ Int U we san see that all d jf , j D 0; : : : ; k 1 (resp. all j 2 N0 if k D 1) are Lipschitz on V . In the next we are interested in general theory of weakly continuous mappings (in several senses, as defined in Section 3.2) which have uniformly continuous derivatives. Definition 2. Let X , Y be normed linear spaces, U X a convex set with non-empty interior, and k 2 N0 . By Cu .U I Y / we denote the subspace of C.U I Y / consisting of mappings that are uniformly continuous on CCB subsets of U . By C k;C .U I Y / we denote the subspace of C k .U I Y / consisting of mappings f such that d kf is uniformly continuous on CCB subsets of U . As usual, in the scalar case we use the notation C k;C .U /.

330

Chapter 6. Structural behaviour of smooth mappings

Notice that by the remarks above if f 2 C k;C .U I Y /, then all d jf , j D 0; : : : ; k are uniformly continuous on CCB subsets of U . In view of Theorem 1.125 we will also call the mappings in C k;C .U I Y / uniformly smooth. Fact 3. Let X, Y be normed linear spaces, U X a convex set with non-empty interior, k 2 N, f 2 C k;C .U /, C U a CCB set satisfying C C \ Int U , and let ! 2 M be a modulus of continuity of d kf on C . Then for any x; y 2 C

k

X 1 1 j

d f .x/Œy x !.ky xk/ky xkk :

f .y/

kŠ

jŠ j D0

Proof. By considering the minimal modulus of continuity of d kf we may assume without loss of generality that ! 2 Ms . For x; y 2 C \ Int U the estimate follows from Corollary 1.108. Then we extend the estimate to the boundary by the continuity. u t Proposition 4. Let X , Y , Z be normed linear spaces, U X and V Y convex sets with non-empty interior, g 2 C 1 .U I V /, and f 2 C 1;C .V I Z/. Suppose further that for any CCB subset C of U the set g.C / is contained in a CCB subset of V (this holds in particular if the set V is closed). Then f B g 2 C 1 .U I Z/ and D.f B g/.x/ D Df .g.x// B Dg.x/ for every x 2 U . If moreover g 2 C 1;C .U I Y /, then f B g 2 C 1;C .U I Z/. Proof. Fix an arbitrary x 2 Int U and let ı > 0 be such that x C h 2 Int U and kg.x C h/ g.x/ Dg.x/Œhk khk for h 2 B.0; ı/. Then kg.x C h/ g.x/k kg.x C h/ g.x/ Dg.x/Œhk C kDg.x/kkhk .kDg.x/k C 1/khk for h 2 B.0; ı/. Let W V be a CCB set satisfying g.B.0; ı// W W \ Int V and let ! be a modulus of continuity of Df on W . Using Fact 3 we obtain

f g.x C h/ f g.x/ Df g.x/ Dg.x/Œh

f g.x C h/ f g.x/ Df g.x/ Œg.x C h/ g.x/

C Df g.x/ g.x C h/ g.x/ Dg.x/Œh ! kg.x C h/ g.x/k kg.x C h/ g.x/k

C Df g.x/ g.x C h/ g.x/ Dg.x/Œh D o.khk/; h ! 0: Thus f B g is Fréchet differentiable at x and its derivative satisfies the Chain rule formula. Now for any x; y 2 U and h 2 BX we have

Df g.y/ Dg.y/Œh Df g.x/ Dg.x/Œh

Df g.y/ Dg.y/Œh Df g.x/ Dg.y/Œh

C Df g.x/ Dg.y/Œh Df g.x/ Dg.x/Œh

Df g.y/ Df g.x/ kDg.y/k C Df g.x/ kDg.y/ Dg.x/k:

Section 1. Weak uniform continuity and higher smoothness

331

Thus the mapping x 7! Df .g.x// B Dg.x/ is continuous on U and consequently f B g 2 C 1 .U I Z/ and the Chain rule formula holds on the whole of U . If moreover g 2 C 1;C .U I Y /, then the estimate above implies that f B g 2 C 1;C .U I Z/. t u Definition 5 ([AP]). Let X be a normed linear space, Y a Banach space, U X k .U I Y / we denote the a convex set with non-empty interior, and k 2 N0 . By Cwu k j subspace of C .U I Y / consisting of all f that satisfy d f 2 Cwu U I Pw . jXI Y / , j D 0; : : : ; k. k .U I Y / if and only if d jf 2 C j We remind that f 2 Cwu wu U I Pwu . X I Y / for j D 0; : : : ; k (Corollary 3.27). Lemma 6. Let X be a normed linear space, Y a Banach space, and let U X be a convex set with non-empty interior. If f 2 C 1;C .U I Y / \ Cwu .U I Y /, then Df 2 Cwu U I Lwu .XI Y / . Proof. The fact that Df .x/ 2 Lwu .X I Y / for every x 2 Int U follows at once combining Proposition 3.70, Lemma 3.18, and Corollary 3.24. For non-interior points we use the fact that Lwu .XI Y / is closed in L.X I Y /. Now let V U be a CCB set and " > 0. First assume that D dist.V; X n U / > 0. U T 1 -smooth Put W D V CU.0; 2 /. Then W is a CCB set with W Int U . Since

f is on W , there is 0 < ı < 2 such that f .x C h/ f .x/ Df .x/Œh 4" khk whenever x 2 V , h 2 B.0; ı/. By the assumptions there exist > 0 and 1 ; : : : ; n 2 X such that kf .x/ f .y/k < "ı y/j < . Thus for 4 whenever x; y 2 W and maxl jl .x x; y 2 V such that maxl jl .x y/j < and h 2 BX we have

Df .x/Œh Df .y/Œh D 1 Df .x/Œıh Df .y/Œıh ı

1

f .x C ıh/ f .x/ Df .x/Œıh C f .y C ıh/ f .y/ Df .y/Œıh ı C kf .y C ıh/ f .x C ıh/k C kf .y/ f .x/k < ": Now assume that V is a general CCB subset of U . Without loss of generality we U . Let r D dist.0; X n U / > 0 and let R > 0 be such that may assume that 0 2 Int S V B.0; R/. Put A D t 2Œ0;1 tV . Then A is a CCB subset of U and so there is ı > 0 such that kDf .x/ Df .y/k < 3" whenever x; y 2 A, kx yk < ı. Choose 2 .0; 1/ so that .1 /R < ı. Since U.0; r/ U , from the convexity we get U.x; .1 /r/ U for any x 2 U . Consequently dist.V; X n U / .1 /r > 0. By the previous part of the proof applied to the CCB set V there are > 0 and 1 ; : : : ; n 2 BX such that kDf .u/ Df .v/k < 3" whenever u; v 2 V and maxl jl .u v/j < . Now pick any x; y 2 V such that maxl jl .x y/j < . Put u D x and v D y. Then kx uk D .1 /kxk .1 /R < ı and similarly ky vk < ı. Clearly u; v 2 V and since maxl jl .u v/j D maxl jl .x y/j < , it follows that kDf .x/ Df .y/k kDf .x/ Df .u/kCkDf .u/ Df .v/kCkDf .v/ Df .y/k < ": t u

332

Chapter 6. Structural behaviour of smooth mappings

Fact 7. Let X be a normed linear space, Y a Banach space, n 2 N, and further let I W L XI L. nXI Y / ! L. nC1X I Y / be the canonical isometry from Fact 1.9. Then Lswu . nC1XI Y / D I Lwu XI Lswu . nXI Y / \ Ls . nC1X I Y / D I LK XI L. nX I Y / \ Ls . nC1X I Y /: Proof. Let M 2 Lwu . nC1XI Y / be given. Denote T D I 1 .M / and choose an arbitrary " > 0. There exist weak neighbourhoods W0 ; : : : ; Wn of the origin in X such that we have kM.x0 ; : : : ; xn / M.y0 ; : : : ; yn /k < " whenever xj ; yj 2 BX and xj yj 2 Wj for j D 0; : : : ; n. Therefore for every fixed x 2 BX we have kT .x/.x1 ; : : : ; xn / T .x/.y1 ; : : : ; yn /kDkM.x; x1 ; : : : ; xn / M.x; y1 ; : : : ; yn /k 0 and a finite set ˚ BX such that kDf .xj /Œhk < 2" , j D 1; : : : ; m whenever h 2 B.0; 2R/ satisfies max2˚ j.h/j < ı. Now if x 2 V is " arbitrary, then there is j 2 f1; : : : ; mg such that kDf .x/ Df .xj /k < 4R and hence kDf .x/Œhk kDf .xj /Œhk C 2RkDf .x/ Df .xj /k < " whenever h 2 B.0; 2R/ satisfies max2˚ j.h/j < ı. So if x; y 2 V \Int U are such that max2˚ j.x y/j < ı, then Theorem 1.107 yields

Z 1

kf .y/ f .x/k D Df x C t.y x/ Œy x dt

": 0

The same inequality for points in the closure is now immediate. The case of general k > 1 follows by induction: Assume that the statement holds for k 1 and set g D D k 1f and Z D Lswu . k 1X I Y /. Then by the assumption and Fact 3.21 we have g W U ! Z and D kf 2 CK U I Lswu . kX I Y / . By Fact 7, Dg 2 CK .U I Lwu .XI Z// and so by the first part of the proof g 2 Cwu .U I Z/. Consequently f satisfies the assumptions of the inductive hypothesis and the proof is finished. t u The previous theorem in general does not hold without the assumption that all the differentials at each point are w–kk uniformly continuous polynomials, as the identity mapping on `2 or c0 clearly shows (its second derivative is a constant zero mapping, but its first derivative at each point is the identity and hence not compact). However, in the special case of polynomials (of degree at least 2) it does hold. Theorem 10. Let X be a normed linear space, Y a Banach space, n 2 N, n 2, and P 2 P . nXI Y /. The following statements are equivalent: (i) P 2 Pwu . nX I Y /. (ii) d kP 2 PK . n kXI P . kX I Y // for some 1 k < n. k .XI Y / for every k 2 N. (iii) P 2 Cwu Proof. (i),(iii) follows from Theorem 8, (iii))(ii) follows from Lemma 3.18. (ii))(i) It suffices to prove this implication for k D 1 and then use induction. So assume that DP 2 PK . n 1X I L.X I Y //. We claim that this in fact means that DP 2 PK . n 1XI LK .XI Y // and so we can apply Theorem 9. By Fact 3.21 we have } 2 LK . n 1X I L.X I Y //. Fix x 2 X . Using Lemma 1.99 and the symmetry of P } DP n 1 n 1 n 2 }. x; BX / D nP }.BX ; } .BX ; we obtain DP .x/ŒBX D nP x/ D DP x/Œx D n 2 } "x B DP .BX ; x/, where "x 2 L.L.X I Y /I Y / is the operator of evaluation at x. It follows that DP .x/ŒBX is relatively compact in Y . t u

2. Bidual extensions In this section we study further the extension of continuous mappings with domain in X to mappings with domain in X initiated in Section 4.1. In particular we concentrate on the behaviour of the derivative of the extension.

334

Chapter 6. Structural behaviour of smooth mappings

Let X , Y be Banach spaces and f W BX ! Y a uniformly continuous mapping. Recall that we can define an extension EU;T .f / W BX ! Y by EU;T .f /.x/ D w -lim f .T .x/˛ / U

for any x 2 BX , where U is the ultrafilter and T 2 L.X I .X/U / is the isometric embedding from Corollary 4.7. (The representation T .x/ D .T .x/˛ /U is chosen so that T .x/˛ 2 BX for all ˛ 2 I , Proposition 4.8.) In general the extension depends on U and T , but as mentioned in Section 4.1 if e.g. f is w–w uniformly continuous, then the extension is uniquely determined. This is the case of an extension of a linear operator L 2 L.X I Y /, and EU;T .L/ D L (the double adjoint operator, in the sense of a restriction), independently of U and T . Our interest lies mostly in some applications of the bidual extension which are independent of the concrete choice of U and T . We are going to assume from now on that U and T have been fixed. In order to simplify the notation we will use the notation E.f / D EU;T .f /, or the more suggestive f D E.f / (in accordance with the extension of linear operators), and in this section also f D E.f / for better readability of some formulae. We remark that if Y is a dual space (e.g. if Y D L.X I Z /), then we can define an extension with range in Y using directly the w topology of Y (see Section 4.1). This extension is in general different from the extension considered above (using the w topology in Y ) and for our applications (mainly Theorem 18) we always need to extend into Y , as the w topology on Y is too weak here. Fact 11. Let X , Y be Banach spaces and f W BX ! Y a uniformly continuous mapping. If f 2 CK .BX I Y /, then f 2 CK .BX I Y /. Similarly, if f 2 CwK .BX I Y /, then f 2 CwK .BX I Y /. w

Proof. This is immediate from the fact that f .BX / f .BX / w

w

resp. f .BX / f .BX / D f .BX / weakest Hausdorff topology.

kk

D f .BX /

Y,

Y , since the compact topology is the t u

Theorem 4.9 has the following corollary. Corollary 12. Let X and Y be Banach spaces and let f 2 C k;C .BX I Y /. Then f 2 C k;C .BX I Y / and d jf .x/Œh D w -limU d jf .T .x/˛ /ŒT .h/˛ for any j D 1; : : : ; k, x 2 BX , and h 2 X . Proof. Theorem 4.9 implies f 2 C k;C .Int BX I Y / and the validity of the formula on Int BX . All the differentials can be continuously extended to BX by the uniform continuity and so f 2 C k;C .BX I Y /. To prove the formula on the whole BX fix j 2 f1; : : : ; kg, x 2 SX , h 2 BX , 2 BX , and " > 0. There is ı > 0 such that kd jf .u/ d jf .v/k < " whenever u; v 2 BX satisfy ku vk ı. Find y 2 Int BX j j such that kx yk < ı and ˇkd ˝ jf .x/Œh d jf .y/Œhk < ". There˛ˇis U 2 U such that ˇ kT .x/˛ T .y/˛ k < ı and d f .y/Œh d f .T .y/˛ /ŒT .h/˛ ; ˇ < " for all ˛ 2 U .

Section 2. Bidual extensions

335

Then for any ˛ 2 U we obtain ˇ˝ j ˛ˇ ˇ d f .x/Œh d jf .T .x/˛ /ŒT .h/˛ ; ˇ

ˇ˝ d jf .x/Œh d jf .y/Œh C ˇ d jf .y/Œh

C d jf .T .y/˛ /ŒT .h/˛ The following fact is immediate using the

w –w

˛ˇ d jf .T .y/˛ /ŒT .h/˛ ; ˇ

d jf .T .x/˛ /ŒT .h/˛ < 3":

t u

continuity of adjoint operators.

Fact 13. Let X, Y , Z be Banach spaces, f W BX ! Y a uniformly continuous mapping, and L 2 L.Y I Z/. Then L B f D L B f . Theorem 14 ([CHL]). Let X be a L1; -space, 1, and let i W X ! Z be an isometric embedding into some Banach space Z. Then for every Banach space Y and f 2 C k;C .BX I Y / there exists a “partial extension” g 2 C k;C . 1 BZ I Y / such that g.i.x// D f .x/ for x 2 1 BX . Proof. The operator i W X ! Z is again an isometry into. Since X is a P -space (Theorem 4.15), there is a projection P W Z ! i .X / of norm at most . We define the extension by g D f B .i / 1 B P . t u We now focus on the relationship between the operations of taking the extension E and taking the derivative D, chiefly as a tool for studying the properties of the derivative of the extension (Theorem 18). Let f 2 C 1;C .BX I Y /. Then D B E.f / D Df W BX ! L.X I Y /; E B D.f / D Df W BX ! L.X I Y / : The ranges of the respective operations are different spaces, so strictly speaking E and D cannot commute. Moreover, the values of the latter composition are not even linear operators. However, there is a very natural way to consider their commutation in a weaker sense which in some sense reflects that both of these compositions somehow coincide as “operators” with respect to evaluations on elements in X . In fact, and somewhat more precisely, we will consider the situation when these two objects coincide with respect to certain duality evaluations involving the tensor product X ˝ Y , which can be paired naturally with both of the spaces in question. Denote by L .X I Y / the subspace of L.X I Y / consisting of all double adjoint bounded linear operators. The operation E restricted to bounded linear operators is an isometry E W L.X I Y / ! L .X I Y / onto. Further, the space L.X I Y / is canonically isometric to a subspace of its double dual space L.XI Y / via i W L.X I Y / ! L.X I Y / such that hi.L/; i D h; Li for L 2 L.X I Y /, 2 L.X I Y / . By Fact 3.8, L.X I Y / D .X ˝ Y / , hence there is a canonical isometric embedding into the double dual space j W X ˝ Y ! L.X I Y / such that hj./; Si D hS; i for 2 X ˝ Y , S 2 L.X I Y /. We consider the following restriction of the quotient operator E adjoint to the embedding E W L.XI Y / ! L.X I Y /: J D E B j W X ˝ Y ! L.XI Y / :

336

Chapter 6. Structural behaviour of smooth mappings

Then J W L.XI Y / ! L.X I Y / satisfies E D J B i. Indeed, hJ B i.L/; i D hi.L/; J./i D hJ./; Li D hE B j./; Li D hj./; E.L/i D hE.L/; i

(1)

for all 2 X ˝ Y , L 2 L.X I Y /. Hence the following diagram commutes: L.X I Y /

E

/ L .X I Y /

i

L.X I Y /

J

(2)

Id

/ L.X I Y /

Definition 15. Let X, Y be Banach spaces and f 2 C 1;C .BX I Y /. We say that the extension and differentiation operations E and D are weakly commuting for f at ´ 2 BX if Df .´/ D J .Df .´//: We say that the operations E and D are weakly commuting for f on a set V BX provided they are weakly commuting at every point ´ 2 V . We note that if Y D K, then the extension operator restricted to X is an identity E W X ! X X and J W X ! X is also an identity. Thus the weak commutation in this case has a natural form Df .´/ D Df .´/. Recall that w -limU T .´/˛ D ´ for every ´ 2 X . Proposition 16. Let X , Y be Banach spaces, f 2 C 1;C .BX I Y /, and ´ 2 BX . The following statements are equivalent: (i) The operations E and D are weakly commuting for f at ´. (ii) hDf .´/; h ˝ i D hJ .Df .´//; h ˝ i for all h 2 X and 2 Y . (iii) w -limU Df .T .´/˛ / .h/ D w -limU Df .T .´/˛ /ŒT .h/˛ for all h 2 X . (iv) The operations E and D are weakly commuting for B f at ´ for all 2 Y . Proof. We have Df .´/ D J .Df .´// if and only if hDf .´/; i D hJ .Df .´//; i for all 2 X ˝ Y (Fact 3.8). It follows that (i) and (ii) are equivalent, because the span of all elementary tensors is norm dense in X ˝ Y . To show (ii),(iii) fix h 2 X and 2 Y . Then, using Corollary 12, ˝ ˛ hDf .´/; h ˝ i D hDf .´/Œh; i D lim Df .T .´/˛ /ŒT .h/˛ ; ; U

while ˝ ˛ hJ .Df .´//; h ˝ i D hDf .´/; J.h ˝ /i D lim Df .T .´/˛ /; E B j.h ˝ / U ˝ ˛ ˝ ˛ D lim j.h ˝ /; E.Df .T .´/˛ // D lim j.h ˝ /; Df .T .´/˛ / U U ˝ ˛ ˝ ˛ D lim Df .T .´/˛ / ; h ˝ D lim Df .T .´/˛ / .h/; : U

If L 2 L.X I Y /, 2

Y ,

and h 2

X ,

then

U L ./

D B L and so

hL .h/; i D hh; L ./i D hh; B Li:

(3)

Section 2. Bidual extensions

Also, for get

2 X we have

337 .h/

D h. /. Combining this with the Chain rule we

hDf .T .´/˛ / .h/; i D hh; B Df .T .´/˛ /i D hh; D. B f /.T .´/˛ /i D D. B f /.T .´/˛ / .h/: On the other hand, we have h; Df .T .´/˛ /ŒT .h/˛ i D B Df .T .´/˛ / .T .h/˛ / D D. B f /.T .´/˛ /ŒT .h/˛ . Thus (iii) is equivalent to lim D. B f /.T .´/˛ / .h/ D lim D. B f /.T .´/˛ /ŒT .h/˛ U

X

U

for all h 2 and 2 B f gives (i),(iv).

Y .

So the equivalence of (i) and (iii) applied to the functions t u

The operations E and D always weakly commute on BX . Indeed, let x 2 BX . Since T .x/˛ D x for all ˛ 2 I , the condition (iii) in Proposition 16 transforms to Df .x/ .h/ D w -limU Df .x/ŒT .h/˛ , which holds by the w –w continuity of adjoint operators. Moreover, Df .x/ D Df .x/ 2 L .X I Y / by Theorem 4.9(iii). In general, however, Df .´/ need not be a double adjoint operator if ´ 2 BX . Indeed, let f W c0 ! c0 be given by f .x0 ; x1 ; : : : / D .x0 ; x0 x1 ; x0 x2 ; : : : /. Then f W `1 ! `1 is given by the same formula and Df .1; 1; : : : /Œh D .h0 ; h0 C h1 ; h0 C h2 ; : : : /. So Df .1; 1; : : : / … L .`1 I `1 /, since it does not map c0 into c0 , although it is a w –w continuous operator and the operations E and D weakly commute on B`1 in this case (Theorem 23 and Theorem 30). Theorem 17 ([CHL]). Let X, Y be Banach spaces, f 2 C 1;C .BX I Y /, and suppose that Df 2 CK BX I L.X I Y / . Then the operations E and D are weakly commuting for f on BX and Df .´/ 2 L .X I Y / for every ´ 2 BX . Proof. Fix ´ 2 BX . Since Df 2 CK BX I L.XI Y / and the compact topology is the weakest Hausdorff topology, Df .´/ D limU Df .T .´/˛ / in the norm topology of L.X I Y / . Hence Df .´/ 2 i.L.X I Y //, i.e. there is L 2 L.XI Y / such that i.L/ D Df .´/. Thus limU kDf .T .´/˛ / L k D limU kDf .T .´/˛ / Lk D 0. Since ˇ ˇ ˇhDf .T .´/˛ / .h/; i h; Df .T .´/˛ /ŒT .h/˛ iˇ

ˇ˝ ˛ˇ Df .T .´/˛ / L kkkhk C ˇ L .h/ L.T .h/˛ /; ˇ C kL

Df .T .´/˛ /kkkkT .h/˛ k

for h 2 X and 2 Y , using the w –w continuity of L we conclude that (iii) in Proposition 16 holds. Finally, by combining the established weak commutation property with the commutative diagram (2) we obtain hDf .´/Œh; i D hJ .Df .´//.h/; i D hL Œh; i for each 2 Y and h 2 X , and so Df .´/ D L 2 L .X I Y /.

t u

338

Chapter 6. Structural behaviour of smooth mappings

Let Y , Z be Banach spaces. We consider two natural topologies on L.ZI Y /. The first one is the weak topology coming from the duality .Z ˝ Y / D L.ZI Y / of Fact 3.8. The second one is the weak operator topology (W OT for short), i.e. the locally convex topology generated by the semi-norms L 7! jhL.x/; ij, x 2 Z, 2 Y . Since the elementary tensors are linearly dense in Z ˝ Y , the W OT and w topologies coincide on bounded subsets of L.ZI Y /. Theorem 18. Let X , Y be Banach spaces and let f 2 C 1;C .BX I Y / be w–w uniformly continuous. Then the bidual extension f 2 C 1;C .BX I Y / is w –w uniformly continuous, and in particular independent of the choice of U and T . Thus for any subspace Z X we have f BZ D f BZ . Moreover, the mapping Df 2 Cu .BX I L.X I Y // is also w –w uniformly continuous. Proof. The first statement is clear (see the discussion preceding Theorem 4.9). If Z X is a subspace, then f BZ is also w–w uniformly continuous. Since the w –w continuous mappings f BZ and f agree on BZ which is w -dense in BZ , they also agree on BZ . To prove the last assertion, fix arbitrary F 2 X and 2 Y . We claim that J.F ˝ / B Df D F B D. B f /:

(4)

Indeed, let D F ˝ . For any x 2 BX we have .J. /BDf /.x/ D hJ. /; Df .x/i D hDf .x/ ; i D hDf .x/ .F /; i D hF; B Df .x/i D hF; D. B f /.x/i D .F B D. B f //.x/, where we successively used (1), Fact 3.8, (3), and the Chain rule. Because by our assumption B f 2 Cwu .BX /, it follows from Lemma 6 that D. B f / 2 Cwu .BX I X /. This combined with (4) gives J. / B Df 2 Cwu .BX /. It also implies using Lemma 3.18 and Theorem 17 that the operations E and D are weakly commuting for B f on BX . Hence by Proposition 16 the operations E and D are weakly commuting for f on BX . So for any ´ 2 BX we have hDf .´/; i D hJ .Df .´//; i D Df .´/.J. // D J. / B Df .´/ D J. / B Df .´/; where the last equality follows from Fact 13. Because the weak uniform continuity of J. / B Df implies the w -uniform continuity of J. / B Df , it follows that the mapping Df W BX ! L.X I Y / is w –W OT uniformly continuous. Since its range is bounded, the uniformities W OT and w on this range coincide and so Df is w –w uniformly continuous. t u

3. Class W The class W of Banach spaces lies at the core of the theory of smooth mappings between Banach spaces. It provides a link between uniform smoothness and weak

Section 3. Class W

339

continuity leading to strong structural results for smooth mappings in the following sections. In some sense, these spaces are the closest to finite-dimensional ones, from the point of view of smoothness. Class W appears naturally in several contexts, including the polynomial approximations studied in Chapter 7, and also the weak commuting of bidual extensions. The pivotal result in this section is the finite-dimensional quantitative Lemma 27, which leads eventually to the proof that c0 , and more generally C.K/ spaces for scattered compacts K, are W1 -spaces. Definition 19. Let 2 .0; 1. We say that a Banach space X is a W -space (or that it belongs to the class W ) if C 1;C .BX / CwsC .BX / (in the sense of restriction). We say that a Banach space is a W -space (or belongs to the class W ) if it is a W -space for some 2 .0; 1. Clearly if X is a W -space, then it is a W -space for every 0 < < . Conversely, using the uniform continuity it is easy to check that if X is a W -space for every 0 < < , then X is a W -space. Every Schur space is trivially a W1 -space. Note also that if X is a W -space, then C 1;C .X/ CwsC .X/. Since the definition of a W -space is sequential, the following observation is immediate: Fact 20. Let X be a Banach space. If for each separable subspace Y of X there is a subspace Z satisfying Y Z X such that Z is a W -space, then X is a W -space. This fact is similar to the usual separable reduction principle, but a warning is in order: The class W is not closed with respect to subspaces as we shall see in Example 39. This is related to the extensions of C 1;C -smooth functions, see also Theorem 34. On the other hand, the class W behaves rather well with respect to quotients. Compare with Proposition 3.87 (especially in view of Fact 22) and Theorem 3.41(v). Proposition 21 ([CHL]). Let X be a W -space and Y X a separable subspace that does not contain `1 . Then X=Y is a W -space. More generally, if Z is a 3 Banach space such that C 1;C .BX I Z/ CwsC .BX I Z/ for some 2 .0; 1, then C 1;C .BX=Y I Z/ CwsC . 3 BX=Y I Z/. Proof. Suppose by contradiction that there is f 2 C 1;C .BX=Y I Z/nCwsC . 3 BX=Y I Z/. By Fact 3.86 there exists a weakly Cauchy sequence fyj g 3 BX=Y such that " D dist ff .y2k /I k 2 Ng; ff .y2k 1 /I k 2 Ng > 0. Since f is uniformly continuous on BX=Y , there is ı > 0 such that kf .x/ f .y/k < 2" whenever x; y 2 BX=Y are such that kx yk < ı. Denote by Q W X ! X=Y the canonical quotient mapping. By Lemma 3.85 there is a weakly Cauchy sequence fxj g X such that limkx2k 1 k 3 , limkx2k k D r , fQ.x2k 1 /g is a subsequence of fy2k 1 g, and fQ.x2k /g is a subsequence of fy ˇ ˇ 2k g. By passing to subsequences we may assume that fx2k 1 g BX and ˇkx2k k r ˇ < ı for all k 2 N. We set ´2k 1 D x2k 1 and ´2k D kxr k x2k for k 2 N. Then 2k

340

Chapter 6. Structural behaviour of smooth mappings

f´j g BX is weakly Cauchy and k´2k x2k k < ı for all k 2 N. However, since by the assumption (and Proposition 4) f B Q 2 CwsC .B X I Z/, the sequence ff .Q.´j //g is Cauchy, but dist ff .Q.´2k 1 //g; ff .Q.´2k //g 2" , which is a contradiction. u t The assumption that Y does not contain `1 in the above proposition cannot be avoided: By Corollary 3.89 the space `1 , which is a W1 -space, has a quotient isomorphic to `2 , which is not a W -space (check the square of the norm). Note also that as a consequence of the bidual extensions (Corollary 12) we obtain that if X is a W -space, then X is a W -space too. Proposition 3.19 and Theorem 3.81 immediately imply the following fact. Fact 22. Let X be a W -space and Y a Banach space. Then each f 2 C 1;C .BX I Y / is w–w sequentially Cauchy-continuous on BX . In particular, P .X/ D PwsC .X/ and PK .X I Y / PwsC .XI Y /. If moreover the space X does not contain `1 , then C 1;C .BX / Cwu .BX /. Consequently, each f 2 C 1;C .BX I Y / is w–w uniformly continuous on BX . In particular, P .X / D Pwu .X/ and PK .X I Y / D Pwu .XI Y /. Theorem 23. Let X be a W -space that does not contain `1 , Y a Banach space, and let f 2 C 1;C .BX I Y /. Then the operations E and D are weakly commuting for f on BX . Proof. By Proposition 16 it suffices to show the result for Y D R. In this case it follows by combining Fact 22, Lemma 6, Lemma 3.18, and Theorem 17. u t Theorem 24 ([Há6]). Let X be a W -space that does not contain `1 , Y a Banach space, k 2 N, and f 2 C k;C .BX I Y /. The following statements are equivalent: (i) f 2 CK .BX I Y /. (ii) f 2 CwsC .BX I Y /. k .B I Y /. (iii) f 2 Cwu X (iv) There exists m 2 f1; : : : ; kg such that d mf 2 CK BX I PK . mX I Y / and d jf .x/ 2 PK . jXI Y / for each x 2 BX and each j 2 f1; : : : ; mg. Proof. (i))(ii) Follows from Fact 22 and Proposition 3.19, (ii))(iii) follows from Theorem 3.81 and Theorem 8, (iii))(iv) follows from Lemma 3.18, and (iv))(i) follows from Fact 22, Theorem 9, and Lemma 3.18. t u By combining Fact 22 with Theorem 18 we obtain immediately the following corollary: Corollary 25. Let X be a W -space that does not contain `1 , Y a Banach space, and f 2 C 1;C .BX I Y /. Then the bidual extension f 2 C 1;C .BX I Y / restricted to BX is w –w uniformly continuous, and in particular this restriction is independent of the choice of U and T . Thus for any subspace Z X we have .f BZ / D f BZ . Moreover, the mapping D.f / 2 Cu BX I L.X I Y / is also w –w uniformly continuous.

Section 3. Class W

341

Later in this section we are going to prove that the spaces C.K/, where K is a scattered compact space, belong to the class W1 . At the core of our arguments lie some finite-dimensional estimates. We denote by fej gjnD1 the canonical basis of the space `n1 and by B`Cn the subset 1 of the unit ball B`n1 consisting of vectors that have non-negative coordinates. The next lemma contains the main idea that is needed for the estimate of the values of uniformly smooth functions on the canonical basis vectors of `n1 . It involves symmetric functions in the sense of Definition 4.20. Compare the lemma below also with Proposition 5.49. Lemma 26 ([Há5]). For every ! 2 M with !.1/ 2 R and every " > 0 there is n.!; "/ 2 N such that if n n.!; "/ and f 2 C 1;C .B`Cn / is a symmetric function 1 such that f .0/ D 0, Df .0/ D 0, and Df has modulus of continuity !, then jf .ej /j < " for j D 1; : : : ; n. Proof. Put D 5" and find k 2 N so that !. k1 / and !.1/ . Further, set k ˚˙ !.1/ 2.k 1/ m D max C 1. Now assume that n n.!; "/ ; 2 and n.!; "/ D m and f 2 C 1;C .B`Cn / satisfies the assumptions of the lemma. For a vector v we 1 denote its pth coordinate by v.p/ and for A f1; P: : : ; ng we denote by PA the projection in `n1 associated with A, i.e. PA .v/ D j 2A v.j /ej . We put x0 D 0, y0 D e1 , A0 D f2; : : : ; n.!; "/g, and define inductively vectors xj ; yj 2 B`Cn and 1 sets Aj A0 , 1 j k 1, satisfying the following: (i) xj .1/ D 0 and yj D e1 C xj ; (ii) for each 0 l j there is p > 1 such that xj .p/ D kl ; (iii) jf .xj /

f .xj

(iv) jAj j D m2.k

1 /j

j 1/

2 k and jf .yj /

f .yj

1 /j

2 k ;

and xj .p/ D j=k for p 2 Aj .

Assume that xj , yj , and Aj are already defined for some 0 j < k 1. Note that Df .xj / 2 `n1 and kPAj .Df .xj //k kDf .xj /k !.1/. Consider splitting the set Aj into m subsets of size m2.k j 1/ 1 . It follows that there is A Aj such that jAj D m2.k j 1/ 1 and kPA .Df .xj //k !.1/ m . Repeating this procedure for yj we conclude that there is B A such that jBj D m2.k j 2/ and kPB .Df .yj //k . We set xj C1 D xj C k1 B , yj C1 D e1 Cxj C1 , and Aj C1 D B. Then (i) and (iv) clearly holds. Since m 2,ˇ we have Aj n B ¤ ;, and so (ii) holdsˇtoo. To see (iii) note ˇ ˇ that ˇDf .xj /Œxj C1 xj ˇ 1 kPB .Df .xj //k and similarly ˇDf .yj /Œyj C1 yj ˇ , k k k and use Fact 3. To finish the proof we put x D xk 1 and y D yk 1 . By (ii) there exists a set C f2; : : : ; ng of cardinality k such that x attains the values 0; k1 ; k2 ; : : : ; k k 1 on C . Note that PC .y/ D PC .x/. It follows that the coordinates of the vector x C k1 C are just a rearrangement of thoseˇof y, and so by the symmetry f .x C k1 C / D f .y/. In ˇ 1 ˇ ˇ particular, jf .y/ f .x/j D f .x C k C / f .x/ !.1/ k1 , as the function f

342

Chapter 6. Structural behaviour of smooth mappings

is !.1/-Lipschitz (Proposition 1.71). Combining this with (iii) we obtain jf .e1 /j D jf .e1 / f .0/j

k X1

jf .yj / f .yj

1 /jCjf .y/

f .x/jC

j D1

k X1

jf .xj / f .xj

1 /j

< 5 D ":

j D1

t u Lemma 27 ([Há5]). For each ! 2 M with !.1/ 2 R, L > 0, and " > 0 there is N.!; L; "/ 2 N such that if n N.!; L; "/ and f 2 C 1;C .B`Cn / is an L-Lipschitz 1 function such that Df has modulus of continuity !, then jf .ej / f .0/j < " for some j 2 f1; : : : ; ng. ˙ Proof. We set N.!; L; "/ D 2n.!; 2" / C 2L " , where n.; / is the function from Lemma 26. Assume that n N.!; L; "/ and f 2 C 1;C .B`Cn / satisfies the as1 sumptions of the lemma. Suppose that jf .ej / f .0/j " for each j 2 f1; : : : ; ng. Since Df .0/ 2 `n1 and kDf .0/k L, setting A D fj I jDf .0/j j 2" g we have " 1 jAj n 2L " 2n.!; 2 /. Further, there is B A such that jBj 2 jAj and either f .ej / f .0/ " for all j 2 B, or f .ej / f .0/ " for all j 2 B. Let D sgn.f .ej / f .0//, j 2 B, and put g.x/ D f .x/ f .0/ Df .0/Œx . Then g 2 C 1;C .B`Cn /, g.0/ D 0, Dg.0/ D 0, g.ej / " jDf .0/Œej j 2" for each j 2 B, 1 and Dg has modulus !. Let N D jBj n.!; 2" /. Finally, we define a symmetric function gs on B CN by `1 ! ! N N X X 1 X gs g aj ej D aj e.j / ; NŠ j D1

2SN

j D1

where SN is the set of all bijections f1; : : : ; N g ! B. Then gs has a uniformly continuous derivative with modulus !, gs .0/ D 0, Dgs .0/ D 0, and gs .ej / 2" for 1 j N . (In fact, gs is a convex combination of functions which satisfy these conditions.) This contradicts Lemma 26. t u The following slightly more general formulation is also useful. Given an ordinal ˛ we denote by C0 .Œ1; ˛/ the subspace of C.Œ1; ˛/ consisting of functions that are zero at ˛. We say that the functions f; g 2 C.Œ1; ˛/ are disjointly supported if f . /g. / D 0 for every 2 Œ1; ˛. Lemma 28 ([Há5]). For each ! 2 M with !.2/ 2 R, L > 0, and " > 0 there is M.!; L; "/ 2 N such that if ˛ is an ordinal, X D C.Œ1; ˛/ or X D C0 .Œ1; ˛/, f 2 C 1;C .BX / is an L-Lipschitz function such that Df has modulus of continuity !, N M.!; L; "/, and if v 2 BX and fuj gjND1 X is a disjointly supported sequence such that fv C uj g BX , then jf .v C uj / f .v/j < " for some j 2 f1; : : : ; N g. Proof. We set !1 .t/ D 2!.2t/ and M.!; L; "/ D N.!1 ; 2L; "/, where N.; ; / is the function from Lemma 27. Assume that N M.!; L; "/ and f 2 C 1;C .BX /, v 2 BX , and fuj gjND1 X satisfy the assumptions of the lemma. We define an affine

Section 3. Class W

343

PN PN operator T W `N 1 ! X by T j D1 aj ej D v C j D1 aj uj . Then T is 2-Lipschitz, T .B CN / BX , T .0/ D v, and T .ej / D v C uj . Further, g D f B T 2 C 1;C .B CN / `1 `1 (Proposition 4), g is 2L-Lipschitz, and Dg has modulus of continuity !1 . So by Lemma 27 there is j 2 f1; : : : ; N g such that jf .v C uj / f .v/j D jg.ej / g.0/j < ". t u Given f 2 C 1;C .BC.Œ1;˛/ /, the preceding lemmata allow us to perturb a vector in the domain of f by a vector with large norm essentially not affecting the value of f . Note that Lemma 27 does not have a general vector-valued analogue (consider the identity on c0 ). Nevertheless, we can apply it at least “componentwise” as in the following result. Lemma 29. Let X D C.Œ1; ˛/ or X D C0 .Œ1; ˛/, where ˛ is an infinite countable ordinal, and let ffn g C 1;C .BX / be an equi-Lipschitz sequence such that all the mappings Dfn have the same finite modulus of continuity. Further, let fxn g BX be a weakly Cauchy sequence and " > 0. Then there are an increasing sequence fnk g N such that nk and k have the same parity for each k 2 N, a function v 2 BX with a finite range, and a sequence fuk g X of disjointly supported functions such that fv C uk g BX and jfnk .v C uk / fnk .xnk /j < " for each k 2 N. Proof. Let L > 0 be the common Lipschitz constant of fn , n 2 N. P Since ˛ is countable, we can fix a system f"ˇ gˇ ˛ of positive numbers such that ˇ ˛ "ˇ < ". The sequence fxn g converges pointwise to F 2 B`1 .Œ1;˛/ which satisfies additionally F .˛/ D 0 in case that X D C0 .Œ1; ˛/. Using a finite induction we are going to construct the following objects: (i) a decreasing sequence of ordinals ˛ D ˇ1 > ˇ2 > > ˇm D 0; (ii) sets N D M1 M2 Mm such that each Mj contains infinitely many odd and even numbers; (iii) v 2 BX defined as v. / D F .ˇj / for 2 Œˇj C1 C 1; ˇj , j D 1; : : : ; m 1; j (iv) a system of sequences fxn gn2Mj BX , j D 1; : : : ; m, satisfying (a) xn1 D xn for all n 2 N D M1 , j j C1 j C1 (b) xn may differ from xn only on Œˇj C1 C 1; ˇj (and consequently xn agrees with xn on Œ1; ˇj C1 ) for each n 2 Mj C1 and j D 1; : : : ; m 1, ˚ j (c) for each j 2 f1; : : : ; mg the system suppo .xn v/ \ Œˇj C 1; ˛ n2M is j pairwise disjoint, ˇ ˇ j C1 j (d) ˇfn .xn / fn .xn /ˇ < "ˇj for all n 2 Mj C1 , j D 1; : : : ; m 1. Note that since the intervals Œˇj C1 C 1; ˇj are clopen, (iii) defines a function in X once all the ˇj are defined. Set ˇ1 D ˛, M1 D N, and xn1 D xn for n 2 N. We proceed by induction. Notice that since Œ0; ˛ is well-ordered, the induction will stop after finitely many steps reaching j ˇm D 0. Now assume that ˇj , Mj , and fxn gn2Mj are already defined. If ˇj is nonlimit, then we set ˇj C1 to be the predecessor of ˇj , find Mj C1 Mj satisfying (ii) such j C1 j that jxn .ˇj / F .ˇj /j < "ˇj =L for all n 2 Mj C1 , and finally set xn . / D xn . /

344

Chapter 6. Structural behaviour of smooth mappings j C1

for ¤ ˇj , and xn .ˇj / D v.ˇj / D F .ˇj /. Condition (c) holds by the inductive j C1 j j C1 j xn k D jxn .ˇj / xn .ˇj /j D hypothesis. Condition (d) holds, since kxn jF .ˇj / xn .ˇj /j < "ˇj =L for each n 2 Mj C1 by (b) of the inductive hypothesis. Assume now that ˇj is a limit ordinal. For < < ˇj we define continuous affine operators P; W X ! X by P; .x/ D .1

ŒC1; / x C F .ˇj /ŒC1; :

Note that P; .BX / BX . We claim that there is ˇj C1 < ˇj such that for each 2 .ˇj C1 ; ˇj / the set ˇ ˇ ˚ n 2 Mj I ˇfn Pˇj C1 ; .xnj / fn .xnj /ˇ < "ˇj =2 contains infinitely many odd numbers and infinitely many even numbers. Assuming the contrary, fix an arbitrary N 2 N. Since ˇj is a limit ordinal, there are ordinals 1 < 1 2 < 2 2N < 2N < ˇj such that for each l 2 f1; : : : ; 2N g ˇ ˚ j j ˇ fn .xn /ˇ < "ˇj =2 contains only finitely many the set n 2 Mj I ˇfn Pl ;l .xn / odd numbers or only finitely many even numbers. By discarding some of the l , l and relabelling we ˇmay assume without loss ofˇ generality that for each l 2 f1; : : : ; N g ˚ j j fn .xn /ˇ < "ˇj =2 contains only finitely many the set n 2 Mj I ˇfn Pl ;l .xn / odd numbers. Thus, using (ii) of the inductive hypothesis, there is n 2 Mj such ˇ j j ˇ j fn .xn /ˇ "ˇj =2 for each l 2 f1; : : : ; N g. Put y D xn and that ˇfn Pl ;l .xn / wl D Œl C1;l .F .ˇj / y/, l D 1; : : : ; N . Then fwl g are disjointly supported, y C wl 2 BX , and jfn .y C wl / fn .y/j "ˇj =2, l D 1; : : : ; N . Consequently for N large enough this contradicts Lemma 28. j C1 We continue by defining Mj C1 and fxn gn2Mj C1 . To this end we construct increasing sequences fkl g Mj and fl g .ˇj C1 ; ˇj / by induction on l. Let j k1 2 Mj be odd such that jxn .ˇj / F .ˇj /j < "ˇj =.2L/ for all n 2 Mj , n k1 j (notice that xn .ˇj / D xn .ˇj / by (b) of the inductive hypothesis). Using the conj j C1 j tinuity of xk1 at ˇj we find 1 2 .ˇj C1 ; ˇj / such that xk1 D P1 ;ˇj .xk1 / satisfies j C1

j

kxk1 xk1 k < "ˇj =L. (This is possible since ˇj is a limit ordinal.) Then (b) and (d) clearly hold for n D k1 . Now assume that kl and l have been defined for some l 2 N. By the above claim there is klC1 2 Mj , klC1 > kl such that klC1 has the same parity as l C 1 and ˇ ˇ j j ˇfk Pˇj C1 ;l .xk / fklC1 .xk /ˇ < "ˇj =2: lC1 lC1 lC1 ˇ ˇ j j Note that ˇPˇj C1 ;l .xk /.ˇj / F .ˇj /ˇ < "ˇj =.2L/. Since Pˇj C1 ;l .xk / is conlC1 lC1 j C1 j tinuous at ˇj , we can find lC1 2 .l ; ˇj / such that xk D PlC1 ;ˇj Pˇj C1 ;l .xk / lC1 lC1

j C1 j Pˇj C1 ;l .xk / < "ˇj =.2L/. Then (b) and (d) clearly hold for satisfies xk lC1 lC1 n D klC1 . Finally, we let Mj C1 D fkl gl2N and note that (c) holds by the construction j C1 above, since suppo .xk v/ \ Œˇj C1 C 1; ˇj Œl C 1; lC1 , and by (b) and (c) lC1 of the inductive hypothesis.

Section 3. Class W

345

This completes the construction of all the aforementioned objects. To finish the proof we put uk D xnmk v, where fnk g Mm is an increasing sequence such that nk and k have the same parity. t u Theorem 30 ([Há5], [Há6]). Let ˛ be a countable ordinal. Then the spaces C.Œ1; ˛/ and C0 .Œ1; ˛/ are W1 -spaces. In particular, c0 is a W1 -space. Proof. Let X D C.Œ1; ˛/ or X D C0 .Œ1; ˛/, where ˛ is an infinite countable ordinal. By contrary, assume that there is f 2 C 1;C .BX /nCwsC .BX /. Without loss of generality (using shifting and scaling in the range) we may assume that there is a weakly Cauchy sequence fxn g BX such that f .x2k 1 / 2 and f .x2k / 2 for all k 2 N. Put fn D f , n 2 N. By Lemma 29 there are a function v 2 BX and a sequence fuk g X of disjointly supported functions such that fv C uk g BX and f .v C uk / < 1 for k odd, f .v C uk / > 1 for k even. If f .v/ < 0, then we set wn D u2n , otherwise we set wn D u2n 1 , n 2 N. Then jf .v C wn / f .v/j > 1 for each n 2 N, a contradiction with Lemma 28. t u Corollary 31 ([Há6]). Let K be a scattered compact space. Then the space C.K/ is a W1 -space. Proof. By Proposition 5.124 every separable subspace of C.K/, K scattered, is contained in a subspace of C.K/ isometric to C.Œ1; ˛/, where ˛ is a countable ordinal. Fact 20 combined with Theorem 30 finishes the proof. t u Theorem 32 ([Há6]). The (original) Tsirelson space T is a W 1 -space. 5

Proof. Let f 2 C 1;C .BT / and let fxn g 15 BT be weakly Cauchy. Since T is w reflexive, there is v 2 51 BT such that xn ! v. We show that f .xn / ! f .v/. If this

is not the case, then we may assume by passing to a subsequence that there is " > 0 such that jf .xn / f .v/j 2" for all n 2 N. Put yn D xn v. Then yn 2 52 BT and w " yn ! 0. Let L > 0 be such that f is L-Lipschitz and put ı D 2L . By Proposition 3.32 we may assume that there is a block basic sequence fun g of the canonical basis of T such that kun yn k < ı for each n 2 N. By scaling the fun g if necessary we get kun k 52 and kun yn k < 2ı. So jf .v C un / f .v/j 2" 2Lı D " for all n 2 N. By [CasaSh, p. 16] we have

N

X

4

max jaj j aj uN Cj 2 max jaj j kuN Cj k

5 1j N 1j N j D1

for all scalars a1 ; : : : ; aN and for every N 2 N. In particular, the continuous affine PN PN given by S ! T a e D v C operators SN W `N j j N 1 j D1 j D1 aj uN Cj are 4 -Lipschitz and satisfy S .B / B , S .0/ D v, and S .e N N T N N j / D v C uN Cj . `1 5 Applying Lemma 27 to f B SN (using also Proposition 4) we arrive at a contradiction for N large enough. t u Being a W -space is invariant under isomorphisms, but the precise value of (such that X is a W -space) may depend on the norm.

346

Chapter 6. Structural behaviour of smooth mappings

Proposition 33 ([D’H]). For every 2 .0; 1/ there is an equivalent norm kk on c0 such that .c0 ; kk / is a W -space but it is not a W -space for any > . In fact if we denote by B the unit ball of .c0 ; kk /, then there is a function ˚; 2 C 1 .B / such that ˚; … Cwsc .B /. Proof. Let f.ej I fj /g be the canonical basis of c0 and let 2 .0; 1/. The renorming kk of c0 is determined by its closed unit ball B D conv Bc0 [ f˙ 1 ej I j 2 Ng : Clearly, Bc0 B 1 Bc0 , and therefore kk is an equivalent norm on c0 and .c0 ; kk / is a W -space. Fix 2 .; 1 and ˇ 2 .1; /. For every x D .xj / 2 B we denote Ax D fj 2 NI jxj j ˇg. We claim that jAx j .ˇ 11/ . Indeed, choose an arbitrary ˛ 2 .1; ˇ/ and let ´ D .´j / 2 c0 be such that kx ´k < ˇ ˛ P P and ´ D a0 y C nkD1 ak 1 ek , where nkD0 jak j 1 and y D .yj / 2 Bc0 . Let C D fkI jak j .˛ 1/g. Then jC j .˛ 11/ . Clearly j´j j 1 for j > n. If j n, then j´j j D ja0 yj C aj 1 j ja0 j C jaj j 1 < ˛, unless j 2 C . Therefore Ax C , and since ˛ can be chosen arbitrarily close to ˇ, the estimate follows. Let ' 2 C 1 .R/ be a function that has all derivatives bounded and satisfies P1 '.Œ ˇ; ˇ/ D f0g and '. / > 0. We set ˚; D j D1 ' B fj . Let x 2 B and D maxj …Ax jxj j < ˇ.PFor any y 2 B with ky xk < ˇ we have Ay Ax and so ˚; .y/ D j 2Ax ' B fj .y/. It follows that ˚; is C 1 -smooth on Int B and it has all derivatives bounded and thus also Lipschitz. Consequently, ˚; 2 C 1 .B /. Finally, ˚; . ek / D '. / > 0 for each k 2 N, while ˚; .0/ D 0, and so ˚; … Cwsc .B /. t u Let X be a W -space. The change of the constant caused by the renorming of the space is intimately connected with the non-existence of extensions of functions with uniformly continuous derivatives from the unit ball into its larger multiple. It is easy to see that the functions ˚; from Proposition 33 cannot be extended to functions belonging to C 1;C . B /. More precisely, there is no function g 2 C 1;C . B / such that gB D ˚; , since C 1;C . B / CwsC .B /. We give an extreme example of this phenomenon next. Theorem 34 ([D’H]). There are a separable Banach space X and f 2 C 1 .BX / such that for any > 1 no function g 2 C 1;C .BX / agrees with f on a neighbourhood of the origin. Proof. Let kk and ˚; be the norm and the functions from Proposition 33. We put L1 XD .c ; kk . Let Pn W X ! .c0 ; kk 1 / be the canonical projections onto / 1 nD2 0 2 n n the direct summands. Put n D cn ˚ 1 ; 1 , where the constants cn are chosen so that n n 1 P kd j n .x/k 1 21n for all x 2 B 1 , j 2 f1; : : : ; ng. Finally, set f D 1 nD2 n B Pn . n

n

Since d k . n B Pn /.x/ D d k n .Pn .x// B Pn for x 2 Int BX (Corollary 1.117), Theorem 1.85 implies that f 2 C 1 .Int BX / with all derivatives bounded and hence also Lipschitz. Thus f 2 C 1 .BX /. Now suppose that > 1 and g 2 C 1;C .BX / is

Section 3. Class W

347

such that g D f on BX for some < 1. If m 2 N is such that m1 1 and mm 1 , then g mm 1 Pm .BX / 2 C 1;C . mm 1 B 1 / and g 1 Pm .BX / D m , a contradiction with m m 1 the remark preceding the theorem. t u There are strong structural limitations for Banach spaces to belong to the class W , especially if they do not contain `1 . The following property has been isolated by Sean Michael Bates in [Bat] in his work on C 1 -smooth surjective mappings between Banach spaces. Definition 35. We say that a Banach space X has property B if X contains a normalised weakly null hereditarily Banach-Saks sequence. Note that a Banach space contains a normalised weakly null sequence if and only if it is not a Schur space. In particular, if X is an infinite-dimensional Banach space such that X has the Banach-Saks property (or more generally if X is not a Schur space and has the weak Banach-Saks property), then X has property B. Using an argument due to E. Odell and H. Rosenthal, S. M. Bates has shown the following characterisation, whose proof is omitted. Lemma 36 ([Bat]). A Banach space X has property B if and only if X contains a such that for every " > 0 there exists k."/ 2 N such normalised sequence ffk g1 kD1 that for each x 2 BX ˇ ˇ ˇfk 2 NI jfk .x/j > "gˇ k."/: Proposition 37 ([Bat]). Let X be a Banach space with property B that does not contain `1 . Then X does not belong to the class W . Proof. Let ffk g1 X be the sequence from Lemma 36. Then ffk g is not relatkD1 ively compact. Indeed, assuming the contrary some subsequence of ffk g converges to f 2 SX . Choosing y 2 BX with f .y/ > 21 and a subsequence ffnk g such that kfnk f k < 41 for all k 2 N we obtain jfnk .y/j jf .y/j kf fnk k > 41 for all k 2 N, a contradiction. Therefore by passing to a subsequence of ffk g using Theorem 3.56 we may assume that there exists a semi-normalised weakly null sequence fxk g1 X biorthogonal to ffk g. kD1 Choose ' 2 C 1 .R/ satisfying '.t/ D 0 for t 2 Œ 1; 1 and '.t/ D 1 for jt j 2, and let f W X ! R be defined by f .x/ D

1 X

' fk .x/ :

kD1 1 For a fixed x 2 B.0; R/ put A D fk 2 NI jfk .x/j > 12 g. Then jAj k. 2R /. For any P 1 y 2 B.x; 2 / we have fk 2 NI jfk .y/j > 1g A and so f .y/ D k2A '.fk .y//. Consequently, f is C 1 -smooth on X and it has all derivatives bounded on bounded sets, i.e. f 2 C 1 .X/. However, limk!1 f .2xk / D 1 ¤ 0 D f .0/, so X is not a W -space. t u

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Chapter 6. Structural behaviour of smooth mappings

In connection with the previous result note that the Tsirelson space T of T. Figiel and W. B. Johnson does not have the Banach-Saks property, but T (the original Tsirelson space) has the Banach-Saks property ([CasaSh, p. 18]) and so T has property B, while T is a W -space (Theorem 32). Proposition 38. Let X be a Banach space with a sub-symmetric Schauder basis. Then X has property B if and only if X is not isomorphic to c0 . Proof. ) If X is isomorphic to c0 , then X is a Schur space and so there is no normalised weakly null sequence in X . ( Let f.ek I fk /g be the sub-symmetric Schauder basis of X. Since a sub-symmetric basis is semi-normalised and unconditional, we may assume without loss of generality that ffk g is normalised. Suppose that X does not have property B. By Lemma 36 there are " > 0 and a sequence fun g BX such that jfk 2 NI jfk .un /j > "gj >P n. Given n 2 N we find A D fk 2 NI jfk .un /j > "g such that jAj D n and put vn D k2A ek . Then kvn k C k 1" un k C" , where C is the unconditional basis constant of fek g. Let

P

K 1 be such that fek g is K-spreading. Then nkD1 ek Kkvn k K C" and therefore

n

n

X

X C2

ak ek C maxjaj j ek K maxjaj j

j " j kD1

kD1

for any a 1P ; : : : ; an 2 R.

The ˇ reverse ˇ is automatic: for each j 2 f1; : : : ; ng Pn inequality ˇ a e we have nkD1 ak ek ˇfj kD1 k k D jaj j. Hence fek g is equivalent to the canonical basis of c0 , a contradiction. t u Example 39 (Schreier’s space). Schreier’s space S is defined as a completion of c00 under the norm X k.xj /k D sup jxj j; A admissible j 2A

where A N is admissible if A D fn1 ; : : : ; nk g, where k n1 < n2 < < nk . It is obvious from the definition that the canonical basis vectors fej g form a normalised 1-unconditional basis of S. The space S has many important properties (it was the first constructed example without the weak Banach-Saks property, [Schr]) of which we only mention the following ones: The space S is isometric to a subspace of some C.K/, K countable ([CaGo]), hence S is isomorphically polyhedral and it is a subspace of a W1 -space (Corollary 31). Denote by ffk g S the functionals biorthogonal to fek g. Obviously ffk g is normalised. We claim that ffk g satisfies the condition in Lemma 36 with k."/ D 2d 1" e 1 and so S has property ˇ ˇ B. Indeed, fix " > 0, x 2PBS , and put m D d " e. Then ˇfk mI jxk j > "gˇ < m, otherwise 1 kxk k2A jxk j > m" 1, where A consists of some m members of the set fk mI jxk j > "g. Since S does not contain `1 , [CasaSh, Theorem 0.5], S is not a W -space (Proposition 37). Thus S is a witness to the fact that the class W is not closed with respect to subspaces. Similarly, as S is c0 -saturated (Theorem 5.103, Corollary 5.92), property B does not pass to subspaces.

Section 4. Uniformly smooth mappings from C.K/, K scattered

349

Note also that S contains `1 by the lifting property of `1 and hence does not have the Banach-Saks property. We close this section by an example of a Banach space that does not belong to the class W , but such that all of its scalar valued polynomials are still weakly continuous (see Fact 22). Let T be the (original) Tsirelson space and let ffk g be its canonical basis. Recall that the symmetrised Tsirelson space S.T / is a reflexive Banach space with a symmetric basis fek g whose norm is given by the formula

1

1

X

X

ak ek D sup a.k/ fk ;

2S

kD1

kD1

T

where S is the set of all permutations of N. Theorem 40 ([CHL]). The symmetrised Tsirelson space S.T / has property B (in particular, it is not a W -space), although P .S.T // D Pwu .S.T //. Proof. The space S.T / has property B by Proposition 38, so it is not a W -space by Proposition 37. By [CasaSh, Proposition V.10] the canonical basis of T has an upper p-estimate for every p < 1 and so by the definition of the norm in S.T / the same holds also for the canonical basis fek g of S.T /. By [CasaSh, Proposition X.b.7] there is K > 0 such that

n

n

X

X

ak uk K ak ek

kD1

kD1

for every normalised sequence fuk g S.T / of disjointly supported vectors and all scalars a1 ; : : : ; an , so each bounded block basis of fek g has an upper p-estimate for every p < 1. Thus S.T / has property Sp for every p < 1 by Proposition 3.32 and we may apply Corollary 3.58 (together with the separability of S.T /). t u

4. Uniformly smooth mappings from C.K /, K scattered In the present section we are going to investigate uniformly smooth mappings from X D C.K/ for K scattered. The results extend the scalar valued theory from the previous section. In particular, under the assumption that the range of f 2 C 1;C .BX I Y / is relatively weakly compact, or alternatively Y has property (WR), we prove that f maps weakly Cauchy sequences to norm convergent ones. Let K be a compact space. Recall that from the Riesz representation theorem and the Lebesgue dominated convergence theorem it immediately follows that a sequence f n g C.K/ is weakly convergent to 2 C.K/ if and only if f n g is bounded and f n .x/g converges to .x/ for each x 2 K, and f n g is weakly Cauchy if and only if it is bounded and f n .x/g is convergent for each x 2 K. Lemma 29 has the following direct consequence (compare also with Lemma 3.31):

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Chapter 6. Structural behaviour of smooth mappings

Lemma 41 ([DeH2]). Let X D C.Œ1; ˛/ or X D C0 .Œ1; ˛/, where ˛ is a countable ordinal, and let Y be a Banach space. If f 2 C 1;C .BX I Y / n CK .BX I Y /, then there exist a function v 2 BX with a finite range and a weakly null sequence fun g X of disjointly supported functions such that fv C un g BX and ff .v C un /g is not relatively compact in Y . Proof. Since X is separable, by passing to span f .BX / we may assume without loss of generality that Y is separable as well. There is a sequence fxn g BX such that ff .xn /g is "-separated for some " > 0. By Rosenthal’s `1 -theorem we may assume that fxn g is weakly Cauchy. Using Theorem 3.56 (applied to ff .xn /g Y ) and passing to a subsequence we find a bounded sequence fn g Y such that f.f .xn /I n /g is a biorthogonal system. Moreover, as .BY ; w / is metrisable, we may assume that fn g is w -convergent. By passing to a biorthogonal system f.f .x2n /I 2n 2n 1 /g we may finally assume that fn g is w -null. By Lemma 29 applied to functions fn D n B f there are an increasing sequence fnk g N, a function v 2 BX with a finite range, and a sequence fuk g X of disjointly supported functions such that fv C uk g BX and fnk .v C uk / 12 for each k 2 N. The sequence fuk g is clearly weakly null. Since nk .f .v C uk // 12 , the set ff .v C uk /g is not relatively compact (Fact 4.90). t u Compare the last statement of the next theorem with Corollary 3.27. Theorem 42 ([DeH2]). Let X D c0 or X D C.K/, where K is a scattered compact space, let Y be a Banach space, and f 2 C 1;C .BX I Y /. Then f 2 CwsC .BX I Y / if and only if f Z 2 Cwsc .BZ I Y / for every subspace Z X isomorphic to c0 . In particular, f 2 CwsC .BX I Y / if and only if f 2 Cwsc .BX I Y /. Proof. The implication ) is trivial. We prove ( by contradiction. As in the proof of Corollary 31 it suffices to consider only K D Œ1; ˛, where ˛ is a countable ordinal. Assuming the contrary suppose by Theorem 24 that f … CK .BX I Y /. By Lemma 41 there are v 2 BX and a weakly null sequence fun g X of disjointly supported functions such that fv C un g BX and ff .v C un /g is not relatively compact. Consequently, Z D span.fvg [ fun I n 2 Ng/ is isomorphic to c0 and f … Cwsc .BZ I Y /. t u Lemma 43. Let X be a Banach space with a shrinking Schauder basis fen g, Y a Banach space, U X a CCB set, and ffn g Cwu .U I Y /. Then there is an increasing sequence fnk g N such that

! k

1 X

aj enj

nk such that nkC1 > max2˚k max supp . P P Now if x D j1D1 aj enj 2 U and y D jkD1 aj enj 2 U , then x y 2 ker for each 2 ˚k and so the estimate follows. t u The following is another type of a diagonalisation lemma. Lemma 44. Let X D C.Œ1; ˛/ or X D C0 .Œ1; ˛/, where ˛ is an ordinal, and let ffn g C 1;C .BX / be an equi-Lipschitz sequence such that all the mappings Dfn have the same finite modulus of continuity. Further, let v 2 BX , let fun g1 nD1 X be a disjointly supported sequence such that fv C un g BX , and let " > 0. Then there is an increasing sequence fnk g N such that ˇ ˇ ! k ˇ ˇ X ˇ ˇ aj unj fnk .v C ak unk /ˇ < " ˇfnk v C ˇ ˇ j D1

for every k 2 N and every a1 ; : : : ; ak 2 Œ0; 1. Proof. Let L > 0 be the Lipschitz constant of fn , n 2 N, let m 2 N satisfy ˚ common " m 1 1 1 L m < 4 , and set S D 0; m ; : : : ; m ; 1 . We will construct the sequence fnk g1 kD1 by induction along with an auxiliary sequence N D M0 M1 M2 of infinite sets so that nk 2 Mk 1 and ˇ ! !ˇ k k ˇ ˇ X X1 " ˇ ˇ aj unj C aun fn v C aj unj C aun ˇ < kC1 ˇfn v C ˇ ˇ 2 j D1 j D1 for every k 2 N, every a1 ; : : : ; ak ; a 2 S , and every n 2 Mk . Set M0 D N, n0 D 0, and assume that Mj and nj , j D 1; : : : ; k 1, are already defined for some k 2 N. Let ! be the modulus of continuity of the mappings Dfn , " /, where the function M.; ; / comes from Lemma 28, and put N D M.!; L; 2kC1 kC1 K D jS j .N 1/ C 1. Further, let I Mk 1 be a set of cardinality K such that min I > nk 1 . Then for each fixed n 2 N, n > max I and each fixed a1 ; : : : ; ak ; a 2 S the set ˇ ) ! !ˇ ( k k ˇ ˇ X1 X1 " ˇ ˇ i 2 I I ˇfn v C aj unj C ak ui C aun fn v C aj unj C aun ˇ kC1 ˇ ˇ 2 j D1 j D1 has cardinality at most N 1. It follows that for each n 2 N, n > max I there is in 2 I such that ˇ ! !ˇ k k ˇ ˇ X1 X1 " ˇ ˇ aj unj C ak uin C aun fn v C aj unj C aun ˇ < kC1 ˇfn v C ˇ ˇ 2 j D1

j D1

for each a1 ; : : : ; ak ; a 2 S . Consequently, there are an infinite set Mk Mk nk 2 I such that in D nk for each n 2 Mk .

1

and

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Chapter 6. Structural behaviour of smooth mappings

For k 2 N and a1 ; : : : ; ak 2 S we can estimate ˇ ˇ ! k ˇ ˇ X ˇ ˇ aj unj fnk .v C ak unk /ˇ ˇfnk v C ˇ ˇ j D1 ˇ !ˇ ! kX l 1 k l k ˇ X X1 ˇˇ ˇ aj unj C ak unk ˇ fnk v C aj unj C ak unk ˇfnk v C ˇ ˇ

4

(7)

j D1

for every k; n 2 N with n k. Assume that a1 ; : : : ; ak 1 are already defined for some fixed k 2 N. Recall that fk .v/ D 0 and fk .v C uk / D 1. Therefore (5) P P implies that fk v C jkD11 aj uj < 14 and fk v C jkD11 aj uj C uk > 43 . Put h.t/ D P P fk v C jkD11 aj uj C tuk for t 2 Œ0; 1. Then h0 .t/ D gk v C jkD11 aj uj C tuk R1 for t 2 .0; 1/. Thus 0 h0 .t/ dt D h.1/ h.0/ > 21 , and so there exists ak 2 .0; 1/ for P which gk v C jkD1 aj uj D h0 .ak / > 21 . Combining this with (6) we obtain (7) for every n k. P Put ´n D v C jnD1 aj uj . Since the sequence f´n g is weakly Cauchy in X , it is w -convergent in the space X . Let ´ 2 BX be its w -limit and let us denote L D D.f /.´/ 2 L.X I Y /. By Corollary 25 and Fact 3.8 hL.w/; i D lim hD.f /.´n /Œw; i D lim h; Df .´n /Œwi n!1 2 Y .

n!1

(8)

for every w 2 X and In particular, hL.uk /; k i D limn!1 gk .´n / for every k 2 N. Therefore hL.uk /; k i 14 by (7). On the other hand, since fun g is weakly null in X and thus also in X , the sequence fL.un /g is weakly null in Y , and in particular limn!1 hL.un /; k i D 0 for every k 2 N. It follows (similarly as in Fact 4.90) that fL.un /g is not relatively compact in Y . The subspace Z D spanfun g X is easily seen to be isometric to c0 . Then LZ … LK .ZI Y / and so LZ … LwK .ZI Y / by Theorem 3.47. Suppose moreover that Y is weakly sequentially complete. For a fixed w 2 X by (8) the sequence fDf .´n /Œwg is weakly Cauchy and hence weakly convergent. It follows that D.f /.´/Œw D w-limn!1 Df .´n /Œw 2 Y . Finally, assume that Y is a dual space and let i W Y ! Y be the canonical embedding. Then hP .L.w//; i D hL.w/; i./i for any w 2 X and 2 Y . Because D.P Bf /.´/Œw D P D.f /.´/Œw D P .L.w// for each w 2 X , from the above ˝ ˛ ˝ ˛ we obtain D.P B f /.´/Œuk ; k 41 and limn!1 D.P B f /.´/Œun ; k D 0. So again, D.P B f /.´/Z … LwK .ZI Y /. t u

354

Chapter 6. Structural behaviour of smooth mappings

We note that passing to the bidual is necessary in the previous theorem: Consider the polynomial P 2 P . 2c0 I c0 / given by P ..xn // D .xn2 / (Theorem 1.29). Then P is clearly not compact, but DP .x/Œh D 2.xn hn /1 nD1 by Lemma 1.99, so D.P /.x/c0 D DP .x/ 2 LK .c0 I c0 / for each x 2 c0 (Theorem 4.9(iii)). Let us now put together the previous results. Theorem 46. Let X D c0 or X D C.K/, where K is a scattered compact space, let Y be a Banach space, and f 2 C 1;C .BX I Y /. The following statements are equivalent: (i) f 2 Cwu .BX I Y /. (ii) f 2 Cw .BX I Y /. (iii) f 2 CwsC .BX I Y /. (iv) f 2 Cwsc .BX I Y /. (v) f 2 CK .BX I Y /. (vi) f 2 CwK .BX I Y /. (vii) D.f /.´/ 2 LK .X I Y / for all ´ 2 BX . (viii) D.f /.´/ 2 LwK .X I Y / for all ´ 2 BX . (ix) f 2 CK .BX I Y /. (x) f 2 CwK .BX I Y /. Proof. (i),(iii),(v) by Theorem 24, (iii),(iv) by Theorem 42, and (ii),(iv) by Theorem 3.79, (v))(ix) by Fact 11, (ix))(vii) by Proposition 3.70, since if (vii) fails, then by the continuity there is ´ 2 Int BX such that D.f /.´/ … LK .X I Y /; (vii))(viii) is trivial, and (viii))(v) by Theorem 45. (v))(vi) is trivial, (vi))(x) by Fact 11, and (x))(viii) again by Proposition 3.70. t u In order to study when the properties of the class W pass to the vector-valued mappings we introduce the following property: Definition 47. We say that a Banach space Y has property (WR) if C 1;C .Bc0 I Y / CwsC .Bc0 I Y /: Note that by Theorem 46 the property wsC in the definition can be equivalently replaced by any of the properties wu, w, wsc, K, wK. Similarly, we can equivalently replace the space c0 in the definition by any C.K/ where K is a scattered compact: Theorem 48 ([DeH2]). Let X D C.K/, where K is a scattered compact space, and let Y be a Banach space with property (WR). Then C 1;C .BX I Y / Cwu .BX I Y /. Proof. Assuming the contrary, by Theorem 46 there is f 2 C 1;C .BX I Y / n CK .BX I Y /. As in the proof of Theorem 45 we may assume that K D Œ1; ˛ for some countable ordinal ˛. By Lemma 41 there are v 2 BX and a disjointly supported sequence fun g X such that fv C un g BX and ffP .v C un /g is in Y . not relatively P1 compact 1 2 Define a polynomial P 2 P 2 .c0 I X/ by P nD1 an en D v C nD1 an un and note that P .Bc0 / BX . Then f B P 2 C 1;C .Bc0 I Y /, but f B P .en / D f .v C un / and hence f BP … CK .Bc0 I Y /. This means that Y fails property (WR), a contradiction. u t

Section 4. Uniformly smooth mappings from C.K/, K scattered

355

Let us now pass to the description of some of the properties of the class of Banach spaces with property (WR). Clearly, property (WR) passes to subspaces and is separably determined, i.e. Y has property (WR) if and only if every separable subspace of Y has property (WR). Moreover, if Y has property (WR), then it does not contain c0 , since every bounded linear operator T W c0 ! Y is compact by Theorem 3.47. Proposition 49 ([DeH2]). Let Y be a Banach space that does not have property (WR). Then Y has the following properties: (i) Y has a complemented subspace isomorphic to `1 and so Y contains `1 . (ii) Y is not of a non-trivial cotype. (iii) Y is not weakly sequentially complete. Proof. By Theorem 46 there is f 2 C 1;C .Bc0 I Y /nCK .Bc0 I Y /. By Theorem 45 there exists a non-compact operator L 2 L.c0 I Y /, so Y contains c0 by Theorem 3.47. (i) then follows from [FHHMZ, Theorem 4.44]. (ii) If Y is of a non-trivial cotype, then the same holds for Y by the principle of local reflexivity. This is not possible, since Y contains c0 . To prove (iii), suppose that Y is weakly sequentially complete. By Theorem 45 there exists a non-compact operator L 2 L.c0 I Y /, so Y contains c0 by Theorem 3.47. This is a contradiction, since c0 is not weakly sequentially complete. t u We say that a Banach space X has property (u) (of Aleksander Pełczy´nski) if for every weakly Cauchy sequence˚fxn g P X there exists a weakly unconditionally Cauchy 1 n sequence fun g X such that xn kD1 uk nD1 is weakly null. Property (u) passes to subspaces, [AK, Proposition 3.5.2]. Banach spaces with an unconditional basis, or more generally order continuous Banach lattices, have property (u), [LiTz3, Proposition 1.c.2]. We point out that a Banach space with property (u) or a complemented subspace of a Banach lattice is weakly sequentially complete if and only if it does not contain c0 , Theorem 3.46 and [LiTz3, Theorem 1.c.7]. Corollary 50 ([DeH2]). Let Y be a Banach space that does not have property (WR) and Y is a dual space, or Y is a complemented subspace of a Banach lattice, or Y has property (u). Then Y contains c0 . Proof. By Theorem 46 there is f 2 C 1;C .Bc0 I Y / n CK .Bc0 I Y /. If Y is a dual space, then by Theorem 45 there exists a non-compact operator L 2 L.c0 I Y /, so Y contains c0 by Theorem 3.47. The rest follows from Proposition 49(iii) and the classical results mentioned above. t u By combining the above with [FHHMZ, Theorem 4.44] we immediately conclude the following characterisation. Corollary 51. A dual Banach space has property (WR) if and only if it does not contain `1 .

356

Chapter 6. Structural behaviour of smooth mappings

Recall that a Banach space X has the point of continuity property (PCP) if for every weakly closed bounded set C X the mapping Id W .C; w/ ! .C; kk/ has a point of continuity. Spaces with the PCP have been extensively studied by many authors. In particular, all spaces with the RNP belong to this class – use [FHHMZ, Theorem 11.15, Corollary 11.4] and the fact that if C is closed, then any strongly exposed point of conv C belongs to C . Theorem 52 ([DeH2]). A Banach space Y has property (WR) whenever Y does not contain `1 , or Y has property (u) and does not contain c0 , or Y is weakly sequentially complete, or Y has the PCP. Proof. The first three cases follow immediately from Proposition 49 and Corollary 50. We proceed with the proof of the last case. Since the PCP is clearly a hereditary property and property (WR) is separably determined, we may assume without loss of generality that Y is separable. By [GhMa, Theorem II.1] there is a finite-dimensional decomposition of Y such that each of its skipped blockings is boundedly complete, i.e. there exists a sequence fYn g of finite-dimensional subspaces of Y satisfying S Y D span S1 nD1 Yn ; Yk \ span n¤k Yn D f0g; if fmk g N and fnk g N are sequences satisfying mk < nk C 1 < mkC1 , k 2 N, and if fxk g P Y is a sequence satisfying xk 2 YP mk ˚Ymk C1 ˚ ˚Ynk ,

k 2 N, and supn nkD1 xk < C1, then the series 1 kD1 xk converges. (In fact the existence of such finite-dimensional decomposition is equivalent to the PCP.) By contradiction, using again Theorem 46 there is f 2 C 1;C .Bc0 I Y / n CK .Bc0 I Y /. By Lemma 41 there are v 2 Bc0 and a disjointly supported weakly null sequence fun g c0 such that fv C un g Bc0 and ff .v C un /g is not relatively compact in Y . Without loss of generality we may assume that f .v/ D 0. By Theorem 3.56 passing to a subsequence we may assume that there is a bounded sequence fn g KBY biorthogonal to ff .v C un /g. By Lemma 44 we may assume that ˇ ˇ ! k ˇ 1 ˇ X ˇ ˇ (9) aj uj k B f .v C ak uk /ˇ < ˇk B f v C ˇ ˇ 4 j D1

for every k 2 N and each a1 ; : : : ; ak 2 f0; 1g. We now proceed by constructing inductively increasing sequences of natural numbers fmk g, fnk g, flk g so that mk < nk C 1 < mkC1 and ! ! ! nk k k X X1 [ 1 dist f v C ulj f vC ulj ; span Yj < k (10) 2 j D1 j D1 j Dm k

for every k 2 N. We put m S1 1D 1 and 1l1 D 1, and we choose n1 2 N such that dist f .vCu1 / f .v/; span jnD1 Yj < 2 . Now assume mj , lj , and nj , 1 j k 1,

Section 5. Uniformly smooth mappings from W -spaces

357

are already defined for some k 2 N, k > 1. Put mk D nk 1 C 2. Since f is w–w ˚ 1 P P continuous (Fact 22), the sequence f v C jkD11 ulj C un f v C jkD11 ulj nD1 S is weakly null. The subspace span j1Dmk Yj is finite-codimensional and hence there is lk > lk 1 such that ! ! ! k k 1 X X1 [ 1 dist f v C ulj f vC ulj ; span Yj < k : 2 j D1 j D1 j Dm k

Finally, we find nk mk for which (10) holds. P Let us denote yk D f v C jkD1 ulj for k 2 N0 . For every k 2 N there is S k wk 2 Y , kwk k < 21k such that ´k D yk yk 1 C wk 2 span jnDm Yj . Then k Pn Pn and since fy g is bounded, by the bounded com´ D y y C w n n 0 kD1 k kD1 k P ´ pleteness the series 1 converges. It follows that fy g is convergent. However, n kD1 k by (9), ln .yn / > ln B f .v C uln / 14 D 43 and ln .yn 1 / < ln B f .v/ C 14 D 41 . 1 1 Thus kyn yn 1 k K ln .yn yn 1 / > 2K for each n 2 N, a contradiction. u t In particular, Banach spaces of non-trivial cotype satisfy the first condition in Theorem 52 and hence they have property (WR).

5. Uniformly smooth mappings from W -spaces In this section we begin by showing that spaces with the hereditary DPP are W -spaces. We proceed by proving that weakly Cauchy sequences in C.K/ spaces are arbitrarily close to weakly Cauchy sequences in C.K/ which are contained in a subspace isometric to some C.Œ1; ˛/ for a countable ordinal ˛. This result will ultimately allow us to bootstrap the results obtained for uniformly smooth mappings from C.K/, K scattered compact, into the case of an arbitrary compact space K. The main result of the theory of W -spaces is Theorem 57, which gives a generalisation of the DunfordPettis property into the setting of uniformly smooth mappings from L1 -spaces. Theorem 53 ([CHL]). Let X be a Banach space with the hereditary DPP. Then X is a W -space for some > 0 and moreover C 1;C .BX I Y / CwsC .BX I Y / for any Banach space Y with property (WR). Proof. The hereditary DPP is equivalent to property S1 ([Dies1], [Cem]). So by Theorem 3.55 there is C > 0 such that for every weakly null sequence fyn g BX , 0 there is a subsequence fynk g satisfying

n

X

ak ynk C max jak j (11)

kD1;:::;n kD1

whenever n 2 N and a1 ; : : : ; an 2 R. Hence by the proof of [KO1, Theorem 2.1] the space X has a uniform version of property (u): If fxn g BX is a weakly Cauchy

358

Chapter 6. Structural behaviour of smooth mappings

sequence, then there are a wuC sequence fuk g 2BX and a weakly null sequence fyn g 2BX such that n X xn D yn C uk (12) kD1

and sup

1 X

j.uk /j .4C C 1/:

(13)

2BX kD1

(In the notation of the proof of [KO1, Theorem 2.1] set um1 D xm1 , umj D xmj xmj 1 , and uk D 0 otherwise.) Put D 8C1C1 . The first statement of the theorem follows from the second one for Y D R. So let Y be a space with property (WR) and f 2 C 1;C .BX I Y /. By contradiction, assume that fxn g BX is weakly Cauchy such that ff .xn /g is not convergent. By Fact 3.86 we may assume that dist ff .x2k 1 /I k 2 Ng; ff .x2k /I k 2 Ng > 0. Let fuk g 2BX be a wuC sequence and let fyn g 2BX be a weakly null sequence satisfying (12) and (13). By passing to further subsequences first in the odd indices and then in the even indices we may assume that (11) holds both for fy2k 1 g and for fy2k g. (Note that simultaneously we do not pass to a subsequence of fuk g but rather to a ˚Pnk 1 1 “subseries” u , which is still wuC and satisfies (13).) j j Dnk 1 kD1 By (13) there is S 2 L.c0 I X/ satisfying S.ek / D uk and kSkP .4C C 1/. Similarly, using (11) separately on the odd and even summands of nkD1 ak yk and recalling that fyk g 2BX we conclude that there exists T 2 L.c0 I X/ satisfying T .ek / D yk and kT k 4C . Finally, we define an operator U 2 L.c0 ˚1 c0 I X/ by U.x; y/ D S.x/ C T .y/. It follows that kU k kSk C kT k .8C C 1/ D 1, 1;C U.B and consequently c0 ˚1 c0 / BX and f B U 2 C .Bc0 ˚1 c0 I Y /. Let us put Pn ´n D kD1 ek ; en 2 c0 ˚1 c0 . Then U.´n / D xn . Because f´n g is a weakly Cauchy sequence in the unit ball of the space c0 ˚1 c0 which is isometric to c0 , and because Y has property (WR), the sequence ff .xn /g D ff B U.´n /g is convergent, which is a contradiction. t u For the general W -spaces not containing `1 we have a slightly more restrictive (in view of Theorem 52 and the remark after it) result. Theorem 54 ([CHL]). Let X be a W -space that does not contain `1 and Y a Banach space of a non-trivial type. Then C 1;C .BX I Y / Cwu .BX I Y /. Proof. By Theorem 24 it suffices to prove that C 1;C .BX I Y / CK .BX I Y /. Let us first prove the theorem in the special case Y D `p , 1 < p < 1. By contradiction, if f 2 C 1;C .BX I `p / is such that f .BX / is not relatively compact, then using Proposition 3.34 and Rosenthal’s `1 -theorem we may assume that there exists a weakly Cauchy sequence fxj g BX such that f .xj / D ej , j 2 N, where fej g is the canonical basis P dpe of `p . Define P 2 P . dpe `p / by P ..aj // D j1D1 . 1/j aj . Then P .ej / D . 1/j and so P B f 2 C 1;C .BX / n CwsC .BX /, a contradiction. Let now Y be a general Banach space of a non-trivial type and suppose by contradiction that there exists f 2 C 1;C .BX I Y / such that S D f .BX / is not relatively

Section 5. Uniformly smooth mappings from W -spaces

359

compact. Since Y is of a non-trivial type, it does not contain `1 and so by Rosenthal’s `1 -theorem S contains a weakly Cauchy ı-separated sequence fyj g. Since Y is also of a non-trivial type, by passing to a subsequence using Corollary 4.43 there is L 2 L.Y I `p / for some 1 < p < 1 such that L.y2j y2j 1 / D ej , where fej g is the canonical basis of `p . In particular, L.S/ is not relatively compact in `p . Indeed, either fL.y2j 1 /g is not relatively compact or there is a convergent subsequence fL.y2kj 1 /g and in this case fL.y2kj /g D fekj C L.y2kj 1 /g is not relatively compact. Hence L B f 2 C 1;C .BX I `p / n CK .BX I `p /, a contradiction with the first part of the proof. t u The assumption that X does not contain `1 cannot be removed from the statement of the result: By Corollary 3.89 there is a quotient mapping T 2 L.`1 I `2 / which is obviously not compact. Our aim now is to generalise our results on C.K/ spaces for scattered compacts K to the general setting of arbitrary compact spaces K. For a sequence a D .ak / 2 RN we define its oscillation on a set I N by osc.a; I / D supj;k2I jak aj j. Lemma 55. Let R be a discrete set, D D N with the product topology, and ı > 0. There is a mapping Q W D ! D with the following properties: (i) If a 2 D is such that there is m 2 N for which osc.a; Œm; 1// ı, then the sequence Q.a/ is eventually constant. (ii) Q is continuous. (iii) kQ.a/ ak1 D supk2N jQ.a/k ak j ı for each a 2 D. Proof. For every a 2 D there exist uniquely determined k0 2 N [ f1g and a (possibly finite) increasing sequence n.a; k/ 2 N, 1 k < k0 such that n.a; 1/ D 1, n.a; k0 / D 1 if k0 2 N, and osc a; n.a; k/; n.a; k C 1/ ı for 1 k < k0 , osc a; n.a; k/; n.a; k C 1/ > ı for 1 k < k0 1. It is important to notice that if a D .aj / 2 D, b D .bj / 2 D, and aj D bj for 1 j n.a; k/, then n.a; j / D n.b; j / for j D 1; : : : ; k. We define Q W D ! D by Q.a/j D an.a;k/ for j 2 n.a; k/; n.a; k C 1/ , 1 k < k0 . Property (iii) is clear from the definition. If a 2 D satisfies the condition in (i), then clearly k0 2 N and Q.a/ is constant on Œn.a; k0 1/; 1/. Finally, to prove (ii) suppose that fam g1 mD1 D converges pointwise to a 2 D. Fix j 2 N. There is 1 k < k0 such that j 2 Œn.a; k/; n.a; k C 1//. Since is discrete, there is m0 2 N such that alm D al for each l 2 Œ1; j , m m0 . Thus n.am ; k/ D n.a; k/ and n.am ; k C1/ > j whenever m m0 and consequently Q.am /j D Q.a/j for m m0 . t u Theorem 56 ([CHL]). Let X D C.K/, where K is a compact space. Let fxk g BX be a weakly Cauchy sequence and " > 0. Then there exist a subspace Z X isometric to some C.Œ1; ˛/, ˛ countable ordinal, and a weakly Cauchy sequence f´k g BZ such that k´k xk k < " for every k 2 N.

360

Chapter 6. Structural behaviour of smooth mappings

Proof. By considering spanfxk g according to Proposition 5.124 we may without loss of generality assume that K is metrisable. Note that the Banach space Y of all bounded Borel functions on K with the supremum norm is isometric R to a subspace of X . Indeed, the isometry S W Y ! X is given by S.f /./ D K f d. Let x 2 BY be a pointwise n 2 N satisfying ˚ j limit of the sequence fxk g. Choose 3 n; n C 1; : : : ; n 1; n . We define a simple n < " and denote D n I j D Borel function y W K ! approximating x within n1 by y.t/ D jn if and only if j j C1 n > x.t / n . Further, we construct a sequence fyk g BY of Borel functions yk W K ! converging pointwise to y and satisfying kyk xk k n1 , k 2 N: We let ( j j if jxk .t/ y.t/j n1 and j C1 n > xk .t/ n for some n j n, yk .t / D n y.t / otherwise. Fix t 2 K. Since jy.t/ x.t/j < n1 and fxk .t/g converges to x.t/, there is m 2 N such that jxk .t / y.t/j < n1 for all k m and so yk .t/ D y.t/ for all k m. Let D D N with the product topology and define ˚ W K ! D by ˚.t/k D yk .t/. Note that D is a metrisable compact space. Put H D ˚.K/. We claim that for every h D .hk / 2 H there is m 2 N for which osc.h; Œm; 1// n2 . Indeed, there is ftj g K such that ˚.tj / ! h. By passing to a subsequence we may assume that tj ! t 2 K. Let m 2 N be such that jxk .t/ y.t/j < n1 for every k m. Fix k m and let > 0 be such that jxk .t/ y.t/j < n1 2. Let j0 2 N be such that jhk yk .tj /j D jhk ˚.tj /k j < for every j j0 . By the continuity of xk there exists j j0 such that jxk .tj / xk .t/j < . Then jhk y.t/j jhk yk .tj /j C jyk .tj / xk .tj /j C jxk .tj / xk .t/j C jxk .t/ y.t/j < n2 . Since y.t / 2 and h 2 D, the claim follows. Let Q W D ! D be the mapping from Lemma 55 for ı D n2 and set D Q B ˚ and L D .K/. Then L is a countable compact set. Indeed, if g 2 L, then there is ftj g K such that .tj / ! g. By passing to a subsequence we may assume that ˚.tj / ! h 2 H . Since Q is continuous, we have .tj / D Q.˚.tj // ! Q.h/ and hence g D Q.h/. By (i) in Lemma 55 and the claim above we conclude that the sequence g is eventually constant. Thus L consists of sequences in D that are eventually constant and so it is countable. Therefore C.L/ is isometric to C.Œ1; ˛/ for some countable ordinal ˛ by the Mazurkiewicz-Sierpi´nski theorem ([HMVZ, Theorem 2.56]). Finally, by Fact 5.123 the mapping ˚ is Borel, and hence so is , and further T W C.L/ ! Y , T .f / D f B is an isometry onto a subspace Z Y . Note that yk D k B ˚, where k is the projection onto the kth coordinate. Put ´k D T .k L /, k 2 N. Then ´k D k B and so k´k yk k D sup t 2K j.Q B ˚.t//k ˚.t/k j n2 by (iii) in Lemma 55. Clearly, ´k 2 BZ and k´k xk k k´k yk kCkyk xk k n3 < " for each k 2 N. Since each g 2 L is eventually constant, fk .g/g is convergent. Hence fk L g is weakly Cauchy in C.L/ and consequently f´k g is weakly Cauchy in Z. u t The next theorem generalises the Dunford-Pettis property of L1 -spaces (Theorem 3.41, see also Theorem 3.47) to the case of uniformly smooth mappings. Recall

Section 5. Uniformly smooth mappings from W -spaces

361

that for any compact space K the space C.K/ is a L1; -space for every > 1 (Theorem 4.15). Theorem 57 ([CHL]). Let X be a L1; -space, 1, and Y a Banach space. Then C 1;C .BX I Y / \ CwK .BX I Y / CwsC . 1 BX I Y /: If Y has property (WR) or if Y has property (WR) and is isomorphic to a complemented subspace of a dual Banach space, then C 1;C .BX I Y / CwsC . 1 BX I Y /: In particular, X is a W 1 -space.

Proof. Embed X isometrically into E D C .BX ; w / . By Theorem 4.15, X is a P -space. Hence there exists a projection P W E ! X with kP k . Assume that f 2 C 1;C .BX I Y / n CwsC . 1 BX I Y / and that fxk g 1 BX is a weakly Cauchy sequence such that ff .xk /g is not convergent. Passing to a subsequence by Fact 3.86 we may assume that dist ff .x2k 1 /I k 2 Ng; ff .x2k /I k 2 Ng > 0. Put g D f B P and note that g.xk / D f .xk /, k 2 N. Then g is uniformly continuous on 1 BE and so by Theorem 56 there exist a subspace Z E isometric to C.Œ1; ˛/, ˛ countable ordinal, and a weakly Cauchy sequence f´k g 1 BZ such that k´k xk k is small enough so that dist fg.´2k 1 /I k 2 Ng; fg.´2k /I k 2 Ng > 0. Thus g 2 C 1;C . 1 BE I Y / n CwsC . 1 BZ I Y /. This contradicts Theorem 48 in case that Y has property (WR). To prove the first part of the theorem, note that by Theorem 46 the set g. 1 BZ / is not relatively weakly compact and so f .BX / is not relatively weakly compact either. Consequently, f … CwK .BX I Y / by Fact 11. Finally, suppose that Y has property (WR) and is isomorphic to a complemented subspace of a dual Banach space. Then there is a projection Q W Y ! Y (use Dixmier’s projection). We put g D Q B f B P and arrive at a contradiction with Theorem 48 similarly as above. t u We remind that Y has property (WR) if and only if Y does not contain `1 (Corollary 51). Recall that the class W behaves rather well with respect to quotients (Proposition 21). Passing to non-linear smooth images, we have the following structure-preserving properties. Proposition 58. Let X be a W -space that does not contain `1 , Y a Banach space, and let f 2 C 1;C .BX I Y / be such that Int f .BX / ¤ ;. Then Y does not contain `1 . Proof. Suppose that Y contains `1 . Then by Corollary 3.89 the space Y has a quotient isomorphic to `2 . Hence there is g 2 C 1;C .BX I `2 / such that Int g.BX / ¤ ;. By Theorem 54, g 2 CK .BX I `2 /, which is a contradiction. t u Proposition 59 ([CHL]). Let X be a reflexive W -space, Y a Banach space, and let f 2 C 1;C .BX I Y / be such that Int f .BX / ¤ ;. Then Y is a reflexive W -space.

362

Chapter 6. Structural behaviour of smooth mappings

Proof. By shifting and scaling in the range we may suppose that there is > 0 such that BY f .BX / and f .BX / BY . To prove that Y is reflexive, by the EberleinŠmulyan theorem it suffices to show that any sequence fyk g BY contains a weakly convergent subsequence. Let fxk g BX be such that f .xk / D yk . As X is reflexive, w there is a subsequence fxnk g such that xnk ! x. The mapping f is w–w sequentially w

continuous on BX (Fact 22) and hence ynk D f .xnk / ! f .x/. We proceed by proving that Y is a W -space. Suppose by contradiction that there exist a function g 2 C 1;C .BY / and a weakly Cauchy sequence fyk g BY such that limk!1 g.y2k 1 / ¤ limk!1 g.y2k /. By the weak compactness of BY there w is y 2 BY such that yk ! y. There is a sequence fxk g BX satisfying f .xk / D yk . By passing to subsequences separately in odd and even indices we w w may assume that x2k 1 ! u and x2k ! v. Clearly, f .u/ D y D f .v/, as f is sequentially w–w continuous on BX . Since g B f 2 C 1;C .BX / CwsC .BX /, we obtain lim g.y2k 1 / D lim g B f .x2k 1 / D g B f .u/ D g.y/ and similarly lim g.y2k / D lim g B f .x2k / D g B f .v/ D g.y/, a contradiction. t u

6. Fixing the canonical basis of c0 In the present section we are going to investigate the phenomenon of fixing the canonical basis of c0 , which holds for non-weakly compact linear operators from C.K/ spaces, in the non-linear setting. We start by showing that in a polynomial version of Theorem 3.47 the implication (iv))(i) remains valid, provided that the compact space K is scattered. However, our main result is a counterexample to this implication for C 1;1 -smooth mappings constructed below. This means that a qualitative change takes place when replacing polynomials with C 1;1 -smooth mappings from c0 . Theorem 60 ([DeH2]). Let X D c0 or X D C.K/, where K is a scattered compact space, let Y be a Banach space, and P 2 P .XI Y / a non-compact polynomial. Then there exist v 2 BX and a sequence fvn g BX such that both fvn vg and fP .vn / P .v/g are equivalent to the canonical basis of c0 . Proof. By Theorems 46 and 42 and Proposition 3.22 it suffices to prove the result for X D c0 . By Lemma 41, Proposition 3.32, and [FHHMZ, Proposition 4.45] there exist v 2 Bc0 and a weakly null sequence fun g c0 such that fv C un g Bc0 , fun g is equivalent to the canonical basis of c0 , and limn!1 P .v C un / does not exist. Put vn D v C un and ´n D P .vn / P .v/. Since P is w–w sequentially continuous (Corollary 3.59), f´n g is weakly null. Thus by passing to a subsequence we may assume that f´n g is a semi-normalised basic sequence. Define a polynomial Q by Q.x/ D P .v C x/ P .v/. Then ´n D Q.un /. Since fun g is a wuC sequence, by Theorem 3.53 the sequence f´n g is also wuC, i.e. it has an upper 1-estimate (Fact 3.51). The is automatic, since f´n g is a semi-normalised basic sequence:

Pmlower 1-estimate

1 maxn kan ´n k 1 infn k´n k maxn jan j, where K is the basis a ´ n n nD1 2K 2K constant of f´n g. t u

Section 6. Fixing the canonical basis of c0

363

The next simple example shows that Theorem 60 is optimal in the sense that the shifting of the basis cannot be avoided, so the result is necessarily P of an affine rather than linear nature. Define P 2 P . 4 c0 I c0 / by P .x/ D x12 j1D1 xj2 ej . Then P is non-compact, but limn!1 P .un / D 0 for every weakly null sequence fun g c0 . Since the image of P is contained in the non-negative cone of c0 , we also see that T .Bc0 / 6 P .Bc0 / for any T 2 L.c0 I c0 /. In the rest of this section we will construct a C 1;1 -smooth and non-compact mapping f W Bc0 ! `1 which has the property that f .Bc0 / does not “contain” the canonical basis of c0 . To this end we first construct separating functions on `n1 with an upper estimate on the second derivative that increases slower than the dimension of the domain. Lemma 61. The functions pn 2 P 4n .`n1 / defined by pn .x/ D 1

n Y

.1

xk4 /

kD1

have the following properties: (i) 0 pn .x/ 1 for every x 2 B`n1 and pn .0/ D 0, (ii) pn .x/ ı 4 for each x 2 B`n1 with kxk ı 0, (iii) pn .x/ 1 .1p 4 /n for each x 2 `n1 with kxk 1, (iv) kDpn .x/k 4 4 n for every x 2 B`n1 , p (v) there is a constant C > 0 such that kD 2pn .x/k C n for every x 2 B`n1 and every n 2 N. Proof. The first three properties Q are clear. We show the estimates on the derivatives by induction on n. Put qn .x/ D nkD1 .1 xk4 / and note that kD jqn .x/k D kD jpn .x/k for all x 2 `n1 , j 2 N. Clearly, supx2B 1 kDq1 .x/k D supjxj1 j4x 3 j D 4. Now `1 p assume that the estimate supx2B`n kDqn .x/k 4 4 n holds for some n 2 N. Since 1 4 qnC1 .x/ D qn Pn .x/ .1 xnC1 /, where Pn is the projection to the first n coordinates, by the Leibniz formula (Corollary 1.116) 4 3 DqnC1 .x/Œh D Dqn Pn .x/ ŒPn .h/ .1 xnC1 / C qn Pn .x/ . 4xnC1 /hnC1 p nC1 4 3 for every x; h 2 `1 . Consider the function '.t/ D 4 4 n.1 t / C 4t , t 2 R. By the inductive hypothesis kDqnC1 .x/k kDqn .Pn .x//k j1

4 3 xnC1 j C jqn .Pn .x//j j 4xnC1 j

'.jxnC1 j/ p for x 2 B`nC1 . Since ' 0 .t/ D 16 4 nt 3 C 12t 2 , the function ' attains its maximum 1 p p 4 27 3 4 p on Œ0; 1 at the point 4 p . So max ' D 4 n C 4 n C 1, where the Œ0;1 4 4 3 < 4 n 256 n last inequality is easy to check. This finishes the proof of the estimate for Dqn . Similarly, we have kD 2q1p .x/k 12 for x 2 B`11 . Now assume that the estimate supx2B`n kD 2qn .x/k C n holds for some n 2 N and C 12. By the Leibniz 1

364

Chapter 6. Structural behaviour of smooth mappings

formula 4 D 2qnC1 .x/Œh; k D D 2qn Pn .x/ ŒPn .h/; Pn .k/ .1 xnC1 / 3 3 C Dqn Pn .x/ ŒPn .h/ . 4xnC1 /knC1 C Dqn Pn .x/ ŒPn .k/ . 4xnC1 /hnC1 2 C qn Pn .x/ . 12xnC1 /hnC1 knC1 p p p 4 4 for x; h; k 2 `nC1 nt 3 C12t 2 , 1 . Consider the function .t/ D C n.1 t /C32 C 2 t 2 R. By the inductive hypothesis kDpqnC1 .x/k .jxnC1 j/ for x 2 B`nC1 1 (we note that the multiplicative constant C was added so that the analysis of the p p p function is much easier). We have 0 .t/ D p4C nt 3 C 96 C 4 nt 2 C 24t and p p 6 . Thus attains its maximum on so 0 .t / D 0 if and only if t D 0 or t D 12˙5 C 4 n p p b Œ0; C1/ at p ap , where a D 12 C 5 6, and hence maxŒ0;1 C n C C p , n C 4 n p 4 3 2 2 where b D a C 32a p C 12a . It is not difficult to check that if C .1 C 2/b, then maxŒ0;1 C n C 1. This finishes the proof. We remark that p the estimate above gives C 533. Had we not artificially inflated the estimate by C , a more tedious computation would have resulted in C 32. u t The next result shows among others that a uniformly smooth version of Theorem 3.53 is false. Theorem 62 ([DeH2]). There is a C 1;1 -smooth mapping f W Bc0 ! `1 that is not weakly compact and has the property that there is no sequence fyn g1 nD0 f .Bc0 / such that fyn y0 g1 is equivalent to the canonical basis of c . 0 nD1 Proof. Let be the set of all finite subsets of N. Then is countable and so `1 . / is isometric to `1 . Let C be the constant and pn the functions from Lemma 61. For each A 2 , A D fk1 ; : : : ; kn g with k1 < < kn we define TA 2 L.c0 I `n1 / by 1 TA .x/ D .xk1 ; : : : ; xkn / and set 'A D C p p B TA . Then kD 2'A .x/k 1 for every n n x 2 Bc0 . Finally we define f W Bc0 ! `1 . / by f .x/ D .'A .x//A2 . Then f is C 1;1 -smooth by Proposition 1.130. Also, kf .ek / f .ej /k j'fkg .ek / 'fkg .ej /j D 1 p1 .0/j D C1 if k ¤ j , where fek g is the canonical basis of c0 . Thus f is C jp1 .1/ not compact, and by Theorem 46 even not weakly compact. We proceed by contradiction. Assume that yn D f .un /, n 2 N0 , and fyn y0 g1 nD1 is equivalent to the canonical basis of c0 . We claim that by passing to a subsequence of fun g1 nD1 we may assume that there exist ı > 0 and an increasing sequence fkn g N such that jun .kn /j ı, n 2 N. Indeed, by passing to a subsequence we may assume that there is u 2 `1 such that w -lim un D u. Since un 6! u in norm (otherwise fyn y0 g would be convergent, which is not possible), we may assume that there is ı > 0 such that kun uk > 2ı for all n 2 N. Suppose that m0 ; n0 2 N are given. If there is m m0 such that ju.m/j > ı, then jun .m/j > ı for sufficiently large n n0 . Otherwise there is n n0 such that jun .j / u.j /j < ı for 1 j m0 . Since kun uk > 2ı, there is m > m0 such that jun .m/ u.m/j > 2ı. Consequently jun .m/j jun .m/ u.m/j ju.m/j > 2ı ı D ı.

Section 7. Ranges of smooth mappings

365

Pn

y0 / K To continue the proof assume that K > 0 is such j D1 .ymCj p that for every m; n 2 N. Let n 2 N be such that 2Cn ı 4 > K. Further, let 2 .0; 1 4 be such that 1 .1 4 /n < ı2 and let m 2 N be such that ju0 .j /j for all j m. Put A D fkmC1 ; kmC2 ; : : : ; kmCn g and note that kmC1 > m. We have 1 1 'A .u0 / D C p p u .k /; : : : ; u0 .kmCn / < 2C1pn ı 4 and 'A .umCj / C p ı4 n n 0 mC1 n

Pn

P for each j D 1; : : : ; n. So j D1 .ymCj y0 / jnD1 ymCj .A/ y0 .A/ D Pn t u 'A .u0 / > n 2C1pn ı 4 > K, a contradiction. j D1 'A .umCj /

7. Ranges of smooth mappings In the present section we are going to investigate in some detail the necessity of the assumption on the initial space to belong to the class W in order to preserve some structural properties through uniformly smooth mappings. In particular, Theorem 69 below shows that property B (which appears to be rather close to the negation of W ) leads to a complete loss of any structural property via C 1 -smooth mappings (or even via homogeneous polynomials). In the case of C 1 -smooth mappings the same is true even for every separable Banach space. These results are due to Sean Michael Bates, [Bat]. We begin with a lemma that allows us to give a joint proof of two types of results regarding the ranges of smooth mappings and the ranges of derivatives of smooth functions. Lemma 63. Let X, Y be Banach spaces, B X a closed convex set containing the origin, 0 < r < R 1, and let ' 2 C k .X I Œ0; 1/ be a function satisfying supp ' RB and '.x/ D 1 whenever x 2 rB. Further, let fx g 2 X be bounded and such that fx C RBg 2 is a ı-separated collection of subsets of B for some ı > 0 and let ff g 2 C k .X I Y /, k 2 N [ f1g, be bounded (in the space C k .XI Y /). Then there are a mapping 2 C k .XI Y / with supp B and a 1 C sequence fın g1 nD0 R with ı0 D 1 and ın nŠ such that for any 0 j < k C 1 1 and any sequence f n gnD0 D

j

1 X nD0

! n

r x n

! 1 1 X 1 X l ın j D f n n r x l : D r r nj nD0

lDnC1

Proof. Since ff g is bounded in C k .XI Y /, for each > 0 and 0 j < k C 1 there is Kj ./ 1 such that kD lg .x/k Kj ./ for each x 2 B.0; /, 2 , and 0 l j , where g .x/ D '.x x /f .x x /. Put K0 .0/ D 1 and Kj ./ D Kk ./ if k 2 N and j > k. We may assume without loss of generality that j 7! Kj ./ and 7! Kj ./ are non-decreasing. We define inductively a sequence of mappings

366

f

1 n gnD0

Chapter 6. Structural behaviour of smooth mappings

C k .XI Y / by 0 .x/

D

X

'.x

x /f .x

x / and

2 n .x/

D

1 nKn

2n rn

X n 2

1 n 1 r .x

x / :

First note that for each n the sum in the definition locally consists only of at most one non-zero summand (and consequently n is C k -smooth) and supp n B. Indeed, for 0 this is clear from the assumptions. Assuming that this statement holds for n 1 , we get supp x 7! n 1 1r .x x / x C rB x C RB B, and so it holds also for n . n n n By induction on n we get kD j n .x/k r j1n nŠ Kn . 2r n n/ 1 Kj . 2r n / r j1n nŠ Kj . 2r n / for every > 0, x 2 B.0; /, n 2 N0 , and 0 j < k C 1. Hence for a fixed 0 j < k C 1 and > 0 we have kD j n .x/k r j1n nŠ for all n maxfj; g P1 k and x 2 B.0; /. Therefore the mapping D nD0 n belongs to C .XI Y /, P 1 supp B, and D j D nD0 D j n (Theorem 1.85). Q m 1 Km . 2r m m/ 1 . We claim that n .x/ D ın f n r1n .x y/ Next, set ın D nmD1 m P and x 2 B whenever n 2 N0 , y D jnD0 r j x j for some 0 ; : : : ; n 2 , and x 2 y Cr nC1 B. Indeed, for n D 0 this is clear. Assuming that the claim holds for n 1 P we put u D 1r .x x 0 / and v D jnD01 r j x j C1 . Then u v D 1r .x y/ and hence u 2 vCr n B. Therefore n 1 .u/ D ın 1 f n r n1 1 .u v/ and u 2 B by the inductive hypothesis. Since x D x 0 Cru 2 x 0 CrB B, it follows that x … x CrB for every n

2 n f 0 g and hence n .x/ D n1 Kn . 2r n n/ 1 n 1 .u/ D ın f n r1n .x y/ . Since by the assumption Int B ¤ ;, we obtain also D j n .x/ D r 1nj ın D jf n r1n .x y/ for 0 j < k C 1. P n We have shown above that m N and every sequence nD0 r x n 2 B for every m 2P P 1 1 n f n gnD0 . Thus nD0 r x n 2 B. So finally if x D 1 r n x n for some Pn P1 nD0 1 j nC1 j nC1 B sequence f n gnD0 , then x j D0 r x j D r j D0 r x nC1Cj 2 r for every n 2 N0 and hence the statement of the lemma follows. t u Fact 64. Let Y be a normed linear space, fy g 2 a dense subset of BY , and let 1 C fın g1 nD1 R be such that ın ! 0. For each P1 y 2 ı0 BY there is a sequence f n gnD0 of distinct elements of such that y D nD0 ın y n . If D N, then f n g can be chosen to be increasing.

Pn such that y Proof. We find a sequence f n g1 ı y < ı kD0 nD0

1 k k nC1 for every n 2 N0 using induction: Let 0 2 be such that ı0 y y 0 < ıı10 . P n 1 By the inductive assumption ´ D ı1n y kD0 ık y k 2 BY and so there exists

n 2 n f 0 ; : : : ; n 1 g such that k´ y n k < ınC1 =ın , which finishes the construction. In case that D N we can of course choose n > n 1 . t u

Section 7. Ranges of smooth mappings

367

Fact 65. Let X be an infinite-dimensional normed linear space. For every " 2 .0; 1/ the sphere SX contains an "-separated set of cardinality dens X. Proof. Let D SX be a maximal "-separated set. By the Riesz lemma card D !. Let E be the set of all linear combinations of vectors in D with rational coefficients. Then card E D card D. Since the set D is maximal, E D X , as otherwise the Riesz lemma would imply the existence of x 2 SX with dist.x; E/ > ". It follows that dens X card E D card D dens X. t u Theorem 66 ([AzDe]). Let X be an infinite-dimensional Banach space that admits a C 1 -smooth Lipschitz bump. Then there exists a Lipschitz bump 2 C 1 .X/ such that BX D .X /. Proof. By Theorem 5.72 we have dens X D dens X. Take a dense ff g 2 BX with card D dens X. By Fact 65 the ball BX contains a 12 -separated set of cardinality card . Thus the result follows from Lemma 63 for Y D R combined with Fact 64. u t Similarly we can obtain that if X is an infinite-dimensional Banach space that admits a C 1 -smooth Lipschitz bump and Y is a Banach space with dens Y dens X , then there is a C 1 -smooth Lipschitz mapping from X onto Y . For Y separable this holds even without the assumption of the existence of the smooth bump. Theorem 67 ([Bat]). Let X be an infinite-dimensional Banach space and Y a separable Banach space. Then there exists a C 1 -smooth Lipschitz mapping from X onto Y . Proof. By the Josefson-Nissenzweig theorem ([Dies2, Chapter XII]) there is a w -null sequence fgn g SX . Define T 2 L.X I c0 ) by T .x/ D .gn .x//1 nD1 . For each n 2 N there is un 2 BX with gn .un / 21 and so T is not compact by Fact 4.90. By Proposition 3.33 we may assume that there is a sequence fxn g BX such that T .xn / D en , n 2 N, where fen g is the canonical basis of c0 . Let fyn g1 nD1 be dense in BY and define constant functions fn .x/ D yn for x 2 c0 , n 2 N. By Theorem 5.94 the space c0 admits a C 1 -smooth Lipschitz bump. By Lemma 63 and Fact 64 there are r 2 .0; 12 / and a Lipschitz mapping 2 C 1 .c0 I Y / such that supp 2Bc0 ˚P1 k and BY .M /, where M D kD0 r enk I fnk g N increasing . Note that M T .2BX /. P 1 3ne1 . Fix an arbitrary x 2 X . Since Let us define .x/ D 1 nD1 n n T .x/ 3n2 C 2n < 3.n C 1/2 2.n C 1/ for each n 2 N, it is easy to see that there is at most one n 2 N such that jT .x/1 3n2 j 2n. Thus there is a neighbourhood U of x 1 1 T .y/ 3me1 m jT .y/1 3m2 j > 2 for every y 2 U and m 2 N, such that m m ¤ n. Hence the sum in the definition of consists locally of at most one summand and consequently 2 C 1 .XI Y / and it has a bounded derivative. For a fixed n 2 N we have n1 T .2nBX C 3n2 x1 / 3ne1 D T .2BX / M . Therefore .X/ D Y . t u The following is a generalisation of the well-known fact that every separable Banach space is a quotient of `1 , [FHHMZ, Theorem 5.1].

368

Chapter 6. Structural behaviour of smooth mappings

Theorem 68 ([Há7]). Let X be a Banach space such that there exists a non-compact operator T 2 L.XI `p /, 1 p < 1. Then for every separable Banach space Y there exists a surjection P 2 P . dpeXI Y /. Proof. By Proposition 3.33 we may assume that there is a sequence fxn g BX such that T .xn / D en , n 2 N, where fen g is the canonical basis of `p . Choose a dense sequence . Denote m D dpe and define a polynomial Q 2 P . m`p I Y / by P1 fyn g BYP m Q nD1 an en D 1 Y , then by Fact 64 there nD1 an yn . Put P D Q B T . If y 2 P 1 km y . Putting is an increasing sequence fnk gkD0 N such that y D kyk 1 nk kD0 2 1 P1 k m xnk we get P .x/ D y. x D kyk t u kD0 2 Theorem 69 ([Bat]). If a Banach space X has property B, then for any separable Banach space Y there exists a surjective mapping f 2 C 1 .X I Y /. Proof. Let fgn g SX be the sequence from Lemma 36. As in the proof of Proposition 37 we may assume that there exists a semi-normalised basic sequence f´n g X which is biorthogonal to fgn g. Put xn D 34 ´n and R D 41 . Let 2 C 1 .RI Œ0; 1/ be R such that .t / D 1 for t 2 Œ R 2 ; 2 and .t/ D 0 for jtj R and let ' W X ! R be defined by '.x/ D

1 Y

gn .x/ :

nD1 R For a fixed x 2 B.0; / put A D fn 2 NI jgn .x/j > R k. 4 /. For any 4 g. Then jAj Q R R y 2 B.x; 4 / we have fn 2 NI jgn .y/j > 2 g A and so '.y/ D n2A .gn .y//. Consequently, ' is C 1 -smooth on X and it has all derivatives bounded on bounded sets, i.e. ' 2 C 1 .XI Œ0; 1/. Denote by B the cube B D fx 2 XI jgn .x/j 1; n 2 Ng and note that rB D fx 2 X I jgn .x/j r; n 2 Ng for r > 0. Put r D R 2 . Then supp ' RB and '.x/ D 1 for x 2 rB. Note that we have xn C RB B for n 2 N and kx yk jgk .x y/j jgk .xk xn /j jgk .x xk /j jgk .y xn /j 34 2R D 41 whenever x 2 xk C RB, y 2 xn C RB, k ¤ n. Choose a dense sequence fyk g BY and define constant functions fk .x/ D yk for x 2 X, k 2 N. By Lemma 63 and Fact 64 there is 2 C 1 .X I Y P / such that supp B and BY .B/. 2 1 Put .x/ D 1 n . Fix x 2 X . Since 2n Cn < 2.nC1/2 .nC1/ x 2n´ 1 nD1 n for every n 2 N, it is easy to see that there exists at most one n 2 N that satis2 j n. Therefore there exists a neighbourhood U of x such that 2nˇ fies 1 .x/ ˇ jg 1 ˇg1 y 2m´1 ˇ > 1 for every y 2 U and m 2 N, m ¤ n. Hence the sum m in the definition of consists of atˇ most one summand and consequently ˇ locally 1 1 ˇ 2 C .X I Y /. Moreover, g1 n x 2n´1 ˇ 2n n1 kxk > 1 for all n > kxk and so n1 x 2n´1 … B for n > kxk. Thus all the derivatives of are bounded on bounded sets. Finally, for a fixed n 2 N we have n1 .nB C 2n2 ´1 / 2n´1 D B and hence .X / D Y . t u

Section 8. Harmonic behaviour of smooth mappings

369

8. Harmonic behaviour of smooth mappings As we have seen in the previous section any separable Banach space can be a smooth image (even by a homogeneous polynomial) of an `p . In order to preserve some structural properties of the initial space `p one needs to consider other properties of the mappings than just surjectivity. The notion which leads to strong properties is the harmonic behaviour of the mapping (more precisely, the lack of it). The harmonic behaviour itself can be viewed as a condition of smallness of the image, in a broad sense. We begin the section by proving a key result (Theorem 71) which properly belongs to the linear structural theory of Banach spaces. Let Y be a Banach space and Z a Banach a Schauder basis fen g. Let us ˚ space withP denote the positive cone of Z by Z C D ´ 2 ZI ´ D 1 nD1 an en ; an 0 . We say that Z C embeds into Y onto a basic sequence fyn g Y if there are A > 0 and C 1 such that

1

1

1

1

X

X

X

X

A an en an yn CA an en for any an en 2 Z C . (14)

nD1

nD1

nD1

nD1

We say that C is an embedding constant. We are interested in the problem whether the fact that Z C embeds into Y already Note that the summing basis fxn g of c0 has the property implies

Pthat Y contains

PZ. 1 C

that 1 a x D nD1 n n nD1 an provided that an 0, which means that `1 embeds into c0 , although c0 does not contain `1 . On the other hand, it is rather easy to see that if c0C embeds into Y onto a basic sequence fyn g, then Y contains c0 , and in fact fyn g is equivalent to the canonical basis of c0 . Indeed, for a 2 R denote aC D maxfa; 0g P and a D maxf a; 0g. If knD1 an yn 2 Y , then

k

k

k

k

k

X

X

X

X

X

an yn D anC yn an yn anC yn C an yn

nD1

nD1

CA maxfanC g

nD1

nD1

nD1

C CA maxfan g 2CA maxfjan jg:

But since fyn g is a semi-normalised basic sequence, the reverse inequality is automatic (see the proof of Theorem 60). In the case of unconditional basic sequences the problem is easy: Fact 70. Let Z be a Banach space with an unconditional Schauder basis fen g, Y a Banach space, and suppose that Z C embeds into Y onto an unconditional basic sequence fyn g. Then Y contains Z and in fact fyn g is equivalent to fen g.

P

P

Proof. There exists K1 1 such that K1 1 knD1 jan jyn knD1 an yn

P

P K1 knD1 jan jyn for any knD1 an yn 2 Y , k 2 N, and there exists K2 1 such that

P

P

P

P K2 1 knD1 jan jen knD1 an en K2 knD1 jan jen for any knD1 an en 2 Z,

P

P P k 2 N. Thus K1 1 AK2 1 knD1 an en knD1 an yn K1 CAK2 knD1 an en P for any knD1 an en 2 Z, k 2 N. t u

370

Chapter 6. Structural behaviour of smooth mappings

Theorem 71 ([HJ2]). Let Y be a Banach space and 1 < p < 1. If `pC embeds into Y , then Y contains `p . Proof. Let fyn g Y be the basic sequence such that `pC embeds onto fyn g. We claim that there is an unconditional normalised block basic sequence of fyn g such that all its vectors have non-negative coordinates with respect to fyn g. Then it is easily seen by Fact 70 and [FHHMZ, Proposition 4.45] that this block basic sequence is equivalent to the canonical basis of `p . P P P For x D knD1 an yn 2 Y we denote x C D knD1 anC yn , x D knD1 an yn , P and b x D knD1 an en 2 `p . Suppose that fyn g is not unconditional and `pC embeds onto fyn g with an embedding constant C . Then for any " > 0 there is y 2 spanfyn g such that ky C k D 1 and kyk < ". If this was not true for some " > 0, then for any x 2 spanfyn g ˚ " " kxk " max kx C k; kx k kx C k C kx k kx C C x k: 2 2 On the other hand C kxk D kx C x k kx C k C kx k CA xc C xc CA21

1 p

c

x C C xc C 21

1 p

kx C C x k;

which means that fyn g would be unconditional. Thus we can find a block basic sequence

c

1

C D 1. fvn g of fyn g such that kvn k < A 2 2n and vn Let fan gknD1 be a finite sequence of non-negative real numbers not all zero. Then

k

! p1 k k k

X

X X A X p

an an vn an kvn k maxfan g kvn k and (15)

2 nD1

nD1

nD1

nD1

k

k

k

k k

X

X

X

X X

an vn D an vnC an vn an vn an vnC

nD1 nD1 nD1 nD1 nD1

k

k

k

! p1 k

X

X

X

X

c p C ; an CA an vc an vn D A A an vc CA n n

nD1

nD1

nD1

nD1

which implies

k

X

1

c an vn C C

2 nD1

k X

! p1 anp

:

C

D 1, clearly As vc n

k

k ! p1 k

X

X X

c : an vnC A an vnC D A anp

nD1

nD1

(16)

nD1

nD1

(17)

Section 8. Harmonic behaviour of smooth mappings

371

Pk Pk C C

For an upper estimate take f 2 S.spanfyn g/ so that f nD1 an vn D nD1 an vn . We will estimate the positive part of the functional f on spanfvnC g using duality S on `p . Let bn D f .yn /, n 2PN and put M D PknD1 supp vn (which P is a finite set). Further, we define g D n2M bn yn , g C D n2M bnC yn , b g D n2M bn en , P C D b C e , where fy g and fe g are the functionals biorthogonal to and gc n2M

n

n

n

n

fyn g and fen g respectively. Note that f .x/ every x 2 spanfyn g with P D g.x/C for 1 1 q 1 supp x M . Let p C q D 1 and put y D n2M .bn / yn . Then !q1 X

c

g C D

.bnC /q

n2M

D

P D P

C q n2M .bn /

C q n2M .bn /

p1

(18)

g.y/ f .y/ g.y/ CA D CA CA: kb yk kyk kyk

Using (17) we have

k

! ! ! k k k

X

X X X

an vn f an vn D f an vnC f a n vn

nD1 nD1 nD1 nD1

k

! k ! k k

X

X

X X

C C C D an vn g an vn an vn g an vn

nD1

A

nD1

k X

! p1 anp

nD1

k X

gC

nD1

! an vn :

nD1

nD1

S S P C C Denote M C D knD1 supp vnC , M D knD1 supp vn , gc MC D n2M C bn en , C and gc M similarly. The last inequality together with (15), the Hölder inequality, and (16) gives A 2

k X

! p1 anp

g

C

nD1

k X

! an vn

nD1

k

X

c C an vc kg M k n

nD1

1 C kgc M k C C 2 C which means that kgc M k obtain

C C q kgc M C k D kgc k

A 2C C1 .

! p1

k X

anp

;

nD1

If we combine this inequality with (18), we

C kgc M kq

q1

q

C A

q

Aq .2C C 1/q

q1 :

372

Chapter 6. Structural behaviour of smooth mappings

This finally allows us to estimate

k

! ! k k

X

X X

an vnC D f an vnC D g an vnC

nD1

nD1

g

C

k X

nD1

! an vnC

nD1

k

X

c c C C kg M C k an vn

nD1

C D kgc M C k

k X

! p1 anp

nD1

CA 1

1 C q .2C C 1/q

q1

k X

! p1 anp

:

nD1

fvnC g

The last inequality and (17) shows that is a semi-normalised block basis such C C that `p embeds onto fvn g with an embedding constant smaller than C . Now either fvnC g is an unconditional basic sequence and we are done, or we can iterate the process to find another block basis. (Notice that in every iteration the constructed block basis is a block basis of fyn g such that all of its vectors have non-negative coordinates with respect to the previous basic sequence and hence with respect to fyn g.) In every iteration 1=q < 1, the embedding constant drops at least by the factor of 1 C q .2C1 C1/q where C is the initial embedding constant corresponding to the basic sequence fyn g. Therefore after finitely many steps we obtain an unconditional block basic sequence as we claimed, otherwise the embedding constant would eventually drop below 1, which is impossible. t u Observe that in the case Z D `p , 1 < p < 1, or Z D c0 if there is an arbitrary sequence fyn g Y (not necessarily basic) such that (14) holds, then Z C embeds be seen as follows: Let f 2P.spanfyn g/ . Similarly as in (18) we into Y . This can q q P1 C < C1 and 1 f .y / < C1. can show that nD1 f .yn / This means n nD1 that fyn g is weakly null, and thus some subsequence of fyn g is a basic sequence (Proposition 3.32). We will make use of yet another diagonalisation result, this time for spaces with perfectly homogeneous basis; compare also with Lemma 4.67, Lemma 4.91, and Lemma 44. A Schauder basis is called perfectly homogeneous if it is equivalent to any of its normalised block bases. By the result of Mordecay Zippin ([Zip1], see also [LiTz2, Theorem 2.a.9]) a Banach space X has a normalised perfectly homogeneous basis if and only if the space X is isomorphic to c0 or `p , 1 p < 1. Recall that a non-constant homogeneous polynomial P W X ! Y separates 0 from SX if infx2SX kP .x/k > 0.

Section 8. Harmonic behaviour of smooth mappings

373

Lemma 72 ([HJ2]). Let X D c0 or X D `p , 1 p < 1, let Y be a Banach space, and k 2 N. Suppose that no polynomial in P . nX I Y /, 1 n < k, separates 0 from SX . Let P 2 P . kX I Y / and " > 0. Then P there is a normalised block basis f´j g of the canonical basis fej g of X such that if j1D1 aj ´j 2 BX and m 2 N, then

P

1 X

! aj ´j

j Dm

1 X j Dm

ajk P .´j /

0. We define the block basis f´j g inductively along with auxiliary block bases fvjn gj1D1 , n 2 N, such that fvjn gj1D1 is a normalised block basis of fvjn 1 gj1D1 , ´n D vnn , and }.´j1 ; : : : ; ´jk /k < kP

" 2nC2 nk

(19)

whenever minfj1 ; : : : ; jk g < maxfj1 ; : : : ; jk g D n > 1. Let us put vj1 D ej for j 2 N, and ´1 D e1 . Assume that ´1 ; : : : ; ´n 1 and fvjn 1 gj1D1 are already defined for some n 2 N, n > 1. Consider the set of polynomials A P k 1 .X I Y /, ˚ }.´j1 ; : : : ; ´jk l ; lx/I 1 j1 jk A D x 7! P

l

n

1; 1 l < k :

Since Z D spanfvjn 1 gj1D1 is isometric to X, by the assumption no homogeneous polynomial on Z of degree at most k 1 separates 0 from SZ . Thus by the inductive hypothesis (on k 1) there is a normalised block basis fvjn gj1D1 of fvjn 1 gj1D1 such " that kQspanfvjn g k < 2nC2 for all Q 2 A. (Note that A is finite and so we can find nk successive block bases, starting with a normalised block basis of fvjn 1 gj1D1 , so that } is the above estimate finally holds for all polynomials in A.) Put ´n D vnn . Since P symmetric, (19) is obviously satisfied.

374

Chapter 6. Structural behaviour of smooth mappings

P Clearly, f´j g is a normalised block basis of fej g. If j1D1 aj ´j 2 BX and m 2 N, then by (19)

! N N

X X X

k }.´j1 ; : : : ; ´jk /k aj ´j aj P .´j / jaj1 ajk j kP

P

j Dm

j Dm

X

j1 ;:::;jk m minfjl g 0. Let P Rk 2 P . kXI Y / be such that R D nkD1 Rk . Clearly it suffices to find a homogeneous Q 2 P n .X I Y / that separates 0 from SX . Assume that no such homogeneous polynomial exists. Then by Lemma 72 there is a non-trivial subspace Z X such that t u kRk Z k < n" for each k D 1; : : : ; n, a contradiction. Recall that a harmonic function f on a bounded open U Rn satisfies the Maximum principle: if f is continuous on U , then f attains its maximum and minimum over U on @U . It follows that f is determined by its values on the boundary (see below). Also, if @U is connected, then f .U / D f .@U /. This motivates the following definition. Definition 74. Let X, Y be normed linear spaces and U X an open set. We say that a mapping f W U ! Y is harmonic-like on U if f .V / f .@V / for every bounded open V U . Note that if all mappings in some linear subspace S Y U are harmonic-like on U , then they are determined by their boundary values: if f; g 2 S are such that f D g on @V for some bounded open V U , then f D g on V . Indeed, f g 2 S and so .f g/.V / .f g/.@V / D f0g. The following Proposition shows a prototype result concerning harmonic-like mappings. Proposition 75 ([BF]). If X is a non-Asplund Banach space, Y is an Asplund space, and U X is open, then every f 2 C.U I Y / that is Fréchet differentiable on U is harmonic-like. Proof. Suppose that there is a bounded open set V U and x 2 V such that dist.f .x/; f .@V // D ı > 0. We may assume without loss of generality that x D 0. By Theorem 5.2 there is a separable subspace Z X that is not Asplund. Put W D V \ Z and E D span f .W /. Then E is a separable Asplund space and so there is ' 2 C 1 .E/ such that '.f .0// D 1 and '.y/ D 0 if ky f .0/k 2ı (Theorem 5.2). Define W Z ! R by .´/ D ' B f .´/ for ´ 2 W and .´/ D 0 otherwise. Since W is bounded, is clearly a bump. Further, is Fréchet differentiable on W and D 0 on a neighbourhood of Z n W , so is Fréchet differentiable. This contradicts Corollary 5.3. t u The following variant is proved by an analogous argument. Proposition 76 ([BF]). Let X, Y be Banach spaces such that Y admits a C k;˛-smooth bump but X does not and let U X be open. Then every f 2 C k;˛ .U I Y / continuously extended to U is harmonic-like. We are going to prove more results of this kind, concerning structural properties related to higher smoothness of the space. It is clear that if X admits a C k;˛ -smooth bump, then for every Banach space Y there exists f 2 C k;˛ .X I Y / that is not harmonic-like (as R Y ). We investigate for a given 1 p < 1 and 1 ˛ > p Œp the structural

376

Chapter 6. Structural behaviour of smooth mappings

conditions on Y which imply that every f 2 C Œp;˛ .U I Y /, U `p is harmonic-like. (Recall that `p has a C Œp;p Œp -smooth bump if p … N, Theorem 5.106.) In particular we show that every such mapping is harmonic-like unless Y contains ` p for some n integer 1 n Œp. It should be noted that the image f .U / may be large: If Y is separable, then there is even a homogeneous polynomial P W `p ! Y such that BY P .B`p / (Theorem 68). Proposition 77 ([HJ2]). Let Y be a Banach space, 1 p < 1, U `p open, and k 2 N, k Œp. Let S D C.U I Y / \ C k;˛ .U I Y / for some ˛ 2 .0; 1 satisfying k C ˛ > p or S D C.U I Y / \ C k;C .U I Y / for k D p. All mappings in S are harmonic-like if and only if there is no polynomial in P . n`p I Y /, n D 1; : : : ; k, that separates 0 from S`p . Proof. Clearly, a polynomial that separates 0 from S`p is not harmonic-like. Conversely, suppose that f 2 C.U I Y / \ C k;˛ .U; Y / is not harmonic-like. There is a bounded open V U and y 2 V such that dist.f .y/; f .@V // D " > 0. We may assume without loss of generality that y D 0. Suppose that no polynomial in P . n`p I Y /, n D 1; : : : ; k separates 0 from S`p . Let C > 0 be such that kd kf .u/ d kf .v/k C ku vk˛ for all u; v 2 V , let kC˛

1 CRkC˛ N 1 p < 2" . R > 0 be such that V B.0; R/, and find N 2 N such that kŠ By induction we find finitely supported vectors x0 ; : : : ; xm 2 `p , m N , such 1 j p R for j D 0; : : : ; m 1, xm 2 @V , and that x0 D 0, xj 2 V and kxj k N kf .xj / f .xj 1 /k < N" for j D 1; : : : ; m. Since V B.0; R/, the requirements on xj ensure that m N . Assume that the vectors x0 ; : : : ; xj are already defined for some 0 j < N . By Lemma 72 there is a normalised block basis ful g of the canonical basis fel g such n=p that kd nf .xj /spanful g k < "nŠN for 1 n k. Thus there is a finitely supported 2kNRn 1 Rn " v 2 S`p with max supp xj < min supp v such that nŠ for kd nf .xj /Œvk < 2kN N n=p R R all 1 n k. If xj C tv 2 V for all t 2 Œ0; N 1=p , then we set xj C1 D xj C N 1=p v. p

C1 p Note that in this case xj C1 2 V and kxj C1 kp D kxj kp C RN j N R . Otherwise we set xj C1 D xj C t0 v, where t0 D inf ft 0I xj C tv … V g, and further we set m D j C 1 and finish the construction. In the latter case clearly xm 2 @V . In both cases by Corollary 1.108

kf .xj C1 /

f .xj /k

k X 1

d nf .xj /Œxj C1 nŠ

nD1

k X nD1

1 xj C C kxj C1 kŠ

xj kkC˛

1 RkC˛ " " C C kC˛ < : 2kN kŠ N p N

P We have " D dist.f .0/; f .@V // kf .xm / f .0/k jmD1 kf .xj / f .xj which is a contradiction. The proof in the C k;C -smooth case is analogous.

1 /k < ",

t u

Section 8. Harmonic behaviour of smooth mappings

377

The next theorem shows that separating vector-valued polynomials on `p or c0 are analogues of linear isomorphisms. Theorem 78 ([HJ2]). Let Y be a Banach space, 1 p < 1, and k 2 N. Suppose that no polynomial in P . n`p I Y /, 1 n < k, separates 0 from S`p . If k is odd and k p, or if k is even and k < p, then there is P 2 P . k`p I Y / that separates 0 from S`p if and only if Y contains ` p . k There is a homogeneous P 2 P .c0 I Y / that separates 0 from Sc0 if and only if Y contains c0 if and only if for every k 2 N there is P 2 P . kc0 I Y / that separates 0 from Sc0 . Proof. By fej g we denote all the canonical bases in the respective spaces. First we prove the `p case. P P ( Clearly, Q W `p ! ` p defined by Q j1D1 aj ej D j1D1 ajk ej is a k-homok geneous polynomial that separates 0 from S`p . Hence if T is an isomorphism from ` p k into Y , then T B Q 2 P . k`p I Y / separates 0 from S`p . ) Let " D infx2S`p kP .x/k > 0 and let f´j g be the normalised block basis of fej g from Lemma 72. Put yj D P .´j /. If k is odd, then for any sequence faj g R P p satisfying j1D1 jaj j k D 1 we have

1

1 ! 1

"

X

X X p p k

k a k a ´ aj yj D

j P .´j / < P j j C

2

j D1 j D1 j D1

1

k

X

" " p

k kP k aj ´j C D kP k C :

2 2 j D1

On the other hand,

1

X

aj yj > P

j D1

1 X p k j D1

!

aj ´j

1

k

X

" p

k a ´ " j j

2

" " D : 2 2

j D1

This implies that spanfyj g Y is a subspace isomorphic to ` p . If k is even, then k p k aj D k aj only if aj 0 and therefore we obtain merely that `C p embeds into Y . k

(In view of the remark after Theorem 71 we do not need fyj g to be a basic sequence.) Now Theorem 71 finishes the proof for k even. In the case of c0 we start by considering the homogeneous polynomial of the smallest degree that separates 0 from Sc0 and analogously as above we conclude P1that ck0 Y . P1 Then we use the fact that Q W c0 ! c0 defined by Q j D1 aj ej D j D1 aj ej is a k-homogeneous polynomial that separates 0 from Sc0 . t u If p 2 N is even, then x 7! kxkp is a separating polynomial on `p and so (using also Corollary 3.59) in this case the statement of Theorem 78 dos not hold for k D p. Putting together Proposition 77 and Theorem 78 we immediately obtain the following result.

378

Chapter 6. Structural behaviour of smooth mappings

Theorem 79 ([HJ2]). Let Y be a Banach space, 1 p < 1, p not an even integer, and U `p open. Let S D C.U I Y / \ C Œp;˛ .U I Y / for some ˛ 2 .0; 1 satisfying Œp C ˛ > p or S D C.U I Y / \ C p;C .U I Y / if p 2 N. Then either all mappings in S are harmonic-like or Y contains ` p for some 1 k Œp. k

9. Notes and remarks Section 1. The results in this section are mostly improvements on the work concerning weakly uniformly continuous and simultaneously uniformly smooth mappings by [AP] and [CHL]. In particular, Theorems 8 and 9 are improvements of the results which appeared many times in the literature, e.g. in [AP], [CHL], [CiGu1], [CiGu2], k , Guillermo Restrepo [Res] seems to be the earliest [CGS]. As for the definition of Cwu reference, in the case of reflexive X and k D 1. Section 2. The bidual extension operation E introduced in this section had many precursors, all in the setting of polynomials or convergent power series. Perhaps one of the earliest versions appears for bilinear mappings in [Are]. Extensions of polynomial mappings between Banach spaces to their biduals have been considered by Aleksander Pełczy´nski in [Peł4], [Peł3], in the special case of C.K/ or spaces with the DPP. For a general Banach space they were described by Richard Martin Aron and Paul D. Berner [ArBer] in their seminal paper. The polynomial extension leads naturally to extensions of holomorphic functions via the Taylor formula. The topic received considerable attention in the literature, e.g. [DT2], [LindsRy], [GGMM], [Za1], [Vi], [Aro3], [Din, p. 79 and Chapter 6]. Our presentation is based on [CHL], where the extensions were obtained for the first time in the setting of uniformly k-times differentiable mappings, which are not necessarily analytic. Problem 80. Find the most general conditions under which E and D are weakly commuting in BX , > 0. A natural and important question is whether the extensions exist without the uniformity assumption, namely: Problem 81. Let f 2 C k .XI Y /, k 2 N [ f1g. Does there exist a bidual extension f 2 C k .X I Y /? The special case with X D c0 , Y D R (a problem posed by R. M. Aron) seems to be also open. It was shown in [AFK1] that C 1 -smooth functions can be extended (preserving C 1 -smoothness) from subspaces of separable Asplund spaces. On the other hand, Václav Zizler [Ziz] proved that extensions preserving higher smoothness are rare. More precisely, let X be a separable Asplund space containing a subspace Y isomorphic to `2 . Then the Hilbertian norm kk of Y can be extended to a real valued function f on X that has Fréchet derivative on BY n f0g and such that Df is locally Lipschitz on SY if and only if Y is complemented.

Section 9. Notes and remarks

379

Problem 82. Can C 1 -smooth norms be extended from subspaces of a separable reflexive (or Asplund) space X to the whole space preserving the C 1 -smoothness? This result is false for Gâteaux smooth norms due to [Ziz]. Theorem 14 generalises the classical results from [Li1] concerning extensions of linear operators into the case of uniformly smooth mappings. For the general problem of extending mappings from subspaces to their superspaces preserving various properties see [BenLi]. Sections 3–6. The theory of the class W was developed in [Há5], [Há6], [DeH2], and [CHL]. In view of Proposition 33 we pose the next problem. Problem 83. Assuming that X belongs to the class W , is there an equivalent renorming of X such that X equipped with the new norm belongs to the class W1 ? Belonging to the class W is not a three space property, as witnessed by the space X D `1 ˚1 S, [CHL]. Problem 84. Let c0 be equipped with the canonical supremum norm. Given a function f 2 C 1;C .Bc0 / and > 1, is there an extension F 2 C 1;C .Bc0 /, F D f on Bc0 ? Problem 85. If X is a W1 -space, is then for some Banach space Y (or at least Y D R) C 1;C .BX I Y / \ Cwsc .BX I Y / D C 1;C .BX I Y / \ CwsC .BX I Y /‹ Perhaps the most interesting open problem of this theory is the following. Problem 86. Is property (WR) for a Banach space Y is equivalent to Y not containing c0 ? Recall that if X is a W -space, then P .X/ D PwsC .X/. The following is a stronger version of Problem 4.103, since the assumptions imply that X does not admit a noncompact operator into `p , 1 < p < 1. Problem 87. Let X be a Banach space failing property B. Is then P . nX/ D PwsC . nX/? Theorem 56 was inspired by the results of Mordecay Zippin [Zip2]. However, our result does not follow from [Zip2] as there are examples of non-Schur Banach spaces that are `1 -saturated, see [AzHa]. In particular, there is a Banach space with a weakly null Schauder basis which contains `1 , and so its dual is non-separable. Recall that X has the DPP and does not contain `1 if and only if X is a Schur space (Theorem 3.41). Problem 88. If X is a Schur space, is then X a W -space? More generally, if X has the DPP, is then X a W -space? In fact, we do not know if L1 .Œ0; 1/ is a W -space. The fixing of the canonical basis of c0 by non-compact polynomials from C.K/, K scattered (Theorem 60) is a direct generalisation of the linear Theorem 3.47. Theorem 62 shows that a similar conclusion fails for uniformly smooth mappings from the unit ball. A more complicated variant of the construction in Theorem 62 leads to a

380

Chapter 6. Structural behaviour of smooth mappings

somewhat stronger statement. Namely, the positive cone generated in `1 by f .Bc0 / does not necessarily contain the canonical basis of c0 , [DeH2]. Let us mention for completeness that in the theory of linear operators on C.K/ spaces many results on the fixing of subspaces (preferably as large as possible) which generalise and strengthen Theorem 3.47 (the c0 -fixing property) have been obtained, see e.g. [BourgJ1], [AlBen], [Wol], [Ros1]. These results are connected with the still open problem of the isomorphic classification of complemented subspaces of C Œ0; 1. For more we refer e.g. to [Ros3]. In should be noted that the highly relevant theory of isomorphic `1 -preduals is rapidly developing. It was shown in [FOS] that every Banach space with a separable dual can be isomorphically embedded into a L1 -space whose dual is isomorphic to `1 . Section 7. The first example of the failure of Rolle’s theorem (even for a polynomial of degree 4 on a Hilbert space) in an arbitrary infinite-dimensional space with a smooth norm is due to Stanislav A. Shkarin [Shk]. This seminal work has lead to a large number of papers studying the ranges of derivatives, e.g. [AzDe], [ADJ], [AFJ], [AJ1], [GHM], [BFKL], [BFL], the survey [Dev3], [DeH1], [GaspaT1], [GaspaT2], [HJ1], [FKK]. In [BenLi, p. 259] there is an example of a pair X, Y of non-isomorphic Banach spaces such that there exist real analytic (even polynomial of degree 3) surjections in both directions which are topological homeomorphisms. Let us also mention a couple of results concerning C k -diffeomorphisms. Parts of this theory have their roots in the work of Czesław Bessaga in [Bes]. It was shown in [Do3] and [Az] that if a Banach space X has a C k -smooth norm, k 2 N [ f1g, then any hyperplane in X is C k -diffeomorphic with the unit sphere SX . If X admits a C k -smooth norm (which need not be equivalent to the original norm), then X is C k -diffeomorphic with X n f0g. These kinds of results play a role in the theory of C k -smooth Banach manifolds. We refer to [Mo1], [BuKu], and [EE] for a sample of classical results in this vast field of research. Section 8. The results in this section come from [HJ2]. In the same paper it was shown that if `pC , 1 < p < 1 embeds into Y onto a basic sequence fyn g with 1 the embedding constant C < 21 p , then fyn g is equivalent to the canonical basis of `p , and this inequality is sharp in the sense that for any 1 < p < 1 there is a space X isomorphic to c0 ˚ `p with a Schauder basis fyn g such that `pC embeds into X 1 onto fyn g with the embedding constant 21 p . The paper [HJ2] also deals with the case of complex `p . In [DM] the following result is proved: Theorem 89. Let X, Y be Banach spaces such that Y is of a non-trivial cotype, while X is not of a non-trivial cotype, and let U X be open. Then all mappings in C.U I Y / \ C 1;C .U I Y / are harmonic-like. The C k -smooth non-linear mappings are of course related with C k -smooth vector fields in Banach spaces, in other words with ODEs corresponding to C k -smooth righthand side. This line of research was pursued in [HV2], and developed in a series of

Section 9. Notes and remarks

381

papers [HJ6], [HV1], and [HV3], where some of the problems regarding the local and global behaviour of abstract ODEs in Banach spaces, e.g. the local existence and global behaviour such as !-limit sets and cross-sections of the solution funnels, have been solved. The general theory of ODEs in infinite-dimensional spaces is treated e.g. in [Dei] or [DMNZ].

Chapter 7

Smooth approximation In the present chapter we are concerned with the general problem of approximating a given mapping from a subset of a Banach space X into a Banach space Y by means of polynomials or C k -smooth mappings. The classical example is the Weierstraß-type Theorem 9, where arbitrary C k -smooth mapping from a compact set K Rn is shown to be approximable by polynomials, uniformly on K together with derivatives of order up to k. The best-known case is when k D 0. Our interest lies primarily in the case when the unit ball BX is non-compact and the Weierstraß theorem for uniformly continuous functions fails. It turns out, quite surprisingly, that in the special case of W -spaces not containing `1 , C k -smooth mappings with uniformly continuous derivatives can be approximated together with their higher derivatives by polynomials uniformly on bounded sets. This result underlines the extremely poor supply of uniformly smooth functions on these spaces rather than the abundance of polynomials. Indeed, all polynomials are in this case weakly uniformly continuous on BX . In Section 3 we prove one of the highlights in the theory of smoothness, Theorem 29 of Jaroslav Kurzweil. This result claims that if a real separable Banach space X admits a separating polynomial, then every continuous mapping from X into a Banach space can be uniformly approximated by real analytic mappings. By adjusting the proof somewhat it can be shown that the result remains true for Banach spaces that admit a separating analytic function with uniform radii of convergence (e.g. for c0 ), provided that the approximated mapping is uniformly continuous. One of the principal tools for obtaining C k -smooth approximations of continuous mappings on infinite-dimensional spaces are partitions of unity, which are studied in Section 5. This is a very powerful tool which leads to general positive results in separable spaces, as well as in non-separable WCG or C.K/ spaces, admitting a C k -smooth bump. The rest of the chapter is devoted to the study of smooth approximations preserving special properties of the approximated mapping. In order to study C k -smooth approximations of Lipschitz mappings preserving the Lipschitz condition we introduce the concept of sup-partitions of unity and characterise it by means of componentwise C k -smooth and bi-Lipschitz embeddings into c0 . /. We show that every separable Banach space admitting a Lipschitz and C k -smooth bump admits C k -smooth suppartitions of unity. This is applied to establish the existence of C k -smooth and Lipschitz approximations of a given Lipschitz function in a separable Banach space X admitting

Section 1. Separation

383

a C k -smooth and Lipschitz bump function. The real analytic case is also included, under the assumption of the existence of a separating polynomial. We also obtain results of this sort for vector valued Lipschitz mappings for certain types of the domain or range spaces. In Section 8 we prove, again for certain types of the domain or range spaces, the existence of approximations of C 1 -smooth mappings by C k -smooth mappings for the mapping and its first derivative. The last section is devoted to C k -smooth (and convex) approximations of convex functions. It is rather easy to see that this problem is essentially equivalent to the statement that every equivalent norm on X can be approximated by C k -smooth norms. A necessary condition for this result is clearly the existence of at least some equivalent C k -smooth norm on the space X (similarly to the existence of a C k -smooth bump in the general case). We prove that this condition is also sufficient for all separable Banach spaces. The methods in this chapter can be essentially divided into two groups: Global methods, in which the approximating mapping is constructed on the whole space at once by a formula – these are represented by integral and infimal convolutions; and local methods, in which we approximate locally and then glue together the approximations using for example partitions of unity. Some of the methods are mixed, for example the real analytic approximations although using the partitions of unity are necessarily retaining the global flavour. A note on the hypotheses on the target space: if the method employed uses a limiting procedure (this includes the Bochner integral, or countable partitions of unity), we need the completeness of the space for the process to converge. This is not necessary if we use a finite procedure such as the locally finite partitions of unity. Likewise, the domain space needs to be complete only in some circumstances, for example when using Schauder bases, or when we are dealing with real analytic mappings. In this chapter all the spaces are real unless stated otherwise.

1. Separation As we shall see, a lot of the approximation methods ultimately boils down to the ability of separation of certain sets by smooth functions. In this section we present some of the separation results that will be used later in the chapter. Some of them are somewhat technical, as we require rather fine separation properties. Nevertheless, we begin with a lemma that serves as a prominent tool for smoothing up mappings on Rn and lies behind most of the approximation results. The method was used already by Karl Weierstraß. Let X be a set, Y a normed linear space, f W X ! Y , and S X . We denote kf kS D supx2S kf .x/k. In the following lemma we consider the space C n with the Euclidean norm.

384

Chapter 7. Smooth approximation

Lemma 1. Let Y be a Banach space, C Y a closed convex set, and let f W Rn ! C be strongly measurable (with respect to the Lebesgue measure) and bounded. Put P .´/ D exp jnD1 ´j2 for ´ 2 C n and 2 RC , and define g W C n ! YQ by the Bochner integral Z 1 g .´/ D .´ y/f .y/ dy; c Rn n R where c D Rn .y/ dy D 2 . Then g 2 H.C n I YQ / and g Rn 2 C ! .Rn I C / for every 2 RC . Further, if f is Bochner integrable, then for every ı > 0 lim kg kGı D 0

!C1

(in case that Gı ¤ ;), where ˚ Gı D ´ 2 C n I kIm ´k2 < dist.Re ´; supp f /2

ı2 :

If f is L-Lipschitz, then so is each g Rn , 2 RC . Finally, if f 2 C k .Rn I Y / for some k 2 N0 [ f1g with all d jf , j D 0; : : : ; k bounded and uniformly continuous on Rn , then for all 0 j k

lim d jg d jf Rn D 0: !C1

Proof. First note that X X n n 2 j .´/j D exp Re.´j / D exp .Re ´j /2 j D1

D exp

kRe ´k2

2

.Im ´j /

j D1

kIm ´k2

P and D .´/Œh D 2 .´/ jnD1 ´j hj . Put F .´; y/ D .´ y/f .y/ and let K > 0 be such that kf kRn K. Then for ´ 2 C n , k´k < r, and y 2 Rn we have kD1 F .´; y/k D 2j .´

y/jk´

ykkf .y/k D 2K exp kIm ´k exp kRe ´ 2

2K exp.r 2 / exp

.maxfkyk

yk2 k´ yk r; 0g/2 .r C kyk/:

Thus we can apply Theorem 1.90 on bounded subsets of C n, which gives g 2 H.C n I YQ / and hence also g Rn 2 C ! .Rn I Y /. If g .x/ … C for some x 2 Rn , then by the separation theorem there are 2 Y and ˛ 2 R such that f .y/ < ˛ < g .x/ for all y 2 Rn . But Z Z 1 1 g .x/ D .x y/ f .y/ dy < ˛ .x y/ dy D ˛; c Rn c Rn which is a contradiction. Hence g Rn 2 C ! .Rn I C /.

Section 1. Separation

385

Further, if f is Bochner integrable, then for a fixed ı > 0 and any ´ 2 Gı we can estimate Z 1 kg .´/k j .´ y/jkf .y/k dy c Rn Z 1 D exp kRe ´ yk2 kIm ´k2 kf .y/k dy c supp f Z 1 2 exp ı kf .y/k dy: c supp f 1 !C1 c

Since lim

exp. ı 2 / D 0, it follows that lim kg kGı D 0. !C1

For any x 2 Rn we can use the substitution y ! x g .x/ D

1 c

y to obtain

Z Rn

.y/f .x

y/ dy:

(1)

Thus if f is L-Lipschitz, then for any u; v 2 Rn we have kg .u/

Z Z 1

.y/f .u y/ dy .y/f .v g .v/k D c Rn Rn Z 1 .y/kf .u y/ f .v y/k dy c Rn Z 1 .y/ dy D Lku vk: Lku vk c Rn

y/ dy

Now suppose that f 2 C k .Rn I Y / for some k 2 N0 [ f1g with all the differentials d jf , j D 0; : : : ; k bounded and uniformly continuous on Rn . Using the formula (1), the boundedness of the differentials, Theorem 1.90, and induction we obtain Z 1 d jg .x/ D .y/d jf .x y/ dy c Rn Z 1 D .x y/d jf .y/ dy c Rn for every x 2 Rn and 1 j k. Fix j 2 f0; : : : ; kg and choose an arbitrary " > 0. Consider the space Rn with the Euclidean norm. By the uniform continuity there exists ı > 0 such that kd jf .x/ d jf .y/k < 2" whenever x; y 2 Rn , kx yk < ı. Moreover there is a constant M > 0 such that kd jf .y/k M for all y 2 Rn . For

386

Chapter 7. Smooth approximation 1

2

n

large enough so that 2Me 2 ı 2 2 < 2" we then have

j

d g .x/ d jf .x/

Z

Z

1

1 j j

D .y/ dy d f .x/ .x y/d f .x y/ dy

c

c Rn Rn

Z

1

j j

D .x y/ d f .y/ d f .x/ dy

c Rn Z

1 .x y/ d jf .y/ d jf .x/ dy c Rn Z Z 1 1 " .x y/ dy C .x y/2M dy c kx yk 0. It is standard to check that 0 2 C 1 .RI Œ0; 1//. Put 1 .t/ D 1 e e 0 1e t and finally D 1 B 0 . t u Lemma 3. Let K Rn be a compact set and U Rn an open neighbourhood of K. Then there is a function ' 2 C 1 .Rn I Œ0; 1/ such that supp ' U and ' D 1 on some neighbourhood of K. 2 C 1 .Rn / Proof. Let be the function from Fact 2 and define the function 2 2 by .x/ D .1 x1 xn /. Notice that supp B.0; 1/ and .0/ D 1. x w Let d D 12 dist.K; Rn n U /. For each w 2 K we define w .x/ D and d 1 n Vw D fx 2 R I w .x/ > 2 g. Since K is a compact set there are w1 ; : : : ; wk 2 K such that K Vw1 [ [ Vwk . Put ! k X '.x/ D 2 t u wj .x/ : j D1

Section 1. Separation

387

Lemma 4. There are functions n 2 H.C/, n 2 N, with the following properties: (T1) (T2) (T3) (T4) (T5)

n R maps into Œ0; 1, n R is 4-Lipschitz, jn .´/j 2 n for every ´ 2 C, j´j 14 , jn .x/ 1j 2 n for every x 2 R, x 1, j.n R /0 .x/j 2 n for every x 2 R, x 21 or x 1.

Proof. Let f W R ! Œ0; 1 be defined as f .t/ D 0 for t 58 , f .t/ D 4t 25 for t 2 58 ; 87 , and f .t/ D 1 for t 87 . Obviously f is a 4-Lipschitz function. We put n D gn from Lemma 1, where n 2 RC is chosen so that (T4) holds and p (2) 2e n =128 2 n ; 2n cn

Z jt j 18

jtje

n t 2

dt D

n =64

2e

cn

2

n

:

(3)

The function n clearly has the properties (T1) and (T2). To prove (T3) we use successively the definition of f , the fact that jIm ´j 41 , Re ´ 41 , and finally (2) to obtain Z 2 Z e n .Im ´/ 1 2 2 f .t/e n Re.´ t/ dt D f .t/e n .Re ´ t/ dt jn .´/j cn R cn R 1

e 16 n cn

Z

C1 5 8

e

n .Re ´ t/2

1 Z e 16 n n . 3 /2 C1 2 8 e e 5 cn 8 p 2e n =128 2 n :

dt

n 2 .Re ´

Finally, we show (T5). Suppose that x integral sign we obtain Z 2n f .t/.t n0 .x/ D cn R Since t 7! t e

t 2

t/2

1 2

dt

or x 1. Differentiating under the

x/e

is odd and f is constant on Œx ˇZ 2n ˇˇ 0 f .t/.t jn .x/j D ˇ cn ˇ jx t j 18 Z 2n jt xje cn jx t j 18

n .t x/2

dt:

C 81 , we get (using also (3)) ˇ ˇ ˇ n .t x/2 x/e dt ˇ ˇ 1 8; x

n .t x/2

dt 2

n

:

t u

388

Chapter 7. Smooth approximation

1 Lemma 5. Let f"n g1 nD1 and fan gnD1 be two sequences of positive real numbers. There n are functions n 2 H.C /, n 2 N, and a sequence fın g1 nD1 of positive real numbers with the following properties: (Z1) n Rn maps into Œ0; 1, (Z2) n Rn is 2-Lipschitz with respect to the maximum norm, (Z3) jn .´/j "n forPevery ´ 2 C n such that there is k 2 f1; : : : ; n 1g for which Re ´k 41 and jnD1 aj .Im ´j /2 ık . (Z4) n .x/ 12 for every x 2 Rn for which xn 1 and xj 1, j D 1; : : : ; n 1, (Z5) n .x/ "n for x 2 Rn satisfying xn 2.

Proof. Let fn W Rn ! Œ0; 1 be a 2-Lipschitz function (with respect to the maximum norm) such that ( 0 whenever xn 2 or 9j 2 f1; : : : ; n 1g W xj 12 , fn .x/ D 1 whenever xn 1 and 8j 2 f1; : : : ; n 1g W xj 1. (See e.g. Lemma 39.) For each n 2 N put ın D an =64 and ! Z n X 1 n .´/ D aj .´j tj /2 dt for ´ 2 C n , fn .t / exp n cn Rn j D1 q Q Pn R 2 n n 1 C where cn D Rn e n j D1 aj tj dt D j D1 aj and n 2 R is chosen so n that (Z4) and (Z5) hold (analogously as in Lemma 1) and n aj =64

e

2

n 2

"n for j D 1; : : : ; n

1.

belongs to H.C n / and has the properties (Z1) and (Z2) (again similarly

The function n as in Lemma 1). To prove (Z3) we use successively use the definition of fn , the fact that Re ´k 14 , and the definition of ık to obtain 1 jn .´/j cn

n P

n

Z fn .t/e

aj Re.´j tj /2

j D1

dt

Rn n P n aj .Im ´j /2

D

e

e n ık cn e n ı k D cn

cn Z t 2Rn tk > 21

e

t 2Rn tk > 12

e

n 1 2 ak 16

fn .t/e

Rn n P

n

Z

e n ık e cn

n

Z

j D1

dt

aj .Re ´j tj /2

n P

dt aj .Re ´j tj /2

e

j D1

Z e Rn

aj .Re ´j tj /2

j D1

j D1

n 2

n P

n 2

n P

n 2

n P

aj .Re ´j tj /2

j D1

aj .Re ´j tj /2

j D1

dt

n

dt D 2 2 e

n ak =64

"n : t u

Section 1. Separation

389

Lemma 6. Let K be a compact space such that C.K/ admits a C k -smooth bump function, k 2 N [ f1g. Then for every ; 2 R, 0 < < , there is a function ˇ; 2 C k .C.K/I Œ0; 1/ such that ( 1 when kf k1 , ˇ; .f / D 0 when kf k1 . Proof. By hypothesis there exists a function ' 2 C k .C.K/I Œ0; 1/ and ˛ 2 R, ˛ > 0, such that '.f / D 1 for kf k1 ˛, while '.f / D 0 for kf k1 1. Choose n 2 N so that . /n ˛ and put n f : ˇ; .f / D ' n Since the mapping f 7! f n is a continuous n-homogeneous polynomial (and in particular it is C 1 -smooth), the function ˇ; has the required properties. t u We remark that the Taylor complexification cQ0 of the real space c0 is isometric to the complex space c0 . Proposition 7. Let q 1. There are an open set W cQ0 and a function 2 H.W / with the following properties: (M1) For every w 2 c0 n f0g there is w > 0 such that UcQ0 .y; w / W for every y 2 c0 satisfying jwj jyj qjwj, where the inequalities are understood in the lattice sense. (M2) .w/ 8 for w 2 c0 , kwk 8, (M3) j.´/j < 2 for ´ 2 UcQ0 .y; w /, where y 2 c0 , kyk 1, and w 2 c0 n f0g, jwj jyj p qjwj, (M4) c0 is 2-Lipschitz and maps into R. ˚ P .xn /2n 1 . Proof. Define on c0 as the Minkowski functional of x 2 c0 I 1 nD1 p Then is an equivalent norm on c0 for which kxk .x/ 2kxk (see also Theorem 5.104 and Example 1.137). This gives property (M2) P and (M4).2n Let f W cQ0 .C n f0g/ ! C be defined as f .´; u/ D 1 1. This nD1 .´n =u/ function is holomorphic on cQ0 .C n f0g/ and for every x 2 c0 n f0g we have f .x; .x// D 0. 2n P1 1 kwk kwk 2q Fix w 2 c0 n f0g. Put R D , S D , M D 1 C jw j , C n nD1 2 4 2 kwk n p o 1 1 aR2 p aD , r D min 2 aRCM ; 2 2 , and w D s as defined in Theorem 1.176. 2qkwk Now choose any y 2 c0 , jwj jyj qjwj. Then R < kwk kyk .y/, thus B..y/; R/ V D C n f0g. For any ´ 2 B.y; S/, u 2 B..y/; R/ we have juj .y/ R kyk R kwk R D kwk yn j n j C j´n 2 and j´n j jy P1 ´n 2n kwk qjwn j C k´ yk qjwn j C hence jf .´; u/jˇ 1 C nD1 j u j M . ˇ 4 , and ˇ 1 P1 yn 2n ˇ yn 2n 1 P1 D Finally, jD2 f .y; .y//j D ˇ .y/ nD1 2n .y/ ˇ .y/ nD1 .y/ 1 .y/

p1 2kyk

a. Thus by Theorem 1.176 the equation f .´; u/ D 0 uniquely

390

Chapter 7. Smooth approximation

determines a holomorphic function yw on UcQ0 .y; w / with values in U..y/; r/ and this holds for every y 2 c0 , jwj jyj qjwj. Take any two functions 1 D yw11 , 2 D yw22 defined on open balls U1 and U2 respectively. If U1 and U2 intersect, then it is easy to check that U1 \ U2 \ c0 is a non-empty set relatively open in c0 . Since 1 D on U1 \ c0 and 2 D on U2 \ c0 , it follows that both holomorphic functions 1 and 2 agree on U1 \ U2 \ c0 and therefore allows us to put S ˚on the whole U1 \ U2 (Corollary 1.158). This observation W D UcQ0 .y; w /I w 2 c0 n f0g; y 2 c0 ; jwj jyj qjwj and define on W by .´/ D yw .´/ whenever ´ 2 U.y; w /. This gives property (M1). To prove (M3) let w 2 c0 n f0g, y 2 c0 , jwj jyj qjwj, kyk 1,p and ´ 2 UcQ0 .y; w /. Then by the choicep of r above we have .´/ 2 U..y/; 2 2/ p p 2 2kyk C 2 2 2. t u and therefore j.´/j < j.y/j C 2 Lemma 8 ([RudinM]). Let .P; / be a metric space and U D fU˛ g˛2 an open covering of P . Then there are open refinements fVn˛ gn2N;˛2 , fWn˛ gn2N;˛2 of U that satisfy the following: Vn˛ Wn˛ U˛ for all n 2 N, ˛ 2 , dist.Vn˛ ; P n Wn˛ / 2 n for all n 2 N, ˛ 2 , dist.Wn˛ ; Wnˇ / 2 n for any n 2 N and ˛; ˇ 2 , ˛ ¤ ˇ. for each x 2 P there are an open neighbourhood Ux of x and a number nx 2 N such that (i) if k > nx , then Ux \ Wk˛ D ; for any ˛ 2 , (ii) if k nx , then Ux \ Wk˛ ¤ ; for at most one ˛ 2 . Proof. Choose some well-ordering of the set . Define the sets Vn˛ by induction on n 2 N: If Vjˇ are already defined for j < n and all ˇ 2 , let Vn˛ be the union of all U.x; 2 n / such that (a) ˛ is the smallest with x 2 U˛ , (b) x … Vjˇ for all j < n, ˇ 2 , (c) U.x; 5 2 n / U˛ . S Further, let Wn˛ D fU.x; 2 n /I x 2 Vn˛ g for all n 2 N, ˛ 2 . Certainly Vn˛ Wn˛ U˛ and dist.Vn˛ ; P n Wn˛ / 2 n for all n 2 N, ˛ 2 . To see that fVn˛ g covers P , observe that, for x 2 P , there is a smallest ˛ 2 such that x 2 U˛ , and n so large that (c) holds. Then, by (b), x 2 Vjˇ for some j n, ˇ 2 . To prove the third property, suppose that n 2 N, ˛; ˇ 2 , ˛ < ˇ, p 2 Wn˛ , and q 2 Wnˇ . There is a ball U.y; 2 n / in the definition of the set Vn˛ such that .p; y/ < 2 2 n , and a ball U.´; 2 n / in the definition of the set Vnˇ such that .q; ´/ < 22 n . By (c), U.y; 52 n / U˛ but, by (a), ´ … U˛ . So .y; ´/ 52 n and .p; q/ .y; ´/ .p; y/ .q; ´/ > 2 n . Finally assume x 2 P . Find some pair n 2 N, ˇ 2 such that x 2 Vnˇ , and choose j 2 N so that U.x; 2 j C1 / Vnˇ . Put nx D n C j 1 and Ux D U.x; 2 n j /. To show (i), suppose that k > nx and choose any ˛ 2 and ´ 2 Wk˛ . It follows that there is a ball U.y; 2 k / in the definition of Vk˛ such that .y; ´/ < 22 k . Since

Section 2. Approximation by polynomials

391

k > n, by (b), y … Vnˇ . And since U.x; 2 .x; ´/ .x; y/ >2

j C1

j C1 /

.y; ´/ 2 2

kC1

2

Vnˇ and k j C 1,

j C1 j C1

.y; ´/ 2

j

D2

j

>2

n j

:

From the definition of Ux and nx it is easy to see that (ii) follows from the third property. t u

2. Approximation by polynomials In this section we begin by proving the classical Weierstraß-type theorem on the density of polynomials among C k -smooth functions in the uniform topology (together with its derivatives) on compact subsets of Rn . Of course, the proof relies heavily on the compactness argument. The result can be extended into infinite-dimensional setting if we are interested in uniform topology on compact sets. We then deal with the approximation in uniform topology on bounded sets, which is not always possible. Applying the theory of W -spaces we give a generalisation of the Weierstraß theorem in some special cases. We finish the section by showing that the assumptions used in order to get positive results are close to being optimal. Theorem 9. Let ˝ Rn be an open set, Y a Banach space, and f 2 C k .˝I Y /, k 2 N0 . For every compact subset K ˝ and every " > 0 there is a polynomial p 2 P .Rn I Y / such that kd jf d jpkK " for 0 j k. Proof. By Lemma 3 there is a function ' 2 C 1 .Rn / such that supp ' ˝, supp ' is compact, and ' D 1 on a neighbourhood of K. Since 'f 2 C k .˝I Y / (Corollary 1.84), 'f has a compact support, and d j .'f /.x/ D d jf .x/ for x 2 K, 0 j k, replacing f by a function defined as 'f on ˝ and 0 on Rn n ˝ we may suppose that f 2 C k .Rn I Y / and S D supp f is compact.

By Lemma 1 there is 2 RC such that d jg d jf Rn < 2" for 0 j k. Put Qm .y/ D

m X l 1 . /l y12 C C yn2 : lŠ

lD0

P .Rn /

Then Qm 2 and limm!1 d jQm D d j locally uniformly on Rn for each 0 j k (Theorem 1.146). Put M D kf kRn . The set L D K S is compact. c 1 " Therefore there is N 2 N such that d jQN d j L M for each 0 j k. .L/ 2 Let Z 1 QN .x y/f .y/ dy: p.x/ D c Rn Then p 2 P .Rn I Y / (e.g. by Theorem 2.49). As Z Z 1 1 j j d .x y/f .y/ dy D d j .y/f .x y/ dy d g .x/ D c Rn c Rn

392

Chapter 7. Smooth approximation

and similarly for d jp, we have for any x 2 K

Z

1

j j

kd jg .x/ d jp.x/k D f .x y/ dy d .y/ d Q .y/ N

c

Rn

Z

1

j j

d .y/ d Q .y/ D f .x y/ dy N

c

L Z

M

d j .y/ d jQN .y/ dy M d jQN d j .L/ " : u t L c L c 2 Definition 10. Let X and Y be normed linear spaces. By Pf . nXI Y / we denote the linear subspace of P . nX I Y / consisting of all polynomials that can be written P in the form P .x/ D jkD1 fj .x/n yj , where fj 2 X and yj 2 Y . We also set S1 Pf .XI Y / D span nD0 Pf . nX I Y /. If X is finite-dimensional, then Pf . nX I Y / D P . nX I Y /. This follows from Proposition 1.23 and the fact that spanfhy; id I y 2 Rm g D P . d Rm / (Section 2.4). Fact 11. Let X , Y be normed linear spaces, P 2 Pf . mX/, and Q 2 Pf . nXI Y /. Then PQ 2 Pf . mCnXI Y /. In particular, Pf .X/ is a subalgebra of the algebra P .X/ and p B R 2 Pf .X / whenever R 2 Pf .X/ and p 2 P .R/. Proof. It is clear that it suffices to show that x 7! f .x/m g.x/n 2 Pf . mCnX/ whenever f; g 2 X . The polynomial q 2 P . mCn R2 / given by q.u; v/ D um v n P can be written as q.u; v/ D jkD1 cj h.aj ; bj /; .u; v/imCn (Section 2.4). Therefore P f .x/m g.x/n D jkD1 cj hj .x/mCn , where hj D aj f C bj g 2 X . t u The following is an extension of the Weierstraß theorem into infinite-dimensional spaces. Of course the usefulness of this theorem is limited by the fact that here the compact sets are very small. Theorem 12. Let X , Y be normed linear spaces, K X compact, f 2 C.KI Y /, and " > 0. Then there is a polynomial P 2 Pf .X I Y / such that kf P kK ". For the proof we need the following lemma on separation of sets by polynomials. Lemma 13. Let X be a normed linear space, ˝ X a bounded set, C ˝ a closed convex set, K ˝ a weakly compact set satisfying C \ K D ;, and ı > 0. Then there is a polynomial P 2 Pf .X/ such that 0 P .x/ 1 for x 2 ˝, P .x/ > 1 ı for x 2 C , and P .x/ < ı for x 2 K. Proof. Without loss of generality we may assume that the set C is non-empty. By the separation theorem for every x 2 K there are fx 2˚ X and bx ; cx 2 R such that fx .y/ < bx < cx < fx .x/ for y 2 C . Since fx 1 .cx ; C1/ x2K is a weakly S open covering of the weak compact K, there are x1 ; : : : ; xn 2 K such that K nkD1 fxk1 .cxk ; C1/ . Denote Mk D maxy2K fxk .y/ and ak D infy2C fxk .y/ for k D 1; : : : ; n, and notice that ak 2 R by the boundedness of C . By Theorem 9 there

Section 2. Approximation by polynomials

393

are polynomials qk 2 P .R/ such that jqk .t/j < for t 2 Œcxk ; Mk , k D 1; : : : ; n. Put Q.x/ D

n X

1 2n

for t 2 Œak ; bxk and qk .t/ > 1

qk2 fxk .x/ :

kD1 1 Then 0 Q.x/ < 4n 14 for Denote m D infy2˝ Q.y/

x 2 C and Q.x/ > 1 for x 2 K. and M D supy2˝ Q.y/. By the boundedness of ˝ both m and M are finite. By Theorem 9 there is a polynomial p 2 P .R/ such that 0 q.t / 1 for t 2 Œm; M , q.t/ > 1 ı for t 2 Œm; 14 , and q.t/ < ı for t 2 Œ1; M . We obtain the desired polynomial P by setting P D p B Q, noting that P 2 Pf .X/ by Fact 11. t u Definition 14. Let X be a set. A collection f partition of unity if P ˛ W X ! Œ0; 1 for all ˛ 2 , ˛ .x/ D 1 for each x 2 X.

˛ g˛2

of functions on X is called a

˛2

Let U be a covering of X. We say that the partition of unity f to U if fsuppo ˛ g˛2 refines U.

˛ g˛2

is subordinated

We note that from the second property it immediately follows that the collection fsuppo ˛ g˛2 is point-countable, i.e. for every x 2 X the set f˛ 2 I ˛ .x/ ¤ 0g is countable. In applications often either the set itself is countable, which allows for “global” constructions (e.g. analytic approximation, Section 3), or the collection fsuppo ˛ g˛2 is locally finite, which then preserves local properties, like C k -smoothness (Section 5). Lemma 15. Let X be a normed linear space, K X compact, fU.xk ; rk /gnkD1 a covering of K, and ı > 0. Then there exists a polynomial partition of unity f k gnkD1 Pf .X/ on K such that k .x/ < ı whenever x 2 K n U.xk ; 2rk /, k D 1; : : : ; n. Proof. Using Lemma 13 we find '1 ; : : : ; 'n 2 Pf .X/ satisfying 0 'k .x/ 1 for x 2 K, 'k .x/ > 1 ı for x 2 B.xk ; rk /, and 'k .x/ < ı for x 2 K n U.xk ; 2rk /, k D 1; : : : ; n. We construct inductively polynomials 1 ; : : : ; n 2 Pf .X/ that will Pk 1 form a partition of unity on K. Put 1 D '1 and k D 'k 1 j D1 j for Pn 1 k D 2; : : : ; n 1. Finally set n D 1 . Notice that 2 P .X/ by Fact 11 f k j D1 j and ! ! k k k X X1 X1 k D 1; : : : ; n 1: j D j 1C 1 j 'k ; j D1

j D1

j D1

Thus we can check by induction that 0 'k .x/

k X j D1

j .x/

1

for x 2 K, k D 1; : : : ; n

1,

(4)

394

Chapter 7. Smooth approximation

and consequently 0

k .x/

'k .x/ 1

for x 2 K, k D 1; : : : ; n

1.

(5)

It follows that 1 ; : : : ; n form a partition of unity on K. Moreover this partition has the property that k .x/ < ı whenever x 2 KnU.xk ; 2rk /, k D 1; : : : ; n. Indeed, for k < n this follows from (5). If k D n and x 2 KnU.xn ; 2rn /, then there is P m 2 f1; : : : ; n 1g such that x 2 U.xm ; rm /. Thus, by (5) and (4), m 'm .x/ < ı. t u n .x/ 1 j D1 j .x/ 1 Proof of Theorem 12. By the compactness there is a covering fU.xk ; rk /gnkD1 of K such that kf .x/ f .xk /k < 2" whenever x 2 U.xk ; 2rk /, k D 1; : : : ; n. Let M > 0 be such that kf kK M and set ı D "=.4nM /. Let f k gnkD1 be the partition of unity from Lemma 15. Put n X P .x/ D x 2 X: k .x/f .xk /; kD1

Obviously P 2 Pf .XI Y /. To show that P approximates f on K fix any x 2 K. Let I D f1 k nI x 2 U.xk ; 2rk /g and J D f1; : : : ; ng n I . Then

n

! n

X

X

kf .x/ P .x/k D k .x/ f .x/ k .x/f .xk /

kD1 n X

kD1

k .x/kf .x/

f .xk /k

k .x/kf .x/

f .xk /k C

kD1

X k2I

" C 2nM ı D ": 2

X

k .x/

kf .x/k C kf .xk /k

k2J

t u

Note that from Theorem 9 it follows that for any mapping f 2 C.˝I Y /, where ˝ Rn is open and Y is a normed linear space, there is a sequence of polynomials fPk g1 P .Rn I Y / such that Pk ! f locally uniformly on ˝. If we are interested kD1 only in the pointwise convergence, then we have the following infinite-dimensional result: Theorem 16. Let X, Y be normed linear spaces, X separable, ˝ X open, and f 2 C.˝I Y /. Then there is a sequence of polynomials fpn g1 nD1 Pf .XI Y / such that lim pn .x/ D f .x/ for every x 2 ˝. n!1

Proof. As X is separable, there is a sequence fxn gn2N X such that spanfxn g D X. Put Kn D fx 2 spanfx1 ; : : : ; xn gI dist.x; X n ˝/ n1 ; kxk ng. Then Kn is a compact subset of ˝. By Theorem 12 there are polynomials pn 2 Pf .XI Y / such that kf pn kKn n1 for every n 2 N. Choose any x 2 ˝ and " > 0. There is ı > 0 such that U.x; 2ı/ ˝ and kf .x/ f .y/k < 2" whenever y 2 U.x; ı/. Further, there is

Section 2. Approximation by polynomials

395

n0 2 N such that n10 < 2" , ı > n10 , kxkCı n0 , and spanfx1 ; : : : ; xn0 g\U.x; ı/ ¤ ;. Choose ´ 2 spanfx1 ; : : : ; xn0 g \ U.x; ı/. It follows that ´ 2 Kn for every n n0 and hence " 1 kf .x/ pn .x/k kf .x/ f .´/kCkf .´/ pn .´/k < C < " for every n n0 . 2 n t u In contrast with that, the norm on c0 . /, uncountable, is not a pointwise limit of a sequence of polynomials on Sc0 . / . Indeed, given any sequence of polynomials fPn g on c0 . /, by Corollary 3.62 there is 2 such that Pn .e / D Pn .0/ for each n 2 N. This was first observed by Aleksander Pełczy´nski. Next, we deal with the problem of uniform approximation by polynomials on bounded sets. The above methods relied on compactness, which is of course not available in the general infinite-dimensional setting. Nevertheless, we show that certain classes of spaces and mappings allow for some compactness argument, which then yields the uniform approximation on bounded sets. Theorem 17 ([AP]). Let X be a normed linear space, Y a Banach space, and U X b a convex set. Then Pf .XI Y / D Cwu .U I Y /. Proof. Let f 2 Cwu .U I Y /, let V U be a CCB set, and " > 0. By the assumption there exist ı > 0 and 1 ; : : : ; n 2 BX such that kf .x/ f .y/k < 2" whenever x; y 2 V , jj .x y/j < 2ı for all j 2 f1; : : : ; ng. Since f is uniformly continuous Rn be defined as on V , there is M > 0 such that kf kV M . Let ˚ W X ! ˚.x/ D 1 .x/; : : : ; n .x/ . We set K D ˚.V /. Consider Rn with the maximum norm. Since ˚.V / is relatively compact, there are points y1 ; : : : ; ym 2 ˚.V / such that fU.yk ; ı/gm is a covering of K. By Lemma 15 there is a partition of unity kD1 " n / on K satisfying f k gm P .R f k .y/ < 4mM whenever y 2 K n U.yk ; 2ı/, kD1 k D 1; : : : ; m. Choose xk 2 V such that yk D ˚.xk /. Finally, put P .x/ D

m X

k

˚.x/ f .xk /:

kD1

Obviously P 2 Pf .X I Y /. To show that P approximates f on V fix any x 2 V . Let I D f1 k mI ˚.x/ 2 U.yk ; 2ı/g and J D f1; : : : ; mg n I . Then

m

! m

X

X

kf .x/ P .x/k D f .x/ k ˚.x/ k ˚.x/ f .xk /

kD1

m X

kD1

k

˚.x/ kf .x/

f .xk /k

k

˚.x/ kf .x/

f .xk /k C

kD1

X k2I

X

k

˚.x/ kf .x/k C kf .xk /k ":

k2J

t u

396

Chapter 7. Smooth approximation

To prove the version of the above theorem for approximating also the derivatives we need the following notion. Definition 18. A Banach space X is said to have the approximation property, if for every compact set K X and every " > 0 there exists a finite rank operator T 2 L.X I X / such that kT .x/ xk < " for all x 2 K. Further, X is said to have the C -bounded approximation property, where C 2 R, C 1, if the operator T can be always chosen so that kT k C . The approximation property, introduced by Alexander Grothendieck, is one of the central concepts in Banach space theory. For some basic properties and equivalent formulations see e.g. [FHHMZ]. We will rely on the following classical result concerning the approximation property in the dual. Theorem 19 ([FHHMZ, Theorem 16.36]). Let X be a Banach space. Then X has the approximation property if and only if for every Banach space Y , every " > 0, and every T 2 LK .X I Y / there exists a finite rank operator S 2 L.X I Y / such that kT Sk < ". Using the principle of local reflexivity (Theorem 4.2) it is possible to obtain the following result (see also [Cas, Proposition 3.5]). Proposition 20. Let X be a Banach space such that X has a C -bounded approximation property. Then for every pair of compact sets K X and L X , and for every " > 0 there exists a finite rank operator T 2 L.XI X/ such that kT k C , kT .x/ xk < " for each x 2 K, and k B T k < " for each 2 L. Theorem 21 ([AP]). Let X be a Banach space. Then X has the approximation property if and only if Pwu . nXI Y / D Pf . nXI Y / for every Banach space Y and every n 2 N. Proof. ( follows from Corollary 3.24 and Theorem 19, which also implies the reverse implication for n D 1. We proceed with the proof of ) by induction on n. Let P 2 Pwu . nXI Y /. Theorem 6.10 implies that d n 1P 2 LK XI Pwu . n 1XI Y / . By }.x; n 1 h/. By Theorem 19 for every " > 0 there Lemma 1.99, d n 1P .x/Œh D nŠP exist 1 ; : : : ; k 2 X and P1 ; : : : ; Pk 2 Pwu . n 1XI Y / such that

k

X

} n 1

h/ j .x/Pj .h/ < ": sup nŠP .x;

x;h2BX j D1

We finish the proof by applying the inductive hypothesis to Pj , j D 1; : : : ; k, and using Fact 11. t u Lemma 22 ([AP]). Let X, Y be Banach spaces such that X has a C -bounded k .X I Y /, k 2 N, and let V X be a CCB set approximation property. Let f 2 Cwu and " > 0. Then there exists a finite rank operator T 2 L.X I X/, kT k C , such that kd jf

d j .f B T /kV ";

j D 0; : : : ; k:

Section 2. Approximation by polynomials

397

Proof. Let R > 0 be such that V B.0; R/. By the assumption there exist ı > 0 and 1 ; : : : ; n 2 BX such that kd jf .x/ d jf .y/k < 4C" k , j D 0; : : : ; k, whenever x; y 2 B.0; CR/, jl .x y/j < ı for all l 2 f1; : : : ; ng. By Lemma 3.18 each set d jf .V /, j D 1; : : : ; k, is relatively compact in Pwu . jXI Y / and so by Theorem 21 for each j 2 f1; : : : ; kg there is a finite set Mj Pf . jX I Y / which is 4C" k -dense in d jf .V /. Next, by Proposition 20 there exists a finite rank operator T 2 L.X I X/, ı kT k C such that kl l B T k < R , l D 1; : : : ; n, and moreover kP P B T k < 4" for all P 2 Mj , j D 1; : : : ; k. ı R D ı. Thus Fix x 2 V . We have T .x/ 2 B.0; CR/ and also jl .x T .x//j < R kf .x/ f B T .x/k < ". Further, fix j 2 f1; : : : ; kg and find P 2 Mj such that kd jf .x/ P k 4C" k . Since T is a linear operator, the Chain rule (Corollary 1.117) implies that d j .f B T /.x/ D d jf .T .x// B T . Thus kd jf .x/

d j .f B T /.x/k

kd jf .x/

d jf .x/ B T k C kd jf .x/ B T

d jf .T .x// B T k

kd jf .x/ P k C kP P B T k C kP B T " " " " < C C kT kj C ": k k 4 4C 4 4C

d jf .x/ B T k C

" kT kj 4C k t u

The next result is another generalisation of the Weierstraß theorem into infinitedimensional spaces. Theorem 23 ([AP]). Let X, Y be Banach spaces and k 2 N. Suppose that X has the bounded approximation property. Then Pf .X I Y /

bk

k .X I Y /. D Cwu

k .X I Y /, let V X be a CCB set, and " > 0. By Lemma 22 Proof. Let f 2 Cwu there is a finite rank operator T 2 L.X I X/ such that kd jf d j .f B T /kV 2" for all j 2 f0; : : : ; kg. Put g D f T .X/ and C D maxfkT k; 1g. Since T .X/ is finitedimensional and K D T .V / T .X/ is compact, by Theorem 9 there is a polynomial p 2 P .T .X /I Y / such that kd jg d jpkK 2C" k , j D 0; : : : ; k. Put P D p B T . Then P 2 Pf .XI Y / (see the remark after Definition 10). Now for any x 2 V and j 2 f1; : : : ; kg we have

j

d f .x/ d jP .x/

d jf .x/ d j .f B T /.x/ C d j .f B T /.x/ d j .p B T /.x/

" C d jf T .x/ B T d jp T .x/ B T 2

" " " kT kj "; D C d jg T .x/ B T d jp T .x/ B T C 2 2 2C k

and similarly for j D 0.

t u

398

Chapter 7. Smooth approximation

The weak uniform topology from the previous result can sometimes be replaced by the uniform topology. The abstract form of this result is contained in the next theorem, which is followed by several concrete examples. Theorem 24 ([CHL]). Let X , Y be Banach spaces such that X does not contain `1 and X has the bounded approximation property. If C 1;C .XI Y / CwsC .X I Y /, then for any k 2 N P .XI Y /

bk

D Pf .XI Y /

bk

D C k;C .XI Y /:

Proof. From Theorem 3.81 it follows that C k;C .X I Y / C 1;C .XI Y / Cwu .X I Y /. k .XI Y / by Theorem 6.8 and the result follows from TheThus C k;C .X I Y / D Cwu orem 23. t u Corollary 25 ([CHL]). Let X, Y be Banach spaces such that X has the bounded approximation property. Then for any k 2 N P .X I Y /

bk

D Pf .XI Y /

bk

D C k;C .XI Y /

provided that one of the following conditions is satisfied: X is a W -space which does not contain `1 and Y is of a non-trivial type, or X has the hereditary DPP and does not contain `1 , and Y has property (WR), or X is a linear quotient of an isomorphic predual Z of `1 , and Y does not contain `1 or Y has property (WR) and is isomorphic to a complemented subspace of a dual Banach space. Proof. In the last case by Theorem 4.16 the space Z is a L1 -space that does not contain `1 . By the lifting property of `1 it follows that X does not contain `1 either. Thus in view of Theorem 24 it suffices to show that C 1;C .X I Y / CwsC .XI Y /. This follows from Theorem 6.54, Theorem 6.53, or Theorem 6.57 and Proposition 6.21 respectively. t u The above results are quite special to the W -spaces. In a general infinite-dimensional normed linear space X there are always continuous functions on X that cannot be uniformly approximated by polynomials on SX . Indeed, while every polynomial is bounded on SX , it is easy to construct a continuous function on X that is unbounded on SX . But typically even much more regular functions cannot be uniformly approximated by polynomials. Theorem 26 ([NS]). Let X be an infinite-dimensional Banach space and let S be one of the spaces C k .X/, C 1 .X/, or C k;˝ .X/, where k 2 N0 and ˝ M is a convex cone. If X admits a bump function from S, then there is a bump function f 2 S such that it cannot be uniformly approximated on BX by polynomials. Proof. The main ingredient of the proof is the fact that for every degree d there is n 2 N such that the unit ball of Rn (with an arbitrary norm) contains a 12 -separated set of cardinality greater than the dimension of the space P d .Rn /. Indeed let A BRn

Section 3. Approximation by real-analytic mappings

399

be a maximal 12 -separated set, i.e. kx yk 12 for every x; y 2 A, x ¤ y. By the S maximality, BRn x2A U.x; 21 /. Therefore X 1 .BRn / U.x; 21 / jAj B.0; 12 / D jAj n .BRn /; 2 x2A and hence jAj 2n . On the other hand, dim P d .Rn / D nCd (Section 2.1). Since d nd Cd n d for every d 2 N there is nd 2 N such that 2 > , there is a 12 -separated set d d n d in BRnd of cardinality greater that dim P .R /. Put ı D 43 . Since the space X is infinite-dimensional, there is a ı-separated set f´d I d 2 Ng BX .0; ı/. Notice that B ´d ; 3ı BX for every d 2 N. Let Xd be some nd -dimensional subspace of X that contains ´d . By the discussion above there is a 6ı -separated subset Ad of BXd ´d ; 3ı satisfying dim P d .Xd / < jAd j < 1. Consider the space RAd with the supremum norm. Define Rd 2 L.P d .Xd /I RAd / by Rd .p/ D pAd . As dim Rd .P d .Xd // dim P d .Xd / < jAd j D dim RAd , the space Rd .P d .Xd // is a proper subspace of a finite-dimensional space RAd and so A d there is fd 2 R d such that dist fd ; Rd .P .Xd // D kfd k D 1. Let ' 2 S be a bump function. By shifting and scaling we may suppose that ı '.0/ D 1 and supp ' BX 0; 18 . Define a function f W X ! R by X X fd .y/'.x y/: f .x/ D d 2N y2Ad

S By the choice of the set f´d g the set d 2N Ad is a 6ı -separated subset of BX . Using this and the fact that jfd .y/j 1 for y 2 Ad it is easy to check that f 2 S . Obviously supp f is bounded. To see that f cannot be approximated on BX by polynomials pick any p 2 P .X/. Let d 2 N be such that p 2 P d .X/. Then pXd 2 P d .Xd / (Fact 1.35). Notice that f Ad D fd . Thus sup jf .x/ x2BX

p.x/j sup jf .x/

p.x/j D kfd

x2Ad

dist fd ; Rd .P d .Xd // D 1:

Rd .pXd /k t u

3. Approximation by real-analytic mappings We begin with a Whitney-type approximation theorem stating that in a finite-dimensional case any mapping in C k .˝I Y /, k 2 N0 , can be approximated on the whole ˝, in a fine topology, and together with its derivatives of order up to k by real analytic mappings. Then we present the famous result of Jaroslav Kurzweil which extends this result (for k D 0) to infinite-dimensional separable Banach spaces X that admit a separating polynomial. We also show that if we only require uniform approximations for uniformly continuous mappings, then it suffices that X admits a separating real analytic function with uniform radii of convergence.

400

Chapter 7. Smooth approximation

Let X , Y be normed linear spaces, ˝ X open, and f 2 C k .˝I Y / for some k 2 N0 . For S ˝ we define kf kS;k D

k X

sup kd jf .x/k:

j D0 x2S

Clearly kkS;k is a semi-norm on the subspace of C k .˝I Y / consisting of mappings with all derivatives up to k bounded on S. Lemma 27. Let X , Y be normed linear spaces over K, ˝ X open, k 2 N0 , ' 2 C k .˝/, f 2 C k .˝I Y /, and S ˝. Then k k'f kS;k k k'kS;k kf kS;k : 2

Proof. Fix x 2 ˝ and 0 j k. By the Leibniz formula (Corollary 1.116) j j X

X

j j j j l l

d j l'.x/ d lf .x/ kd .'f /.x/k d '.x/ d f .x/ l l lD0

lD0

X j

j l

j

d '.x/ d lf .x/ : j 2

lD0

Therefore k'f kS;k

X j j k X k X

j l l

j X k j l l

d ' d f

d ' S df S k j S S j D0

2

2

lD0

j D0 lD0

X k X k

j l k

d ' d f D k k'kS;k kf kS;k : k k S S 2

2

j D0 lD0

In the next theorem we consider

Cn

t u

with the Euclidean norm.

Theorem 28. Let Y be a Banach space, ˝˚ Rn an open set, k 2 N0 [ f1g, f 2 C k .˝I Y /, and " 2 C.˝I RC /. Put G D ´ 2 C n I kIm ´k < dist.Re ´; Rn n ˝/ . Then there exists a mapping g 2 H.GI YQ / such that g˝ 2 C ! .˝I Y / and satisfies kd jf .x/ d j .g˝ /.x/k < ".x/ for all x 2 ˝, 0 j minfk; 1=".x/g. Notice in particular, that if in the preceding theorem ˝ D Rn , then the approximating mapping g will be an entire mapping, i.e. a mapping that is holomorphic on the whole C n . Proof. Define K 1 D K0 D ;, Kj D fx 2 Rn I dist.x; Rn n ˝/ 2 j g \ B.0; j /, Lj D Kj n Int Kj 1 , and Uj D .Int Kj C1 / n Kj 2 for j 2 N. Note that SKj Kj C1 , Lj is compact, Uj ˝ is an open neighbourhood of Lj , ˝ D j1D1 Lj , and Lj \ Ul D ; for l > j C 1. By Lemma 3 there are functions 'j 2 C 1 .Rn I Œ0; 1/, j 2 N, satisfying supp 'j Uj (hence supp 'j is compact) and 'j D 1 on a neighbourhood of Lj .

Section 3. Approximation by real-analytic mappings

401

Further, we put "0 D 1, "j D minf"j 1 ; minx2Lj ".x/g, k0 D 0, kj D k if k < 1, 1 and finally kj D maxfkj 1 ; Œmaxx2Lj ".x/ g if k D 1. Notice that the sequence 1 f"j gj D1 is non-increasing, while the sequence fkj gj1D1 is non-decreasing. We set Mj D kj k'j kRn;kj , where l D Œ ll . For each j 2 N let ıj > 0 be such that 2

"j : (6) 2j To slightly shorten our notation we denote gN D gM \Rn for g W M ! Y , where M C n . For each j 2 N we define inductively mappings fj 2 C k .Rn I Y / and gj 2 H.C n I YQ / such that gxj maps into Y as follows: We put fj D 0 on Rn n ˝ and ıj .1 C Mj C1 /

0. Then there is g 2 C ! .˝I Y / such that kf gk˝ ". We prove both Theorem 29 and Theorem 31 together, with the help of the next two lemmata. Lemma 32. Let X be a separable Banach space and ˝ X open. Suppose there is ˛ > 0 such that for any open covering fU.xn ; rn /g1 nD1 (or for any uniform open covering fU.xn ; r/g1 , i.e. r D r for all n 2 N) of ˝ and any sequence fwn g1 n nD1 nD1 of positive real numbers there exists an open neighbourhood V XQ of ˝ and a sequence of functions f'n g1 nD1 H.V / with the following properties: P1 (i) The sum nD1 wn 'n converges absolutely locally uniformly on V , (ii) 'n ˝ maps into Œ0; C1/ for every n 2 N, (iii) wn 'n .x/ 41 2 n for every x 2 ˝ n U.xn ; ˛rn /, n 2 N, and (iv) for every x 2 ˝ there is k 2 N such that x 2 U.xk ; ˛rk / and 'k .x/ wk . Then for every mapping f 2 C.˝I C /, where C is a closed convex subset of a Banach space Y , (resp. for every f 2 Cu .˝I C /) there is a mapping g 2 C ! .˝I C / satisfying kf gk˝ 1. Proof. Using the separability of ˝ and the continuity (resp. uniform continuity) of f we find a covering (resp. uniform covering) fU.xn ; rn /g1 nD1 of ˝ such that 1 for x 2 U.xn ; ˛rn / \ ˝. (11) kf .x/ f .xn /k < 2 Put wn D 1 C kf .xn /k. Let f'n g be the sequence of functions satisfying (i)–(iv). P1 The function '.´/ D nD1 'n .´/ is well-defined for every ´ 2 V by (i) and moreover ' 2 H.V /. Further, by (ii) and (iv), for every x 2 ˝ '.x/ 'k .x/ 1: (12) Q Hence there is an open neighbourhood W of ˝ in X such that W V and ' ¤ 0 on W . Define n .´/ D 'n .´/='.´/ for ´ 2 W . Then the functions n are holomorphic on W and have the following properties:

404

Chapter 7. Smooth approximation

(a) f n ˝ g is a partition of unity on ˝, (b) n .x/kf .xn /k 14 2 n for every x 2 ˝ n U.xn ; ˛rn /, n 2 N, and (c) n .x/kf .x/k 41 2 n for every x 2 ˝ n U.xn ; ˛rn /, n 2 N. Indeed, property (a) follows from (ii) and the definition of n and ', and property (b) follows from (iii) and (12). To prove (c) choose n 2 N and x 2 ˝ n U.xn ; ˛rn /. Then '.x/ 'k .x/ wk D 1 C kf .xk /k > kf .x/k by (iv) and (11). Thus ' .x/ 41 2 n by (iii). n .x/kf .x/k n .x/'.x/ D 'n .x/ w Pn1n To finish the proof nD1 n .´/f .xn / for ´ 2 W . Because P we put g.´/ D g.´/ D .1='.´// 1 ' .´/f .x / and the sum converges locally uniformly on W n n nD1 by (i), we obtain g 2 H.W I YQ /. Clearly g˝ 2 C ! .˝I C /. Further, choose an arbitrary x 2 ˝. Put I D fn 2 NI x 2 U.xn ; ˛rn /g. Then using (a), (11), (b), and (c) we obtain

1

X

kf .x/ g.x/k D f .xn / n .x/ f .x/

nD1 X X f .xn /k C n .x/kf .x/ n .x/ kf .x/k C kf .xn /k < 1: n2I

n2NnI

t u

Lemma 33. Let X be a Banach space, ˝ X open, and fU.xn ; rn /g1 nD1 X an open covering of ˝. Suppose that there are a function q 2 H.G/ and ˛ > 0 such Q kIm ´k < supn2N 1 g for some > 0, qX maps into Œ0; C1/, that G D f´ 2 XI rn q.x/ 2 for x 2 X n UX , Re q.´/ 41 for ´ 2 UXQ .0; 1=˛/, and suppose there is a sequence fan g1 nD1 of positive real numbers such that for each x 2 ˝ the function ´ 7!

1 X

an Im q .x

2 xn C ´/=.˛rn /

(13)

nD1

is defined on some neighbourhood of 0 in XQ and is continuous at 0. Then for every sequence fwn g1 nD1 of positive real numbers there are an open neighbourhood V XQ of ˝ and a sequence of functions f'n g1 nD1 H.V / satisfying the properties (i)–(iv) in Lemma 32. Proof. Put "n D 1 2 14 2 n and let n be the functions and fın g the sequence from 2wn Q kIm ´k < ˛g and put Lemma 5. Denote ˛n D 1=.˛rn / and G˛ D f´ 2 XI 'n .´/ D 2wn n q ˛1 .´ x1 / ; : : : ; q ˛n .´ xn / for ´ 2 G˛ , n 2 N. Then 'n 2 H.G˛ / and by (Z1), 'n X maps into Œ0; C1/. Pick any x 2 ˝. Then there exists j 2 N such that x 2 U.xj ; rj / and hence q.˛j .x xj // 14 < 1. Let k 2 N be the smallest index such that q.˛k .x xk // < 1. Then x 2 U.xk ; ˛rk / and property (Z4) implies that 'k .x/ wk . 2 P ıj Let ıx > 0 be such that kx xj C´k < rj and 1 nD1 an Im q.˛n .x xn C´// Q k´k ıx . Then Re q.˛j .x xj C ´// 1 and hence, by (Z3), whenever ´ 2 X, 4

Section 3. Approximation by real-analytic mappings

405

P that 1 jwn 'n .x C ´/j < 2 n for n > j . It followsS nD1 wn 'n converges absolutely uniformly on UXQ .x; ıx /. We put V D G˛ \ x2˝ UXQ .x; ıx /. Finally we show that (iii) is satisfied. Fix n 2 N. For x 2 ˝ n U.xn ; ˛rn / we have q.˛n .x xn // 2, hence, by (Z5), wn 'n .x/ 41 2 n . t u The next lemma shows that in certain circumstances it is possible to pass from uniform approximations to fine approximations. Lemma 34 ([Kur1]). Let ˝ be a topological space and Y a normed linear space. Let S C.˝I Y / and S1 C.˝/ be such that h= 2 S for any positive function 2 S1 and any mapping h 2 S . Suppose that for any f 2 C.˝I Y / there is h 2 S such that kf hk˝ 1 and for any ' 2 C.˝/ there is 2 S1 such that j' j˝ 1. Then for any f 2 C.˝I Y / and any positive function " 2 C.˝/ there is g 2 S such that kf .x/ g.x/k < ".x/ for every x 2 ˝. Proof. Define ' 2 C.˝/ by ' D 1 C 2=". According to the assumptions there is 2 S1 such that j'.x/ .x/j 1 for every x 2 ˝. Since f 2 C.˝I Y /, there is h 2 S such that k.x/f .x/ h.x/k 1 for every x 2 ˝. Notice that .x/ '.x/ 1 D 2=".x/ > 1=".x/ > 0 for every x 2 ˝. Thus g D h= 2 S and kf .x/ g.x/k 1=.x/ < ".x/ for every x 2 ˝. t u Proof of Theorem 29. By Fact 4.45 we may assume that there is an m-homogeneous polynomial p on X such that p.x/ 2 for x 2 X n UX . Let q D p. Q Because q.0/ D 0, from the continuity there is ˛ > 0 such that Re q.´/ 41 for ´ 2 UXQ .0; 1=˛/. Suppose that fU.xn ; rn /g1 nD1 is an open covering of ˝. Put an D

rn2m 2n .1 C kxn k/2m

Then an Im q .x

ˇ 2 xn C ´/=.˛rn / an ˇq .x

ˇ2 xn C ´/=.˛rn / ˇ

kqk2 kx xn C ´k2m xn C ´k2m .˛rn /2m ˛ 2m 2n .1 C kxn k/2m 2 2m 1 kqk 2m 1 C kxk C k´k ˛ 2n and hence for every x 2 X the sum in (13) converges absolutely locally uniformly Q Thus the hypotheses of Lemma 33 are satisfied and to a continuous function on X. using it together with Lemma 32 we can conclude that for any Banach space Z and any continuous mapping f 2 C.˝I Z/ there is a mapping h 2 C ! .˝I Z/ satisfying kf hk˝ 1. Finally Lemma 34 applied to S D C ! .˝I Y / and S1 D C ! .˝/ finishes the proof. t u an kqk2

kx

Proof of Theorem 31. By Theorem 1.171 there are d > 0 and a function q 2 H.G/, G D f´ 2 XQ I kIm ´k < d g, such that qX W X ! Œ0; C1/, q.0/ D 0, q.x/ 2 for x 2 X n UX , and the radius of norm convergence of the Taylor series of q at every

406

Chapter 7. Smooth approximation

point x 2 X is at least d . Let ˛ > 0 be such that Re q.´/ 14 for ´ 2 UXQ .0; 1=˛/ and 1 2 ˛d > 1. Suppose fU.xn ; r/g1 nD1 is a uniform open covering of ˝. Put nˇ o ˇ Q kwk 1 ˛rd; 1 j n Mn D sup ˇq .xj xn C w/=.˛r/ ˇ I w 2 X; 2 and an D 1=.2n Mn2 /. (Note that by the assumption on the radius of the Taylor series Q Mn < C1.) Fix x 2 ˝. There is k 2 N such that x 2 U.xk ; r/. For ´ 2 X, 1 1 k´k r. 2 ˛d 1/ we have kx xk C ´k 2 ˛rd and hence for n k an Im q .x

ˇ 2 xn C ´/=.˛r/ an ˇq .xk an Mn2 D

xn C x

ˇ2 xk C ´/=.˛r/ ˇ

1 : 2n

Therefore the sum in (13) converges absolutely uniformly on BXQ 0; r. 21 ˛d 1/ to a continuous function. Using Lemma 33 together with Lemma 32 and a suitable scaling finishes the proof. t u The space c0 does not admit a separating polynomial (Proposition 5.49 or Corol2n in Example 1.137 and lary 3.59), but it has property (K) (take Pn .x/ D en .x/ combine it with Corollary 1.165). The property (K) is inherited by subspaces and finite direct sums. In certain circumstances it can also pass to infinite direct sums: Assume that all members of a sequence of Banach spaces fXn g have property (K) witnessed by non-negative functions qn with radii at least dn and satisfying qn .0/ D 0 and qn .x/ 1 whenever kxk 1. Suppose that there are 0 < d 21 infn2N dn and a sequence f˛n g N such that supn2N sup´2B z .0;d / jqQn .´/j˛n < 1, where qQn is Xn the analytic extension of q to a neighbourhood of Xn in Xzn (Theorem n L1 P1 1.171). Then 2n˛n nD1 Xn c0 has property (K) witnessed by q.x1 ; x2 ; : : : / D L nD1 qn .xn / 1 with radii at least d (use Corollary 1.165). Thus for example c0 ˚ nD1 `2n c0 has property (K). By Theorem 5.64 a space with property (K) that does not contain c0 admits a separating polynomial. By Corollary 5.68 every space with (K) is saturated by spaces from f`p I p eveng [ fc0 g. Let us mention without proof the next result, which should be compared with Corollary 5.105. Proposition 35 ([CH]). Let X be a Banach space with property .K/ and such that P .X / D Pwsc .X/. Then X is isomorphic to a subspace of c0 . Whence all Banach spaces with the Dunford-Pettis property and (K) are isomorphic to subspaces of c0 (Theorem 3.68). In particular, since every C.K/ space which is isomorphic to a subspace of c0 is isomorphic to c0 [LP], we have the following corollary: Corollary 36. If the Banach space C.K/ has property (K), then it is isomorphic to c0 .

Section 4. Infimal convolution

407

4. Infimal convolution The infimal convolution is another global approximation technique, which similarly to the integral convolution preserves certain regularity properties of the approximated function, like for example the Lipschitzness. The undisputable advantage is that it does not need any finite-dimensional structure and it works equally well even on nonseparable spaces. The drawback is that this technique is fundamentally scalar and also it usually produces only smoothness of the first order. The notion goes back to Felix Hausdorff around 1919. Definition 37. Let X be a set, f W X ! R [ fC1g, and K W X 2 ! R [ fC1g. We define the infimal convolution of f and K by .f K/.x/ D inf f .y/ C K.x; y/ ; x 2 X: y2X

The function K is called a kernel. If .X; C/ is a commutative group, then we associate with each g W X ! R [ fC1g the kernel Kg .x; y/ D g.x y/. We may then define the infimal convolution of f and g as f g D f Kg , i.e. .f g/.x/ D inf f .y/ C g.x y/ : y2X

Note that in this case f g D g f . Fact 38. Let X be a set, f W X ! R [ fC1g, and K W X 2 ! R. (i) If K.x; x/ D 0 for every x 2 X , then f K f . (ii) If f is proper, then f K < C1 everywhere. (iii) If X is a metric space, f is proper, and the functions x 7! K.x; y/, y 2 X, are uniformly continuous with modulus ! 2 M, then either f K is identically 1, or f K is real-valued and uniformly continuous with modulus !. Proof. Both (i) and (ii) are obvious. (iii) Let be the metric on X. Suppose there is ´ 2 X such that .f K/.´/ D 1. Then there exists a sequence fyn g in X satisfying f .yn / C K.´; yn / < n for each n 2 N . Now for any x 2 X we have .f K/.x/ f .yn / C K.x; yn / D f .yn / C K.´; yn / K.´; yn / C K.x; yn / < n C !..x; ´//, which implies that .f K/.x/ D 1. Therefore f K is either identically 1, or f K is realvalued and uniformly continuous with modulus !, as it is an infimum of a family of uniformly continuous functions with modulus !. t u The following extension lemma is useful for example when we deal with smooth approximations of Lipschitz functions defined on some subset of a normed linear space X: It suffices to formulate the approximation results for functions defined on the whole of X . Lemma 39. Let .P; / be a metric space, ; ¤ A P , and f W A ! R a uniformly continuous function with modulus ! 2 Ms . Then there is an extension of f to the whole of P which is uniformly continuous with modulus !.

408

Chapter 7. Smooth approximation

Proof. Define fN W P ! R by fN D f on A and fN D C1 on P n A. Further, put g D fN .! B /. Then g.x/ D inf f .y/ C ! .x; y/ y2A

For any x; y 2 A we have f .x/ f .y/ !..x; y// and so f .x/ f .y/C!..x; y//. It follows that f g on A. This together with Fact 38(i) implies that g D f on A. Consequently by Fact 38(iii) g is real-valued and uniformly continuous with modulus !. t u The next lemma tells us that the results on uniform approximation of Lipschitz functions immediately give also approximation of uniformly continuous functions. Lemma 40. Let .P; / be a metric space, f W P ! R a uniformly continuous function with modulus ! 2 Ms , and " > 0. Further, let a 2 RC be such that !.a/ ". Then there is an a" -Lipschitz function g W P ! R such that jf gjP ". Proof. We let g D f a" . Fix x 2 P . Clearly g.x/ f .x/ (Fact 38(i)). From the sub-additivity of ! it follows that for any y 2 P .x; y/ f .x/ f .y/ ! .x; y/ ! a a .x; y/ .x; y/ !.a/ C 1 ": a a Thus f .y/ C a" .x; y/ f .x/ " a -Lipschitz by Fact 38(iii).

", which implies that g.x/ f .x/

". Finally, g is t u

Theorem 41 (Jean-Michel Lasry and Pierre-Louis Lions, [LL]). Let H be a Hilbert space, f W H ! R an L-Lipschitz function, and " > 0. Then there is an L-Lipschitz function g 2 C 1;1 .H / satisfying jf gjH ". For the proof we need a few auxiliary results concerning convex functions. Lemma 42. Let X be a normed linear space, f 2 C.X/, and suppose there exist functions ; 2 C 1;˛ .X/, ˛ 2 .0; 1, such that f C is convex and f is concave. Then f 2 C 1;˛ .X/. Proof. It clearly suffices to show that f C 2 C 1;˛ .X/ which we show using Lemma 5.20. Notice that C D .f C / C . f C / is necessarily convex. From the concavity of f it follows that .f /.x Ch/C.f /.x h/ 2.f /.x/ 0 for any x; h 2 X and hence .f C /.x C h/ C .f C /.x h/ 2.f C /.x/ .f C /.x C h/ C .f C /.x .f

h/

/.x C h/ C .f

D . C /.x C h/ C . C /.x

2.f C /.x/

/.x h/

h/

2.f

/.x/

2. C /.x/ C khk1C˛ ;

where the last inequality follows from Lemma 5.20 used on C .

t u

Section 4. Infimal convolution

409

Let X be a normed linear space. For any f W X ! R [ fC1g and t > 0 we define 1 kk2 . We note that the constant 12 is the Moreau-Yosida regularisation f t D f 2t useless (and perhaps even annoying) in our proofs but using this particular kernel is customary in convex analysis for many good reasons. Fact 43. Let H be a Hilbert space and f W H ! R [ fC1g a proper function. 1 (i) The extended real-valued function f t C 2t kk2 is convex for every t > 0. 1 (ii) Suppose that f is real-valued and f C 2t kk2 is convex for some t > 0. Then 1 2 fs C 2.t s/ kk is convex for every 0 < s < t .

Proof. (i) This follows from the fact that 1 1 1 2 2 f t .x/ C kxk D inf f .y/ C kx yk C kxk2 y2H 2t 2t 2t 1 1 1 D sup f .y/ D sup kxk2 kx yk2 hx; yi kyk2 2t t 2t y2H y2H

f .y/ ;

which is a supremum of affine functions. (ii) We have 1 1 1 kxk2 D inf f .y/ C kx yk2 C kxk2 fs .x/ C y2H 2.t s/ 2s 2.t s/ 1 1 1 1 2 2 2 2 D inf f .y/ C kyk C kx yk C kxk kyk y2H 2t 2s 2.t s/ 2t !

2

1 t s t 2

: D inf f .y/ C kyk C x y

y2H 2t 2st t s It is easy to verify that for any convex function W X Y ! R, where X , Y are vector spaces, the function x 7! infy2Y .x; y/ is also convex (in other words, “a convex body casts a convex shadow”), from which the result follows if we set 1 .x; y/ D f .y/ C 2t kyk2 C t2sts k t t s x yk2 . t u Proof of Theorem 41. For every function h W H ! R that is C -Lipschitz we have lim jh t hjH D 0. Indeed, t!0C 1 2 0 h.x/ h t .x/ D h.x/ inf h.y/ C kx yk y2H 2t 1 1 2 2 kx yk D sup h.x/ h.y/ kx yk D h.x/ C sup h.y/ 2t 2t y2H y2H 1 1 2 C 2t 2 sup C kx yk Cı kx yk D sup ı D 2t 2t 2 y2H ı2Œ0;C1/ for any x 2 H , where the first inequality follows from Fact 38(i). Moreover, as 1 kk2 h, each h t is C -Lipschitz by Fact 38(iii). h t D 2t

410

Chapter 7. Smooth approximation

So choose t > 0ˇ such that jf t ˇ f jH 2" . Then f t is L-Lipschitz. Next, find 0 < s < t such that ˇ. f t /s . f t /ˇH 2" and put g D . f t /s . Then jf gjH " and the function g is L-Lipschitz. 1 kk2 is convex by Fact 43(i), and hence the function Further, the function f t C 2t 1 2 g 2.t s/ kk is concave by Fact 43(ii). Using Fact 43(i) again, this time on the 1 kk2 is convex. Since the function kk2 is function f t , we can conclude that g C 2s a 2-homogeneous polynomial, it belongs to C 1;1 .H /, and so Lemma 42 finishes the proof. t u Another nice application of the infimal convolution gives the next result. Proposition 44 ([We2]). Let A be a closed subset of a Hilbert space H . Then there is a function f 2 C 1;1 .H / such that A D f 1 .f0g/. Proof. Let IA W H ! R [ fC1g be the indicator function of the set A, i.e. IA .x/ D 0 for x 2 A and IA D C1 for x 2 H n A. We let f D .IA /1 1 . Without loss of 2 generality we may assume that A is non-empty and hence IA is proper. It is easy to see 1 dist2 .x; A/ for every x 2 H , t > 0. Thus .IA /1 is real-valued and that .IA / t .x/ D 2t using Fact 43 similarly as in the proof of Theorem 41 we can conclude that f C kk2 is convex and f kk2 is concave. Next, notice the following observation: Suppose that h W X ! R [ fC1g is a proper function on a normed linear space X and 0 < s < t . For any x; y; ´ 2 X we have 1 k´ 2t

1

s t .´ yk D 2t t s

1 s

t .´ 2t t s

x/ C 1

2

D

1 k´ 2s

2

x/

C 1

xk2 C

1 2.t

s/

kx

s t .x t t s

s

t .x t t s

2

y/

2 !

y/

yk2 ;

where we used the convexity of kk2 . From this and Fact 38(i) we obtain h t .x/

. h t /s .x/ D sup inf

´2X y2X

sup inf

´2X y2X

h.y/ C

1 2.t

1 1 k´ yk2 kx ´k2 2t 2s yk2 D h t s .x/ for every x 2 X.

h.y/ C

s/

kx

This gives us 12 dist2 .x; A/ f .x/ dist2 .x; A/ for every x 2 H . It follows that f 1 .f0g/ D A and that f C kk2 is locally bounded and thus convex continuous. Now Lemma 42 implies that f 2 C 1;1 .H /. t u

Section 5. Approximation of continuous mappings and partitions of unity

411

5. Approximation of continuous mappings and partitions of unity In this section we investigate smooth partitions of unity, the main tool for obtaining C k -smooth approximations of continuous mappings in Banach spaces. We show that several rather general classes of Banach spaces admit C k -smooth approximations provided they have a C k -smooth bump. This applies especially to separable spaces, WCG spaces, or C.K/ spaces. We finish by showing that super-reflexive spaces admit partitions of unity consisting of functions with Hölder derivative. Definition 45. Let be a class of functions. We say that a topological space X admits -partitions of unity if for any open covering U of X there is a partition of unity on X subordinated to U such that each member of the partition belongs to . Definition 46. A family of subsets of a topological space X is called locally finite if for each point x 2 X there is a neighbourhood of x that meets only finitely many members of this family; discrete if for each point x 2 X there is a neighbourhood of x that meets at most one member of this family; -locally finite if it can be decomposed into countably many locally finite families; -discrete if it can be decomposed into countably many discrete families. A family of subsets of a metric space P is called uniformly discrete if there is d > 0 such that the distance of any two members of this family is at least d ; -uniformly discrete if it can be decomposed into countably many uniformly discrete families. A partition of unity f ˛ g˛2 is called locally finite if fsuppo ˛ g˛2 is locally finite, it is called -discrete if fsuppo ˛ g˛2 is -discrete, and it is called -uniformly discrete if fsuppo ˛ g˛2 is -uniformly discrete. If is a class of mappings, then we use the notation S.X I Y / D \ Y X , i.e. S.XI Y / is the set of mappings from X to Y that belong to . A class of C k -smooth mappings will be denoted by C k and similarly for other smoothness classes from Section 1.5. Definition 47. Let P be a metric space and S C.P / a ring of functions. We say that S is a partition ring if it satisfies the following conditions: (i) For each S0 S with fsuppo f I f 2 S0 g uniformly discrete S in P and suppo f bounded for each f 2 S0 there is a g 2 S with suppo g D f 2S0 suppo f . (ii) Let f 2 S and suppo f D U1 [ U2 , where U1 and U2 are open subsets of P with dist.U1 ; U2 / > 0. Then U1 f 2 S . (iii) For each f 2 S bounded below and " > 0 thereis a g 2 S such that 0 g 1, f 1 . 1; " g 1 .f0g/ and f 1 Œ2"; C1/ g 1 .f1g/.

412

Chapter 7. Smooth approximation

Examples of partition rings: C k -smooth functions on normed linear spaces, smooth bounded Lipschitz functions, or smooth bounded functions with bounded Hölder derivatives (see the proof of Theorem 56). Definition 48. Let be a class of mappings defined on a topological space X. We say that is determined locally if whenever f is a mapping defined on X such that for every x 2 X there are a neighbourhood U of x and a mapping g 2 such that f D g on U , then f 2 . Examples of classes determined locally are C k classes or class of continuous Gâteaux differentiable mappings. Note that if a ring of functions on a metric space is determined locally then conditions (i) and (ii) in the definition of a partition ring are automatically satisfied. Lemma 49. Let P be a metric space and S a partition ring of functions on P . Consider the following statements. (i) For every A W P , A closed and W open there is ' 2 S such that ' D 1 on A and suppo ' W . (ii) For every V W P bounded open sets satisfying dist.V; P n W / > 0 there is ' 2 S such that V suppo ' W . (iii) For every V W P bounded open S sets satisfying dist.V; P n W / > 0 there are 'n 2 S, n 2 N, such that V n2N suppo 'n W . (iv) The family fsuppo f I f 2 Sg contains a -uniformly discrete basis for the topology of P . (v) The space P admits locally finite and -uniformly discrete S -partitions of unity. (vi) The space P admits locally finite S-partitions of unity. (vii) The family fsuppo f I f 2 S g contains a -locally finite basis for the topology of P . Then (i))(ii))(iii))(iv),(v))(vi))(vii). If S is moreover determined locally, then all seven statements are equivalent. We note that the -uniformly discrete partitions of unity will prove very useful in Sections 7 and 8, as they allow us to use certain separable techniques in a non-separable setting. Proof. (i))(ii))(iii) is obvious. (iii))(iv) Let Um D fU˛m g˛2m be a uniform covering of P by open balls with 1 mg m radius m . By Lemma 8 there are open refinements fVn˛ n2N;˛2m , fWn˛ gn2N;˛2m m W m U m , dist.V m ; P n W m / 2 n and the famof Um such that Vn˛ n˛ ˛ n˛ n˛ m n 2 N. Therefore, by (iii), there exily fWn˛ g˛2m is uniformly discrete for all S m m m m ist functions 'n˛k 2 S such that Vn˛ k2N suppo 'n˛k Wn˛ . The family m fsuppo 'n˛k I m; n; k 2 N; ˛ 2 m g is therefore a -uniformly discrete basis for the topology of P . (iv))(v) Let U be an open covering of P . We construct a locally finite and -uniformly discrete S -partition of unity subordinated to U. Without loss of generality we may assume that U consists of bounded sets. By (iv) there are Sj S, j 2 N, such

Section 5. Approximation of continuous mappings and partitions of unity

413

that fsuppo f I f 2 Sj g are uniformly discrete and fsuppo f I f 2 Sj ; j 2 Ng is an open covering of P that refines S U. By property (i) of a partition ring there are functions fj 2 S such that suppo fj D f 2Sj suppo f . Replacing fj by fj2 if necessary we may assume that fj 0. By property (iii) of a partition ring there are functions gj k 2 S such that 0 gj k 1, suppo gj k suppo fj , and fj 1 Œ k1 ; C1/ gj k1 .f1g/. Let n 7! .jn ; kn / be a bijection of N onto N N and put 'n D gjn kn . Q 1 Now for n 2 N let n D 'n nkD1 .1 'k /. Then n 2 S (as S is a ring) and f n gn2N is a locally finite partition of unity on P . Indeed, for any x 2 P there is j 2 N such that x 2 suppo fj and hence there are a neighbourhood U of x and k 2 N such that fj .y/ > k1 for y 2 U . It follows that gj k .y/ D 1 for y 2 U . Let m 2 N be such that j D jm and k D km . Choose any y 2 U . Then 'm .y/ D gj k .y/ D 1 and hence n .y/ D 0 for n > m. Since .1 '1 /.1 '2 / .1 'm / D 1 1 m; P Pm it follows that 1 nD1 n .y/ D nD1 n .y/ D 1. Finally, let us define n;f D suppo f n for n 2 N and f 2 Sjn . Using the fact that suppo n suppo 'n suppo fP jn and the uniform discreteness of the family fsuppo f I f 2 Sjn g it follows that f 2Sjn n;f D n and from property (ii) of a partition ring also that n;f 2 S . As moreover suppo n;f suppo f , we can conclude that f n;f gn2N;f 2Sjn is a locally finite, -uniformly discrete S -partition of unity on P subordinated to U. 1 (v))(iv) Let Um be a uniform covering of P by open balls with radius m . By (v) m there exists an S -partition of unity f n˛ gn2N;˛2m subordinated to Um such that fsuppo n˛ g˛2m is uniformly discrete for each n 2 N. It follows that the family m I m 2 N; n 2 N; ˛ 2 g is a -uniformly discrete basis for the topology fsuppo n˛ m of P . (v))(vi) is obvious. 1 (vi))(vii) Let Um be a uniform covering of P by open balls with radius m . By (vi) m there is a locally finite S-partition of unity f ˛ g˛2m subordinated to Um . It follows that the family fsuppo ˛m I m 2 N; ˛ 2 m g is a -locally finite basis for the topology of P . Now suppose that S is determined locally. (vi))(i) Let f ˛ g˛2 be a locally finite S -partition of unity subordinated to the open P covering fW; P n Ag of P . Let 1 D f˛ 2 I suppo ˛ W g and put ' D ˛21 ˛ . As the sum is locally finite and S is determined locally, ' 2 S. Further, suppo ˛ P n A for ˛ 2 n 1 and hence Obviously P suppo ' W .P '.x/ D ˛21 ˛ .x/ D ˛2 ˛ .x/ D 1 for x 2 A. (vii))(iii) By (vii) there is a family f'n˛ I n 2 N; ˛ 2 n g S such that S V D n2N;˛2n suppo 'n˛ and fsuppo 'n˛ g˛2n is locally finite for each n 2 N. P 2 . As the sum is locally finite and S is a ring determined locally, Put 'n D ˛2n 'n˛ S 'n 2 S . Further, suppo 'n D ˛2n suppo 'n˛ , hence (iii) follows. t u

414

Chapter 7. Smooth approximation

Corollary 50 ([BF]). Let X be a separable normed linear space and S a partition ring on X such that for every f 2 S , a 2 R, and b 2 X the function g.x/ D f .ax C b/ belongs to S . Then X admits locally finite and -uniformly discrete S -partitions of unity if and only there is a bump function in S . Proof. Suppose ' 2 S is a bump function. Since S is stable under shifts and scaling, we may suppose that '.0/ > 0 and suppo ' UX . By the continuity of ' there is 0 < r < 1 such that ' > 0 on U.0; r/. We show that the statement (iii) in Lemma 49 is satisfied. So let us suppose that V W X are bounded open sets satisfying dist.V; P n W / D ı > 0. By the Lindelöf Sproperty of the set V there is a countable subset fxn gn2N of V such D ' .x xn /=ı . Then 'n 2 S and that VS n2N U.xn ; ır/. S We put 'n .x/ S V n2N U.xn ; ır/ n2N suppo 'n n2N U.xn ; ı/ W . The reverse implication is clear for example from Lemma 49(vii). t u Definition 51. Let Y be a class of normed linear spaces and be a class of mappings from a metric space P into spaces from Y. We say that is an approximation class if is determined locally, S.P I R/ is a partition ring, f C g 2 whenever f; g 2 map into the same space, for every Y 2 Y, every y 2 Y , and every function ' 2 S.P I Œ0; 1/ the mapping x 7! '.x/y belongs to . Notice that the second property implies that the class Y must contain at least R. The following theorem goes back to Kazimierz Kuratowski around 1922. Theorem 52. Let P be a metric space and an approximation class on P . Then the following statements are equivalent: (i) P admits locally finite -partitions of unity. (ii) For any convex subset C of a normed linear space of class Y, any f 2 C.P I C /, and any " 2 C.P I RC / there is g 2 S.P I C / such that kf .x/ g.x/k < ".x/ for every x 2 P . (iii) For any 1-Lipschitz f W P ! Œ0; 1 and any " > 0 there is g 2 S.P I R/ such that jf gjP ". Proof. (i))(ii) For each x 2 P find r.x/ > 0 such that ".x/ ".x/ and kf .y/ f .x/k < for each y 2 U.x; r.x//. ".y/ > 2 2 It follows that kf .y/

f .x/k < ".y/ for each y 2 U.x; r.x//.

(14)

By (i) there is a locally finite -partition of unity f ˛ g˛2 on P subordinated to fU.x; r.x//I x 2 P g. For each ˛ 2 let U˛ D U.x˛ ; r.x˛ // be such that suppo ˛ U˛ . Define X g.x/ D (15) ˛ .x/f .x˛ /: ˛2

Section 5. Approximation of continuous mappings and partitions of unity

415

By the properties of each mapping ˛ f .x˛ / belongs to as well as finite sums of these mappings. Since the sum in the definition of g is locally finite and is determined locally, g 2 . Moreover, as g.x/ is a convex combination of points from C for every x 2 P , g 2 S.P I C /. Finally, choose any x 2 P . Then

X X

kf .x/ g.x/k D f .x˛ / f .x˛ /k ˛ .x/ f .x/ ˛ .x/kf .x/

˛2 ˛2 W x2U˛ X < ".x/ ˛ .x/ D ".x/; ˛2 W x2U˛

where the last inequality follows from (14). (ii))(iii) is obvious. (iii))(i) We show that the condition (ii) in Lemma 49 is satisfied. Let V W P be bounded open sets satisfying dist.V; P n W / > 3ı for some 0 < ı < 31 . Put f .x/ D minfdist.x; P n W /; 1g. By (iii) there is g 2 S.P I R/ such that jf gjP ı. Then g ı on P n W and g > 2ı on V . By property (iii) of a partition ring there is ' 2 S.P I R/ such that ' D 0 on P n W and ' > 0 on V . t u Next we show how to construct smooth partitions of unity on various classes of Banach spaces. In the following theorem the mapping ˚ introduces a “coordinate system” on X, while the mappings PF serve as the “projections” associated to this “coordinate system”. The requirement is that for every x 2 X if we take “large coordinates of x”, then the associated “projection” approximates x well. Theorem 53 ([Hay2]). Let X be a normed linear space that admits a C k -smooth bump function, k 2 N [f1g. Let be a set and ˚ W X ! c0 . / a continuous mapping such that for every 2 the function e B ˚ is C k -smooth on the set where it is non-zero. For each finite F let PF 2 C k .XI X/ be such that the space span PF .X/ admits locally finite C k -partitions of unity. Assume that for each x 2 X and each " > 0 there exists ı > 0 such that kx PF .x/k < " if we set F D f 2 I j˚.x/. /j ıg. Then X admits locally finite and -uniformly discrete C k -partitions of unity. Proof. Denote by F a set of all finite subsets of (including an empty set). For any q 2 RC let q 2 C 1 .RI Œ0; 1/ be such that q .t/ D 0 for jt j q, 0 < q .t/ < 1 for q q q C 2 < jtj < q, and q .t/ D 1 for t 2 Œ 2 ; 2 (Fact 2). For each F 2 F and q; r 2 R we define a function 'F ;q;r W c0 . / ! R by Y Y 'F ;q;r .x/ D 1 2r x. / q x. / :

2F

2 nF

For x 2 c0 . / let H D f 2 I jx. /j q4 g. Then H 2 F and jy. /j < q2 for y 2 U.x; q4 /, 2 n H . Thus Y Y 1 2r y. / q y. / (16) 'F ;q;r .y/ D

2F

2H nF

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Chapter 7. Smooth approximation

for y 2 U.x; q4 /, which implies that the function 'F ;q;r is LFC-fe g 2 and also 'F ;q;r 2 C 1 .c0 . /I Œ0; 1/. It is easy to check that suppo 'F ;q;r D WF ;q;r , where n o WF ;q;r D x 2 c0 . /I min jx. /j > r; sup jx. /j < q :

2F

2 nF

Notice that dist.WF ;q;r ; WH;q;r / r q whenever F; H 2 F , F ¤ H , and r > q. Therefore the family fWF ;q;r I F 2 F ; q; r 2 Q; r > q > 0g is -uniformly discrete. Further, 'F ;q;r B ˚ 2 C k .X/. This follows from the formula (16), the fact that s B e B ˚ 2 C k .X/ for each 2 and s 2 RC , and the continuity of ˚. Note that the ring C k .Z/, where Z is a normed linear space, is a partition ring determined locally. By the hypothesis and Lemma 49 for each F 2 F there exists a -discrete basis VF for the topology of span PF .X/ formed by the sets in the family fsuppo f I f 2 C k .span PF .X//g. Further, as X admits a C k -smooth bump function, the family fsuppo f I f 2 C k .X/g contains a neighbourhood basis of 0, say fUm gm2N . In X consider the family ˚ 1 ˚ .WF ;q;r / \ PF 1 .V / \ .Id PF / 1 .Um /I F 2 F ; q; r 2 Q; 0 < q < r; V 2 VF ; m 2 N : Using the continuity of the mapping ˚ it is easy to verify that this is a -discrete (and in particular -locally finite) subfamily of fsuppo f I f 2 C k .X/g (notice that ˚ 1 .WF ;q;r / D suppo 'F ;q;r B ˚). To finish the proof using Lemma 49 we need to show that this family forms a basis for the topology of X . To this end choose x 2 X and " > 0. Let m 2 N be such that Um U.0; 6" / and further let ı > 0 be such that x

PF .x/ 2 Um

when we set F D f 2 I j˚.x/. /j ıg. Because ˚.x/ 2 c0 . /, there exist q; r 2 Q with 0 < q < r < ı satisfying j˚.x/. /j < q whenever 2 n F . Thus x 2 ˚ 1 .WF ;q;r /. Since VF is a basis for the topology of span PF .X/, there exists V 2 VF such that " : PF .x/ 2 V U PF .x/; 3 It follows that x 2 ˚ 1 .WF ;q;r / \ PF 1 .V / \ .Id PF / 1 .Um /. If y is any other member of this set, then we have kPF .x/ PF .y/k < 3" because PF .y/ 2 V , while kPF .y/ yk < 3" because y PF .y/ 2 Um . Thus kx yk < ", which is what we wanted to prove. t u Corollary 54 ([Hay2]). Let X be a normed linear space that admits a C k -smooth bump, k 2 N [ f1g. Let be a limit ordinal and let fP˛ 2 C k .XI X/I ˛ < g be an equi-continuous family of mappings having the property that for every x 2 X the mapping Px W Œ0; ! X defined by Px .˛/ D P˛ .x/ for ˛ < , Px ./ D x, is continuous. If for each ˛ < the space span P˛ .X/ admits locally finite C k -partitions of unity, then so does X.

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417

Proof. Since X admits a C k -smooth bump, there exist a function h 2 C k .X I Œ0; 1/ and > 0 such that h.x/ D 0 for kxk , while h.x/ D 1 whenever kxk 1. We set D Œ0; / N and define ˚ W X ! `1 . / by ˚.x/.˛; n/ D 2

n

h 2n .P˛C1 .x/

P˛ .x// :

We note that the enlargement of the index set by the factor N would not be necessary if we knew that h is zero only at the origin. Such function however may not exist, cf. Theorem 5.161. Given x 2 X and " > 0 we fix m 2 N such that 2 m < " and note that the quantity kP˛C1 .x/ P˛ .x/k can exceed 2 m 2 only for ˛ in some finite H Œ0; /. Indeed, otherwise there would be an increasing sequence f˛n g of ordinals such that kP˛n C1 .x/ P˛n .x/k > 2 m 2 , which contradicts the continuity of Px at the ordinal sup ˛n D lim ˛n . Thus 0 ˚.x/. / < " except when 2 K D H f0; 1; 2; : : : ; m 1g. This shows that ˚ maps into c0 . /. Furthermore, by the equi-continuity of fP˛ g˛ ˛. For finite subsets B of K we set VB D t 2B V t . We shall say that a finite subset A of K is admissible if s … V t whenever s; t 2 A, s ¤ t. Suppose that H is a closed subset of K. We claim that there is a unique admissible set A with the property that A H VA . If H D ;, then this is obviously satisfied with A D ; and no other. For a non-empty H we construct an admissible A with the required property. Let ˛0 D maxf˛I H \ K .˛/ ¤ ;g; thus H \ K .˛0 / is a non-empty

Section 5. Approximation of continuous mappings and partitions of unity

419

finite set, which we shall call A0 . If H VA0 , we set A D A0 and stop. Otherwise, we set H1 D H n VA0 , which is a closed set, ˛1 D maxf˛I H1 \ K .˛/ ¤ ;g, and A1 D H1 \ K .˛1 / , and repeat the procedure. In this way we construct a decreasing ˛0 > ˛1 > > ˛l of ordinals, and finite sets (and so necessarily finite) sequence Aj D H n .VA0 [ [ VAj 1 / \ K .˛j / , j D 1; : : : ; l, such that H VA0 [ [ VAl . By construction, the set A D A0 [ [ Al is admissible and A H VA . Now suppose that there are two different admissible sets B and D satisfying B H VB and D H VD . Let ˇ D maxf˛I B \ K .˛/ ¤ D \ K .˛/ g. Without loss of generality we may assume that there is u 2 .B n D/ \ K .ˇ / . Since u 2 B H VD , there is s 2 D such that u 2 Vs . Because u 2 Vs \ K .ˇ / n D and K .˛.s// \ Vs D fsg, it must be that ˛.s/ > ˇ. By the maximality of ˇ we have B \ K ˛.s/ D D \ K ˛.s/ , whence s 2 B, which contradicts the admissibility of B. We now pass to the construction of partitions of unity. We shall proceed by transfinite induction on the height of K. Let be an ordinal satisfying ht.K/ D C 1, i.e. K ./ is finite and non-empty. If D 0, then C.K/ is finite-dimensional, and so has the required partitions of unity for example by Corollary 50. For > 0 we assume inductively that if L is a compact space with ht.L/ < C 1 and such that C.L/ has a C k -smooth bump function, then C.L/ admits locally finite C k -partitions of unity. To show that C.K/ also admits locally finite C k -partitions of unity it will be enough to construct the partitions of unity on the finite-codimensional subspace Z D ff 2 C.K/I f .t/ D 0 for all t 2 K ./ g. (Using Lemma 49 it is not difficult to ascertain that whenever some normed linear spaces Y and Z admit locally finite C k -partitions of unity, then so does the space Y ˚ Z.) To this end we construct the mappings required by Theorem 53. Put D Q A, where Q is the set of all triples .; ; / 2 Q3 with 0 < < < , and A consists of the admissible subsets A of K for which A\K ./ D ;. Let nW Q ! N be some one-to-one mapping, ˇ; be as in Lemma 6, and 2 C 1 .RI Œ0; 1/ be such that 1 .f0g/ D Œ 1; 1 (Fact 2). We define ˚ W Z ! `1 . / by Y f .t/ 1 ˇ; .KnVA f / : ˚.f /.; ; ; A/ D n.; ; / t2A

Notice that since the set VA is clopen, KnVA 2 C.K/. We shall show that ˚ is actually a continuous mapping into c0 . /. To do so fix a function f 2 Z and " > 0. The quantity n.; ; / 1 is greater than " only for .; ; / in some finite ˚ subset R of Q. Let us put D min 21 . /I .; ; / 2 R > 0. For a given triple ˚.; ; / 2 R we have ˚.g/.; ; ; A/ D 0 for each g 2 U.f; / unless A t 2 KI jf .t/j 12 . C / VA , which can happen for at most one set A 2 A, as we have shown earlier. It follows easily that ˚ is a continuous mapping into c0 . /. Moreover, f 7! KnVA f and f 7! f .t/ are bounded linear operators, whence each e B ˚, 2 , is C k -smooth. Finally we define the associated projections PF W Z ! Z as follows: if FS is a finite subset with elements .j ; j ; j ; Aj /, j D 1; : : : ; m, we set V .F / D jmD1 VAj ,

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Chapter 7. Smooth approximation

and define PF .f / D V .F / f . Because V .F / is a clopen set with V .F / \ K ./ D ;, PF is a well-defined linear projection of norm 1, the image PF .Z/ is isometric to the space C.V .F //, and ht.V .F // < C 1. Hence by the inductive hypothesis PF .Z/ admits locally finite C k -partitions of unity. It only remains to check the required relation between ˚ and the projections. So let f 2 Z and " > 0 be given. Let H D ft 2 KI jf .t/j "g and let A be the admissible set such that A H VA . Then A 2 A, since H \ K ./ D ; by the definition of Z. There is .; ; / 2 Q satisfying kKnVA f k1 < < < ". It follows that ˚.f /.; ; ; A/ > 0. We set ı D ˚.f /.; ; ; A/, F D f 2 I j˚.x/. /j ıg, and note that V .F / VA . So kf

PF .f /k1 D kKnV .F / f k1 kKnVA f k1 < ":

For the proof of the case of X being WCG see [McL].

t u

Theorem 56 ([JTZ]). Let X be a super-reflexive Banach space. Then X admits locally finite and -uniformly discrete C 1;˛ -partitions of unity for some ˛ 2 .0; 1. For the proof we need two auxiliary statements. Lemma 57. Let be a non-empty set, p; q 2 Œ1; C1/, and r 2 N odd such that rq p. Then the Mazur mapping ˚r W `p . / ! `q . / defined by ˚r .x/. / D x. /r is a one-to-one r-homogenous polynomial with k˚r k D 1. Proof. Define M W `p . /r ! `1 . / by M.x1 ; : : : ; xr /. / D x1 . / xr . /. Obviously M is an r-linear mapping and ˚r .x/ D M.x; : : : ; x/. Hence ˚r is an r-homogeneous polynomial. Moreover, for any x 2 `p . /, !1 !1 ! q1 X ˇˇ x. / ˇˇrq q X ˇˇ x. / ˇˇp q X ˇ ˇ ˇ ˇ D kxkpr kxkpr D kxkpr ; jx. /r jq ˇ kxk ˇ ˇ kxk ˇ p p

2

2

2

and hence ˚r maps into `q . / and k˚r k 1. Further, ˚r .e / D e , which implies that k˚r k D 1. As r is odd, ˚r is obviously one-to-one. t u Proposition 58. For every super-reflexive Banach space X there is ˛ 2 .0; 1 and a homeomorphic embedding of the space X into `2 . / for some set such that 2 C 1 .XI `2 . // with D ˛-Hölder on bounded sets. Proof. Let kk be an equivalent norm on X which is uniformly rotund and its derivative is ˛-Hölder on the unit sphere for some ˛ 2 .0; 1, see e.g. [DGZ, Proposition IV.5.2]. It is easy to check that then the function kk2 2 C 1 .X/ with its derivative ˛-Hölder on bounded sets. Further, there are p 2 .1; C1/ and a one-to-one bounded linear operator T W X ! `p ./ for some set . This a folklore result that follows from the results of Robert Clarke James [Ja2] and Joram Lindenstrauss [AL], see also [JTZ]. Let r 2 N be odd and satisfy 2r p. By Lemma 57 the Mazur mapping ˚r W `p ./ ! `2 ./ is a one-to-one continuous polynomial, in particular D˚r is Lipschitz on bounded sets and hence also ˛-Hölder on bounded sets. Assume that 0 … and put D [f0g. Define

Section 5. Approximation of continuous mappings and partitions of unity

421

W X ! `2 . / by .x/ D ˚r .T .x//; kxk2 . Then obviously is one-to-one and 2 C 1 .X I `2 . // with D ˛-Hölder on bounded sets. It remains to show that 1 is continuous. Let xn ; x 2 X be such that .xn / ! .x/. Then kxn k ! kxk and T .xn /. / ! T .x/. / for all 2 . It follows that fxn g is w bounded and hence T .xn / ! T .x/. Using the weak sequential compactness of closed w balls in X and the fact that T is one-to-one, we obtain that xn ! x. The weak lower semi-continuity of kk implies kx Cxk lim infkxn Cxk lim supkxn Cxk 2kxk, t hence limkxn C xk D 2kxk, and by the uniform rotundity we finally get xn ! x. u Proof of Theorem 56. First we recall the following easy fact used several times in this proof: A bounded Lipschitz mapping is ˛-Hölder for every ˛ 2 .0; 1. Let S be the set of functions from C 1;˛ .X / that are bounded and have bounded derivative. Noting that functions in S are Lipschitz, Proposition 1.129 implies that S is a ring. We show that S is a partition ring. Property (i): Let ff g 2 S be such that fsuppo f g 2 is uniformly discrete. Let g D c f for some suitable constants c ¤ 0 chosen so that jgP

jX 1, kDg kX 1, and Dg is ˛-Hölder with constant 1 for all 2 . Put g D 2 g . It is obvious that g 2 C 1 .X/ and both g and Dg are bounded by 1. To see that Dg is ˛-Hölder, pick any x; y 2 X . Suppose there are ; ˇ 2 , ¤ ˇ, such that x 2 suppo g and y 2 suppo gˇ . Then kDg.x/ Dg.y/k D kDg .x/ Dgˇ .y/k D kDg .x/ Dg .y/k C kDgˇ .x/ DgS yk˛ . The other cases are ˇ .y/k 2kx obvious. So g 2 S and clearly suppo g D 2 suppo f . Property (ii): Let f 2 S and suppo f D U1 [ U2 , where U1 ; U2 X are open with d D dist.U1 ; U2 / > 0. Consider the function g D U1 f . Then g D f on an open set X nU2 and g D 0 on some neighbourhood of U2 , hence g 2 C 1 .X/ and both g and Dg are bounded, say by M . To see that Dg is ˛-Hölder, pick any x; y 2 X. Suppose that x 2 U1 and y 2 U2 . Then kDg.x/ Dg.y/k D kDg.x/ 0k M dM˛ kx yk˛ . The other cases are obvious and so g 2 S. Property (iii): Let f 2 S and " > 0. Let 2 C 1 .RI Œ0; 1/ be such that .t/ D 0 for t " and .t/ D 1 for t 2" (Fact 2). Put g D B f . Since 0 is bounded and Lipschitz and f is Lipschitz, g 2 S by Proposition 1.128, and has the properties required in (iii). To finish the proof we show that (ii) of Lemma 49 is satisfied. Let be the embedding of X into `2 . / from Proposition 58. Let W X be an arbitrary bounded open set. Then .W / is open in .X/ and so there is an open U `2 . / such that .W / D U \ .X/. By Proposition 44 there is f 2 C 1;1 .`2 . // such that suppo f D U . Put ' D f B . The mapping D is Hölder on bounded sets, therefore bounded on bounded sets. Consequently, is Lipschitz on bounded sets, hence bounded on bounded sets. Further, Df is Lipschitz and hence bounded on bounded sets. Therefore D' is bounded and ˛-Hölder on bounded sets by Proposition 1.128. Finally, as is one-to-one, suppo ' D W , which is a bounded set and so D' is globally ˛-Hölder and bounded. So we have found a function ' 2 S for which suppo ' D W . t u

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We remark that the last part of the proof, namely the fact that for any open W X there is ' 2 C 1;˛ .X/ with suppo ' D W , can be shown directly without embedding X into `2 . /. The proof is similar in spirit to that of Proposition 44 but technically much more involved, see [Cep, Corollary 2].

6. Non-linear embeddings into c0 . / We begin by giving a characterisation of the existence of C k -partitions of unity on a normed linear space X by means of non-linear componentwise C k -smooth embeddings of X into c0 . /. This result is not essential in our approach to smooth partitions of unity, but it nicely completes the picture in view of the main result of this section: In our aim towards the approximation of Lipschitz functions by smooth functions preserving the Lipschitz property we introduce an important technique of supremal partitions and characterise it again by means of bi-Lipschitz componentwise C k -smooth embeddings into c0 . /. We show that every separable normed linear space with a C k -smooth Lipschitz bump admits C k -smooth Lipschitz sup-partitions of unity (and a bi-Lipschitz componentwise C k -smooth embedding into c0 ). It is useful to explicitly state the following fact. Fact 59. For any set the space c0 . / admits locally finite and -uniformly discrete C 1 -smooth and LFC-fe g 2 partitions of unity. Proof. The family fWF ;q;r I F 2 F ; q; r 2 Q; r > q > 0g from the proof of Theorem 53 is a -uniformly discrete basis for the topology of c0 . / such that WF ;q;r D suppo 'F ;q;r and each 'F ;q;r is C 1 -smooth and LFC-fe g 2 , so we can use Lemma 49. It is also instructive to notice that the uniform refinements from Fact 62 for r D n1 , n 2 N, form a -locally finite basis for the topology of c0 . /. Thus combined with the following observation it gives another proof: For any x 2 c0 . / and r > 0 there is ' 2 C 1 .c0 . // which is LFC-fe g 2 and such that suppo ' D U.x; r/. Indeed, it Q suffices to take '.y/ D 2 y. / x. / , where 2 C 1 .R; Œ0; 1/ is such that .t / D 1 whenever jtj 2r and .t/ D 0 if and only if jt j r. t u Proposition 60 ([Tor]). Let X be a normed linear space and k 2 N0 [ f1g. The space X admits locally finite C k -partitions of unity if and only if there are a set and a homeomorphism ˚ W X ! c0 . / such that e B ˚ 2 C k .X/ for every 2 . k Proof. ) By Lemma S49 there is a basis V fsuppo f W f 2 C .X/g for the topology of X such that V D n2N Vn , where each Vn is discrete and Vn \Vm D ; for m ¤ n. For every V 2 V we choose 'V 2 C k .XI Œ0; 1/ such that V D suppo 'V . We put D V and define ˚ W X ! `1 . / by

˚.x/.V / D

1 'V .x/; n

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423

where n 2 N is the uniquely determined number for which V 2 Vn . The mapping ˚ is one-to-one. Indeed, if x; y 2 X, x ¤ y, then there is V 2 V such that x 2 V , y ¤ V . It follows that 'V .x/ > 0, while 'V .y/ D 0, and consequently ˚.x/.V / ¤ ˚.y/.V /. Moreover, ˚ is a continuous mapping into c0 . /. To see this, for a given x 2 X and " > 0 we find n0 2 N such that n10 < ". Then 0 ˚.x/.V / < " and also j˚.x/.V / ˚.y/.V /j < " whenever y 2 X and V 2 Vn for n n0 . Further, by the discreteness of the families Vn , there is a neighbourhood U of x such that U meets only finitely many sets in W D V1 [ [ Vn0 , say V1 ; : : : ; Vm , m 2 N0 . Then ˚.y/.V / D 0 whenever y 2 U and V 2 W n fV1 ; : : : ; Vm g. It follows that ˚ maps into c0 . /. Finally, using the continuity of 'V1 ; : : : ; 'Vm we can find a neighbourhood W U of x such that j˚.x/.Vn / ˚.y/.Vn /j < " whenever y 2 W , n 2 f1; : : : ; mg. Thus k˚.x/ ˚.y/k1 < " for each y 2 W . This shows the continuity of ˚. To show that ˚ 1 continuous fix x 2 X and " > 0. Since V is a basis for the topology of X , there is V 2 V such that x 2 V U.x; "/. Let n 2 N be such that V 2 Vn and choose some 0 < ı < n1 'V .x/. Suppose y 2 X is such that k˚.x/ ˚.y/k1 < ı. Then n1 'V .x/ n1 'V .y/ < ı and hence 'V .y/ > 0. It follows that y 2 V U.x; "/. ( Denote S D ff 2 C 1 .c0 . //I f is LFC-fe g 2 g. By Fact 59 and Lemma 49 there is a -locally finite basis V for the topology of c0 . / consisting of the sets suppo f with f 2 S . Using the homeomorphism ˚ we pull this basis back onto X. Moreover, if f 2 S , then f B ˚ 2 C k .X/ (Lemma 5.81). Lemma 49 now finishes the proof. t u In particular, when k D 0, the previous result together with Lemma 8 implies that any normed linear space is homeomorphic to a subset of c0 . / for some set . This is no longer true for uniform homeomorphisms: the space C.Œ0; !1 / is not uniformly homeomorphic to a subset of any c0 . /. This is a result of Jan Pelant, [PHK]. However, for any separable normed linear space X there is a bi-Lipschitz homeomorphism ˚ W X ! c0 . This result of Israel Aharoni, [Ah], can be recovered from Corollary 65 when k D 0. Definition 61. Let X be a set. A collection f ˛ g˛2 of functions on X is called a supremal partition (sup-partition) if ˛ W X ! Œ0; 1 for all ˛ 2 , there is a Q > 0 such that sup˛2 ˛ .x/ Q for each x 2 X , for each x 2 X and for each " > 0 the set f˛ 2 I ˛ .x/ > "g is finite (or in other words . ˛ .x//˛2 2 c0 ./). If in the second property Q D 1, then f ˛ g˛2 is called a sup-partition of unity. Let U be a covering of X. We say that the sup-partition f ˛ g˛2 is subordinated to U if fsuppo ˛ g˛2 refines U. We say that f ˛ g˛2 is locally finite if fsuppo ˛ g˛2 is locally finite. Notice that in fact in the above definition for each x 2 X there is ˛ 2 such that ˛ .x/ Q.

424

Chapter 7. Smooth approximation

For a metric space P we denote by U.r/ D fU.x; r/I x 2 P g the full uniform covering of P . Fact 62. Let be an infinite set, r > 0, and 0 < ı < 2r . There is an open locally finite uniform refinement V D fV g 2 of the uniform covering U.r/ of c0 . / such that U. 2r ı/ refines V. Moreover, V is formed by the translates of the open ball U.0; r ı/. Further, there is a C 1 -smooth LFC-fe g 2 and . 2r Cı/-Lipschitz locally finite sup-partition of unity f g 2 on c0 . / subordinated to U.r/. Proof. Notice that by the homogeneity it suffices to prove all the statements only for r D 1. Let fa g 2 be the set of all vectors in c0 . / with coordinates in Z. (Notice that the cardinality of such set is j j and so we may index its points by .) We claim that V D fU.a ; 1 ı/g 2 is the desired refinement. Clearly, V is an open refinement of U.1/. To see that it is locally finite, pick any x 2 c0 . / and find a finite F such that jx. /j < 2ı whenever 2 n F. Suppose that ˛ 2 is such that y 2 U.a˛ ; 1 ı/ for some y 2 U.x; 2ı /. If 2 n F , then ja˛ . /j ja˛ . / y. /j C jy. / x. /j C jx. /j < 1 ı C 2ı C 2ı D 1 and so a˛ . / D 0. From jx. / a˛ . /j < 1 2ı and a˛ . / 2 Z it follows that there are at a˛ . /ˇ for each 2 F . From this we can conclude that ˇ most two possibilities for ˇf˛I U.a˛ ; 1 ı/ \ U.x; ı / ¤ ;gˇ 2jF j . 2 Finally, we show that U. 12 ı/ refines V . Choose any x 2 c0 . / and find ˇ 2 such that kx aˇ k 12 . This is always possible, since there is a finite F such that jx. /j < 21 whenever … F , and so aˇ . / D 0 for such . Suppose ´ 2 U.x; 12 ı/. Then kaˇ ´k1 kaˇ xk1 C kx ´k1 < 21 C 12 ı D 1 ı, which implies that U.x; 12 ı/ U.aˇ ; 1 ı/. To construct the supremal partition of unity subordinated to U.1/ find " > 0 and ı ı < 2 C 0 < < 12 such that 0 < 1= 1 1C" and .1 C "/ 2 C 2 4 2 2 C ı. Let W D fU.a ; 1 /g 2 be the locally finite refinement of U.1/ from the first part of the proof such that U. 12 / refines W . Further, let kk be an equivalent C 1 -smooth LFC-fe g 2 norm kk on c0 . / such that kxk1 kxk .1 C "/kxk1 for all x 2 c0 . /. (To construct P such a norm, take for example the Minkowski functional 1 of the set fx 2 c0 . /I

2 '.x. // 1g, where ' 2 C .R/, ' is convex 1 1 and even, '.1/ D 1, and '.t/ D 0 for t 2 Œ 1C" ; 1C" .) For each 2 we put ı 1 a k , where q 2 C .RI Œ0; 1/, q is 2 C 2 -Lipschitz, q.t/ D 0

.x/ D q kx for t 1 , and q.t/ D 1 for t 1C" 2 . The collection f g 2 is a locally finite sup-partition of unity. Indeed, clearly suppo U.a ; 1 / for each 2 . It also follows that the set f 2 I .x/ > 0g is finite for each x 2 X. Further, fix any x 2 X . There is an ˛ 2 such that U.x; 21 / U.a˛ ; 1 /, which gives kx a˛ k1 12 . Hence kx a˛ k .1 C "/kx a˛ k1 1C" 2 , which in turn implies that ˛ .x/ D 1. As the function q is 2 C 2ı -Lipschitz and the function kk is .1 C "/-Lipschitz (with respect to the norm kk1 ), the functions are .2 C ı/-Lipschitz according to the choice of ". The rest of the properties of the functions is obvious. t u

Section 6. Non-linear embeddings into c0 . /

425

Theorem 63. Let X be a normed linear space, an infinite set, and k 2 N0 [ f1g. Then the following statements are equivalent: (i) There is M 2 R such that there is a C k -smooth and M -Lipschitz sup-partition f' g 2 on X subordinated to U.1/. (ii) There is M 2 R such that there is a C k -smooth and M -Lipschitz locally finite sup-partition of unity f' g 2 on X subordinated to U.1/. (iii) X is uniformly homeomorphic to a subset of c0 . / and for each " > 0 there is K > 0 such that for each 1-Lipschitz function f W X ! Œ0; 1 there is a K-Lipschitz function g 2 C k .X/ such that jg f jX ". (iv) There is a bi-Lipschitz homeomorphism ˚ W X ! c0 . / such that the component functions e B ˚ 2 C k .X/ for every 2 . Proof. (ii))(i) is obvious. (i))(iv) Let Q be the quantity from the definition of the sup-partition. Then there is ˇ 2 such that 'ˇ .0/ Q. By scaling and composing 'ˇ with a suitable function we construct a C -Lipschitz function h 2 C k .XI Œ0; 1/ such that h D 0 on B.0; r/ and h D 1 outside U.0; 1/ for some constants C; r 2 R, r > 0. (We may for example choose r such that Q 2M r > 0 and take h.x/ D q.'ˇ .2x//, where q 2 C 1 .R/, q is Lipschitz, q.Œ0; 1/ D Œ0; 1, q.0/ D 1, and q.s/ D 0 for s Q 2M r.) Choose t > 1 and for each n 2 Z and 2 define functions ' n 2 C k .X/ by x x ' n .x/ D t n ' n h n : t t The properties of the functions ' and h guarantee that each ' n is .M C C /-Lipschitz. Let d W Z ! be some one-to-one mapping and define ˚ W X ! R by ˚.x/.˛/ D ' n .x/ if ˛ D d.n; / for some n 2 Z, 2 ; ˚.x/.˛/ D 0 otherwise. We show that ˚ actually maps into c0 . /. Choose an arbitrary x 2 X and " > 0. There is n0 2 Z such that t n < " for all n < n0 and n1 2 Z such that kxk rt n for all n > n1 . It follows that j' n .x/j < " for all n < n0 and 2 , and, by the properties of h, ' n .x/ D 0 for all n > n1 and 2 . As for each n0 n n1 , ' .x=t n / > "=t n only for finitely many 2 , we can conclude that ˚ W X ! c0 . /. Since each ' n is .M C C /-Lipschitz, the mapping ˚ is .M C C /-Lipschitz as well. To prove that ˚ is one-to-one and ˚ 1 is Lipschitz too, choose any two points x; y 2 X, x ¤ y, and find m 2 Z such that 2t m kx yk < 2t mC1 . Without loss of generality we may assume that kxk t m . Then h.x=t m / D 1 and so there is

2 such that ' m .x/ Qt m . Now suppose that ´ 2 X is such that ' m .´/ > 0. As suppo ' U.w; 1/ for some w 2 X, k txm t´m k < 2 and consequently kx ´k < 2t m . But this means that ' m .y/ D 0 and therefore Q k˚.x/ ˚.y/k1 j' m .x/ ' m .y/j D ' m .x/ Qt m > kx yk: 2t (iv))(ii) Let A; B > 0 be such that Akx yk k˚.x/ ˚.y/k1 Bkx yk for all x; y 2 X. By Fact 62 there are C > 0 and a C 1 -smooth LFC-fe g 2 and C -Lipschitz locally finite sup-partition of unity f g 2 on c0 . / subordinated to U. A 2 /. Putting ' D B ˚, f' g 2 is a BC -Lipschitz locally finite sup-partition of unity subordinated to U.1/. Moreover, each ' is C k -smooth by Lemma 5.81.

426

Chapter 7. Smooth approximation

(ii))(iii) We already know that (iv) holds and from this the first part of (iii) follows immediately. To prove the second part of (iii), let " > 0. The basic idea of the proof is that Lipschitz functions are stable under the operation of pointwise supremum. To preserve the smoothness, we will use a “smoothened supremum”, or an equivalent smooth norm on c0 . /. Let kk be an equivalent C 1 -smooth LFC-fe g 2 norm on c0 . / and let C > 0 be such that kxk1 kxk C kxk1 for all x 2 c0 . /. We will show that K D 4C 3 M=" satisfies our claim. By adding the constant 1 we may and do assume that f maps into Œ1; 2. Put ı D C" and .x/ D ' . xı / for all x 2 X, 2 . It follows that f g 2 is a C k -smooth and M=ı-Lipschitz sup-partition of unity subordinated to U.ı/. Recall that . .x// 2 2 c0 . / for each x 2 X. For each 2 there is a point x 2 X such that suppo U.x ; ı/. The boundedness of the function f guarantees that also f .x / .x/ 2 2 c0 . / for each x 2 X. Hence we can define g W X ! R by

f .x / .x/

2

: g.x/ D

.x/

2 As k. .x//k k. .x//k1 D sup .x/ D 1 for each x 2 X, (17)

2

the function g is well-defined on all of X. The mapping x 7!

.x/ and, by the boundedness of f , also the mapping x 7! f .x / .x/ are Lipschitz mappings from X into c0 . / n U.0; 1/. (Notice that for each x 2 X there is 2 such that .x/ D 1 and f .x / .x/ 1.) Since kk is C 1 -smooth and depends locally on finitely many coordinates away from the origin, and since 2 C k .X/ and f .x / 2 C k .X/ for each 2 , using Lemma 5.81 we infer that g 2 C k .X/. Using the facts that f maps into Œ1; 2, the functions are M=ı-Lipschitz and map into Œ0; 1, and kk is C -Lipschitz as a function on .c0 . /; kk1 /, we obtain that the

function x 7! f .x / .x/ is 2CM=ı-Lipschitz and bounded by 2C . Similarly,

the function x 7! is CM=ı-Lipschitz and bounded below by 1. It follows

.x/ that the function g is K-Lipschitz. Finally, to show that g approximates f, pick any x 2 X . Applying successively the inequality (17) and the facts that suppo U.x ; ı/ and f is 1-Lipschitz, we obtain ˇ

ˇˇ ˇ f .x / .x/

ˇ

.x/

ˇ ˇ f .x/ jg.x/ f .x/j D ˇ

.x/ ˇ ˇ .x/

.f .x / f .x// .x/

C .f .x / f .x// .x/ 1

.x/ ˚ ˚ D C sup jf .x / f .x/j .x/ D C sup jf .x / f .x/j .x/

2

˚ C sup kx

2 x2U.x ;ı/

xk C ı D ":

2 x2U.x ;ı/

Section 6. Non-linear embeddings into c0 . /

427

(iii))(ii) Let ˚ be the uniform homeomorphism and let > 0 be such that k˚ 1 .x/ ˚ 1 .y/k < 1 whenever x; y 2 ˚.X/ are such that kx yk < 2. Take an open locally finite uniform refinement of the uniform covering U./ of c0 . / from Fact 62 and pull it back onto X using ˚. We obtain an open locally finite uniform refinement V D fV g 2 of the covering U.1/ of X. Let 0 < ı 1 be such that U.ı/ refines V. For each 2 we define the function f W X ! Œ0; 1 by f .x/ D minfdist.x; X n V /; ıg. Choose some 0 < < 2ı . For each 2 the function f is 1-Lipschitz and so, by (iii), there is a K-Lipschitz function g 2 C k .X/ such that jg f jX . Let q 2 C k .RI Œ0; 1/ be a C -Lipschitz function for some C 2 R, such that q.t/ D 0 for t and q.t/ D 1 for t ı . Finally, we let ' .x/ D q.g .x// for each

2 . Clearly, each function ' belongs to C k .XI Œ0; 1/ and is M -Lipschitz, where M D CK. Further, for any x 2 X there is ˛ 2 such that U.x; ı/ V˛ , hence f˛ .x/ D ı and consequently '˛ .x/ D 1. As suppo ' V for all 2 and V is locally finite, f' g 2 is a locally finite sup-partition of unity subordinated to U.1/. t u We note that the proof could be made considerably shorter by proving (iv))(iii) directly using Theorem 79 instead of (ii))(iii) and (iv))(ii). However, the reasons for our strategy of the proof were two: First, we do not need the full generality (and associated machinery) of Theorem 79 (or Theorem 74) and second, the proof of (ii))(iii) shows an interesting technique for constructing smooth Lipschitz approximations (due to Robb Fry, [Fry2]), and in fact shows the reason for the definition of the notion of sup-partition of unity. Theorem 64 (Robb Fry, [Fry2]). Let X be a separable normed linear space that admits a C k -smooth Lipschitz bump function, k 2 N0 [ f1g. Then there is M 2 R such that there is a C k -smooth M -Lipschitz sup-partition of unity f j gj1D1 on X subordinated to U.1/. Proof. Using the C k -smooth Lipschitz bump on X as a start, by shifting, scaling, and composing with a suitable real function we construct two functions f; g 2 C k .X I Œ0; 1/ along with real numbers C > 0 and 0 < ı < r < 1 such that f .x/ D 0 for all x 2 X n U.0; 1/, f .x/ D 1 for all x 2 B.0; r/, g.x/ D 1 for all x 2 X n U.0; r/, g.x/ D 0 for all x 2 B.0; ı/, and both functions are C -Lipschitz (see also the proof of Theorem 63). Let fxj gj1D1 X be a sequence such that fU.xj ; ı/gj1D1 is a covering of X . We put fj .x/ D f .x xj / and gj .x/ D g.x xj / for each x 2 X , j 2 N. Choose 0 < < 1 and for each j 2 N let 'j 2 C k .Rj / be a 1-Lipschitz function (with respect to the maximum norm) such that minfw1 ; : : : ; wj g 'j .w/ minfw1 ; : : : ; wj g C

for each w 2 Œ0; 1j

(use Lemma 1). We note that the functions 'j will serve as a “smoothened minimum”. Finally, to confine the sup-partition into the interval Œ0; 1, let h 2 C k .RI Œ0; 1/ be a D-Lipschitz function such that h.t/ D 0 for t and h.t/ D 1 for t 1.

428

For each j 2 N we define j .x/ D h 'j g1 .x/; : : : ; gj

Chapter 7. Smooth approximation

1 .x/; fj .x/

for each x 2 X.

Clearly, j 2 C k .XI Œ0; 1/ and j is M -Lipschitz for each j 2 N, where M D CD. Moreover, f j gj1D1 is a sup-partition of unity. Indeed, choose an arbitrary x 2 X . Let k 2 N be the smallest index for which x 2 U.xk ; ı/. Then gk .x/ D 0, which implies that for j > k, 'j g1 .x/; : : : ; gj 1 .x/; fj .x/ and so j .x/ D 0. Therefore the set fj 2 NI j .x/ > 0g is finite. Further, let n 2 N be the smallest index for which x 2 U.xn ; r/. It follows that gj .x/ D 1 for each j < n and fn .x/ D 1, hence n .x/ D 1. Finally, if kx xj k 1, then fj .x/ D 0 and hence j .x/ D 0, which shows that f j gj1D1 is subordinated to U.1/. t u Corollary 65. Let X be a separable normed linear space that admits a C k-smooth Lipschitz bump function, k 2 N0 [ f1g. Then there is a bi-Lipschitz homeomorphism ˚ W X ! c0 such that the component functions ej B ˚ 2 C k .X/ for every j 2 N.

7. Approximation of Lipschitz mappings In this section we turn our attention to the problem of approximating Lipschitz mappings by smooth Lipschitz mappings, preferably keeping the control over the Lipschitz constant. Such approximations have applications for example in the theory of Banach manifolds. The finite-dimensional case is easy – the integral convolution respects the Lipschitz property and in fact it preserves the Lipschitz constant. In the infinite-dimensional case, the infimal convolution preserves the Lipschitz constant too, but unfortunately it gives only the first order smoothness and works only for functions (i.e. mappings into R). Using the local techniques (partitions of unity) alone to obtain the global property (Lipschitzness) presents some insurmountable obstacles. First, it is essentially impossible to gain any global control over the Lipschitz constant of the individual functions in the partition and regardless, there is no control over the cardinality of the (locally finite) sum in (15). Therefore we have to develop several alternative approaches to this problem. The first one is using the integral convolution even in the infinite-dimensional setting. We show two cases when this is possible. The first one is for separable spaces, where we can use “convolution in a dense set of directions” and then exploit the Lipschitz property of the approximated mapping. The gain is however not particularly strong, as we obtain merely Gâteaux (or uniformly Gâteaux) smooth approximation. More interesting is the use of the integral convolution in the space c0 . /, which is possible thanks to the very strong LFC structure in this space. The latter result has interesting corollaries when either the source or the target space have certain special properties. The above techniques are somewhere in-between local and global – they use approximation on finite-dimensional subspaces, which are then somehow “glued together”.

Section 7. Approximation of Lipschitz mappings

429

Another example of this approach is a technique of Nicole Moulis that uses an unconditional basis for gluing together the finite-dimensional approximations. There is also a “local” procedure: the supremal partitions, developed in the previous section, which essentially replace the sum in (15) by supremum, which preserves the Lipschitz property. All the above results give approximations in the uniform topology. Using the -discrete partitions of unity we show how to proceed from uniform approximations to the approximation in fine topology. Finally, we prove an analogue of Theorem 31 (the real analytic approximation) for Lipschitz functions. We start with a notion of a uniform Gâteaux differentiability. If f is Gâteaux differentiable and for a fixed x in the domain we require the uniformity of the limit defining @f .x/ in h 2 BX , we obtain the notion of Fréchet differentiability. If, on @h the other hand, for each fixed h 2 BX we require the uniformity in x, then we obtain uniform Gâteaux differentiability. Uniformity of this type will prove important later, for example in the applications of Theorem 84. Definition 66. Let X, Y be normed linear spaces, U X open, and f W U ! Y a Gâteaux differentiable mapping. We say that f is uniformly Gâteaux differentiable .x/ is uniform for x 2 U . (UG for short) if for each fixed h 2 SX the limit defining @f @h Lemma 67. Let X, Y be normed linear spaces, U X open, and let f W U ! Y be a Gâteaux differentiable mapping. If for each h 2 SX the mapping x 7! ıf .x/Œh is uniformly continuous on U , then f is uniformly Gâteaux differentiable on U provided that U is convex; otherwise f is uniformly Gâteaux differentiable on any open V U satisfying dist.V; X n U / > 0. Conversely, if f is uniformly Gâteaux differentiable and uniformly continuous on U , then for each h 2 X the mapping x 7! ıf .x/Œh is uniformly continuous on any A U satisfying dist.A; X n U / > 0. Proof. Choose an arbitrary direction h 2 SX and " > 0, and find > 0 such that kıf .x C t h/Œh ıf .x/Œhk < " for all x 2 U and t 2 . ; / satisfying x C th 2 U . If U is convex we set V D U and D , otherwise we let D minf; dist.V; X n U /g. Fix x 2 V and define a mapping g W I ! Y by g.t/ D f .x C th/ tıf .x/Œh, where I D ft 2 . ; /I x C th 2 U g. Notice that I is an open interval containing 0 and g 0 .t / D ıf .x C th/Œh ıf .x/Œh for t 2 I . By the assumption we have 0 kg on the interval I . Therefore

1 .t /k " for every t 2 I and hence

g 1is "-Lipschitz

f .x C th/ f .x/

D g.t/ g.0/ " for all t 2 I . ıf .x/Œh t t To prove the converse statement, choose h 2 X, h ¤ 0, a subset A U for dist.A; X n U / > 0, and " > 0. Find 0 < < dist.A; X n U /=khk such that which

1

f .x C h/ f .x/ ıf .x/Œh < 4" for any x 2 A. Let > 0 be such that kf .x/ f .y/k < 4" whenever x; y 2 A are such that kx yk < . Then, for such x; y, we have

ıf .x/Œh ıf .y/Œh < " C 1 f .x C h/ f .x/ f .y C h/ C f .y/ < ": 2 t u

430

Chapter 7. Smooth approximation

We remark that if f W U ! Y , U X open, is such that for each h 2 SX the .x/ is uniformly continuous on U , then f E \U is C 1;C -smooth for mapping x 7! @f @h each finite-dimensional affine subspace E X (Theorem 1.96). In particular, if X is a Banach space and f is Baire measurable, then in view of Theorem 1.101 we do not need to assume that f is Gâteaux differentiable in Lemma 67. Lemma 68. Let X, Y be normed linear spaces, H a dense subset of X , U X open, and let f W U ! Y be a Gâteaux differentiable Lipschitz mapping such that for each h 2 H the mapping x 7! ıf .x/Œh is uniformly continuous on U . Then the mapping x 7! ıf .x/Œh is uniformly continuous on U for every h 2 X . Proof. Let L be a Lipschitz constant of f . Pick an arbitrary h 2 X and let " > 0. Find " h0 2 H such that kh h0 k < 4L . By the uniform continuity of x 7! ıf .x/Œh0 there is > 0 such that kıf .x/Œh0 ıf .y/Œh0 k < 2" whenever x; y 2 U , kx yk < . Then

ıf .x/Œh ıf .y/Œh

ıf .x/Œh0

2L . Let let 'j 2 C 1 .R/, j 2 N, R fhj g be a dense subset ofSX and " ; " . Extend f to the be such that 'j 0, R 'j D 1, and supp 'j 2L2j 2L2j whole of X by setting f .x/ D 0 for x 2 X n U and define gn W V ! Y , n 2 N, by f x

Z gn .x/ D

Rn

n X j D1

tj hj

Y n

'j .tj / dn .t/;

(18)

j D1

where n denotes the n-dimensional Lebesgue measure. Then gn ! g uniformly on V and the mapping g W V ! Y has the following properties: It is L-Lipschitz, Gâteaux differentiable, satisfies kf gkV < ", and for each h 2 X the mapping x 7! ıg.x/Œh is uniformly continuous on V . Moreover, if Y D R and U , V , f are convex, then so is g. Q Pm " " Rm . Since x Proof. Denote Km D jmD1 2L2 j 2 U for j ; 2L2j j D1 tj hR x 2 V and .t1 ; : : : ; tm / 2 Km , using the Fubini theorem and the fact that R 'j D 1

Section 7. Approximation of Lipschitz mappings

431

we obtain for m > n and any x 2 V

Z

X X ! Y m n m

kgm .x/ gn .x/k D f x f x tj hj tj hj 'j .tj / dm

Rm

j D1 j D1 j D1

m ! m Z Z m m

Y

X X Y

'j .tj / dm L jtj j 'j .tj / dm L tj hj

Km j DnC1 Km j DnC1 j D1 j D1 !Z m m Y X " " 'j .tj / dm < L : 2 2n 2L2j Km j D1

j DnC1

It follows that there is g W V ! Y such that gn ! g uniformly on V . The mappings gn are L-Lipschitz on V . Indeed, for any x; y 2 V we have Z kgn .x/

gn .y/k

Kn

n X

f x

tj hj

f y

j D1 n Y

Z Lkx

yk

Kn j D1

n X j D1

Y

n tj hj 'j .tj / dn

'j .tj / dn D Lkx

j D1

yk:

Therefore g is also L-Lipschitz. Similarly we can check that the functions gn and g are convex under the additional convexity assumptions. Moreover, kf gkV < ". Indeed, pick n 2 N such that kgn gkV < 2" . Then kf .x/

g.x/k kf .x/ gn .x/k C kgn .x/ g.x/k Y Z n X

n "

tj hj 'j .tj / dn C

0 be such that T . ; /n V . Using substitution t ! s t we obtain Z gn B T .s/ D

Rn

f B T .s

t/

n Y

Z 'j .tj / dn .t/ D

j D1

Qn

K

f B T .t/

n Y

'j .sj

tj / dn .t/

j D1

" " ; 2L2 for any s 2 . ; /n , where K D j D1 2L2 j j C . It follows from Corollary 1.91 that the mapping gn B T is C 1 -smooth on . ; /n . Since by the n definition @g .x/ D D.gn B T /.0/Œ.s1 ; : : : ; sn / for all h D s1 h1 C C sn hn , it @h

432

Chapter 7. Smooth approximation

follows that h 7!

@gn .x/ @h

@gn .x/ D @hi

is linear on spanfh1 ; : : : ; hn g and

Z K

f B T .t/'i0 . ti /

'j . tj / dn .t/

j D1 j ¤i

f x

Z D

n Y

Rn

n X

n Y tj hj 'i0 .ti / 'j .tj / dn .t/

j D1

j D1 j ¤i

for each i 2 f1; : : : ; ng. n 1 Further, f @g converges uniformly for x 2 V . Indeed, using the Fubini theorem g @hi nDi R and the fact that R 'j D 1 we have for m > n i and any x 2 V

@gm

@h .x/ i

@gn .x/

@h

Z i

D f x

Rm Z L

Km

m X

m X

tj hj

f x

!

j DnC1

! tj hj

'i0 .ti /

j D1

j D1

jtj j

n X

j'i0 .ti /j

m Y j D1 j ¤i

" 'j .tj / dm 2 2n

m Y j D1 j ¤i

Z R

'j .tj / dm

j'i0 .t/j dt:

By Theorem 1.85 (used on the restrictions to xCspanfhi g) we obtain that the directional @g derivative @h .x/ exists for all x 2 V , i 2 N. From the above it also follows that i

@g @g D @h .x/ C @h .x/ for all x 2 V , i; j 2 N. i j To see that for given x 2 V the derivative @g .x/ exists for all h 2 SX choose > 0 @h and let i 2 N be such that kh hi k < 3L . Then for any 2 R n f0g small enough so that x C h 2 V , x C hi 2 V we have

1 1 L

g.x C h/ g.x/ g.x C hi / g.x/ k.h hi /k < :

j j 3 @g .x/ @.hi Chj /

Thus there is > 0 such that

1

g.x C 1 h/ 1

1 g.x/ g.x C 2 h/ g.x/

2

1 1 2 < C g.x C 1 hi / g.x/ g.x C 2 hi /

3 1 2

for 0 < j1 j < , 0 < j2 j < .

g.x/

0 we have

@g

.x/ @g .x/ @g .x/

@u

@u @v

@g .x/ @h

433

is L-Lipschitz. For arbitrary u; v 2 X

1 g.x C u/ g.x/

1

C

g.x C u/ g.x C v/

@g 1

g.x C v/ g.x/ C .x/

@v C Lku vk

@g for small enough. Thus @u .x/ @g .x/ Lku vk. It follows that the mapping @v h 7! @g .x/ belongs to L.X I Y /, since it is a Lipschitz mapping that is linear on a @h dense subset of X. Therefore g is Gâteaux differentiable on V . It remains to prove that x 7! ıg.x/Œh is uniformly continuous on V for any h 2 X . n To this end, first that the mapping x 7! @g .x/ is Li -Lipschitz for any n i , @hi R note 0 where Li D L R j'i .t/j dt :

@gn

@gn

@h .x/ @h .y/ i i

Z ! n n n

X Y X

tj hj f y tj hj 'i0 .ti / 'j .tj / dn f x

Rn j D1

j D1

Z Lkx

yk R

j'i0 .t/j dt D Li kx

j D1 j ¤i

yk:

Thus the mapping x 7! ıg.x/Œhi is Li -Lipschitz for each i 2 N. It follows from Lemma 68 that x 7! ıg.x/Œh is uniformly continuous on V for any h 2 X . t u Corollary 70. Let X be a separable normed linear space, Y a Banach space, U X open, k 2 N0 , f 2 C k .U I Y / such that d jf is Lj -Lipschitz for j D 0; : : : ; k, " > 0, and let V U be open such that dist.V; X nU / > 0. Then there is g 2 C k .V I Y / such that d jg is Lj -Lipschitz for j D 0; : : : ; k, d kg is uniformly Gâteaux differentiable (in particular, g is G kC1 -smooth), and kd jg d jf kV < " for all j 2 f0; : : : ; kg. Proof. Let W U be open such that dist.W; X n U / > 0 and dist.V; X n W / > 0. We define mappings gn W W ! Y by formula (18). By Corollary 1.91 we have gn 2 C k .W I Y / and Y Z n n X d jgn .x/ D d jf x tl hl 'l .tl / dn .t/ (19) Rn

lD1

d jf

lD1

for x 2 W , j D 0; : : : ; k. Since each is Lj -Lipschitz, by Lemma 69 used on (19) we obtain that there are Lj -Lipschitz mappings qj W W ! P . jXI Y / such that d jgn ! qj uniformly on W and kd jf qj kW < ". Moreover, qk is Gâteaux

434

Chapter 7. Smooth approximation

differentiable on W and x 7! ıqk Œh is uniformly continuous on W for each h 2 X. Therefore qk is uniformly Gâteaux differentiable on V by Lemma 67. Theorem 1.85 implies that gn ! g 2 C k .W I Y / uniformly on W and d jg D qj , j D 0; : : : ; k. u t The following version of Lemma 69 is for mappings that are only locally Lipschitz. Lemma 71. Let X be a separable Banach space, Y a Banach space, U X an open set, f W U ! Y a locally Lipschitz mapping, and let V U be open such that ı D dist.V; X n U / > 0. Let fhj g be a dense subset of SX and let 'j 2 C 1 .R/, ı ı R ; j 2 N, be such that 'j 0, R 'j D 1, and supp 'j . Extend f to the 2j 2j whole of X by setting f .x/ D 0 for x 2 X n U and define gn W V ! Y , n 2 N, by formula (18). Then gn ! g locally uniformly on V and the mapping g W V ! Y is locally Lipschitz and Gâteaux differentiable. Moreover, if Y D R and U , V , f are convex, then so is g. ˚P1 ı Proof. Let K D j D1 tj hj I jtj j 2j . Then K is a compact subset of X and so it is easy to show that for each x 2 V there is a neighbourhood Vx V of x such that f is Lipschitz on Vx K. Note that for y 2 Vx each gn .y/ is defined using values of f on Vx K only. So we may repeat the proof of Lemma 69 with the following differences: gn ! g only locally uniformly on V . gn are only locally Lipschitz and so is g. n 1 g converges only locally uniformly on V . f @g @hi nDi In the proof of the Gâteaux differentiability of g we use the fact that g is locally Lipschitz. t u k;1 Corollary 72. Let X be a separable Banach space that admits a Cloc -smooth bump k;1 for some k 2 N0 . Then X admits a Cloc -smooth bump with Gâteaux differentiable kth derivative (in particular it admits G kC1 -smooth bump). k;1 Proof. Let f 2 Cloc be a non-negative bump function with supp f B.0; 1/. Let 1 1 R 1 ; 'j 2 C .R/, j 2 N, be such that 'j 0, R 'j D 1, and supp 'j . 2j 2j Define mappings gn W X ! Y by formula (18). By Corollary 1.91 we have gn 2 C k .X/ and (19) holds for x 2 X , j D 0; : : : ; k. Since each d jf is locally Lipschitz, by Lemma 71 used on (19) we obtain that there are locally Lipschitz mappings qj W X ! P . jX/ such that d jgn ! qj locally uniformly on X . Moreover, qk is Gâteaux differentiable on X. Theorem 1.85 implies that gn ! g 2 C k .X/ locally uniformly on X and d jg D qj , j D 0; : : : ; k. Finally, since by the definition each gn is zero outside B.0; 2/, g is a bump. t u

To proceed to integral convolutions in c0 . / we need an auxiliary notion. Let X be a topological vector space, ˝ X an open subset, E an arbitrary set, M X , and g W ˝ ! E. Let U be a neighbourhood of zero in X . We say that g depends U -uniformly locally on finitely many coordinates from M (U -ULFC-M for short) if for each x 2 ˝ there is a finite subset F M such that g depends only on F on .x C U / \ ˝ (cf. Definition 5.78).

Section 7. Approximation of Lipschitz mappings

435

For F P we denote the associated coordinate projection in c0 . / by PF , i.e. PF .x/ D 2F e .x/e for x 2 c0 . /. By c00 . / we denote the linear subspace of c0 . / consisting of finitely supported vectors. Lemma 73. Let be an arbitrary set, Y a Banach space, and let f W c0 . / ! Y be a mapping that is U.0; r/-ULFC-fe g 2 for some r > 0. Further, let ˝ c0 . / be open, let f be uniformly continuous on ˝ with modulus !, and suppose that f D 0 on c0 . / n ˝. Then for every V ˝ with dist.V; c0 . / n ˝/ > 0 and for every " > 0 there is a U.0; 2r /-ULFC-fe g 2 mapping g 2 C 1 .c0 . /I Y / such that kf gkV ", g is uniformly continuous on V with modulus !, and the mapping x 7! Dg.x/Œh is uniformly continuous on V for any h 2 c00 . /. If f is even, then so is g. If moreover Y D R and f is convex, then so is g. Proof. Let D dist.V; c0 . / n ˝/ and find 0 < ı < minf; 2r g such that !.ı/ < ". 1 Choose function ' on R such that supp ' Œ ı; ı R R an even C -smooth non-negative and R ' D 1. We denote C D R j' 0 .t/j d. Let F 2 be a partially ordered set of non-empty finite subsets of ordered by inclusion. For any F 2 F we define the mapping gF W c0 . / ! Y by Y Z X gF .x/ D f x t e '.t / djF j .t/: RjF j

2F

2F

Notice that the integral is well-defined, since f D 0 on the closed set c0 . / n ˝ and f is uniformly continuous on ˝ and so it is bounded on totally bounded sets. The net fgF gF converges on c0 . / to a mapping g W c0 . / ! Y . In fact, we claim that for any x 2 c0 . / there is an F 2 F such that gF .y/ D gH .y/ for any F H 2 F and any y 2 U.x; 2r /. Indeed, for a fixed x 2 c0 . / let F 2 F be such that the mapping f depends only on the functionals fe g 2F on U.x; r/ and r kx PF .x/k < 2r . Choose any y 2 U.x; H 2 F satisfying H F . Suppose

2 / andP r r

and consequently that t 2 Œ 2 ; 2 for all 2 H . Then x y t e < r

2H P P f y

2H t e D f y

2F t e . Thus, by the Fubini theorem, Y Z X gH .y/ D f y t e '.t / djH j .t/ Œ ı;ıjH j

Z D

f y Œ ı;ıjF j

2H

X

2F

t e

2H

Y

2F

'.t / djF j .t/

Y

2H nF

Z Œ ı;ı

'.t / d D gF .y/:

Moreover, kx PF .y/k kx PF .x/k C kPF kkx yk < r and so we can easily see that gF .y/ D gF .PF .y//. The mapping gF PF .c0 . // is in fact a finitedimensional convolution with a smooth kernel on RjF j , and so gF is a C 1 -smooth mapping on U.x; 2r / (Corollary 1.91; recall that a uniformly continuous mapping is bounded on totally bounded sets). The mapping g is therefore U.0; 2r /-ULFC-fe g 2 and g 2 C 1 .c0 . /I Y /, as for any x 2 c0 . /, g D gF B PF on U.x; 2r / for some F 2F.

436

Chapter 7. Smooth approximation

To show that kf gkV " choose anParbitrary x 2 V . Let

such that P F 2 F be

D

ı < g.x/ D gF .x/. Notice that x x t e t e

2F P

2F whenever t 2 Œ ı; ı for all 2 F . Hence x

2F t e 2 ˝ and kf .x/ g.x/k D kf .x/ gF .x/k

Z

Y Z

Y X

D f .x/ '.t / djF j .t/ f x t e '.t / djF j .t/

RjF j

jF j R

2F

2F

2F

Y Z X

f .x/ f x

t e '.t / djF j .t/

Œ ı;ıjF j

2F

Z

!.ı/ Œ ı;ıjF j

Y

2F

'.t / djF j .t/ D !.ı/ < ":

2F

To see that the mapping g is uniformly continuous on V with modulus !, choose x; y 2 V and find F; H 2 F such that g.x/ D gF .x/ and g.y/ D P gH .y/. Then for K DF [ H we have g.x/ D g .x/ and g.y/ D g .y/. As x K K

2K t e 2 ˝ P and y t e 2 ˝ whenever t 2 . ; / for all

2 K,

2K kg.x/

g.y/k D kgK .x/ gK .y/k

Z X

t e

f x Œ ı;ıjKj

f y

2K

X

2K

Y

t e '.t / djKj .t/

2K

yk :

! kx

Similarly we can check that g is even if f is even and g is convex under the additional assumptions that Y D R and f is convex. We finish the proof by showing that the directional derivatives of g in the directions of c00 . / are uniformly continuous on V . So first, choose any ˛ 2 . For x; y 2 V find F; H 2 F such that g.x/ D gF .x/ on U.x; 2r / and g.y/ D gH .y/ on U.y; 2r /. Put K D F [ H [ f˛g. Using Corollary 1.91 and substitution we obtain Z X Y DgK .x/Œe˛ D f x t e ' 0 .t˛ / '.t / djKj .t/: RjKj

Hence, similarly as above,

Dg.x/Œe˛ Dg.y/Œe˛

Z X

f x t e

RjKj

! kx

2K

f y

2K

yk

Z

j' 0 .t/j d D C ! kx