Analyticity in Infinite Dimensional Spaces [Reprint 2011 ed.] 9783110856941, 9783110109955

Table of contents :
Chapter 1 Some topological preliminaries
Summary
1.1 Locally convex spaces
1.2 Vector valued infinite sums and integrals
1.3 Baire spaces
1.4 Barrelled spaces
1.5 Inductive limits
Chapter 2 Gâteaux-analyticity
Summary
2.1 Vector valued functions of several complex variables
2.2 Polynomials and polynomial maps
2.3 Gâteaux-analyticity
2.4 Boundedness and continuity of Gâteaux-analytic maps
Exercises
Chapter 3 Analyticity, or Fréchet-analyticity
Summary
3.1 Equivalent definitions
3.2 Separate analyticity
3.3 Entire maps and functions
3.4 Bounding sets
Exercises
Chapter 4 Plurisubharmonic functions
Summary
4.1 Plurisubharmonic functions on an open set Ω in a I.c. space X
4.2 The finite dimensional case
4.3 Back to the infinite dimensional case
4.4 Analytic maps and pluriharmonic functions
4.5 Polar subsets
4.6 A fine maximum principle
Exercises
Chapter 5 Problems involving plurisubharmonic functions
Summary
5.1 Pseudoconvexity in a I.c. space X
5.2 The Levi problem
5.3 Boundedness of p.s.h. functions and entire maps
5.4 The growth of p.s.h. functions and entire maps
5.5 The density number for a p.s.h. function
Exercises
Chapter 6 Analytic maps from a given domain to another one
Summary
6.1 A generalization of the Lindelöf principle
6.2 Intrinsic pseudodistances
6.3 Complex geodesics and complex extremal points
6.4 Automorphisms and fixed points
Exercises
Bibliography
Glossary of Notations
Subject Index

Citation preview

de Gruyter Studies in Mathematics 10 Edtiors: Heinz Bauer · Peter Gabriel

Michel Herve

Analyticity in Infinite Dimensional Spaces

w DE

G

Walter de Gruyter Berlin · New York 1989

Author

Michel Herve Mathematiques Universite de Paris VI F-75252 Paris Cedex 05

Library of Congress Cataloging-in-Publication

Data

Herve, Michel (Michel A.) Analyticity in infinite dimensional spaces / Michel Herve. p. c m . - ( D e Gruyter studies in mathematics : 10) Bibliography: p. Includes index. ISBN 0-89925-205-2 : 1. Analytic functions. 2. Harmonic functions. 3. Analytic mappings. I. Title. II. Series. QA331.7.H47 1989 88-31955 515.7'3-dc 19 CIP

Deutsche Bibliothek Cataloging-in-Publication

Data

Herv6, Michel: Analyticity in infinite dimensional spaces / Michel Herve. Berlin ; New York : de Gruyter, 1989 (De Gruyter studies in mathematics ; 10) ISBN 3-11-010995-6 (Berlin) Gewebe ISBN 0-89925-205-2 (New York) Gewebe NE: GT

© Copyright 1989 by Walter de Gruyter & Co., Berlin 30. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form - by photoprint, microfilm or any other means nor transmitted nor translated into a machine language without written permission from the publisher. Printed in Germany. Cover design: Rudolf Hübler, Berlin. Typesetting: Asco Trade Typesetting Ltd., Hong Kong. Printing: Druckerei Gerike G m b H , Berlin. - Binding: Dieter Mikolai, Berlin.

Foreword

During the last twenty years, the theory of analyticity in infinite dimensions has developed from its foundations into a structure which may be termed harmonious, provided that one accepts to do without some features of the finite dimensional case. This harmony is of course favoured by the choice of a unique setting - locally convex spaces over the complex field, analytic maps into sequentially complete spaces - and a central topic: plurisubharmonicity, where a multitude of results obtained by different authors Pierre Lelong, the founder of the notion, Gerard Coeure, Christer Kiselman and others - deserved to be brought together. The reader will find the precise contents of each chapter in the summary which opens it. The concern for unity has inevitably led to the omission of several other topics in spite of their indisputable interest, and among these I insist on the local theory of analytic sets. But the methods used have little in common with those used in this book; the material consists essentially in two theses: "Sous-ensembles analytiques d'une variete analytique banachique" (Paris, 1969) by Jean-Pierre Ramis, "Ensembles analytiques complexes dans les espaces localement con vexes" (Paris, 1969) by Pierre Mazet, and for the time being there is little more to say on the subject. This book is a tribute to all the authors it mentions. The first, chapters owe much to the paper [Boc Sic]; comparatively short sections are devoted to some topics extensively developed in the excellent monographs published as North Holland mathematics studies: "Pseudo-convexite, convexite polynomiale et domaines d'holomorphie" by Philippe Noverraz (n° 3); "Analytic functions and manifolds in infinite dimensional spaces" by Gerard Coeure (n° 11); "Holomorphic maps and invariant distances" by Franzoni and Vesentini (n° 40); "Complex analysis in locally complex spaces" by Sean Dineen (n° 57); "Complex analysis in Banach spaces" by Jorge Mujica (n° 120). To bring a new contribution was no easy task, but it seemed to me that classical potential theory deserved a better place, as a special tribute to the recently deceased Marcel Brelot. Finally I express my gratitude to Professors Heinz Bauer and Peter Gabriel who accepted this book in their renowned series. November 1988

Michel Herve

Contents

Chapter 1 Some topological preliminaries

1

Summary 1.1 Locally convex spaces 1.2 Vector valued infinite sums and integrals 1.3 Baire spaces 1.4 Barrelled spaces 1.5 Inductive limits

1 2 6 9 11 13

Chapter 2 Gäteaux-analyticity

19

Summary 2.1 Vector valued functions of several complex variables 2.2 Polynomials and polynomial maps 2.3 Gäteaux-analyticity 2.4 Boundedness and continuity of Gateaux-analytic maps . . . . Exercises Chapter 3 Analyticity, or Frichet-analyticity Summary 3.1 Equivalent definitions 3.2 Separate analyticity 3.3 Entire maps and functions 3.4 Bounding sets Exercises Chapter 4 Ρlurisubharmonic functions Summary 4.1 Plurisubharmonic functions on an open set Ω in a I.e. space X 4.2 The finite dimensional case 4.3 Back to the infinite dimensional case 4.4 Analytic maps and pluriharmonic functions

19 20 28 35 43 50 51

..

51 52 58 65 73 79 81 81 82 87 94 104

VIII

Contents

4.5 Polar subsets 4.6 A fine maximum principle Exercises

107 120 127

Chapter 5 Problems involving plurisubharmonic functions

129

Summary 5.1 Pseudoconvexity in a I.e. space X 5.2 The Levi problem 5.3 Boundedness of p.s.h. functions and entire maps 5.4 The growth of p.s.h. functions and entire maps 5.5 The density number for a p.s.h. function Exercises

129 130 135 144 146 154 162

Chapter 6 Analytic maps from a given domain to another one Summary 6.1 A generalization of the Lindelöf principle 6.2 Intrinsic pseudodistances 6.3 Complex geodesies and complex extremal points 6.4 Automorphisms and fixed points Exercises

163 163 164 168 179 184 194

Bibliography

195

Glossary of Notations

201

Subject Index

205

Chapter 1

Some topological preliminaries

Summary The aim of this first chapter is only to present the topological tools and set the notation which will be used throughout the book.

1.1. A. B. C. D.

Locally convex topologies; the metrizable case. Bounded sets. Complete and quasi-complete spaces. The adjoint space; the weakened topology. Strong and weak* topologies on the adjoint space. Reflexivity and semi-reflexivity.

1.2. A. Summable families. B. The integral of a vector valued function. 1.3. A. Meagre sets and Baire spaces. B. Topological properties involving the Baire property. C. Banach and F rechet spaces. 1.4. Barrelled spaces. Prop. 1.4.2. The bounded sets are the same for the weakened topology and the initial one. Rem. 1.4.3. Semireflexive spaces. 1.5. A. The gauge of an absorbent balanced convex set. B. Inductive limits. C. Strict inductive limits.

Chapter 1

Some topological preliminaries

1.1 Locally convex spaces All topologies considered throughout the book shall be Hausdorff; therefore, any finite dimensional vector space will be endowed with its euclidean topology ([B0] 2 , Chap. I, §2, Th. 2). (A) All topological vector spaces considered will be over C and locally convex; in this chapter, (AT, Π) will denote a vector space X with the HausdorfT topology defined by some family Π of seminorms such that s u p p ( x ) > 0 for all χ e I \ { 0 } ; a fundamental system of neighbourpen

hoods of χ in this topology is made up of the finite intersections of sets χ + ρ " 1 ( [ 0 , α ] ) , ρ ε Π , 0 < α < oo. Then all seminorms in Π are continuous, but the family csn(X, Π) of all continuous seminorms on (Χ, Π) is generally larger than Π: pe csn(X, Π) if and only if ρ is majorized by a finite linear combination, with positive coefficients, of seminorms in Π; another fundamental system of neighbourhoods of χ is made up of the sets χ + ρ-^ΕΟ,Ι^ρεοβη^,ΙΙ). If Π is countable, namely Π = {p n : η e N*}, then (Χ,Π) is metrizable since the formula »1 pn(x-y) d(x,y) = Σ -ΎΓ~, ; 7 „ti η 1 + p„(x - y) provides a distance defining the same topology as Π, and obviously invariant under translations: d(x + z,y + z) = d(x, y). Conversely, if (Χ, Π) is metrizable, then its topology can be defined by a countable family of seminorms (since the origin has a countable fundamental system of neighbourhoods) and also by an increasing sequence of seminorms. A linear map λ\ X Y is a continuous map (Χ, Π) (Υ, Π') if and only if q ο λ e csn(X, Π) for all q e ΓΓ; in particular, the topology of (Χ, Π) is finer than the topology of {Χ, ΓΓ) if and only if IT c csn(X, Π), and both topologies coincide if and only if csn(X, Π') = csn(X, Π). (B) A set Β in the locally convex space (.Χ, Π) is bounded if p(B) is bounded,

1.1 Locally convex s paces i.e. s u p p ( x ) < +00, for all p e l l xeB

3

or for all p e c s n ^ , Π ) . Then the

_

closure Β of Β is bounded too. Whenever useful in order to avoid unessential difficulties (See lor example 2.1.8 or 2.2.2.B), analytic maps will be assumed to have their values in complete (in some sense) locally convex (I.e.) spaces. A filter on X is a Cauchy filter in ( Χ , Π) if for each ρ e Π and each ε > 0 there is an A e SF such that p(x — y) ^ ε for all x,yeA; ( Χ , Π) is complete if any Cauchy filter converges, and quasi-complete if any Cauchy filter on a closed bounded subset of (Χ, Π) converges. In other words, (Χ, Π) is quasi-complete if any Cauchy filter 3F, for which there is a bounded subset Β of (Χ, Π) such that Α η Β φ 0 for all A e converges (to some point e B). Since any Cauchy sequence is bounded, quasi-complete (q.c.) implies sequentially complete (s.c.); if (Χ, Π) is metrizable, then complete, quasicomplete, sequentially complete, are equivalent properties. A Frichet space is a metrizable space (Χ, Π) in which they hold. The topology of (Χ, Π) can be defined by one norm if and only if the origin has bounded neighbourhoods. In fact, if Π contains pl,..., pn such that the set {x e X: p,(x) ^ 1, i = 1 , . . . , « } is bounded, then each ρ e II satisfies ρ ^ const. s u p ( p j , . . . , p„); therefore s u p ( p 1 , . . . , p„) is a norm defining the same topology as Π. The space (Χ, Π) is bornological if all seminorms on X which are bounded on each bounded set are continuous. Given any I.e. space {Χ, Π), there always exists a unique topology on X for which the bounded sets are the same and which makes the space bornological: it is defined by the family of all seminorms on X which are bounded on each bounded set, and of course finer than the given one. If a linear m a p λ: X -* Y is continuous from (Χ, Π) to {Υ, Π'), Β bounded in (Χ, Π) implies λ(Β) bounded in (7,11'). The converse statement is true only if (Χ, Π) is bornological. (C) Given a locally convex topology on the space Χ, X' or C) will denote the adjoint space, the set of all linear maps χ ' : X -> C which are continuous, i.e. |x'| is a continuous seminorm, and x'(x) or x '> the image of χ under x'. The family of seminorms |x'| defines the so-called weakened topology of X, which is coarser than the initial one, but related to it as follows. Proposition 1.1.1. a) The weakened topology is Hausdorff. b) A convex set in X, if closed for the initial topology, is also closed for the weakened one.

4

1 Some topological preliminaries

c) The continuous linear maps X -* C are the same for the initial topology and for the weakened one. Proof, a) Given x 0 e X\{0}, there is a ρ e Π such that p(x 0 ) > 0; then, by the Hahn-Banach theorem ([Bo] 2 , Chap. II, §6, Th. 1), the linear map Cx 0 3 Cx0t—*ρ(χο)ζ can be extended into x' e X' such that | < x , x ' ) | ^ p(x) for all χ e X and 0 such that | |

for all x' e X'.

Hence follows that x* vanishes on p) ker(x' x'>) and therefore, by j an algebraic argument ([Ta] 2 , Th. 3.5.C), x* is a linear combination of the Xj. b) First assume that X is the adjoint space to X' for the strong topology. Given a set C, let c 0 (x') = s u p | < x , x ' ) | ^ c(x') for all x' e X', so that we xeC also have C={xeX: | < x , x ' ) | ^ c 0 (x') for all x' 6 X'}. Then any linear map / : X' C satisfying |/(x')| < c 0 (x') for all x' e X' is strongly continuous and C (with the weakened topology) is exactly the set (with the topology of pointwise convergence) of all linear maps / satisfying |/(x')| < c 0 (x') for all x' e X', a closed subset (for the same topology) of the set of all maps f: X' -*• C satisfying the same inequalities. The latter set (with the topology of pointwise convergence) can be identified with the product space Δ[0, c 0 (x')] of compact discs, x' e X'

which is compact by the Tychonov theorem ( [ B o ] l 5 Chap. I, §10, Th. 3). Conversely, assume that each C is compact for the weakened topology, and let X" be the adjoint space to X' for the strong topology. Given Xo e X", choose C so that ||, induces the weakened topology on the subspace X of X", and for this topology the convex set C is compact. Then ( [ B o ] 2 , Chap. II, §3, n° 3): if χό' φ C, by a) above there is an x' e X' such that ^2e

where F1 runs through tF, make up the basis of

a Cauchy filter converging to S e X. If / is countable, i.e. I = {in: η e N*}, η the sums yn = £ x ik make up a Cauchy sequence converging to S. If Ft η F = 0 implies Ρ ^ Σ

^ ε»

F

13

implies p^S —

Σ x« JJ ^ ε: hence follows the sufficiency of the Cauchy condition for the ieF' uniform summability of a family of maps. • Example 1.2.2. [Coe]. Let (ΑΤ,Π) be q.c. (or s.c. with / countable), (x,) i e / a summable family of vectors in X and the a, complex variables, i.e. α = (4/6C'. a) The family (ajX*),·^ is summable for ||a|| = sup|a, | < oo and uniformly ie/ summable on A = { a e C : ||a|| < 1}. b) The map As α i—• £ α,χ, is continuous for the product topology on A iel and therefore its image in X is compact.

8

1 Some topological preliminaries

Proof, a) Since it is enough to prove the uniform summability on A of the families {{0te α;)χ{) and ( ( J m a^Xf), we may assume af e [ — 1,1] for all i e I. Let Ft η F = 0 imply ρ ( Σ xt I ^ any such that FinF = 0 can VieF / 3 be written as F = {i l s ...,i„} with — 1 ^ α^ ^ ··· ^ ain ^ 1, and from Σ 0} of X' is aBanach space with the norm ||x'|| = inf{a > 0: |x'| < ap}. Since | continuous; then it makes the (finite by the assumption) seminorm Ε s x' h-» sup | 0, and let ζ be a complex variable: by the Hahn-Banach theorem ([Bo] 2 , Chap. II, §6, Th. 1), the linear map ζα\-+ p(a). ζ can be extended into an α' ε X' such that \a'\ ^ p, hence α' ε Ε, \\a'\\ ^ 1, sup| q(x), β > q(y) implies — ε α y A, — e A, hence χ + j; e (α + β)A. If A is moreover balanced (i.e. xe A, ß |a| ^ 1 imply ax e A), then q is a seminorm. Conversely, any seminorm ρ on X is the gauge of the absorbent, balanced, convex (a.b.c.) sets {x e X: p(x) < 1 } and {χ ε X: p(x) < 1}. Therefore the semi-norms in a family π defining a locally convex topology on X can be given as the gauges of a family si of a.b.c. sets A, and this topology is HausdorfT if and only if Q A does not contain any 1-dimensional AE subspace of Λ'. ** When the space X is endowed with a topology: a) if q is the gauge of A, A absorbent and star-shaped with respect to the origin: A closed [resp. open] implies A = 4 _ 1 ([0,1]), [resp. 1 [)] and f l o w e r [resp. upper] semi-continuous; b) if A is a.c.: A a neighbourhood of the origin implies q continuous, and conversely; c) X is barrelled if and only if any closed a.b.c. set (i.e. barrel) is a neighbourhood of the origin. (B) Let I be a preordered set in which any two elements i, Γ have a common majorant j (i 0}, which are Banach spaces with the norms ||x'||„ = inf{a > 0: |x'| ^ ctp„}. Each injection X'„ -> X'n+i is continuous, and the inductive limit topology finer than the strong topology on X': in fact, if Β is bounded in X, then

If X is also a Schwartz space ([Ho], Chap. 3, §15, especially Prop. 5), to each integer η corresponds an integer m> η such that the ball {x' e X': |x'| ^ pn} of X'n is relatively compact in X'm. In the next chapters, since the interest will no longer be in topological matters, a simpler notation will be used: X instead of (Χ, Π) for a I.e. space, c s n ^ ) for the set of all continuous seminorms on X.

Chapter 2

Gäteaux-analyticity

Summary Analyticity, holomorphy, scalar analyticity of a map U -* Ζ (U a finite dimensional open set, Ζ a I.e. space). Th. 2.1.2. The m-uple Cauchy formula; the inequalities of Cauchy. Th. 2.1.3. The above three notions are equivalent if Ζ is s.c. Th. 2.1.5. Bounded subsets of s/(U,Z). Prop. 2.1.7. The Vitali theorem [HiPh], Prop. 2.2.1. Polynomials in one complex variable with coefficients in Z. Prop. 2.2.4. The Bernstein-Walsh inequality [Sic]. Coroll. 2.2.5. The Leja polynomial lemma [Sic]. Def. 2.2.6. Polynomial maps. Def. 2.2.8. Homogeneous polynomial maps. Th. 2.2.9. The relation between «-homogeneous polynomial maps and «-linear maps. Prop. 2.2.11. A translation lemma, ^-analytic maps. Def. 2.3.1. The homogeneous polynomial expansion of a ^-analytic map; the Th. 2.3.5. generalized inequalities of Cauchy. Prop. 2.3.6. A converse statement [Sic]. Prop. 2.3.8. The Hartogs property for ^-analytic maps. Prop. 2.3.10. ^-analytic maps do not enjoy the composition property. Prop. 2.4.1(2) Boundedness and continuity of polynomial (^-analytic) maps. Th. 2.4.4. Zorn's theorem [Zo]. Prop. 2.4.7,8,9. Discontinuous ^-analytic functions.

Def. 2.1.1.

Chapter 2

Gäteaux-analyticity

2.1 Vector valued functions of several complex variables In this section, the data are an open set U in Cm and a vector space Ζ with a given locally convex topology. When needed, the generic point in U will m be written as χ = £ where the ξ} are the complex coordinates of χ in the canonical basis {ex,..., m x = Σ a = j=ι

em) of C m ; for m Σ ajeP k = j=i

(ki,...,km)eNm,

we write (x - a)k instead of {ξι - a x ) kl ...(£ m - a m ) S and set fc! =

Definition 2.1.1. A map / : U

Ζ is

a) analytic i f , for each a e U, there is a family (c k ) ke ^ m of vectors in Ζ such that the power series (cfc(x — a)fc)i(e^m is summable to f(x) for χ sufficiently near a\ b) holomorphic i f , at any point ae U, each first order partial derivative lim yζj[ f(a + ξ}β}) - /(α)] exists; 0Mj^O c) scalarly analytic or holomorphic if each complex valued function z' ο f z' e Z', is analytic or holomorphic on U: these are equivalent properties by the classical Hartogs theorem ([He]!, Th. 2 in II.2.1) The analytic (resp. holomorphic, scalarly analytic) maps U Ζ are the elements of a vector space over C; the first vector space thus defined will be denoted by Z). (A) Obviously a) or b) implies c); for more information, we first investigate the properties of a power series (ck(x — a) k ) fce ^ m . If this series is summable to /(x) for I ξj — ocj I ^ pj,j = 1 ,...,m, then (Prop. 1.2.1. a) for each q e csn(Z) there is a number M(q) such that pk q(ck) = q(ckpk) < M(q) for all k e Nm, and \ ξj — otjl ίξ 9ppj = 1 , . . . , m, 0 < θ < 1, implies

2.1 Vector valued functions of several complex variables

«[/(*) - /(e)]
• U, where V is open in C"; in particular, the definition of stf(U,Z) does not depend on the basis chosen for the finite dimensional space in which U is an open set; b) to prove the Liouville theorem for a map C -»· Z; c) to find cases when an analytic map U -*• Ζ remains analytic for a finer topology on Ζ (it obviously remains analytic for a coarser one): here are two such cases. If / is analytic U -*• Ζ for the weakened topology on Ζ and Ζ is s.c. for the initial one, then / is also analytic for the initial topology on Ζ since the adjoint space to Ζ is the same (Prop. 1.1.l.c). If / is analytic U Z' for the weak* topology on Z' and Ζ is barrelled (which implies Z' strongly quasi-complete by Th. 1.4.4.b), then / is also

2.1 Vector valued functions of several complex variables

25

analytic for the strong topology on Z', although the adjoint space to Z ' is not always the same. In fact, since [ / a x i - > ( z , / ( x ) > is analytic for all ζ e Ζ , the sets J3, in the proof of Theorem 2.1.3. are weak* bounded, hence also strongly bounded (Th. 1.4.4.b),/is strongly continuous and the power series (ck(x — a f ) in the proof of Theorem 2.1.3. strongly summable to f(x). (E) A set 38 in s/(U,Z) is said to be bounded if it is bounded for the topology of compact convergence (defined in Th. 2.1.2.c), i.e.: for each q e csn(Z) and each compact set Κ c U, there is a number M{q,K) > 0 such that sup ( g o / ) ^ M(q,Κ) for a l l f e 38. κ By Theorem 2.1.2.C, 38 bounded implies Dk38 = {Dkf:fe38} bounded for all k e Theorem 2.1.5. Let 38 c si(U, Ζ) be bounded and Ζ s.c. a) 38 is equicontinuous; therefore, if a sequence (/„) c: 38 converges pointwise to /, it actually converges uniformly on compact sets, and f e s/(U,Z) by Coroll 2.1.4, (Dkfn) converges to Dkf, uniformly on compact sets, by Theorem 2.I.2.C.

b) Let Ζ be not only s.c. but s.r. too (see Rem. 1.4.3.). Given a sequence (/„) c= 38, there is a g e jtf{U,Z) with the property: for each z' e Z ' and a subsequence (/„J depending on ζ', ζ' ο (g — fnJ 0 uniformly on compact sets. More generally, given a filter on 38, there is a g e srf(U,Z) with the property : for each z' e Z', each compact set Κ c U and ε > 0, the set f e 38: sup Iζ' ο (g — /)| ^ ε I meets all sets e 3F. κ J If only one g has the above property, then g = lim f„ or g = l i n v / for the weakened topology, uniformly on compact sets. c) Let μ be a real or complex measure on a compact metric space T. If / ( . , i ) e 38 for all t e Τ and f(x,.) is continuous for all χ e U, then the integral formula F(x) = j / ( x , ήάμ(ί) defines a F e s/{U,Z), Dkf{x,.) is k k m continuous and D F(x) = $D f{x, ήάμ(ί) for allxeU,keN . m _ Proof, a) Let Κ = A(ocp p}) c U; for / e j=ι

b e K, we have

f(b) - f(a) = J £ (ßj - aj)%- [(1 - t)a + tb] dt 0 j=1 Οζj since for all z' e Z ' this is true with z' ο f instead of / ; hence follows q[f(b)

- /(α)] ^ f

Ißj ~ a ;l S " P

° Jf )

for

a11

4e

csn(Z)>

and

boundedness oiDk38 for \k\ = 1 implies the equicontinuity of 38.

the

26

2 Gäteaux-analyticity

b) Let Φ be any ultrafilter on f^ finer than the Frechet filter; for all x e U , since the set {/„(χ): η e N} is relatively compact for the weakened topology, there is a g(x) e Ζ such that each weakened neighbourhood of g(x) contains {/„(χ): η e Ω} for some Ω e Φ. Since Μ is equicontinuous: given z ' e Z ' , a compact set Κ cz U and ε > 0, there is an Ω e Φ such that sup |z' ο (g — fn)\ ^ ε for all η e Ω, and g e Ζ) by Corollary 2.1.4. κ The ultrafilter Φ can be chosen as follows. Given χ e X, if ζ e Ζ is such that each weakened neighbourhood V of ζ meets {/„(χ): η ^ n 0 } for all n 0 , then the sets A(n0, V) = {n ^ n0: fn(x) e V} are the basis of a filter finer than the Frechet; if Φ is chosen still finer, then g(x) = z. So the uniqueness of g implies the uniqueness of z, i.e. ζ = lim /„(χ) for the weakened topology, and this holds for all χ e X. The argument may be repeated with an ultrafilter Φ on finer than the given for all χ e U, g(x) is such that each weakened neighbourhood of gf(x) contains {/(x):/ e Ω} for some Ω e Φ; given ζ', Κ and ε, there is an Ω e Φ such that sup|z' ο (g — f)\ ^ ε for all f eil; the uniqueness of g κ implies the uniqueness of ζ such that each weakened neighbourhood V of ζ meets {/(x): / e A} for all A e Φ, i.e. ζ = lim^r f(x). c) F is the pointwise limit of the sequence J(n) F„(x)= Χ μ ( Τ Μ χ , Φ used in the proof of Proposition 1.2.3., which is in the bounded set & ={fe

Z): sup (q ο f ) ^ ||μ||. M{q, Κ) for all q and K} κ

with M(q, Κ) as in the beginning of (E). Then Fes/(U, of the sequence J(n) DkF„(x) = Σ μ(ΤΜ/{χ,φ j=ι and f(x,.)

Z), Dk F is the limit

continuous, /(., i) e & entail D^(x,.) continuous for all χ e U.

Proposition 2.1.6. Conversely,

•

let & κ k=i\

Ρ

/

Ρ

- «ι

if Choosing m so that \ ζ„ — oc| ^ q(c°„ - c„°.) ^ 4

we have

— ICm - 0C| + qlfniU Ρ

- /„'(CJ]

28

2 Gäteaux-analyticity

which proves that (c°) is a Cauchy sequence in Ζ since |Cm — a| is arbitrarily small: c? -> c°, q(c°) < M(q). Now gn(0 =

f (η _ co

- defines another bounded sequence (gn) in

s&{U, Z), which again converges at each point Cm; so by the same argument, cln -* c\ qic1)
—te~tx. If / were analytic, the function t • t2 would lie in Z, hence a relation m t2 = Σ (aj + ßjt)e~tXj f ° r all ί e C, which is impossible: this is left to the 7=1 reader.

2.2 Polynomials and polynomial maps In this section, we consider polynomials with coefficients in a vector space Ζ over C, which needs not be endowed with a topology. Proposition 2.2.1. Let f be a map C m -> Ζ with the property: for any x, y e Cm, f(x + Cy) is a polynomial in the complex variable ζ. Then f(x) is a polynomial in the coordinates ξ} of χ (the converse statement being obvious), and the degree (denoted by d°f) of this polynomial is the upper bound of the degree of ζ η-»· /(χ + Cy) as χ and y run through C m .

2.2 Polynomials and polynomial maps

29

Proof by induction on m. Let m ^ 2, let / satisfy the above assumption, and set χ = x' + where (eif.. .,em) is the canonical basis of Cm. If the proposition is true for a map C m _ 1 Z, then X ' I - » f(x' + £ M E M ) is a polynomial Ρξ in ξί, ... , ξ„-ι, whose degree and coefficients depend on ξ„. First assume that X0 = {ξΜ e C: P im constant} is infinite: then, given a' e Cel + --- + Cem_1? we have /(*' + = /(a' + £ m eJ for each x' and for all X0, hence for all ξη e C since both members are polynomials in ξη. Now let ρ be the smallest positive integer such that Xp = e C : Ρξηι has degree p} is infinite: ρ exists because C is uncountable. One can find Ν = points x j e + · · · + Cem_u j = 1, ... , N, and Ν polynomials Aj(x'), with degrees ^p and complex coefficients, such that Ν P(x') = Σ P(Xj)Aj(x') for any polynomial Ρ with degree ^ p and coeffij=ι cients in Z: for the sake of completeness, a proof of this algebraic fact is given in Remark 2.2.2. (A) below. In particular: f(x' + ξ„β„) = Σ /(*; + j=ι

tmeJAj(x')

for each x' and for all ξη e Xp, hence for all ξη e C since both members are polynomials in ξη. Now let η be the degree of the polynomial /(x): the degree of ζι—• f(x + Cy) is with equality if and only if the sum of the terms of degree η in f{x) does not vanish at y. • Remark 2.2.2. (A) The generic polynomial in m variables, with degree < p, Ν can be written as Pix) = Σ c k M k (x), where Ν = C™+p and the Mk are all k=l

monomials with degrees < ρ in the m coordinates of χ e P(x), χ e m C , are linear maps CN -> C which generate the N-dimensional vector space of all linear maps CN -*•C; then Ν points x} can be found in Cm so that the Ν linear maps c t-> Ρ(χ 7 ) are independent, i.e. det Mkixj) Φ 0. Denoting by (a,· K)JTK =i ,N ^ e inverse of the matrix (M,(x k )), we have Ν c f c = Σ a j k P(xj\ these relations still hold if the ck lie in any vector space 7=1 over C, and they imply P(x) = f k=1

= Σ P(Xj)Aj(x), j=1

Ajix) = Σ *j,kMkix). k= 1

(B) If the space Ζ is I.e. and s.c., the following simpler proof of Proposition

30

2 Gäteaux-analyticity

2.2.1 proceeds from the results of Section 2.1. The assumption implies that / is holomorphic, hence analytic; for all χ e C m , - — 1- (x) = 0 for any integer νζΐ ρί greater than the degree of the polynomial ξί \->f(x + Ci^i), so that C m Γ dPlf ") m is the union of the closed sets < x e C : - r — (x) = 0 >, and one of these must I "fi 1 3 QPif have a non empty interior, hence an integer such that -r—vanishes ^Sl1 identically. We obtain the other pj in the same way, and then / ( x ) = £ pkm k\

*k%

k x

•

η Proposition 2.2.3. Let /(C) = X

C

*M0> where each cke Ζ and the hk are

k= 0

complex valued holomorphic functions on a connected open set U in C: for any seminorm q on Z, q ο f and In q ο f are continuous functions on U, with values in [0, +oo [ and [—oo, +oo [ respectively, q° f subharmonic, In q ο f subharmonic or = - c o . For subharmonic functions, more generally for potential theory on the complex plane, we refer to [Brel] 3 . Proof, q ο / is continuous on U because Iq ο /(C) - q ο /(Co)I < qim

~ /(Co)] < Σ Λ ) Ι Μ 0 - MCo)l· fc = 0

π For any linear map ζ*: Ζ -*• C: ζ* ο /(C) = ]Γ z*(ck)hk{C) is a holok=ο morphic function on U, therefore \z* o / | subharmonic, ln|z* ο / 1 subharmonic or ξ —oo. By the proof of Proposition 1.1.l.a, q is a supremum of moduli of linear maps Ζ -» C; then q ο / , a continuous supremum of subharmonic functions, is subharmonic and In q ο / subharmonic or ΞΞ - 0 0 .

•

Proposition 2.2.4. Let f be a polynomial C Ζ with degree n,q a seminorm οηΖ,Κ α nonpolar compact set in C, U the unbounded connected component of C\K, G the Green function of U with pole at infinity. Then (BernsteinWalsh inequality): qof(C)^Sup(qof)enG^ κ q ο /(C) < sup {q ο f )

for all ζ eU, for all ζ e ZU.

2.2 Polynomials and polynomial maps

31

G can be defined as the smallest positive harmonic function h on U such that h(C) — In I CI has a finite limit as ζ oo; y = lim [G(C) — ln|C|] is ζ-00 known as the Robin constant of the compact set ZU. Proof. Since q ο / is subharmonic (Prop. 2.2.3), the second inequality is an immediate consequence of the maximum principle: Κ contains the boundary of the open set ZU\K. In order to prove the first one, we may assume 0 e Κ and consider V = \ ξ =

/1\

ζ

ζ ε U or C = oo 1, an open set in C: ξ ι->

J

G l - I + In I ξ I is a harmonic function on V, therefore, on each open disc Ö cn ^ Proposition 2.2.3 also asserts that fc = 0 .exp is a subharmonic function of ξ e By the local nature of subharmonicity, the right hand member is a subharmonic function of ξ Ε V, which by the maximum principle is smaller than the maximum of q

Γ "

ΛΥΊ

£ c I - I for ξ Lfc=o k \QJ J on the boundary of V (oo included if V is unbounded), i.e. sup( 1 and a family of polynomials C -> Ζ such that sup q ο /(C) < +oo for all ζ Ε K, there is a number m such that q ο / ^ m(AL)d°f everywhere for all f Ε Μ (the Leja polynomial lemma), in particular q Ο f ^ mld"s on Κ for all / e A Proof. Since In L is continuous, In L is subharmonic on C, there is a positive Radon measure μ, carried by the boundary of ZU, hence by K, and a harmonic function h on C such that l n L ( f ) = Ηξ) + J In\ξ - η\ άμ{η)

for all ξ e C;

a comparison with ln|£| as ξ -> oo shows that μ(Κ) = 1 and actually h is the constant y: therefore J In | ξ — η | άμ(η) = — y for all ξ Ε CU. Without loss of generality, we may assume that diam(CC/) ^ 1.

32

2 Gäteaux-analyticity

For all m e N·, Km = {ξ e K: q ο / ( ξ ) ^ m for all f e is a compact subset of Κ, and the sequence (Km) increases to K; we consider only the indeces m for which μη = μ\Κτη is not 0. Since J In | — η\ άμ(η) is continuous, the upper semicontinuous functions ξι—• Jln| JinIξ — η\ά(μ — μη)(η) are continuous too; the first one decreases on ZU to |1η|ξ — η\άμ(η), uniformly on ZU by the classical Dini lemma: let em — γ be its lower bound on ZU, em decreasing to 0. Then I n L J f H μ{Κ ) η

y-Sm + J l n l f - f l ^ m f a )

defines a positive continuous function lnL m , harmonic on Um (the unbounded connected component of C\Km), whose difference with 1η|ξ| has a finite limit as ξ oo: by Proposition 2.2.4 we have qo f ίζ. m. Ldmf

for all / g ^ everywhere.

But l n ^ tends to 0 uniformly on ZU, hence everywhere since it is harmonic on U and has a finite limit at oo: given λ > 1, we can choose m so that Lm ίζ XL everywhere. • Definition 2.2.6. Let X be an infinite dimensional vector space. A map φ: X -*• Ζ is a polynomial map i f , for one or any integer me Ν*, φ has the property: for any a, al,...,ame Χ, φ (a + ζ ιαί + ··· + ζ„α„) is a polynomial in the complex variables ζρ and the degree (depending on a, ay,..., am) of this polynomial has a finite upper bound n, which by definition is the degree ά°φ of the polynomial map φ. By Proposition 2.2.1: this property does not depend on m, nor does the integer η under the assumption of bounded degrees, which is not superfluous. Consider, for example, the space c00 (f^*) of sequences χ = (ξ ΐ 5 ξ2,. ·.) of complex numbers ξη with ξη = 0 for sufficiently large η: φ{χ) = 00 Σ ξ" is a map c 0 0 C such that ζ -* φ(χ + ζα) is a polynomial with n=l

unbounded degree. If ψ is a polynomial map Υ -*• Ζ, then \J/(b + C^i + ''" + (mbm) is also a polynomial in the complex variable ζ. Therefore the composed map of polynomial maps X Υ and Υ -*• Ζ is a polynomial map X -*• Z. Proposition 2.2.7. a) If φ is a polynomial map of degree η, φ(χ) is a linear combination, with rational coefficients depending only on n, of φ(χ + a), φ(χ + 2a),..., φ(χ + (η + 1) a). b) If φ is a polynomial map of degree 1, φ — φ(0) is linear.

2.2 Polynomials and polynomial maps

33

Proof, a) Let φ(χ + ζα)=

J ] ckCk: given φ(χ + α), φ(χ + 2a), ..., φ(χ + k=0 (η 4- l)a), the ck are the solution of a linear system with coefficients e N*, whose Vandermonde determinant is not 0. b) Let • Ζ is n-homogeneous, φ(ζχ) = ζηφ(χ) for all χ e Χ, ζ e C.

η e N*, if

Theorem 2.2.9. a) Any n-homogeneous polynomial map has degree η unless it vanishes identically. b) For η eN*, any n-linear map λ: X" Ζ generates an n-homogeneous polynomial map φ: X Ζ (which may vanish identically even if λ does not) by the relation (1)

φ(χ) —

. . . , x) (x written η times).

c) Conversely, given an n-homogeneous polynomial map φ: X -*• Ζ, there is a unique symmetric n-linear map λ: X" -*• Ζ associated to φ by the relation (1)· Proof, a) Any n-homogeneous polynomial C 2 Ζ has degree η or vanishes identically; since this is true for the polynomial (ξ,ζ)\-*φ(ξχ + ζά), the polynomial ζ ι-» φ(χ + ζα) has a degree < n. But the polynomial ζ (-• φ(ζα) has degree η unless φ(α) = 0. c) Let Λ be a symmetric n-linear map X" Ζ and >ί(χ) the right hand member of (1). For a x , . . . , a„ e Χ, Χ{ζίαι + · • • + ζηα„) is a n-homogeneous polynomial in the complex variables ...,£„, in which the coefficient of Ck,keN",\k\ = k1+--- + k„ = n, is — λ(αι written kj times,... ,a„ written k„ times). Now replace ζ = (C ls ...,C„) e C by ε = (8lt...,eJ e { - 1 , +1}" and (n ,£,)/2 remark that = ( - l) ^~ ; then, for all k e N", \k\ = n, ε

ε

is 0 if some kj is even, 2" if all kj are odd, which happens only if all kj are 1. Therefore a symmetric n-linear λ satisfying (1) must be

34 (2)

2 Gäteaux-analyticity A(alf. · ·, α„) = - J - Σ ( - 1) ( π - , £ | ) / 2 ?Μι + · • · + ε„α„). Lm ΐ Ι·,

£

This formula defines a symmetric map X" - » Z ; but is it n-linear and does it satisfy (1)? In order to prove that it depends linearly on set ε' = (ε2,...,£„)ε { - 1 , + 1}"-1 and ψ(χ) = Σ ε2... ε„φ(χ + Η^ι + ''" + Wn) = Δβ2... Α„ηφ(χ) ε' with the notation ΔΛ->(c k _ 1 ) e (x 1 ,...,χ^); secondly, that (c k ) a (x,x 1 ,...,x k _ 1 ) is the coefficient of ζ in the Taylor expansion around0 of(cn-^.+^ixj,.. . , x - ) . Then Xj ^(x^..., x k _ x ) linear implies x 1 i-^(c k ) 0 (x,x 1 ,...,x k _ 1 ) linear, hence (ck)a actually fc-linear since (ck)a is symmetric. ι

k

ι

t

If |ζ| ^ ρ implies csn(Z), is summable by Theorem 2.1.2.b and therefore, by Theorem 1.2.l.c, is summable to f(a + αχ + ζχ) — f{a) for |α| + |ζ| ^ p. By considering the subfamily in which the degree of

40

2 Gäteaux-analyticity

ζ is k - 1 (k > 2), we get φ^/)(χ)

and (/5 k /)(x) is

= £

the coefficient of α in the Taylor expansion around 0 of 0 a + l x f )(x). Putting all results together, we see that (c^X, (k — l)-linear for all a ε Ω implies (c k ) a /c-linear for all α ε Ω, and (5) for all a ε Ω for the degree k — 1 implies (5) for all a ε Ω for the degree k. b) Since Ζ) contains each map a * (ck)a (xl,..., xk), by (5) ^(Ω, Ζ) also contains ah->0kf)(x). The fact that (/> k /)(x 0 ) is the coefficient of ζ in the expansion of 0 ^ ο / ) ( χ ο ) is now written Dkf(a) = ß fl 1 (D k_1 /)(x0 ) or Dkf = Dl(Dk~1f); the general formula follows by an induction on j. • Proposition 2.3.6. Conversely: let Ω be finitely open and balanced, (bk) — ck for all k e N, which by the inequalities of Cauchy implies the boundedness of the sequence in fact, one can choose a sequence (YjJ of distinct factors in the product, and bke Y with prikbk Φ 0, prjbk = 0 for all i / ik. On the contrary, a bounded sequence (bk) with the property of linear independence cannot exist if the topology of Y is the finest I.e. one, defined by all possible seminorms: if it did exist, it could be completed into a Hamel basis by other linearly independent vectors 6/, i e /, each y sY could be uniquely written as y =

Σ ßk(y)h JceN

+

Σ iel

with only a finite number of nonvanishing ßk(y) and #(}>); the seminorm ρ — Σ k\ßk\ would be unbounded on the sequence (bk). keM

2.4 Boundedness and continuity of Gateaux-analytic maps In this section, I.e. topologies are given on both spaces X and Z. In view of the expansion (1) in Theorem 2.3.5, we begin with polynomial maps X -> Z;

44

2 Gäteaux-analyticity

their fc-homogeneous parts were defined by formula (3) in Proposition 2.2.10. Proposition 2.4.1. Let φ be a polynomial map X -> Z. a) For each a e X, the following properties are equivalent: (i) for all q e csn(Z), q ο φ is bounded on some neighbourhood of (ii) for all q e csn(Z), q ο φ is continuous at the point a; (iii) φ is continuous at the point a. b) for all q e csn(Z), q° φ continuous at one point implies q° φ everywhere; consequently, φ continuous at one point implies φ everywhere. c) φ is (everywhere) continuous if and only if its k-homogeneous (everywhere) continuous.

a;

continuous continuous parts are

Proof, a) In order to show that (i) implies (iii), write formula (3) in Proposition 2.2.10 for the polynomial map χι—κρ(α 4- χ): φ(α + χ) = φ(α) + η

£ 1: ~q ο 0kf){x)

^ M[p 0 (x)]* < [Mp 0 (x)] fc ,

i.e. (iv) with ρ = Mp 0 . So it only remains to show that (v) implies (i). Let V cz Ω — a be a balanced neighbourhood of the origin in X: by formula (1) in Theorem 2.3.5, -j-q ο 0

k

ki

q\_f(a + χ) - / ( α ) ]

f ) ^ Μ for all keN*

on V implies

Μ— 1 9 — for all χ e 0F, 0 < Θ < 1.

1 b) g(x) = sup — qo 0kf)(x) fceN* k\

is finite for all χ e ω(α) since the expansion

(1) in Theorem 2.3.5 is summable; 0 k f ) continuous for all keN* entails g lower semi-continuous. As an open set in Χ, ω(α) is Baire and there is a set b + V α ω(α), where V is a balanced neighbourhood of the origin in X, such that ^ A i o n H F o r ^ o 0kf)(b + x) < Mk\ for all k e N* χ e V, hence also q ο 0af)(x)

^ Mk\

for all k e Ν*, χ e V,

by the translation lemma 2.2.1 l.b.

•

Remarks 2.4.3. (A) / bounded on some neighbourhood of the point a is equivalent to the properties in a) if and only if the topology of Ζ can be defined by a norm, i.e. the origin has bounded neighbourhoods: if not, the identity map Ζ Ζ is continuous but not locally bounded. (B) Statement b) may no longer hold without the Baire assumption, and once more counter-examples are easily obtained in the space X = Cqo(^I). With the notation χ = {ζοΛιΛζ,···), ψ( χ )

=

00

Σ

n=1

a ma

P X

46

2 Gäteaux-analyticity

such that ζ\->φ{α + ζχ) is a polynomial with unbounded degree, and 0% C: / and φ ° f are continuous at the same points e Ω. Proof, a) Since Ω is connected: given a e Ω, b e Ω, there are a finite number of points a0 = a, at, ap = b such that Ω contains each line segment {(1 1 + taq\ 0 < t ίζ 1}, q = 1, p. From this follow that /(Ω) is connected and also, by Proposition 2.3.7.a, that / cannot be constant on any open nonempty set ω c Ω. Now assume that, for some a e ω, /(ω) is not a neighbourhood of f(a): then, for all xe X, f{a + = /(a) for sufficiently small and / is constant on Ω by Proposition 2.3.7.b. b) Let φ ο f be continuous at a e Ω. Since φ is continuous and locally injective, for any sufficiently small closed disc Δ e: /(Ω), with centre f(a), there is an α > 0 such that \ψ — φ ο f{a)\ 3s α on the circumference of Δ; the point a has an open connected neighbourhood ω c Ω where | ψ ο f — φ ο /(α) I < α, which implies /(ω) cz Δ since /(ω) is connected. • The following Proposition and Theorem show that ^-analytic scalar functions which are not continuous behave as badly ([Ta] 2 , Th. 3.5.E) as linear functionals which are not continuous. Again we begin with polynomial maps, which by Proposition 2.4.l.b are continuous either everywhere or nowhere. Proposition 2.4.8. Let φ be a polynomial map X -* €: either φ is continuous or f(a + ζχ), χ e V, which therefore form a normal family in the sense of Montel. Since they assume the same value f(a) for ζ = 0, they are bounded together for ζ = V a+ J .

which means / bounded on

The omission of one value is not a sufficient condition since, by Proposition 2.4.7.b, / and ef are simultaneously continuous. The same counterexample serves for statement b). b) We may assume that g = (f — α)(/ — β) is not identically 0; 0 so that Κ — {a + Cb: Id ^ r}