A powerful introduction to one of the most active areas of theoretical and applied mathematics This distinctive introduc

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*76*
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*English*
*Pages 370*
*Year 2000*

- Author / Uploaded
- Terry J. Morrison

**Commentary**- Improvements with respect to 77F9130556D5E829177CA78D2505C905: pagenated and added bookmarks

*Table of contents : PrefaceIntroduction1. Basic Definitions and Examples 1.1 Examples of Banach Spaces 1.2 Examples and Calculation of Dual Spaces2. Basic Principles with Applications 2.1 The Hahn-Banach Theorem 2.2 The Banach-Steinhaus Theorem 2.3 The Open-Mapping and Closed-Graph Theorems 2.4 Applications of the Basic Principles3. Weak Topologies and Applications 3.1 Convex Sets and Minkowski Functionals 3.2 Dual Systems and Weak Topologies 3.3 Convergence and Compactness in Weak Topologies 3.4 The Krein-Milman Theorem4. Operators on Banach Spaces 4.1 Preliminary Facts and Linear Projections 4.2 Adjoint Operators 4.3 Weakly Compact Operators 4.4 Compact Operators 4.5 The Riesz-Schauder Theory 4.6 Strictly Singular and Strictly Cosingular Operators 4.7 Reflexivtty and Factoring Weakly Compact Operators5. Bases in Banach Spaces 5.1 Introductory Concepts 5.2 Bases in Some Special Spaces 5.3 Equivalent Bases and Complemented Subspaces 5.4 Basic Selection Principles6. Sequences, Series, and a Little Geometry in Banach Spaces 6.1 Phillips' Lemma 6.2 Special Bases and Reflexivity in Banach Spaces 6.3 Unconditionally Converging and Dunford-Pettis Operators 6.4 Support Functionals and Convex Sets 6.5 Convexity and the Differentiability of NormsBibliographyDiestel J. -EFGHJJohnson W.B -KLMNPPefczynski A. -RSTVWYAuthor Index ABCDEFGH IJKLMNOPRS TUVWYZSubject Index a bc d efgh ijkl mno pqrs tuvw zSymbol Index*

FUNCTIONAL ANALYSIS

PURE AND APPLIED MATHEMATICS A Wiley-Interscience Series of Texts, Monographs, and Tracts Founded by RICHARD COURANT Editors Emeriti: PETER HILTON and HARRY HOCHSTADT, JOHN TOLAND Editors: MYRON B. ALLEN III, DAVID A. COX, PETER LAX A complete list of the titles in this series appears at the end of this volume.

FUNCTIONAL ANALYSIS An Introduction to Banach Space Theory

TERRY J. MORRISON Gustavus Adolphus College St. Peter, Minnesota

A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York · Chichester · Weinheim · Brisbane · Singapore · Toronto

This book is printed on acid-free paper. @ Copyright © 2001 by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4744. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: PERMREQ @ WILEY.COM. For ordering and customer service, all 1-800-CALL-WILEY. Library of Congress Cataloging-in-Publication Data: Morrison, T.J. (Terry J.), 1947Functional analysis. An introduction to Banach space theory / T.J. Morrison. p. cm.—(Pure and applied mathematics) Includes bibliographical references and indexes. ISBN 0-471-37214-5 (cloth: alk. paper) 1. Functional analysis. I. Title. II. Pure and applied mathematics (John Wiley & Sons: Unnumbered) QA320.M69 2000 515.7—dc21 00-036822

10

9 8 7 6 5 4 3 2 1

For Kathy and Cheri

CONTENTS

Preface

ix

Introduction

1

Notation and Conventions, 2 Products and the Product Topology, 4 Finite-Dimensional Spaces and Riesz's Lemma, 8 The Daniell Integral, 11 1. Basic Definitions and Examples

17

1.1 Examples of Banach Spaces, 17 1.2 Examples and Calculation of Dual Spaces, 36 2. Basic Principles with Applications 2.1 2.2 2.3 2.4

The Hahn-Banach Theorem, 64 The Banach-Steinhaus Theorem, 75 The Open-Mapping and Closed-Graph Theorems, 78 Applications of the Basic Principles, 82

3. Weak Topologies and Applications 3.1 3.2 3.3 3.4

63

107

Convex Sets and Minkowski Functionals, 108 Dual Systems and Weak Topologies, 117 Convergence and Compactness in Weak Topologies, 124 The Krein-Milman Theorem, 144

4. Operators on Banach Spaces 4.1 Preliminary Facts and Linear Projections, 156 4.2 Adjoint Operators, 164 4.3 Weakly Compact Operators, 170

155

viii

CONTENTS

4.4 4.5 4.6 4.7

Compact Operators, 181 The Riesz-Schauder Theory, 186 Strictly Singular and Strictly Cosingular Operators, 201 Reflexivity and Factoring Weakly Compact Operators, 210

5. Bases in Banach Spaces 5.1 5.2 5.3 5.4

Introductory Concepts, 220 Bases in Some Special Spaces, 232 Equivalent Bases and Complemented Subspaces, 238 Basic Selection Principles, 246

6. Sequences, Series, and a Little Geometry in Banach Spaces 6.1 6.2 6.3 6.4 6.5

217

269

Phillips' Lemma, 270 Special Bases and Reflexivity in Banach Spaces, 285 Unconditionally Converging and Dunford-Pettis Operators, 303 Support Functionals and Convex Sets, 310 Convexity and the Differentiability of Norms, 321

Bibliography

341

Author/Name Index

345

Subject Index

348

Symbol Index

358

PREFACE

A mathematician is a person who canfindanalogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories; and one can imagine that the ultimate mathematician is one who can see analogies between analogies. —S. Banach The theory of Banach spaces really began with the 1922 publication of Stefan Banach's doctoral dissertation "Sur les operations dans les ensembles abstraits et leur application aux equations intégrales" in Fundamenta Mathematicae, followed in 1932 by his famous monograph Theorie des Operations Linéaires in Warsaw, Poland. In the minds of a majority of mathematicians, the appearance of these two publications also signaled the onset of "modern" functional analysis as an independent discipline. Through the influential work of Banach, M. Fréchet, J. Hadamard, H. Hahn, F. Hausdorff, D. Hubert, S. Mazur, J. von Neumann, F. Riesz, and M.H. Stone, to name but a few, mathematics was changed; there was no looking back. It is hoped that by studying the ideas and techniques presented in this text, by following through on the directions indicated by many of the fundamental results presented here, and by gaining a deeper understanding of the beauty and subtlety underlying most of Banach's work and legacy, you will develop an appreciation for and understanding of this rich area of mathematics. Much is to be gained from mastering the basic ideas you will be exposed to in the material that follows, and in then continuing to pursue both the theoretical avenues they open and the many applications they represent in mathematics and science. Consider this book a beginning point only. What I have attempted to do here is to gather and organize the work of those mathematicians that has formed the basis for the discipline of functional analysis as it is known today. There is, of course, some arguably fundamental material omitted from this book; you are naturally seeing my personal bias as to what is most important and most memorable. However, you will be exposed to the basic ideas, techniques, and methods that form the underpinnings of this ix

X

PREFACE

discipline. Primarily through the study of Banach spaces (with the occasional side trip into general topological vector spaces), you should gain the necessary tools and insight to successfully investigate whatever area of mathematics you have chosen (or which has chosen you). Ideally, you will be sufficiently excited and motivated to make Banach spaces themselves an integral part of your mathematics. A FEW NOTES TO THE STUDENT This book is meant to be more than just a reference source, a repository of definitions and theorems; it is meant to be read. In the commentary between theorems I have attempted to help motivate you and explain why certain results are important, when particular attention should be paid to their method of proof, and when further exploration beyond the text itself is recommended or even needed. I have often included more detail than one might expect to see in a book at this level; on the other hand, you will note a fair number of familiar phrases such as "the details are left to the reader." These have been chosen with some purpose in mind; more detail where the author feels initial proding and aid in reaching the heart of the argument is necessary, less when you should be adequately prepared to proceed on your own. My goal is for you to gain understanding and insight from the presentation; I hope the sometimes less formal nature of the arguments will help, and not hinder, this process. You will soon realize that there are no exercises at the end of sections, as you will find in most books; don't be misled, however, there are exercises and problems embedded in the text itself. I have chosen to present these "in context," as the results they give often have immediate benefit. Other times, as when having to recreate the context in which they are found could be distracting, they are where they belong most naturally. These problems, while they are not delineated or numbered in any special way, should be recognizable when you meet them, and are introduced by such phrases as " . . . as the student should verify . . . , " or " . . . it is straightforward to see . . . , " or " . . . as a moment's thought reveals . . . , " each such phrase a clue to the student. There are also more explicit exercises and problems whose solution has been left to you, and it is expected you will "fill in the missing details." It is the author's intention that all these exercises be completed and the ideas internalized. Understanding does not come passively. A FEW NOTES TO THE INSTRUCTOR The book is designed for a two-semester first course in functional analysis and should allow time for topics presented here to be explored in further detail, or new material to be introduced if desired. The introductory material is for

PREFACE

xi

the convenience of the student; Chapter 1 is where it all begins. While not all the examples of the first chapter need be presented, there are many ideas surrounding them to which the student should be exposed. While it is not strictly necessary that all topics in every chapter be presented and thoroughly understood by the students before proceeding to the next, the author feels the majority of the material plays an integral role in what every young analyst should know and master. ACKNOWLEDGMENTS The author is deeply indebted to many people who helped (and sometimes literally forced) this book to come into existence. Undoubtedly, the two most important and influential are my best friend, largest supporter in every possible meaning of that word, partner, most successful motivator, and wife, Kathy; and, my advisor, mentor, and long-time friend, Dr. Professor Joseph Diestel of Kent State University. There is absolutely no question that you would not be reading this now without their presence in my life; my appreciation and gratitude are beyond words. There are, of course, many others whose suggestions, ideas and remarks, and general support mathematically as well as personally during the long development of this book have been greatly appreciated, and who must be acknowledged; particularly all my colleagues in the Mathematics Department at Gustavus Adolphus College; I must especially mention R. Rietz, J. Rosoff, and L. Hewitt. Additionally, my thanks and appreciation are extended to M. DiBattista, N. Miller, M. Gaviano, G. Georgacarakos, R. Hubert, D. Kelley, T. Henry, and R.H. Lohman. Thanks must also be given to my former colleagues at Addis Ababa University, and particularly my students there, who were subject to early versions of the lecture notes that eventually led to this manuscript, especially Fereja Tahir and Negash Begashaw, who have become close friends and colleagues. Acknowledgment must also be given to the staff at John Wiley & Sons, particularly Lisa Van Horn, Heather Haselkorn, Andrew Prince, and Steve Quigley, for their support and help in making this process as smooth and easy as possible. Finally, a special debt must be acknowledged to B.J. Pettis who, late one night over a mostly empty bottle of Virginia Gentlemen some years ago, advised me that if I was going to dedicate my dissertation to my wife, not to, but to wait and save that for my first real book; while it has taken some time, I have done so. St. Peter, Minnesota October, 2000

TERRY J. MORRISON

FUNCTIONAL ANALYSIS

Functional Analysis: An Introduction to Banach Space Theory by Terry J. Morrison Copyright © 2001 John Wiley & Sons, Inc.

INTRODUCTION

Before entering into the formal material of the text, there are always ground rules that need to be understood so that the reader and the author know they will progress down the same path and understand they share the same goal. While the direction of the road and recognition of the purpose in this case are not difficult to comprehend (after all, the student assumes the author is devoted to presenting and explaining the fundamental concepts of the discipline to the uninitiated, and the author assumes the reader is approaching the material eager to learn, motivated to pursue the ideas beyond the meager boundaries of the text itself, and is more than adequately prepared for this journey), all involved realize this works well theoretically. More often than not misunderstandings arise early: How in the world can they expect we've seen this stuff before!, "as the reader should recall..." and "from the student's real analysis course, one has ..." become roadblocks (or at least at times substantial barriers) to achieving the desired ends. While there is no practical way to completely avoid these problems, since any two different readers have not only different backgrounds but different purposes for reading this text, they can to some extent, be alleviated. At least this is the author's purported rationale behind including these opening comments, remarks, and observations before formally engaging in the adventure that follows; and it is hoped it will be an adventure, a remarkable experience. After all, contained herein are some of the basic ideas and techniques that lead to the theory of Banach spaces, one of the most beautiful and profound disciplines in all of mathematics, certainly within the realm of functional analysis. What more need be said? Beyond the reasonable expectation that students wishing to partake of this material have been exposed to the standard material one normally encounters in any appropriate sequences in real analysis and topology, the author tries to assume only a fledging mathematical maturity and the openmindedness to give beauty and subtleness a chance to work its magic when encountered. You don't have to be a true believer to begin with; this i

2

INTRODUCTION

will be a natural outgrowth of the exposure to the work of Stefan Banach and his followers. Of course, practically speaking, assumptions must be made and reliance on background must be assumed. While what immediately follows is not by any means all-inclusive, the author has chosen a few short topics that may aid readers as they begin their perusal of these topics. They need not be considered in the order given, or for that matter closely considered at all. They are included because the author assumes there are always a few facts and ideas whose understanding will help to ease the transition into the new material, and that the reader may or may not have within easy reach. The first short section on notational conventions and standard symbology used throughout the text should, of course, be quickly scanned. The first topic presented is a brief look at product spaces at a very basic level. This can more than likely be ignored until one is ready to begin Chapter 3 and perhaps even then; its inclusion here is to ensure that the reasons behind changing topologies on a given space are understood and realized to be reasonable in the given context. The second short section concerning finite-dimensional spaces is ideally totally unneeded; the notion of a Hamel basis and the role it plays in understanding these spaces are indistinguishable algebraically and topologically, underlies much of what we will encounter in the near future. The only possible exception is the inclusion here of a pretty result by F. Riesz on approximations in a normed linear space, which often proves useful to have at hand. The last material in the Introduction will prove to be superfluous to any student with a firm grounding in abstract measure and integration theory. Because the Daniell approach to integration theory is not necessarily a standard topic in real analysis, it is included here for those readers not fortunate enough to have been exposed to it. While undoubtedly insufficient to completely prepare the student for its use in Chapter 1, it is hoped the brief exposure to the basic concepts will allow the reader to achieve an adequate facility with the central ideas to either be able to fill in the missing details, or at least locate and prepare for its use. So, we begin. NOTATION AND CONVENTIONS As always whenever one encounters a new book or set of notes, there are notations and conventions that the author uses, but with which the student is not familiar, and, of course, these can lead to confusion. We include a number of the more common of these here; other more specialized notation will be encountered in the text as they are needed. A comprehensive list of symbols and notation can also be found in the appropriate index at the end of the text. We start with a list of some basics.

3

NOTATION AND CONVENTIONS

1. We will use the following designations for some standard collections of numbers: R for the real numbers; C for the complex numbers; S for an arbitrary scalar field (when the specific use of either IR or C is immaterial); N for the natural numbers {1, 2, 3,...); Ö for the collection of all integers. Furthermore, we will always use the symbol Θ to denote the zero vector in an arbitrary space rather than the number 0 so that no confusion arises. 2. Anytime we are considering a singly indexed object such as a sequence, a sum, or a limit (as long as the underlying index set is countable; that is, essentially the natural numbers), if the beginning index value does not matter or is unimportant to the meaning of the expression, this value will be omitted in our representation. Similarly, as typically the upper limit of our indexing is °°, we will omit this as well and write these expressions as follows: (χη)„ instead of (*X,

or Σηαη

or limny,,

or Σ^,α,,

or

lim n ^„v n

In case we wish to allow our index set to be uncountable (or at least not restrict ourselves to only countable sets), we will change the symbol used for our underlying index set and assume it is some arbitrary directed set Γ. Thus, the expressions just listed will usually be written as (a x ) r

or lim r y y

[or perhaps {αγ)γ£Γ

or lim,,Er yr]

to distinguish this. In other words, we consider (ar)r to be a net. In a similar manner, to indicate that a function (or operator)/is the pointwise limit of the sequence (/„)„, we write / = lim„/„, rather than/(jc) = lim„f„(x) for all x, when no confusion should arise. 3. Anytime we wish to make it overtly clear the choice of a particular constant depends upon a previously determined value, we subscript that constant with the dependent value; that is, "... there exists an Nc e N such that.. ." means that the choice of the integer JV depends on the value of ε given, while "... choose δ„ ,ε > 0 with ..."

4

INTRODUCTION

means that the value of δ depends on the values of both n and ε, and so forth. 4. The usual symbols for set operations will be used, so if we let {Α^76Γ be a (nonempty) family of sets indexed by the directed set Γ, then: U r6l - Ar ={a:aeAY

for some γ e Γ},

( l ^ r AY = {a: a e AY for some γ e Γ}, rirer A7 = {(aY)r : ay e A r for each γ € Γ}. We will further use A\B to denote the set-theoretic difference of A and B, that is, A\B = {x:xe A but x e B), and Ac to denote the complement of the set A (relative to some universal set, of course). 5. Another convention that will be followed is the occasional use of the symbol = in place of the usual =. This will be employed primarily to stress that we are defining a particular object by this description. That is, if we wish to define the function f:U—>U by f(x) = sin(3;t), we might write "let /:(R-»R be given by f(x) = sin(3x)" to further emphasize that / i s being defined here. This will often be used when we define an object in the middle of a proof or in an explanatory paragraph where it is inconvenient to interrupt theflowof the text. It should also aid readers in realizing they are encountering a particular object for the first time, and have not inadvertently overlooked its meaning. 6. In conjunction with remark 5, the reader should note that definitions or descriptive titles that will be used and referred to through the text are often embedded in an explanation, remark, or proof of a statement. In order to make these easier to recognize and locate for later reference, they will be written in boldface lettering to improve visibility. As with any special symbols used, all can be found in the symbol or subject index. 7. As the final comment, the student should note that in order to more clearly specify the actual end of a proof, we will always use the symbol ■. No special meaning should be attached to this symbol, as it merely serves as the obvious visual indicator of the end of an argument.

PRODUCTS AND THE PRODUCT TOPOLOGY While products and the product topology are not difficult concepts, but are standard fare in any course in general topology, somehow it often appears to be one of those topics quickly lost at the conclusion of the course itself. While it is certainly assumed the readers are capable of reexamining some of this material on their own when it is necessary, we include here a short overview of this material as both a ready reference and particularly to remind the student of the concept of weak topologies. A good grasp of these general ideas will serve the reader well when we encounter them in Chapter 3.

PRODUCTS AND THE PRODUCT TOPOLOGY

5

Of course, at this point there is no reason, a priori, to simply read through the material that follows. Scan it now, remember it is here, and then return for a closer look should this become necessary at some point in the future. In any case, we now give a quick walk through the basic ideas. Let Γ be an arbitrary directed set and for each ye Γ, let (Χγ,τγ) be a topological space.The Cartesian product (or just product) of the family {(Χγ,τγ)}γ£Γ is the set of all functions / : Γ-> [}γΐτΧΥ, where for each ye Γ we have /(y) e Xy. For notational purposes, we will denote this collection of functions by ΠΤεΓΧγ or by n r X r when this will cause no confusion. It is easy to see that there is another way to view the elements of a product; namely, each element x e nrXr can be realized as a net in the following way: xeYlf-Χγ

if and only if x = {xY) e r where xr eXr for every y eT.

Hence statements about the elements of an (arbitrary) product space are automatically statements about nets. In addition to the preceding remarks, we employ the following conventions both to simplify the notation used and because the reader is likely to encounter (or has encountered) these same concepts in other books and contexts where they are frequently found: 1. If the family Γ is countable, say we have X\,Xi,... ,X„,..., then we will write the product as UnX„ to conform with our other notations. 2. If for each ye Γ we have XY=X (that is, all the spaces XY are the same space X), then we will write the product n r X x as Xr; often as Χω should Γ = Ν. 3. Finally, in the case that we have a finite product of spaces that are all the same, say X,; = X for i = 1,2,... ,n, we will write Π"=λ X¡ = X". We now indicate how to topologize the product ΠΓΧγ of a number of topological spaces. For each y0e Γ, define the map πΥο: n r X r -> Χϊο by TCY(f)=fr. That is, πΎο is the map that selects the y0th-coordinate of /, for each / in the product. Each such map πΥ is called the y-coordinate map or the y-projection map. By definition (that is, we are defining it here) the product topology on UrXr will be the weakest topology on the product for which each of the maps πγ, for ye Γ, is continuous, that is, it is the smallest topology for which all of the projection maps are continuous. It can be seen that a local base of open sets for the product topology is the family of all sets of the form n*=i7r^(Urt), where each set \JYk is a τ^-open set in XYk, for k = 1,2,...,« and n s N. That is, each such set is a finite intersection of inverse images of open sets under the (appropriate) projection map. Thus, if / e ΠΓΧγ, then a fundamental system of neighborhoods of / is the family of all sets of the form:

6

INTRODUCTION

{g e Ilr^r : g(Tk) ε Ur„ for k = 1,2,... ,n, where U /t is Tn-open}

At this point it is well worth pausing to consider that there is another way both to "think" about the product topology and to describe it. This is as follows: Take as a base for the open sets of the product all sets of the form n r U r , where 1. for each ye Γ, U r is open in Xr; 2. for all but finitely many ye Γ, we have U r = Xr; that is, sets like n r U r , where U r =X r for all y except some finite number 7= Yuft, ■ ■ ■ ,7n- Note that this set can now be written as

n r u r - π;ΐ (ur,) n < (ur2) n · · · n < (ur.) = Π2=.ΠΓΧγ, where the product has the product topology. Then / i s continuous if and only if ;r y o /is continuous for each ye Γ. Proof. Clearly if/is continuous, so is Ky°f for every ye Γ, since the composition of continuous functions is continuous. On the other hand, suppose that nr°f is continuous for all ye Γ. Since sets of the form πγ\υγ) for ye Γ and U r open in Xrform a subbase for the topology on UrXr, to show that/is continuous, it suffices to show thatf~l(Kyl(Ur)) is open for every subbasic open set π^(υ 7 ). But note that this is clear since f'l(Ky\Ur)) - (Kr°f)~l(\Jy) and ^ r ° / i s continuous by assumption.· Finally, we now make the connection between the product topology and "weak topologies" a little more explicit and, hopefully, a little clearer. First

PRODUCTS AND THE PRODUCT TOPOLOGY

7

recall that it is the topology of a space X that determines which functions are continuous on that space (remember, a continuous function is characterized by having inverse images of open sets being open; that is, being in the topology o/X).Thus the topology one might impose on a particular set determines the continuous functions on that space, or put another way, if you know which functions you want to be continuous in advance, then you can (theoretically at least) give Xa proper topology so as to ensure your collection has that property. A reasonable question to ask at this point is: Given a space X, what functions might we want to be continuous? Well, as you might guess, while this depends on the circumstances involved, we can at least give some idea some of the time. For example, suppose X is Euclidean «-space R". The most natural set of functions we might want to be continuous is the collection of coordinate functions /„; that is, the set of functional that assign to any n-dimensional vector its «th-coordinate. Now, as we all know, the natural topology on R" does make each of these continuous; in fact, any function whose range is R" is continuous exactly when its composition with each of these coordinate functionals is continuous. Of course, considering what we have said so far about product topologies (and about what you already know about R"), it should be clear this natural topology on R" is exactly the product topology. In fact, it can be shown, without too much work, that this topology is the weakest (or smallest) topology that can be put on R", which makes each coordinate functional continuous. This idea can very naturally be carried a bit further, not only to products of sets besides R, but to arbitrary products of any sets. That is, given a collection of spaces Xy, for yin some arbitrary index set Γ, we want to put a topology on the product of the Xr that will make each of the coordinate (or projection) maps πγ continuous, this being as natural in this context as it is in U". Accordingly, the product topology on these sets is called the weakest topology for which each projection map is continuous, a natural enough idea. So, what we basically have is that in the setting of taking a product of spaces, where the maps we want to be continuous are fairly evident, the product topology is the weakest topology we can put on the product making each of the given maps continuous. But suppose now we are in a setting where we have a space X (not necessarily a product), and a collection of functions on X (perhaps each mapping into a different space); we should still be able to talk about the weakest topology on X that makes each of these maps continuous, and in fact we can; the definition embedded in the next paragraph makes this explicit. Suppose we are given a set X and a topological space Xr with a map fy: X —> Xy for each ye Γ. We will call the weak topology induced on X by the collection {/,,: ye Γ} the smallest topology one can place on Xfor which each /,, is continuous. It should be evident that this topology is that for which the sets fr' (Ux) for y e Γ and U r open in Xy

8

INTRODUCTION

form a subbase. Clearly, the product topology on ΠΓΧγ is the weak topology induced by the collection \ny: ye Γ) of projections. It should also be clear that Proposition 1 carries over to any weak topology in this sense, without any essential change in the proof. That is, we have the following proposition. Proposition 2. If X has the weak topology induced by a collection [fy: ye Γ) of functions / r : X - > X r , then / : \) —>X is continuous if and only if fr°f: y —> XY is continuous for each ye Γ. One final comment here before turning to some of the basics of finitedimensional spaces. These relatively simple ideas concerning weak topologies will be just as relevant for us in considering normed linear spaces. Here, one of our concerns will be to take a given space X endowed with the topology it naturally inherits from its norm and seek to find the weakest topology that can be given to Xthat still yields its topological dual space remains unchanged. In other words, we want the weakest topology for which all the (norm) continuous linear real-valued functions on X are still continuous. The full resolution to this search is the primary content of Chapter 3. FINITE-DIMENSIONAL SPACES AND RIESZ'S LEMMA Our purpose here is simply to remind the reader of a few of the basic ideas concerning spaces of finite dimension with which they should be familiar; no attempt is made at completeness in any sense. In fact, our primary concern will be to make clear to the reader that linear spaces of the same (finite) dimension are always algebraically isomorphic and that if endowed with a norm, are isomorphic topologically as well. The easiest way to realize this, and one we will look closely at in Chapter 5, comes from considering the concept of a basis in this setting. Definition 1. Let X be a (nontrivial) linear space. A collection H of vectors from X is called a Hamel basis (or often just a basis) for X if H is a linearly independent set in X and the subspace of X generated by H is all of X (that is, span (H) - X). A particular consequence of the definition itself is that every element of the space has a unique representation as a (finite) linear combination of basis elements. That every linear space (regardless of its "dimensionality") has such a basis is a direct result of Zorn's Lemma, whose proof will not be given here. The reader who has somehow missed (or simply misplaced) these results is urged to spend a short time looking at the ideas inherent here; there are some very nice, and eminently useful, techniques that are well worth adding to one's repertoire. While any reasonable linear algebra text will yield such a presentation, one good source is Friedberg et al. (1989).

FINITE-DIMENSIONAL SPACES AND RIESZ'S LEMMA

9

As indicated earlier, the following well-known result is of some importance to us because of its use and implications for more general normed linear spaces and hence for Banach spaces. Proposition 3. If X and y are both «-dimensional linear spaces with the same scalar field, then they are (algebraically) isomorphic. Proof. Letting S denote the underlying scalar field, we will prove that X (and hence A/ as well) is isomorphic to S" (and so isomorphic to each other by the symmetric and transitive nature of the "isomorphic" relation). To see this, let x]yx2,... Λ be a basis for X so we have that every x s X has a unique representation as x - s,*, + s2Xi + · · · + s„x„, where each s, is in S. We now define the operator T : X - > S" by T(x) = (s]rs2, · · · s„). It is straightforward to verify that T is linear, bijective, and hence that T -1 exists (and is by necessity also a linear bijection). The details we leave to the student.■ Before we give our next result, the student should recall that we often consider linear spaces with a topological structure as well as just an algebraic one. In particular, as previously indicated, this topological structure results from endowing the space with a norm; that is, if X is a linear space, a norm on X is a mapping ||-||: X -> [0, °°), satisying the properties (i) ||x|| = 0 if and only if x= Θ (the zero vector in X)\ (ii) for any scalar λ and any x € X, ||λϊ|| = |λ| ||.x||; and (iii) for any x,y e X, we have ¡JC + y\\ < \\x\\ + \\y\. In this case, Proposition 3 can be extended to give finite-dimensional spaces that are topologically as well as algebraically isomorphic. Proposition 4. If X and \f are «-dimensional normed linear spaces with the same scalar field, then they are topologically isomorphic. Proof. Again we show that if S is the underlying scalar field, then both X and y are isomorphic to §", allowing us to conclude our result as before. So, let X be an «-dimensional normed linear space over § with \x\,xi, ■ ■. ,xn\ a basis for X. Define the operator T : S" —> X b y

Then, by Proposition 3, we know that T is a linear (algebraic) isomorphism. That T is continuous follows immediately from the fact that the addition and scalar multiplication operations on such spaces are themselves continuous maps; as the student should verify (for pedagogical reasons, we prove this shortly as Proposition 1.1 in Chapter 1, the needful student may merely "look ahead"), T is continuous. Thus, we need only show that T" 1 is continuous.

10

INTRODUCTION

To do this,first note that S s ( j e S": ||s|| = 1},being closed and bounded,is compact in S" by the Heine-Borel Theorem. Hence T(S) is also compact, and hence closed, in X. Since T is an isomorphism, Θ i T(S), and so there must be some open set U, containing Θ, such that U Γ) T(S) = 0 . Now choose δ> 0 so that if V B [x e X: ||*| < δ], then V c U. We claim that V c T(B), where B = {s e S": ||s| < 1). In fact, if x e T(B), then x = T(z) for some z e §", with ||z|| > 1. Note if x e V, then JC/||Z|| e V, which is impossible since

¡Hil> T < S ) and we know V D T(S) = 0 . Thus it must be that V e T(B). But then we have T ' ( V ) c B and T _ 1 ( ( 1 / Ä ) V ) c(l/5)B. From this, in turn, we get (1/S)V = \x& X: |[x|| < 1}, and so T _1 maps the open unit ball in X into a bounded set in S" (thus every bounded set of X into a bounded subset of S") and hence T "' is continuous, as was needed.· For the sake of completeness here, you should note (and naturally be able to verify) the following useful facts all come directly from this theorem: 1. In finite-dimensional normed linear spaces, closed bounded sets are always compact. 2. Any finite-dimensional normed linear space is complete. 3. In a general normed linear space, all finite-dimensional linear subspaces are closed. As a final result in this section we present the very general and useful classic result of F. Riesz, now known as Riesz's Lemma; sufficiently interested readers should see Riesz (1918) for his original presentation. You will note that it is not restricted tofinite-dimensionalspaces at all, but holds in any normed linear space setting. We will initially encounter its use when we first consider the idea of approximation in Chapter 4. Theorem 1 (Riesz's Lemma). Let X be a normed linear space and X be a proper closed subspace of X. Then for each real number a with 0 < a < 1, there is an xa e X such that ||χ0|| = 1 and |ΛΓ - *„|| > a for all x e X,. Proof. Let Xi be any element of XOQ and let rf = inf{|[jc — JCI|| : Λ: e X,}. Since X, is closed, we know that d > 0. Now, since (1/a) d>d,we know there is some x0 e X, such that |JC0 - *i|| < (1/a) d. For notational purposes, we will let h s||*0-Xif1, and choose xa = h (x0-xx). Then HJC^JI = 1, and if we let x e Xo, so is h~lx + x0, and so ||* - ocJI = \\x- hxx + hxa\ = h KA"1* + * 0 ) -xt\\>hd.

THE DANIELL INTEGRAL

11

But, h d = \\x0 - jCi|| ld > a by the way in which x0 was chosen, and so |JC - JC„|J > a for all x e Xo, and we are done.B We can restate Riesz's Lemma as: Given any closed, proper subspace X, of a normed linear space X, there exist on the surface of the unit ball of X points whose distance from X0 is as near to 1 as we wish. This will prove useful to recall later in these notes. One should note that while we can always find points as close to a distance of 1 from Xo as we want, it is not true that we can necessarily find points on the surface of the ball whose distance from X0 is exactly 1. A relatively simple example of this can be found in the space C ([0,1]) of continuous real-valued functions on [0,1] [see, for example, Taylor (1958)]. THE DANIELL INTEGRAL Typically, when we think of developing a general theory of integration, we fall back on our experience in dealing with the Lebesgue integral, or perhaps even the Riemann integral. We begin with some notion of the measure of nice sets, and then extend this idea to include a more complex collection of sets on which we have a structure that allows us some control over their interrelationships. We then balance this with wanting to have a sufficiently rich collection of sets so that our notion of measure is not only "natural" in some sense, but allows the flexibility we will need to accomplish our goals; that is, be able to work with the broad range of functions we will need or want to be able to "integrate". Of course, this naturally leads us to considering what collection of functions our notion of measure will need or be able to "handle", and of what natural idea of integration will be compatible with these restrictions. Thus, often enough to make the point at least a valid one, we begin with measuring sets in such a way it directly generalizes our elementary notion of length, use this notion to generate a broad class of functions that will be "nice" (i.e., measurable) in this context, and then develop an integration theory consistent with our old Riemann and Lebesgue integrals and hope for the best. Instead, what we will do here—and the rationale for including this material in this text at all—is a little different and may not have been encountered by the typical student. We will begin with some kind of simple or "elementary" integral defined on a small collection of "elementary functions" and then work toward enlarging this set of functions (and consequently extending our integral) to larger collections in such a way that the result has all the properties we want to be able to retain from the Lebesgue integral. Of course, coming with this will be a resultant concept of measure, so that eventually we arrive at the same end as before. The first person who really successfully carried out this process was the English mathematician RJ. Daniell, who did most of his work in the early 1900s, and thus the basic integral obtained in this fashion is usually called the Daniell integral (or Daniell functional). The development we sketch next,

12

INTRODUCTION

while far from all-encompassing, will roughly follow his lead. The reader should note that there is an assumption here that full details need not be given and that general ideas together with indications of techniques will be sufficient either to allow the reader to supply the missing justifications on their own, or find the motivation to seek more comprehensive and complete developments. Such can be found, for example, in the well-written Real Analysis by Royden (1963), whose general presentation we follow here. We begin by introducing the appropriate setting and putting forth some fundamental definitions. You should note that our immediate focus is on a fairly general collection of functions over which we have a moderate degree of control and on how we might define their integral in a reasonable fashion; ideally, the rest will follow. Let Ω be a set and L be a family of real-valued functions defined on Ω closed under finite linear combinations and with fv g (s max(/,g}) and / Λ g (= min{ f,g}) in L whenever / and g are in L (you should recall that such a family is called a vector lattice). While this may initially seem to be a very abstract collection of functions to be concerned with, it is easy to see that any linear space L of functions is a vector lattice, provided we require / v 0 to be in L whenever/is (just n o t e / v g = ( / - g) vO + g, w h i l e / A g = / + g - (/vg)). So, those linear spaces of functions that include the "positive" part of each of their members, or / + = / v 0 for each /, are always vector lattices. Of course, as l/l = / + + (-/)+> these spaces are closed under the taking of absolute values. On the other hand, if we have a vector space L with |/| e L when/e L, it is always a vector lattice, as / + can be realized as y(/+1/|), so these restrictions are the same. Now let / be a (real-valued) linear functional on L. Here / is called positive if /(/) > 0 for each nonnegative function fe L. Note that such functiona l always preserve the order of L; that is, if f 0 and /,ge Le. Moreover, if/is any nonnegative function, with (φ„)„ an increasing sequence from L with / as their limit, by replacing each 1, we have/= Σ„ψη and that /(/) = lim„ Ι(φη) = lim„ I(lUWk) = lim„ ΣΖ=1/(ψ*) = ΣηΙ(ψη), so that/E Le exactly when there are nonnegative functions (φ„)„ c L with/= Σ„φ„. Of course, with this we get not only that the sum of any sequence of nonnegative functions from Le is still in Le, but that Ι(Σ„/„) = Σ„Ι(/„). The extension of the Daniell integral to an arbitrary function on Ω is now pretty standard: one defines the upper and lower Daniell integrals by / ( / ) a inf{/(g): g > / and g e Le} and /(/) = - / ( - / ) ) , and declares a function on Ω to be Daniell integrable whenever /(/) = / ( / ) = /(/) < °°; the class of all Daniell integrable functions on Ω is denoted by Li. Of course, we need to know we have a legitimate extension of / to the collection L], and the basic properties listed below guarantee this. As their verification is straightforward, we leave these to be supplied by the student: 1. 2. 3. 4.

Haf+ßg) < al(f) + ßl(g) for α,β > 0; Both / and / preserve order; /(/) < / ( / ) , and they agree on / e Le; Ί(Σ„/„) < Σ„/(/„) for any nonnegative functions (/„)„ on Ω; (given ε> 0, for each n just choose g„ e Le, with g„ >f„ and I(gn) < I (f„) + ε/2").

It should be noted that L] is a vector lattice; while this is not difficult to see (one just shows that Li is a linear space with / + e L] for each fe L,), it

14

INTRODUCTION

involves enough manipulation to be a bit messy. We will avoid the technicalities here. Not surprisingly, the Daniell integral does satisfy appropriate versions of the major convergence theorems enjoyed by integrals arising from more standard measures. Again, we will not give full details here, but try to indicate the primary direction and idea needed. We begin with the Daniell counterpart to the monotone convergence theorem. Proposition 5. If (/„)„ is an increasing sequence in Li with /=lim„/ n , then / e L] exactly when lim„ /(/„) < °o; in this case / ( / ) = lim„ /(/„). Proof. The necessity of the condition is easily seen, sufficiency follows from letting g s / - / i , and noting g = Σ„(/„+1 -/„) so that we have / (g) < lim„ /(/„) /(/i). This immediately yields /(/) < limn /(/„), with the other half of the inequality following from noting /(/) > lim„ /(/„)■■ Fatou's lemma in the Daniell context is given by the following proposition. Proposition 6. If (/„)„ ς; Li are nonnegative, then inf„/„ e L! with Hm „/„ e L t whenever lirn„ /(/„) < «>. In this case /dim, /„) < lim„ /(/„). Proof. If, for each n e M, we let g„ =f Λ/ 2 Λ · · · Λ/„, then (g„)„ c Li, which decreases to infn /„. Noting (~gn)„ increases to -inf„ /„, a moment's reflection on Proposition 5 yields inf„ /„ e Li. Now, for each n, setting h„ = inf{fk: k > n) gives us a (nonnegative) sequence in L[ increasing to lim„/n, which must be in Li, as limn/(/i„) < Urn „/(/„) < °°. The final inequality is immediate from Proposition 5M The final result in this context is just the Lebesgue dominated convergence theorem, which also tells us / really is a Daniell integral on Lj. Proposition 7. Let (/„)„ c L, with, for all ne N, \f„\ < g for some g e Li. Then /(lim„ /„) = lim„ /(/„). Proof. As in the standard case, this proposition follows almost directly from Proposition 6. Just note that (/„ + g)n is a nonnegative sequence in Lt with /(/„ + g) < 21(g), so that lim„ /„ + g e Lu with /(lim„/„ + g) lim„/(/„) gives us all we need.B The last result that we need here is due to Marshall Stone. In some ways it pulls all of this together and ties our new Daniell integral back to the integration theory developed in the more standard manner. In order to see this,

THE DANIELL INTEGRAL

15

we must first introduce the concept of measurable functions, and of measures, as they arise in this context. Thus, we say a nonnegative function / on Ω is measurable (with respect to /) if / Λ g € Li for all g e L,. Because Lj is a vector lattice, it follows that / v g and/Ag are measurable whenever/and g are nonnegative measurable functions; from this it is not hard to see that lim„/n is measurable as long as the/„ are. Measurable (and integrable) functions naturally give rise to sets with these properties; in fact, E c Ω is called a measurable set if its characteristic function χΕ is measurable, and an integrable set if χΕ is integrable. It is not difficult to see (and a straightforward exercise to prove) that provided Ω is measurable (that is, 1 is a measurable function), the collection of all such measurable sets is a σ-algebra. We make one final comment before taking the last step we really need here. Suppose our measurable sets are a σ-algebra and/is a nonnegative integrable function. Note if aα} = Ω and so is a measurable set. If cr>0, consider the function g = (\la)f-{{lla)f/\\), and note we have g e Li with g(ß))>0 for all ωε Ε. Since it should be clear that (lA/ig) n is a sequence from Li increasing to χΕ, we have that χΒ is measurable so that E = {ω:/(ω) > a) is measurable (for any as R) as well. So, measurable functions? Measurable sets? There should be a measure somewhere in sight. Well, let's define the set function μ on the σ-algebra of measurable sets by (E\ _ Í 7 ^ E ) [sup{/i(A): A is integrable with A c E }

if E is integrable otherwise.

It should be clear that μ(0) = 0 and that μ(Α) < μ(Β) for A and B integrable sets, so it must hold for measurable sets as well. If E = UnE„, where the E„ are pairwise disjoint measurable sets, then given any integrable A ς: Ε, if we let A„ = A (Ί E„, then each A„ is integrable and, by Proposition 5, μ(Α) = Σ„μ(Α„) < Σ„μ(Εη) so that μ(Ε) < ΣΛμ(Ε„). On the other hand, if μ(Ε) < °o, then given ε > 0, for each n we can find an integrable A„ c E„ with μ(Α„) > μ(Ε„) - ε/2". But this means that μ(Ε) > Σπμ(Α„) > Σπμ(Ε„) - ε, so that μ(Ε) > Σ„μ(Ε„). As this inequality holds even if μ(Ε) = °°, we actually have that μ is countably additive and hence is a measure as we wanted. We are now ready for the main result toward which we have been laboring; namely, the beautiful theorem of M.H. Stone that tells us that the natural integral with respect to this measure μ is exactly the Daniell integral / on L^ Theorem 2 (Stone's Theorem). Let L be a vector lattice of functions on a set Ω with the property that / e L implies that 1 Λ / e L, and let / be a Daniell integral on L. Then there is a σ-algebra Σ of subsets of Ω, and a measure μ on Σ, such that each / on Ω is integrable with respect to / if and only if it is integrable with respect to μ; moreover, we have /(/) = ί/άμ.

16

INTRODUCTION

Proof. Note that by the preceding discussion, we have that the family Σ of measurable sets with respect to / do form a σ-algebra, and that each nonnegative /-integrable function is measurable on Σ. Since each such /-integrable function is the difference of two nonnegative /-integrable functions, every /integrable function must be measurable on Σ. Moreover, if we deñne μ as in the last paragraph, and let/be any nonnegative /-integrable function, then for any n and k in N we have Ekjl = [ω& Ω: f(aj) > kin) is measurable. Also, since

# E M e Li with μ(Ε*„) < °°. If we now let gn = (1//ι)ΣΖ.1^Ε^ for each n e N , then (gnh C Li is an increasing sequence that converges pointwise to /, so that /(/) = lim„/(#„). But

Since ί/άμ = lim„ j g„ άμ, Proposition 5 yields /(/) = ί/άμ and / is integrable with respect to μ. Finally, as any /that is /-integrable is the difference of two nonnegative /-integrable functions, / is also μ-integrable with I(f) = ¡fdμ as desired. The difficult half of our result now holds. On the other hand, let / be any nonnegative function on Ω integrable with respect to μ. As before, we construct the sets Ek„ and the functions g„, and note that since each Ε*„ has finite measure, each g„ e L,. But then Proposition 5 yields that we have / e L, [after all, (g„)n increases to / and lim„/(^„) = ¡f άμ < «>!], that is, / i s integrable with respect to /.■ As a final comment, it is worth noting that it is not all that difficult to show that if we take Σ to be the smallest σ-algebra for which each / e Lj is measurable, then the measure μ corresponding to each Daniell integral is unique.

Functional Analysis: An Introduction to Banach Space Theory by Terry J. Morrison Copyright © 2001 John Wiley & Sons, Inc.

1

BASIC DEFINITIONS AND EXAMPLES

In this chapter we present many of the fundamental examples of Banach spaces that should serve as an indication of the type and broad range of spaces we will be concerned with throughout most of the remainder of this book. While we will, in fact, encounter some other examples and some arguably rather basic results in later chapters, most of what are now considered to be the elementary "classic" Banach spaces are contained herein. The chapter is divided into two sections: the first presents the basic examples of the spaces themselves, as well as some of the standard fundamental ideas and results we will need in our subsequent work; the second section is devoted primarily to calculation of the "dual space" (or space of continuous linear scalar-valued functions defined on the original space) of many of our examples from the first section. The importance of understanding the relationship between a space and its dual will initially become most apparent beginning in Chapter 3, where we first begin to really develop some of the deeper consequences of this relationship. For now, however, we begin building our collection of concrete spaces and elementary facts. 1.1 EXAMPLES OF BANACH SPACES While it should be clear that the reader of this text should already have a passing familiarity with normed linear spaces, without belaboring the point we begin with the relevant definitions and ideas. Definition 1.1. Let Xbe a linear space (that is, a vector space). By a norm on Xwe will mean a mapping |||: X —> [0,«>) such that 17

18

BASIC DEFINITIONS AND EXAMPLES

(i) \\x\\ = 0 if and only if x = Θ (the zero vector in X); (ii) ||·|| is positive homogeneous; that is, for any scalar λ and any x e X, we have|A*| = W|x|; (iii) ||·|| is subadditive; that is, for any x,y e X, we have | * + y\\ < \\x\\ + ||y||. In this case, the pair (·Χ»||Ί|) is called a normed linear space. Sometimes, if it is clear what the norm is for a particular space X (such as the norms defined for the spaces in all of Examples 1.1-1.12 in this section), or if Xis just an arbitrary normed linear space with whatever generic norm "||||"Xmust have, we will drop any specific representation of the norm and just refer to the pair (X,||||) as the normed linear space X. Before considering our first examples, it is important to notice that all normed linear spaces (X||||) have the more familiar property of being metric spaces. That is, one can always induce a metric structure on X via the formula p(x,y) = \\x- y\\- Thus, this functional p as just given, is a well-defined metric on the linear space X [so that (X,p) is always a metric space, as the student should be able to readily verify for himself]. This leads us to the next definition. Definition 1.2. We say that a normed linear space (X,||||) is a Banach space whenever the metric space (X,p) derived from (Χ,||·||) as before is a complete metric space. Thus, in a Banach space X unless otherwise noted, convergence is always with respect to the metric induced by the particular norm for X. Before proceeding further, we give some examples and simple consequences of these definitions. We start with a list of basic, and what should be familiar, examples. Example 1.1: IR and C. The real number system, IR, with its usual linear structure and norm defined to be "absolute value" is a Banach space, as the student should know from any typical undergraduate analysis text. Likewise, the complex numbers, C, together with its usual linear structure and absolute value for a norm, constitutes a complex Banach space. (We shall agree to use the word "complex" whenever the underlying scalar field of a Banach space is specifically the complex number system, otherwise, all Banach spaces referred to are real Banach spaces.) Example 1.2: W and C . Let n e N, then Euclidean it-space, IR", with the usual linear operations of vector addition and scalar multiplication together with the Euclidean norm for x - (xhx2,x3,... ,xn) e IR" defined by IWI=IK*I , *2, *3 ,...,*„ )ii=(Σ ι = ]

xf)

EXAMPLES OF BANACH SPACES

19

is a Banach space that is sometimes denoted by the symbol l\ (note that here, the 2 refers to the "/2-norm", which will be defined in Example 1.9, while the n refers to the dimension of the space and signifies that we are looking at some sort of finite-dimensional space). In proving that IR" with this norm is a Banach space, the only seemingly difficult step is in establishing Property (iii) of the norm. While the student should either already know or be able to verify this directly, we will postpone our proof until Example 1.9, where it will be established in a more general setting. The completeness of the Euclidean spaces comes, of course, from the fact we know convergence here is just "coordinatewise" convergence and, since IR itself is complete, verifying that this property holds should represent no problem. One should note that similarly, C , complex Euclidean π-space, is also a Banach space. Our next example, standard fare in any reasonable beginning analysis course, is given here for good reason. The basic idea and techniques presented serve as a model (in fact, often with only minimal modification) for how to deal with a variety of similar, yet important, examples that will be encountered later. While this particular example is very specific, as it is worked through, the student should look for places where the specificity is not really used. Thinking about what properties of the underlying real interval and of the actual collection of functions defined on that interval actually possess and of how these are really used to achieve the desired results will provide an insight into not only the following example, but into a number of other spaces, beginning with Example 1.7. Grasping this general idea of deducing more abstract-oriented information from particular more familiar examples is one that will not only arise often in one's mathematical career, but often provides the basis on which valuable research is conducted. Example 1.3: C([0,1]). The space C([0,1]) of continuous real-valued functions (or, complex-valued functions) on the interval [0,1] with the usual linear operations of pointwise addition and scalar multiplication is a Banach space when endowed with the norm: ||/||~ = sup{|/"(r)|: 0 < i < 1) f o r / e C([0,1]). (Recall that we know this supremum exists since any continuous function defined on a compact set, such as [0,1], is bounded, and hence ||/||„ < °° for any such/.) To see that this is indeed a norm as claimed, first note that Properties (i) and (ii) of the norm are readily verified for ||||«,.To establish Property (iii), let /] and f2 be in C([0,1]) and observe that for each t e [0,1], l(/+/2XOI = D5(0 + /2(0|á|/WI+l/ 2 (0|. Now note that by definition of the norm, we have |/i(r)| < ||/i||,„ and |/2(f)| < ||/2||„„, so that for each t, |(/, +f2)(t)\ < ||/,||M + ||/2||„. From this it follows that 11/, + ΛIL = sup{| (/, + /2 XOI: 0 < r < 1} < 11/! |L -H ||/2 |L

20

BASIC DEFINITIONS AND EXAMPLES

and Property (iii) is established. Finally, to see that C([0,1]) is a Banach space, we must show that Cauchy sequences in C([0,1]) (with respect to the metric induced by ||·||_) converge in C([0,1]). So, let /„ e C([0,1]) for ne N, and suppose that ||/„ -/„,||~ -> 0 as n,m -> °°. We will first construct what turns out to be the "appropriate" function / : [0,1] -> U, then show that our sequence (/„)„ actually converges to this function, and finally conclude with showing this function is in C([0,1]). To this end, we first observe that since (/„)„ is a Cauchy sequence, there is for each ε > 0 a natural number Nc such that \f„(t) -fm(t)\ Νε and all / e [0,1]. In particular, for each fixed t e [0,1] (f„(t))„ is a Cauchy sequence of real numbers and hence convergent. Now define for each te [0,1] :/(í) = limn/„(í) and observe that for all te [0,1], |/{í)-/ m (í)| = lim„ |/ π (/)-/ m (f)|á ε/3 whenever n,m>Ne. From this it follows that given ε > 0, there is a positive integer Ne such that for m>Ne ||/ - / J L - sup{|/(/) - fm(t)\: 0 < t < 1} < e/3 < ε so that (/m)m converges (uniformly, in fact!) to /. We need only show the continuity of / to be through. So, using the same Νε as before, for η = Ν& we have by the continuity of each /„ that given t e [0,1] there exists a neighborhood U, of t such that if s e U„ then |/„(i) -f„(s)\ < ε/3. From this we have that if s e U„ then \f(t)-f(s)\ X defined by (p(jrlrx2) = *ι +*2 and ψ(λ,χ) = λχ are continuous maps for any normed linear space X Proof. A quick thought about the definition of continuity together with a careful choice of constants yields our theorem as an easy consequence of the following two inequalities: yix^xj-

φ(χι,χ2% = \\{χι -χΊ) + (χ2 -xil^hi

-*ill + lk> -*2II

and \\ψ(λ,,χ])-ψ{λ2,χ2)\\

=

\\λ,χι-λλχ2+λ]χ2-λ2χ2\\

¿W^Xi - λ^Χ2\\ + \\λ1Χ2 - λ2Χ2\\ =\λ,\\\χ>-χ2\\+\^-λ2\\\χ2\\.

m

Guided in part by this result, we can make the following definition, which distinguishes these kinds of "nice" topologies from others. Definition 1.3. Let X be a linear space endowed with a topology τ. We call τ a linear topology on X whenever the operations of addition and scalar multiplication are continuous functions from X x X and S x X into X, respectively. In order to adhere to historical distinctions, if T is also a T r topology (i.e., if singleton sets are closed in X), then the pair (Χ,τ) (or sometimes it is written Χ(τ)) is called a linear topological space (or a topological vector space). Thus, another way to state Theorem 1.1 is that every normed linear space is a topological vector space (although the converse to this is false, as a little

22

BASIC DEFINITIONS AND EXAMPLES

thought about topologies and what it might take to be "normable" should quickly reveal). Using Theorem 1.1, an easy induction argument (which we leave to the student) will now yield a proof of our next fact. Proposition 1.1. In a topological vector space (Χ,τ), all (finite) linear combinations of the scalars λι,λ 2 ,... ,λ„ and vectors ;clrx2, · · ■ ■PCn determine continuous linear mappings of the product space Π"=ι§ χ ITiiXinto X. Moreover, the mappings ψ and φ from X to X itself and defined by ψ(χ) = λχ (for a fixed scalar λ * 0) and φ(χ) s χ + y (for a fixed vector v in X ) are homeomorphisms of X onto itself. (Recall that this means both ψ and φ are one-to-one, onto, continuous mappings with a continuous inverse.) A particular consequence of the continuity of these linear operations is our next fact. The student should pay close attention to the proof itself, as we will encounter this same technique in the future. Theorem 1.2. The closure of a linear subspace y of a topological vector space (Χ,τ) is still a (closed) linear subspace of X. Proof. Given a subset A of X, let us use the notation cl(A) to denote the closure of the set A in the topology τ. Now, consider the map ξ: XxX-^X given by ξ{χ\^ι) = λχΧχ + λ2Χζ for fixed scalars kuX2. Note that since y is a linear subspace of X, we have

and further that ξ~\ν) c |"'(cl(y)). Moreover, since ξ is continuous, ^ _1 (cl(y)) is a closed set and thus cl(y) xcl(y) c ^'(cl(y)). From this it readily follows that cl( y ) is also "closed" under the operations of addition and scalar multiplication, and hence is a linear subspace of X as claimed.■ We now continue with our examples by presenting a common way in which new normed linear spaces arise from known spaces. Example 1.5. Let X be any normed linear space and y be a linear subspace of X. Then note that the restriction of the norm on X to y is clearly a norm on y. Under normal circumstances, this "restricted" norm is denoted with the same symbols as the original (the usage will thus be clear from context). So, we have that linear subspaces of normed linear spaces are themselves, in a natural way, normed linear spaces. Again, if we let X be a normed linear space and y be a linear subspace of X, then by Theorem 1.2 the closure of y with respect to the norm topology (that is, the topology induced by the metric derived from the norm) is a linear

EXAMPLES OF BANACH SPACES

23

subspace of X. By the previous paragraph, this closure is a normed linear space, which we denote by y (thus, the overbar will always be reserved for closure in the norm topology). This notation should be kept in mind, as later (in Chapter 3 and in what follows) we will encounter other topologies useful to consider on a normed linear space and we will continue to use this means of distinguishing the norm closure of a set from other topological closures. Finally, then, if X is a Banach space and V is a linear subspace of X, we have that y is a closed linear subspace of X. Thus y is a complete normed linear space (recall that closedness and completeness are equivalent for subsets of complete metric spaces) and hence a Banach space. In particular, we have that closed linear subspaces of Banach spaces are themselves Banach spaces. Before proceeding to our next specific example of a Banach space, in the spirit of using subspaces as a means of creating "new" Banach spaces, this seems an appropriate place to introduce using quotients of known spaces to accomplish this same task. So, to begin, let X be a normed linear space and y be a (not necessarily closed) subspace of X We will use the familiar notation X / y to denote the collection of all cosets {x+y-.xe X), as the student has undoubtedly seen before. Using the standard means for defining an addition and scalar multiplication on X / y (i.e., define (x¡ + y) + {x2 +y) = (xi + x2) + y and a(x + \f) = ax + y for all x,X\jc2 e X and a e S), it should be evident that X / y forms a linear space. That it also (in a natural way) becomes a normed linear space is perhaps not so immediately obvious, but certainly not difficult either. Indeed, with a little thought, one can imagine that a straightforward way to do this might be, as we did with subspaces, to again try to use the norm already given for X. Let us consider defining, for any fixed x e X, ||* + y\\ = inffll* + y||: y e A/). Now using the fact that y is a subspace, we can see that inf(||;t + y\: y e yj = inf{|jc - _y||: v e y}, so that this potential norm represents the "distance' from the point x to y. While this appears to be very good, the observant student should have seen a possible difficulty. In fact, as should easily be verified by this student, this does come very close to yielding a norm on X / y as desired, lacking only (in general) a single necessary property, namely, while for JC e y we have |p: + y || (= ¡0 + y ||) is 0, ||J: + y || = 0 does not necessarily yield that x e y ( y may contain points arbitrarily close to JC, but not x itself). The obvious remedy to this circumstance is, of course, to require that y be a closed linear subspace of X to begin with. The student should now be able to quickly show that if y is such a (closed) linear subspace of X, X / y becomes a normed linear space using the norm II* + y II - inf{||* + y\\:y e\f\. Thus, for our purposes throughout the remainder of this text, X / y will only "make sense" when y is given to be a closed linear subspace of X. Anticipating the (soon to be realized) fact that if we begin with X a Banach space, then X / y retains this property, we will record the result of this discussion as follows.

24

BASIC DEFINITIONS AND EXAMPLES

Example 1.6: Let X be a Banach space and y be a closed linear subspace of X. The quotient space Xl\j of X, consisting of all cosets [x + \/: x e X) together with the norm ||* + y || = inf{||jc + _y||: y e y) for any x e X, is a Banach space. The natural map q : X - » X / y given by q(x) s x + y for all x e X, is called the natural quotient map of X onto X / y Before continuing from here to develop the proposition giving us the basic facts about quotient spaces we will need in the future (including the promised completeness of X / y when X is), one comment should be made about the "potential norm" we found for Xl\) before restricting \J to being a closed subspace of X. At the time we noted that we had "almost" a norm. This phenomenon of a functional being near to achieving the status of a norm is not unusual and will arise later in a number of circumstances. For these reasons, together with the standardization of such terminology, we will call such a nonnegatively valued functional p on a linear space L a seminorm if p is positive homogeneous, subadditive, and has the property that for the zero vector Θ of L, p(0) = O. The student interested in further exploring the notion of a seminorm, and the use and structure of spaces endowed with such, has a vast amount of material available to consider. Several good general choices include Köthe (1969) and Schaefer (1971). Returning to the topic at hand, the following proposition yields the relevant information we are now able and willing to derive about such spaces. One need only recall that between two metric spaces, Mi and M2, a map φ : Mx —» 3Í 2 being continuous is equivalent to asking that φ preserve the convergence of sequences; that is, if xn -» x0 in Mu then φ(χ„) —» φ(χο) in M2. In other words, we would need to be able to force \\φ(χη) - φ(χ0)\\ t o be arbitrarily small whenever ||X„-JC0|| is small. It should be evident the inequality provided by part (i) in the following proposition yields exactly such a mechanism. Proposition 1.2. Let X be a normed linear space, y be a closed linear subspace of X, and q : X-» Xl\) be the natural quotient map, then: (i) ||q(jt)|| < \x\ for all x e X (that is, q is continuous); (ii) If X is a Banach space, then X / y is a Banach space; (iii) A subset V of X / y is open in X / y if and only if the set q"1 (V) is open in X; (iv) If U is an open subset of X, then q(U) is an open subset of X / y (that is, q, mapping open sets to open sets, is an open mapping). Proof. The fact that Θ e y immediately yields \\q(x)\\ = II* + yil = infill* + yll: y € y} < |W| for all x € X

EXAMPLES OF BANACH SPACES

25

which gives the desired inequality for part (i). To see that part (ii) holds, we will show that every Cauchy sequence in Xiy converges (to a point in Xiy). So, let (x„ + y)„ be a Cauchy sequence in Xiy, and note we can select a subsequence {x„k + y)k of (x„ + y)n, so that

IK +y)-u,.., +y)HK -o+yii R . For any fe Έ(Ω), let ||/||„ = sup{|/(ö))|: ωe Ω}, and note that this norm is well-defined by the bounded nature of these functions. Then, using exactly the same idea as in the first part of the proof of Example 1.3, one can readily verify that 2?(Ω) is a normed linear space (of course, if the functions are complex-valued, 2(Ω) becomes a complex normed linear space). A closer look at the second part of the proof in Example 1.3 should reveal that the continuity assumption for the functions involved was never used here; that is, we can conclude that any Cauchy sequence of bounded functions converges to some function /. While, in our current case we do not need to try to evoke the third part of the aforementioned proof (as we only need our limit function to be bounded), from what we have just done it should be easy to see that this is indeed the case. (In fact, the use of the inequality ||/|| < ||/-/„|| + ||/n|| for the appropriate choice of n should do the trick.) That is, we have easily been able to realize that 2?(Ω) is a Banach space. One should also notice here, by again emulating the proof of Example 1.3, that if we endow Ω with any Hausdorff topology τ under which Ω is compact, then (Γ(Ω), the linear space of all continuous real- (or complex-) valued maps on Ω, is clearly seen to be a closed linear subspace of 2?(Ω), and hence a (complex) Banach space through an appeal to Example 1.5. Remark. One should realize that the way in which the norm on 2?(Ω) is defined, in a manner identical to that defined for C(X )-spaces, is not a mere coincidence nor is it accidental. In fact, 2?(Ω) can be identified as belonging to the general class of C(X )-type spaces; indeed, it can be shown rather easily that 2?(Ω) is completely identifiable with the space (Γ(βΩ), where Ω is endowed with the (obviously completely regular) discrete topology, and βΩ. denotes the Stone-Céch compactification of Ω. If necessary the student can review, for example, the development given in Chapter 19 of Willard (1968). Before leaving our discussion of 2?(ß)-spaces and picking up another useful general property of Banach spaces, it is worth noting that we can use 2(Ω) to give us what will prove to be a useful generalization of the C(X )spaces themselves. Namely, instead of requiring X to be compact, we will here only ask that X be a locally compact Hausdorff topological space (that is, that each point of X has a compact neighborhood). In this context we will consider the collection C0{3C) of all continuous functions on X that "vanish at infinity"; that is, those scalar-valued functions/on Xthat are continuous and for which, given any ε > 0, we have that the set [x e X: |/(JC)| > ε) is compact. Note here in the special case that X= R, C„(R) = {/: R -» S | / i s continuous and limx_,±«,/(jt) - 0}. In any case, it should be clear that CC{X) is a linear subspace of 2(JC), and, in fact, it is not difficult to see it is also closed. Indeed, let (/„)„ be a sequence in Ca{X) with/„ - » / i n Έ(Χ).Then given any ε> 0, choose Ne N so that ||/-/„ll < ε/2 for all n > N. If x e X with \flx)\ > ε, then

27

EXAMPLES OF BANACH SPACES

ε < \f(x)\ < \f(x) - fn (x)\ + |/„ {x)\ < ε/2+1/„ U)| for all n > N\ and so we have \f„(x)\ > ε/2 of all n>N.Oi course, in particular, this means [x e X: \f(x)\ > ε\ c [x e X: |/ΛΓ(Λ·)| > ε/2), and it immediately follows that/e C0(JC), as desired. Thus C0(X) is also a Banach space, a fact we record as our next example. Example 1.8: Let Xbe a locally compact Hausdorff topological space, and let Co(X) = {/: X -> § I / e C(X) and for each ε > 0, the set {x € X : |/(.r)| ^ ε} is compact}. With the supremum norm inherited from Έ(Χ), C0(X) is a Banach space. As a final remark, it should be noted that if X happens to be compact to begin with, C„{X) is just the space C{X) of Example 1.4. As promised, we now return to get another general property of Banach spaces. You should realize that this theorem, as well as many other now fundamental results, appeared in Banach's dissertation, where it was used to good effect. For us, it will often be useful in proofs of completeness, where having the given criterion available can sometimes give a quick insight into that aspect of the space; we will see other interesting uses as well later. First, we should recall the following definitions, which are the obvious modifications of the more familiar concepts defined for the real numbers. Definition 1.4. Let X be a normal linear space and (*„)„ be a sequence in X. We say the sequence (xn)„ is (a) Absolutely summable if Σ„||.Ϊ„|| < °°; (b) Summable if lim„_,_L2=i^ exists in X. Theorem 1.3. Let X be a normed linear space, then X is a Banach space (that is, it is complete) if and only if every absolutely summable sequence in X is summable. Proof. First suppose that X is a Banach space, and let (xn)n be an absolutely summable sequence in X. We must show that (xn)n is summable. So, consider the sequence {Σ?=1^)η=ι; we assert that this sequence is Cauchy in X. In fact, for any positive integers m and n (with, say m < n), we have £¡5=1** — "L^\Xk = Y.nk=m+\Xk, s o that

«

X"" 1

^-"n

II

UV"1"

||

V "

II

II

^"""

i)

|i

Lk=í xk - 2-Λ=, * * l r ¡ 2 , *=„,+, **|| - Lk=m+1 IWI - Lk=m+i IW1But this last term goes to zero a s m - * » , since Ση\\χ„\\ < =», and so our sequence is Cauchy. Thus by the completeness of X, it converges; that is, lim„ li=xxk exists in X, and so (JC„)„ is summable.

28

BASIC DEFINITIONS AND EXAMPLES

The converse is a bit trickier, but nonetheless not too bad. We begin by observing (or recalling) that any Cauchy sequence with a convergent subsequence is itself convergent (to the same limit point as the subsequence, of course). So, now suppose that X is a normed linear space with the property that every absolutely summable sequence (y„)n in X is summable in X. Let (x„)n be a Cauchy sequence. Our idea will be to manufacture from (x„)„ a subsequence that itself leads to an absolutely summable (and hence summable) sequence of some sort with the limit of its partial sums the same as what the limit of the subsequence "should" be. More precisely, we will manufacture from (xn)n a subsequence that is expressible as a telescopic (absolutely convergent) series. We do this as follows. The sequence (x„)n is Cauchy, hence for each positive integer k, the numbers pk = sup{||;c„ - xm\\: m,n > k] go to zero. We now choose « , E N such that p„, < 1; then choose n2e N such that n2 > n, and p„2 n2 and p„3 < (γ)2; and continue selecting natural numbers nk in this pattern. In this way we obtain a subsequence (x„k)k of (x„)„ such that for each ke N, p„k < (y)*"1. Now observe that for each k, Xnk = (X„t - Xnk_, ) + (*„ 1 is nontrivial and involves some important classic inequalities. We start with the following result, which while in itself is somewhat interesting, gives a nice tool to use for our first inequality: Lemma 1.1. Let 0 < a < 1 and a,b > 0, then a" ¿1_a < a a + (1 - a)b. Proof. Note if either a or b is 0, the inequality is clearly true; while if a = b, we get equality in the preceding expression. Thus we can assume that a,b > 0 with, say, a< b. The whole trick here is to apply the Mean-Value Theorem (from standard calculus) to the function xl~"; this useful theorem yields the existence of a real number ξ such that a < ξ < b, with b

^~a^ b-a

=(1-α)ξ-" yields

anc

· using 1 - (l/^) = (l//>),

(Σ;>.^Ι'Γ*(Σ;>/)'''*(Σ>.Ι'Γ· If we how proceed as in Holder's Inequality to extend this finite inequality to infinite sums, we have our result. Let us now take a look at what we have so far. Note that Properties (i) and (ii) of the norm are easily shown for any 1

°° (for each m, of course) to some real number that we will denote by x„- If we now let x0 be the sequence defined by *o = (Xo)m, then we get that for each k e N,

32

BASIC DEFINITIONS AND EXAMPLES

Σΐ.>:-*:Γ=Σ1.,Ν^Γ-*:Ι'=^>Σ1.>7-':Ι' the last term of which is no larger than ερ for all n > Νε. But by making the obvious step, this yields ||JC0 - JC„||£ < ε" for all n > Νε. Hence x0-x„e lp for all η>Νε, which, since x„ e lp for all nsN, yields x0 = (x0 - xn) + xn e lp by linearity (of lp). But also notice that by the preceding, not only is x0 e lp but, for n>Newe have \\x0-x„\\px0 in /p and thus (/ρ,||·||ρ) is a Banach space, as we earlier asserted.· Our next two classic sequence spaces arise in somewhat the same manner as the previously defined /p-spaces; we only need to make a small change in the norm from a summability criterion to a more "uniform" one. Example 1.10: ca. Let c0 denote the collection of all null sequences of real numbers; that is, c„ = [x = (x„)n c R : xn -> 0}. For each x = (*„)„ e ca, define the norm \\x\\„ = sup{|x„| :ne N}. Then, as ca is clearly a linear subspace of ®(N), with the usual pointwise operations, and as ||·||„ is obviously the restriction of the norm of 2?(N) to c0, we have (referring to Example 1.5) that (c D ,|||P is a normed linear space. We will now show that ca is a Banach space by showing that it is closed as a linear subspace of 2?(N). Indeed, toward this end let (xn)n be a sequence of members of c„ that converges (in the S(M)-norm) to some element x0 e Έ(Ν). Using the notation previously introduced in Example 1.9, for each n, let x„ = (x™)m and note that \\x0 - xn\\„ -> 0 implies that given ε > 0, there exists an Ne e Ñ such that η>Νε yields \x% - x™\ < ε/2 for all m e N. But now each xn e ca, so that there exists an Με/Ι e N such that for m > Mejl we have \x™\ < ε/2. If we use the inequality |;C| < \x" - x™\ + \x"\, and choose n sufficiently large so as to ensure that the first term on the right side is small (say, < ε/2 for n > Νε), and then choose Mt„ such that m > Mejl yields \x™\ < ε/2 for the given (fixed) n, we get |;C| < (ε/2) + (ε/2) = ε for m > Λ/βπ; that is, x0 ε ca, as desired. Example 1.11: c. In light of Example 1.10 we can quickly and easily get another example of a Banach space from this source. In fact, let c denote the collection of all real sequences that are convergent (to any real number); that is, c = [x = (x„)„ c R : (JC„)„ converges). If we give c the same norm as cc, then since c can also be realized as a linear subspace of Έ(Ν), (c,||-||>) is clearly a normed linear space. If one now looks at the proof of the completeness of ca, it should be quickly seen that the same argument (with only a slight modification) gives the completeness of c as well. As usual, this should be carried out by the student. Remark: L. Because of its natural representation in terms of sequences, it has become standard in the area of Banach spaces to denote the space Έ(Ν) by the symbol L. Thus we can consider the space L to be com-

EXAMPLES OF BANACH SPACES

33

posed of the collection of all sequences of real numbers that are bounded; that is, L = \{x„)n £ R : sup(|x„| :ne N} 0, ( i e T : |/(*)| > ε\ isfinite}.Of course, in case Γ = N, Q(N) or L(N) is just our "standard" L, while C„(N) is just the "standard" c0. At this point it seems worthwhile to introduce the student to, or perhaps merely remind him or her of, what will turn out to be one of the more important concepts in Banach space theory in terms of giving us additional information about the internal structure of certain spaces. The student should have seen this idea in any general topology course, but we give the definition here again for the sake of completeness. Definition 1.5. A normed linear space X is called separable if X has a countable dense subset, that is, if there is a sequence {x„)„ in X such that every element of X can be realized as the limit of some sequence formed from the elements of (JC„)„. The student should realize we have already been introduced to several familiar separable Banach spaces, namely, W and C" for any n e N (just consider rational combinations of the usual unit vectors) and C([0,1]) (this is just Weierstrass' Approximation Theorem with the polynomials restricted to those with rational coefficients). Somewhat less obvious (although certainly not a great deal harder to verify) is the fact that the /^-spaces for 1

0 be given. As (x„)„ e lp, we know l„\x„f < °°, and thus Σ"=Ν+ι\χ„\ρ - ^ 0 a s « - > « . Choose N > 0

34

BASIC DEFIMTIONS AND EXAMPLES

so that I^=tM\xJC < ερ/2. Using this N we select our approximating element of D as follows: for each ne N, choose a rational number y„ so that if 1 < n < N, then \xn - y„\ < (2N)-Vp ε, while if n > N, then yn = 0. Clearly, y = (y„)„ e D, and we have II

\\P

V

^

I

\\χ -y\\p = LnJxn

V * °°

\P

+

-y»\

I

\P

Σ„=* + >" -^-l

= ΣΓ=>» -^-Ι' + Σ^ιΐ'-Ι' r*,l*l HU*)XII(***)JI]·

(i·*)

Thus, each (xf)k e l¡ yields a continuous linear map x* on ce via the formula In other words, the mapping of c* to l\ given by JC* —>(xX)k, where each x\ = x*(ek) (which we can easily check to be a linear mapping), is an isometry [simply combine Equations (1.3) and (1.4)] of the spaces c* and Ιγ. Summarizing the preceding example, we have the following theorem. Theorem 1.6. The dual space of c0 is identifiable in a linear isometric manner with the Banach space lx\ the correspondence may (and for our purposes always will) be given by the formula: x* e c* corresponds to (x*)k e l\ via x*{(xk)k) = Σ*ΛΓ£(**)» where the series on the right-hand side converges absolutely, and the JCJ can be obtained from x* by evaluation at the fcth-unit vector ek e cg. Example 1.16: (l{)* = L. The dual space of /i is L, that is, we have the following theorem. Theorem 1.7. The mapping u(x) = Σ„χ„ιιη, where u e /f and (u„)n e L with u„ = u(e„) for each n & N (here e„, as usual, will denote the nth-unit vector in /i) establishes an isometric isomorphism between the spaces /fand L. Proof. We start by considering u e /f. Then, as in the case of c*, we observe that if x = (x„)„ e lu then | * - Σ Ι = ι ■****!, = ΙΚΟ,Ο, - - -,

0,xn+uxn+2,...)l

->0

as n —> °° so that x = \\m„Lk^xkek, or, JC = Σ„χηβ„. From this, using the linearity and continuity of the functional u e If, we get that u(x) - L„x„u(en). Again if we make the identification «„ = u(en) for each n, we have that u(x) = Σ„χ„ιι„. It remains then to show that the sequence (u„)„ just defined has the desired properties. First, it should be noted that the map u —> (u„)„ is linear, and we now consider showing that (u„)„ € L. So,fixn e N and note that

46

BASIC DEFINITIONS AND EXAMPLES

ΙΜ«Ι = Ί7- Μ " ■■-¡f-u(e„) = u\

( Α . · ) | = ||ν(Α)\υ?=.Α < ||δΑμ(Α\υ?..Α ί )^0

as

/I-*~].

(ii) ||v)|„, defined by the formula ||v|L = inf {K > 0: ||v(A)|| < K μ(Α) for all A e Σ} satisfies, ||v(A)|| < M-MA) for all A el. (iii) If χ:Ω—>[R is a simple function and μ-integrable, say s((o) = Σ£ιΛ#Άι·(β>), then ¡s dv= Σ," I ||u„|| - 1. Thus, letting x„ = (\)"z„ we get

\K(xj\=H(\)"zn)\\>ty"\K(zj >(})"(lkll-i)

* Φ"ΗΙ-φ" > 4M. Also note that u„ has been chosen with \\x„\\ = \\(\)"z„\\ < (\)n and J||UB(JC„)|| > M > Σ^",1 ||uA(jcn)|| so that we have parts (ii), (iii), and (iv) holding. But now if we consider z = Σ„χπ, then we have

Ση\\χη\\=Ση(ϋη\\ζΛ y where y„ = U(JC„) implies that u(x) = y], then u is continuous. Proof. Define for each x e X, the new norm ||·||ι for the space X by Wli - IMI + ||U(JC)||. Then it is easy to show that (Χ,||·||ι) is a normed linear space with ||·|| < ||·||,. We assert that (X,||||i) is a Banach space. Indeed, suppose that (x„)„ is a Cauchy sequence in (Χ,||·||,), then (x„)n is also Cauchy in (X,||||), and so (u(*„))„ is Cauchy in y.Thus by the completeness of both X and y, we have that x„ -> x0 and y„ = u(x„) —»y0 for some x0 e (Χ,||·||) and y0 e y. But now look at what we have: |k,-.ro||-»0

and

||u(xJ-yo||->0

with u being an operator with a closed graph. Thus, by hypothesis, u(* 0 ) = y 0 and ||U(A:„) - u(x0)|| -> 0. Hence, we have that

82

BASIC PRINCIPLES WITH APPLICATIONS

B*n-*oBi=lk-*oll+l|u(*,,-*o)ll-»o and so (xn)n is a convergent sequence (in ||||,), and thus (Χ,||·||ι) is a Banach space. Now,as we noted earlier,||·|| 0 for whichfcfa< M ||·||, whence we have \\u(x% < \\x\l < M\\x\\ holding for all

xeX.

Thus u is bounded and must be a continuous operator, as claimed.· Remark. A few final comments are needed here before we move on to look at some applications of these basic results. You should note that it is possible for a linear operator to be discontinuous and yet still have a closed graph. In fact, let X= y~ C ([0,1]) and let D denote the set of all functions x(t) e X such that the derivative x\t) exists and is continuous. If we define T : D -» y by T(x) = x', then we claim that T is not continuous, but does have a closed graph. Hmmm . . . In fact, note that the sequence (x„(t))„ C X given by x„(t) = f for each ne N and t e [0,1], satisfies ||JC„|| = 1 for all n, and ¡TOOII = s u p í K ^ : 0 < t < 1} = sup{K- ] |: 0 < t < 1} for all n e N. So,T is not bounded and thus is not continuous. To see that the graph of T is closed, let (*„)„ c D with lim„ xn - x and lim„ T(x„) = y. Then x'n(t) converges uniformly to y(t), and x„(t) converges uniformly to *(/). Hence, x(t) must be differentiable with a continuous derivative given by y(t). Thus x e D and T(x) = y as claimed. 2.4 APPLICATIONS OF THE BASIC PRINCIPLES We now look at some applications of our basic principles. We will concentrate mostly on the Hahn-Banach and Banach-Steinhaus Theorems, as these give us some quick insights into other areas of functional analysis, which we will not pursue in any depth in this text. Since we will see many applications of the Open-Mapping and Closed-Graph Theorems as we go along, we give here but a small indication of the uses to which they can be put. First we consider some consequences of the Hahn-Banach Theorem. As a first application of this principle we prove the following theorem, which gives a partial solution to what is usually called the lifting problem (that is, when, in general, can a continuous linear operator on a subspace of a Banach space be "lifted", or extended, to the whole space?). Note that the theorem below gives conditions on the range, not on the operator or the space themselves.

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Theorem 2.5. Let X be a linear subspace of the normed linear space X, and suppose that u is a continuous linear operator from X, into L. Then u has a continuous linear extension U : X—> L such that ||U|| = ||u||. Proof. Note that for any x¡ e Xu \i{xx) e L, and hence is representable as a sequence; so for each n B N, denote by u„ the functional on Xt given by \in{xx) = u(xi)„, that is, the nth-coordinate of u(x,) for any *ι € X,. Now observe that KMHuOOJ^supiKuU.flJrneN}

= M* i.

s|Mllk||.

Thus, as the functional p defined by p(x) = ||u|| ||JC|| clearly satisfies the conditions set forth in the Hahn-Banach Theorem, we can extend u„ to a functional on all of X, which we will denote by U„. Note that this extended functional is linear on X and satisfies |U„W| L given by (U(x))n = \Jn(x), that is, the nth-coordinate of V(x) is Vn(x). It is readily verified that U is linear (since each U„ is linear), it is well-defined [i.e., its values are in L, which we can see by looking at Equation (2.6)], and it is also continuous [again just look at Equation (2.6)]. Moreover, U is clearly an extension of u since each U„ extends u„.Thus the proof of the result is complete.· Before proceeding to our next main application of the Hahn-Banach Theorem, we will prove the complex form of this theorem, which interestingly enough is itself an application of the original theorem for the real case, so we are not getting too far afield. The proof given here is due to Bohnenblust and Sobczyck (1938). Theorem 2.6 (Complex Hahn-Banach Theorem). Let X! be a linear subspace of the complex linear space X and let p : X -> R be subadditive and have the following property: ρ(λχ) = \λ\ p(x) for all x e X and λ€ C. Suppose further that / e X\ (i.e., / is a linear, not necessarily continuous, functional on Xi) and satisfies \f{x\)\< p(xx) for all x, e X,. Then there exists a linear functional /-"that extends/, is defined on all of X, and that satisfies \F{x)\0) is just the nth-derivative of/). By the continuity of the members of X* we have the following important observation. Observation. If / : Ω -> X is analytic at ω0 and x* e X* then x* ° f: Ω -» C is analytic at ω0. Indeed, note that

hm

— —¡rs;—=,lm~* —^^— = χ*(Γ(ω0))

with the appropriate steps being justified by the linearity and continuity of x*. Utilizing this observation, we can immediately prove many Banach spacevalued versions of the classic complex variable results. The whole idea, of course, is to take a complex Banach space-valued function and consider the composition of this function with any element of the dual space. We then apply the classic complex variable result to each of these and then,finally,"piece" them back together to get the desired result for the original function itself. One such result is indicated here to give an example of the general idea. We refer the interested student to Nagumo (1965). The particular result that we will prove has some interesting consequences in the area of Banach algebras; it is also the natural generalization of the classic (and very beautiful!) theorem of J. Liouville. Theorem 2.7. Let X be a Banach space, / : C -> X be analytic on the entire complex plane, and suppose that /(C) is bounded in X. Then / is a constant function. Proof. As we have observed, x* ° f is analytic for each x* e X* whenever / is analytic. Consequently, given any x* e X*, each x* °/is an entire function of a complex variable with complex values. Moreover, if ||/(ft))|| < M for all toe C and some given M > 0, then |χ*(/(ω))| < ||χ*|| ||/(ω)|| < ||**|| Μ, so each x* °/is bounded. Thus by the classic Liouville Theorem, we have that x* °f is a constant for every x* e X*. Now, if / itself is not a constant, then there exist 0)χ, [)-/(ω2)φ0. Consequently, by Corollary 2.3 of the

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Hahn-Banach Theorem, there is an x* e X* with ¿♦(/(«O -f(ah)) * 0; in fact, ** can be chosen so that k*(/(ö>,) - /(ω 2 ))| = ||/(ω,) - /(ω 2 )|| > 0. But then x*(f(a)x)) ϊχ^βα^)), contradicting the deduction of the first paragraph. Hence,/must be constant as claimed.· Remark. Since we are considering various types of applications of the HahnBanach Theorem to different aspects of the area of analysis, this seems an appropriate point at which to fulfill our promise made in Chapter 1 to supply the arguments justifying the representations of the dual of a subspace of a Banach space and of the related quotient space. You will recall these were originally given as Propositions 1.4 and 1.5. So without further ado we restate, and prove, our advertised results. Proposition 1.4. If X is a Banach space, y a closed linear subspace of X and y 1 = \x* € X* : x{y)* < 0 for all y e y), then the map ψ: X*/y x -> y* given by φ(χ* + y 1 ) =x*\y is an isometric isomorphism of X*/yx onto y*. Proof. The idea here is pretty straightforward. You should first notice that since y 1 is a closed subspace of X*, considering X*/y x is legitimate. Further as any x* + y x in X*/yx induces the linear functional **|y, our map is welldefined. Since it should be clear that φ is linear, as well as being one-to-one, we can move directly to establishing the needed bounds on its norm. First note if we let x* e X* and / e y 1 , then

|^* + y i )I=lk*l y ||=ll(x* + /)l y ll \\x* + yx||.H As you might have anticipated, Proposition 1.5 takes a little more care, but presents no real roadblock. Proposition 1.5. If X is a Banach space, y a closed linear subspace of X and q : X-> X/y is the natural quotient map, then ψ: (Χ/\/)* -> y1 given by i//(/) = / ° q for all / e (X/y)* is an isometric isomorphism of (X/y)* onto y1. Proof. Note that ι/ns well-defined since if we l e t / e (X/y)* then (f° q)(y) = 0 for all y e y and we have f° q e y \ It should be clear that ψ is linear, one-to-one, and remembering that q is a quotient map, || 11/11· Choosing a sequence (y„)„ in V Ik« + >Ή|| < 1 for each n e N we have

87 witn

II v(/)ll * M/))(*» + yn )l=l/(*„ + ^ )l ->11/11 yielding \ψ\ = 1 as desired. It only remains to show that ψ is onto, which is not too hard, but not trivial either. So, suppose we let g e V 1 a °d then construct the appropriate element of (XIV)* we need by defining / : XIV -> U by /(* + V) = g(*) for each * e X. Note that / i s clearly well-defined (remember that g annihilates ~\f). Further, for any x e X, we have \f(x + V)l = lsWI = |g(* + y)l ^ llgll II* + )1I for any y € y, so that

\f(x + yi< M{\\g\\\\x + y\\:y e VI = «glllk + yl But now we're done as this gives u s / e (Xl)/)* withg= ψ{/) and ||/|| 0 whenever x„ > 0 for all n e N; (ii) LIM(*) = LIM(ot*)), where σ denotes a left shift operator, that is, φ) = ot (*„)„) = (Λ.)"=2;

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(iii) LIM(5) = 1, where 2) < LIM(ox) = LIM(x). That is, we can see, through repeated use of property (ii), that inf(*„ :n>k) LIM(JC), then sup{;t„: n > k] > LIM(*) for all k e N, so that LlM(x) < limM xn. Thus we need only show inf„ x„ < LIM(*) < sup„ xn to be done, and this isn't too bad, as we now see. Let ε > 0 be given and choose an index N so that inf„ x„ < xN < inf„ xn + ε. Note that from this we have xn + ε - xN > inf„ xn + ε - xN > 0 for all n e ΛΙ. Thus, from property (i), we have 0 < LIM(x + ε - xN) = LIM(o:) + ε - xN (by linearity and property (iii)), so that xN < UM(x) + e. Now since we clearly have inf„ x„ 0, inf„ xn < LIM(^) as needed. Finally, since inf„(-;t„) < LIM(-^:) = -LIM(^), it follows that sup„ xn = -inin(-xn) > LIM(JC), and we are done.B We now turn our attention to the issue you might have noticed was sidestepped earlier; namely, now that we have produced a legitimate generalization of the usual notion of a limit, does such an object as LIM actually exist? Banach, naturally, provides us with the answer. Theorem 2.8. Banach limits exist. Proof. We begin, perhaps a bit obliquely, working toward using the HahnBanach Extension Theorem to give us what we need; in this case, we let, for each x = (xn)n e L,

ρ(χ)=Μη(χ>+χ*+η-+χή. It should be clear that p{x + y) 0, Z„|umn| < K for all m e N; (b) limm umn = 0 for all n e N; (c) limm S„wm„ = 1. Proof. First note that condition (i) implies condition (ii) is easy. In fact, condition (ii) (a) follows from condition (i) and Theorem 2.14; condition (ii) (b) follows from the regularity of u, the identity (um„)m = (u(e„)), and the fact that e„ e c0 for all n e N; finally, condition (ii) (c) follows similarly from

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101

the regularly of u, the identity (Z„Mmn)m = u(5), and noting 0, which yields /„,(λ) - lim„A„ -> 0 or that fm{X) -» lim„A„, which is what we wanted to have, since the nth-coordinate of u(A) is nothing but UA)\m As another application of the Banach-Steinhaus Theorem to summability theory, we present the classic Knopp-Lorentz Theorem on what are called "/-/"-summability techniques, this time asking that our map u take absolutely summable sequences into absolutely summable sequences. You should notice here that the condition for this to happen involves explicitly only the columns of u. Theorem 2.16 (Knopp-Lorentz Summability Theorem). A necessary and sufficient condition that lx Q (/))„ for a given matrix map u is that there exists a K > 0 such that lm\umn\ < K for all neN. Proof. One can easily, and of course should, show (using Theorem 2.10, as we did in Theorem 2.11) that {μn)n = μe L if and only if for each (λ„)η = λ e l\, the series Σημ„λ„ converges (of course, it follows that the series is absolutely convergent).This fact will prove useful in the development given below. Now let u be a matrix map such that /, c (/,)u; then, in particular, /, c (s)u, so for each m e M, Σηιι„„λ„ converges for all (λ„)„ = X e /,. Consequently, from the preceding paragraph, for each m e N there is an Mm > 0 such that \umn\ < Mm for all n e N. Now for each k e N, define the operator ak: /j -> /) by

a* W) = (Σ„ «i A 'Σ„ "2 A .···>£„ "*A ,0,0,...). Note that each ak is clearly linear, and since

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each ak is continuous. But α*(λ) -> (L„umnXJm as k -> °° for each λ e /,, so that by Theorem 2.10 the operator u : /; -> lx must be bounded, that is, there must exist an M > 0 such that IC^AJmli * M||(>UJi for all (*,)„ e /,. Now this, in turn, yields for each m e N, lm\umn\ = \\(Tnumnenm)m\\u where (enm)m = enelx is the «th-unit vector (using the same notation as we did in Theorem 2.13). This now forces Σ^„„ε„„ = um„, and so Zm|um„| < M||e„||i = M. Fortunately, sufficiency is easy; indeed, from the fact Σ„|Μ„„| < M, it follows that for all m,n e N, \umn\ < M. So that for each fce^we have

Σΐ =1 ΙΣ„«™ K N Σΐ =1 Σ„ I««.« KI =Σ Π Σ1>».ΑΙ ^Σ„Λ/|Λ Λ Ι = ΛίΙΐΛΐΙ, for any λ e /,. Thus, as this inequality holds for all k e Fy, we have Zm|Z„umn/y < Μ1|λ||ι, and so u maps /rsequences to /rsequences, as we desired. The sufficiency is complete.! Remark. If we proceed similarly to Theorems 2.12 through 2.14, one can prove the following theorem (although obviously we do not explicitly do so here). Theorem 2.17. The matrix map u maps Ιλ to Ιλ with preservation of sums if and only if there exists an M > 0 such that for all n e N, Em|wmn| < M and for all m e N , Z„Hmn = 1. Example. Consider the following operation of a sequence λ - (λ„)π: For each ne N, define σ„ to be (1/n) Σ£=ιλ*, that is, σ„ is just the average of the first n terms of λ. We then generate a new sequence (σ„)„ = σ. Now observe that if (C,l) is the matrix map given by 1 0 1/2 1/2 (C,l) = 1/3 1/3 1/4 1/4

0 0 0 0 0 0 0 0 0 0 1/3 0 0 0 0 1/4 1/4 0 0 0

then it is readily verified [because (C,l) is "row-finite"] that we have s c (s)(ci). Moreover, observe that (C,l) satisfies the criterion of Theorem 2.15. Thus (C,l) maps convergent sequences to convergent sequences with preservation of limits, that is, in particular we get that the sequence of averages of a convergent sequence converges to the same limit.

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103

Example 2.2. One further example seems to be in order here. Let u be a matrix map of nonnegative entries, that is, um„ > 0 for all m,n e N. We call u doubly stochastic if and only if Xn«mn = Imum„ = 1 holds for all m,n € N. Observe that by Theorem 2.14, doubly stochastic matrices map convergent sequences into convergent sequences with preservation of limits. This is an easy calculation and the student should verify it. Also, by Theorem 2.16, doubly stochastic matrices map absolutely summable sequences into absolutely summable sequences with preservation of norm. We shall see many more applications of the Banach-Steinhaus Theorem and of both the Open-Mapping and Closed-Graph Theorems in the future (actually, the near future, in fact, in this text). So, rather than dwell on these now, we give just a couple here to get some idea of what these may be like. The theorem given just below is perhaps somewhat indicative of what we might expect. Theorem 2.18. Let (Ω,Σ,μ) be a finite measure space, X be a Banach space, and u : X -> Ζ,^μ) be a linear map. Then, u is continuous if and only if for each A € Σ, the functional uA : X -» R given by uA(x) = iAu(x) άμ is continuous. Proof. Sufficiency is clear from Holder's Inequality. To prove necessity, we proceed by the Closed-Graph Theorem, that is, we will show that u's graph is closed. So, let (x„)„ Q X and denote by /„ the function u(xn) for each ne M. Suppose that x„ -»x 0 and that /„ -»f0. We must show that U(JC0) = f0. Now, x„ —> x0, so for each A e Σ, we have that uA is continuous. Thus uA(xn) -> uA(x0). On the other hand, as \\4αάμ-\4„άμ\< ||/ 0 -/ η ||ι, we have ν\{χ„) = \/^η(1μ^>\ρ$0άμ. It now follows that for each A e Σ, we have UA(*O) = ÍA/OÍ^, that is, for each A e Σ, ¡Α\ι(χ0)άμ = ίΑ/0άμ. This yields, by the uniqueness of derivatives given by the Radon-Nikodym Theorem, that u(x0) =/„ and u's graph is closed. Of course, the continuity of u follows.· What we have next is not only a nice application of the Open-Mapping Theorem, but gives us some information about separable Banach spaces, and indicates the central role played by the Banach space /,. As should be clear by the statement of the theorem, it obviously tells us that if one is interested in looking at separable Banach spaces, the quotients of (and thus closed subspaces of) lx cannot be ignored. Theorem 2.19. Let X be a separable Banach space, then X is topologically isomorphic to a quotient of /t. Proof. Since X is separable, let (xn)„ be a dense sequence in Ux. Define the operator T : /, -> X by Τ(λ) = Σηλ^χη for λ = {λ„)η € Ιλ. Note that T is certainly well-defined, since Σ„||λ„Λ:„|| < Σ„|Λ,| < e», and so Σ„λ^„ (being absolutely summable, thus summable), exists in X. Also note that T is clearly a linear map, and is continuous since

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|τ(λ)ΗΣΛ*π||< IJPU.N ΣΛΗΙ4· Finally, T is also onto, although this is a little harder to see. In fact, to see this, let xe \JX and choose nx e N such that HJC-JC„,|| nu and with ||2(JC - xni) - jt„2|| n2, and such that \\x - χηχ -\x„2 - (\)2xnJi < φ 3 · It should be clear that if we continue this process, we will be able to select an increasing sequence of indices ri\ < n2 < n3 < · ■ · so that for any ke Nv/e have |k - *«, - 2 Xn2 - (i)' Xn,

(i)*"' Xnt \\ < (^Ϋ ■

It is not hard to see (and the student should check it!) that if we let (a„)n be the sequence in l¡ defined by the following, each an - 0 except a„, = 1, a„2 =\, fl„3 = (\)2,... ,a„k = φ*" 1 ,..., we will have Τ((α„)„) = x, and thus T is onto. We are finally in a position to apply the Open-Mapping Theorem to get that T is open (i.e., that T is a quotient map). It should now be clear that Xis topologically isomorphic to the quotient space X^T^O)). [In fact, for any such operator T, range(T) is (algebraically) isomorphic to domain(T)/nullity(T) (as you should recall from your undergraduate linear algebra course), and so if we let S: (//Γ'ίΟ)) -> X [which is range(T)], then T = S°TT, where π: lt —> (Λ/Τ^Ο)) is the obvious quotient map]. Since S is clearly one-to-one and onto, and since Τ is open and continuous, so is S. Thus S is the desired topological isomorphism.· Our next result is an old theorem that was originally meant to give conditions on when certain types of finite collections of equations had a solution in linear spaces.The original theorem wasfirstdue, in this setting, to Hahn (1927). An earlier result valid for Lp- and /,,-spaces, and in many ways closer to the version we give here, was proved by F. Riesz (1910). However, the result closest to the "modern day" version of this theorem was shown to hold in the space C([0,1]) by Helly (1912). The result we give below is in some sense a mixture and refinement of the results of Hahn, Riesz, and Helly, even though only one of them is today honored in its title. Theorem 2.20 (Helly's Theorem). Let X be a Banach space and the set {/1/2, .../„} be a finite collection of members of X*.Then given any collection [aucx2,... ,a„) of scalars, a necessary and sufficient condition that for each given ε > 0 there exist an element χε of X such that f¡(xe) = cc¡ for all 1 < i < n and H-JCel < M + ε for some M > 0, is that the inequality

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holds for any finite set of scalars [β\,βι,... ,ß„). Proof. You should first notice that from the definition of the norm of a continuous linear functional, the necessity of the condition follows easily.Thus, we prove the sufficiency only. Note first that without loss of generality, we can assume that the collection [f,\t\ is linearly independent (if it isn't, then some subset of these are linearly independent and span the same subspace as the original collection). Now consider the map φ: X -> R" given by the formula 0, we let Sf = [x e X: ||JC|| < M + ε}, then since φ is clearly continuous and onto, by the Open-Mapping Theorem we have that (p(Se) contains the zero vector Θ of R" as an interior point for any such ε. Let us now suppose our condition is not sufficient, that is, the vector (αι,α 2 ,... ,α„) e 0 for each /, Σ,^λ, = 1, and the a, are in A. It should also be clear that a set A c £ is convex if and only if any translate of it is convex (i.e., if and only if A + x0 = [a + x0: a e A) is convex for Xo € £ ) ·

Of course, any linear subspace of £ (including £ itself) is convex. And in the case that X is a normed linear space, any open or closed sphere centered at a point x0 is a convex set, that is, every set of the form jxe X: |JC - x01| < ε} or [x e X: ||* - x0 \\ < ε) for a given ε > 0 and x0 e X is convex. Finally, it can easily be seen that an intersection of (any number of) convex sets is still convex, and equally easily recognized that a union of convex sets need not have this property. Definition 3.2. Let £ be a linear space and A £ £. By the convex hull of A, which we will denote by co(A), we mean the smallest convex subset of £ containing A. Note that such a set always exists since the whole space £ is convex and we simply intersect all convex sets containing A to get the desired set. Moreover, it is easily shown (and, of course, should be by the student) that co(A) = j X " λ,α,: A, > 0 and a, e A for all Í < n, and "^" λ, = 1] where n e N. Definition 3.3. Let £ be a linear space and A C £. We say that A is circled whenever λ A £ A for all scalars λ with |λ| < 1. You should be aware that in many texts such a set is called balanced instead of circled. While convex sets are closed under translation, it is clear that circled sets are not; indeed, every circled set must contain Θ. Of course, all linear subspaces

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of £ (including £ itself) are circled, and any intersection of circled sets is again a circled set. Thus, as with the preceding convex sets, every set A £ £ is contained in a smallest circled set, which we denote by c(A), and call the circled hull of A. It should be obvious to the reader that c(A) is simply the intersection of all circled sets that contain A. The student easily should be able to verify that a subset A of a linear space £ is convex and circled if and only if for any n e N , Σ"=1λ,α, e A whenever Σ,Ίι|λ,| < 1 and a, € A for all i < n. Similarly, if A £ £, then the circled convex hull of A (which should obviously denote the smallest circled and convex set containing A) coincides with the set

{ Σ Η λ·α': Σ Η Ιλ Πν Χυ by the following: for each xe Έ, the Uth-coordinate of τ(χ) is ru(x). With a little thought, we can see that r is continuous and linear as well. Furthermore, if we have τ(χ) = Θ for some x e Έ, then τυ(χ) = Θ for all U e 11, that is, x e ||·||υ(0) for each U e 11, and S O J I E U for each such U. But this means that x e flu U, and so x = Θ, since Έ is a Hausdorff space; of course, this yields that r is one-to-one. We now only need show that r is almost open to be done. Toward this end, let V be any neighborhood of Θ, then since 11 is a neighborhood base for Έ, we can assume that V e 11. If, for notational purposes, we let U(lW) denote the unit ball of any space 1/V, then we have

= tf(*v'(iU(Qv))) civ'OrvWUdv)))) = i v I ( U ( I v ) + Mv(e))

= iv,(u(rv)) = U(£V) = {xeT:\\x\\w(x) X** be the natural injection of the Banach space Xinto its bidual X**.Then j(Ux) is a(X**,X*)-dense in Ux«. That is, we have the unit ball of X is weak*-dense in the unit ball of X**, so that it follows that X itself is weak*-dense in all of X**. Proof. Here we prove the theorem in the real case; the complex case is left as an (easy) exercise for the student.

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Let S, be the a(X**,X*)-closure of j(U x ). Then since Ux.. is σ(Χ**,Χ*)closed [in fact, it is ff(X**,X*)-compact!] we have that Si c Ux... Moreover, S, is convex. Note that σ(Χ**,Χ*) is a locally convex linear topology; hence, the closure of a convex set in σ{Χ**,Χ*) is still convex. Thus we have that Si is a closed convex subset of U x .. in the a(X**,X*)-topology. Now suppose that S, £ \JX„ but is not equal to it, then there must be some x** in Ux-ASi. But now, by Theorem 3.3, there is a CT(X**,X*)-continuous linear functional F on X** such that sup{F(s): s e Si) \ (in fact, we can choose this difference as close to 1 as we wish by Theorem 1 in the Introduction). If we let 22 be the subspace generated by *, and x2, then as before, Z2 is a proper closed subspace of X as well. Thus we can select an x, in X with ||JC3|| = 1 such that ||*3-x,|| >-j for /= 1,2. Proceeding in this manner, by induction, we are able to select an infinite sequence (x„)„ in Ux (in fact, all of norm 1) such that \\x„ - xm\ >\ as long as n*m. Clearly this sequence can have no convergent subsequence, and hence Ux is not compact. This contradiction yields that X must be finite-dimensional as claimed.· As we shall see in the not too distant future, the use of this fact will give a construction that will be of central importance for the main construction of the proof that assertion (ii) implies assertion (i).

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Now, however, let us suppose we have that J is a finite-dimensional subspace of the dual space 2* of a Banach space Z. Then by Fact 3.1, the surface of the unit ball (sometimes called the unit sphere and denoted by Sz) given by {/E J\ ||/|| = 1} is norm-compact. Consequently, there exist points z*,z*, ■ ■ ■ ,z*e J, with ||z*|| = 1 for i = 1,2,... ,n such that given any z* e J with ||z*|| = 1, there is an integer p (\||z*(zj-|z*(z„)-z*(zj| which, if p is chosen so that ||z* - z*|| \a-ß\ where α,β>0 are such that a > { a n d ß\ and therefore by "normalizing" these vectors, we have that ||z*|| < 2max{|z*(zJ: 1 < p < n} for z* e J. We state this result here as follows. Fact 3.2. Given a finite-dimensional subspace JF of the dual space of a Banach space Z, there exist vectors Z\,Z2, ■ ■ ■ ,z„ e Z, with ||z,|| = 1 such that for all z*e Z*, \\z*\\ < 2 max{|z*(zp)| : 1 < p < n] for z* e J. We now return to our proof and apply this result as follows: let x\ e X* be any vector of norm one, and JCJ* e w*[j(A)]. We want to show that x** e j(A), so note that the weak* neighborhood of *** generated by JC* of "radius 1" intersects j(A), that is, there exists an ax e A such that |(ac** - j(ai))(**)| < 1. Let Jx be the linear span of the vectors *** and x**-j(fli). Then Jx is a finite-dimensional subspace of X**, and thus by Fact 3.2, there are vectors x*,... ,x*2 e X* such that |JC?|| = · · · = ||JC*2|| = 1 and such that ||y**|| < 2max{|y**U,-)|: 2 0 be given, then choose k e N so that U* is in the base for X with φ(χ) e Uk and diam(U*) < ε. Then if y = .yws... is in K with \x - y\ < l/3 2 \ we have yk = xk and

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Thus, letting p denote the metric on X, we have ρ(φ(γ),φ(χ)) < ε, and φ is continuous as claimed. The map φ ° φ: C - » X yields our conclusion.· Theorem 3.12 (Banach-Mazur Theorem). Every separable normed linear space A" is linearly isometric to a subspace of C([0,1]). Proof. First we consider U*. in its weak* topology, so that it is compact and metrizable by Corollary 3.7. Recall that with each xe Xv/e can associate the element Jc : U*. -» R given by x(x*) sx*(x). As we have seen, x is weak*continuous and ||JC|| = sup{| x(x*)\: x* e Ux*}, so that the association of x with x establishes a linear isometry of X into C(UX.). Now since U*. is compact and metrizable, we know from Lemma 3.4 that Ux. is the continuous image of the Cantor set C, and thus, as is not hard to show, C(Ux.) (and therefore X itself) is linearly isometric to a closed linear subspace of C(C). Finally, we need only note that if we take a n y / e C(C), it can be extended to all of [0,1] simply by defining its extension linearly on each interval in [0,1]\C. Since it is clear that this extension is both linear and has the same supremum norm as / did, we get that C(C) (and hence X itself) is linearly isometric to a subspace of C([0,1]).B Before closing this section, we give two final consequences of the results we have developed here, and of the Eberlein-Smulian Theorem in particular. Both of these results, as well as the included corollary, go under the name of the Krein-Smulian Theorem and as we shall see, especially in Chapter 4, can be very useful in achieving some deep facts about operators on Banach spaces. These facts all come from some early work on the geometry of abstract spaces, and can be found in a paper by Krein and Smulian (1940). It is worth recognizing that the first theorem essentially gives us a converse to the Banach-Alaoglu Theorem. Here we present only an outline of the proof with the expectation that the student will be sufficiently motivated (and able) to fill in the details. For a complete proof the student should refer to the excellent, if somewhat encyclopedic, book by Dunford and Schwartz (1958). Theorem 3.13 (Krein-Smulian Theorem). Let X be a Banach space and K £ X* a convex set with K Π aU x . a weak*-compact set for every a > 0, then K is weak*-closed. Proof. As just indicated, we will only indicate the major steps here; the student should endeavor to fill in the missing details.

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(i) K is a (norm) closed subset of X*. (ii) For each subset F of X, we can associate with F another set called its polar and defined by P = {x* e x* : \x*(x)\ < 1 for all x e F}. For any a>0, aU x . = n { P : F is a finite subset of iU x }. (iii) The theorem itself now follows from the following proposition: If, in addition to the given hypothesis, K Π U x . = 0 , then there is an x ε X such that x*(x) > 1 for all x* e X*. (iv) Now to prove the proposition, put Fo = {0) and suppose finite sets F 0 ,F),..., F„_! have been chosen so that iF, £ Ux for / = 0,1,. . . , n - 1, and so that

Π;>,οηκηηυχ.=0.

(3.1)

Note that Equation (3.1) is true for n = 1, and let Q = DS/FTn K Π (n + 1)UX.. If F° Π Q * 0 for any finite subset F £ (l/n)Ux, then the weak* compactness of Q and (ii) together imply that n\Jx. Π Q * 0 , contradicting Equation (3.1). Thus there is a finite set F„ £ (1/«)UX such that Equation (3.1) holds with "n + 1" replacing «.This construction can now proceed, yielding Kn

rCo F «°= 0 ·

(3·2)

Now arrange the members of U„F„ in a sequence (x„)„, then ||JC„|| -> 0 as n -> °°. Define T: X* -»c 0 by T(x*) = (**(*„)),,. Then T(K) is convex, contained in ca, and by Equation (3.2), ||T*(x)|| = sup{|x*(jcn)|: n e N) > 1 for all x* e K. Hence, there is a sequence (c^)„ C R such that Σ„|α„| < °° and Σηα„χ*(χη) > 1 for all x* e K. Put * = Σ„α^κ„ to complete the proof.· A little thought should yield the following as an immediate corollary to Theorem 3.13. Corollary 3.8. A linear subspace X0 of X* is weak*-closed if and only if X, Π U x . is weak*-compact. The "second" Krein-Smulian Theorem gives us the often useful fact that if a set A in a Banach space is weakly compact, so is its (norm) closed convex hull, which we will denote in what follows by co(A). Theorem 3.14 (Krein-Smulian Theorem). If A C X is weakly compact, then cö"(A) is weakly compact. Proof. Let A be a weakly compact subset of X and note that by Mazur's Theorem we know co(A) is weakly closed. It now follows, from the

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Eberlein-Smulian Theorem, that it suffices to show that co(A) is weakly sequentially compact. So, toward this end, let (x„)„ be a sequence in co(A) and note that each x„ is a convex combination of a finite set of points B„ in A. Let B 0 denote UnBn and Xo s [BJ, where this denotes, as usual, the closed linear span of B0. You should note that Xo is easily seen to be separable. Now let A0 = A Π Xo and note that A0 is weakly compact and, since (x„)„ Q co(A0), the theorem will be complete if we can show that cö(A0) is weakly compact. Now by the Eberlein-Smulian Theorem, A0 is weakly sequentially compact, and since sup{|jc*(;t)|: x e A0} < «> for every x* e X*, we know by the BanachSteinhaus Theorem that there is a constant K > 0 such that sup(||jc||: x e A„} < K. Moreover, in the relative weak topology, A0 is a compact Hausdorff space, and so we will consider the collection of all continuous functions on A0 given by C(A0). Note that by the Riesz-Kakutani Theorem, we know C(A