Introduction to Banach spaces analysis and probability. Vol.1 9781107162631, 9781107160514, 9781316675762

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C A M B R I D G E S T U D I E S I N A DVA N C E D M AT H E M AT I C S 1 6 6 Editorial Board B . B O L L O B Á S , W. F U LTO N , F. K I RWA N , P. S A R NA K , B . S I M O N , B . TOTA RO

INTRODUCTION TO BANACH SPACES: ANALYSIS AND PROBABILITY This two-volume text provides a complete overview of the theory of Banach spaces, emphasising its interplay with classical and harmonic analysis (particularly Sidon sets) and probability. The authors give a full exposition of all results, as well as numerous exercises and comments to complement the text and aid graduate students in functional analysis. The book will also be an invaluable reference volume for researchers in analysis. Volume 1 covers the basics of Banach space theory, operator theory in Banach spaces, harmonic analysis and probability. The authors also provide an annex devoted to compact Abelian groups. Volume 2 focuses on applications of the tools presented in the first volume, including Dvoretzky’s theorem, spaces without the approximation property, Gaussian processes and more. Four leading experts also provide surveys outlining major developments in the field since the publication of the original French edition. Daniel Li is Emeritus Professor at Artois University, France. He has published over 40 papers and two textbooks. Hervé Queffélec is Emeritus Professor at Lille 1 University. He has published over 60 papers, two research books and four textbooks, including Twelve Landmarks of Twentieth-Century Analysis (2015).

C A M B R I D G E S T U D I E S I N A DVA N C E D M AT H E M AT I C S Editorial Board: B. Bollobás, W. Fulton, F. Kirwan, P. Sarnak, B. Simon, B. Totaro All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit: www.cambridge.org/mathematics. Already published 131 D. A. Craven The theory of fusion systems 132 J.Väänänen Models and games 133 G. Malle & D. Testerman Linear algebraic groups and finite groups of Lie type 134 P. Li Geometric analysis 135 F. Maggi Sets of finite perimeter and geometric variational problems 136 M. Brodmann & R. Y. Sharp Local cohomology (2nd Edition) 137 C. Muscalu & W. Schlag Classical and multilinear harmonic analysis, I 138 C. Muscalu & W. Schlag Classical and multilinear harmonic analysis, II 139 B. Helffer Spectral theory and its applications 140 R. Pemantle & M. C. Wilson Analytic combinatorics in several variables 141 B. Branner & N. Fagella Quasiconformal surgery in holomorphic dynamics 142 R. M. Dudley Uniform central limit theorems (2nd Edition) 143 T. Leinster Basic category theory 144 I. Arzhantsev, U. Derenthal, J. Hausen & A. Laface Cox rings 145 M. Viana Lectures on Lyapunov exponents 146 J.-H. Evertse & K. Gy˝ory Unit equations in Diophantine number theory 147 A. Prasad Representation theory 148 S. R. Garcia, J. Mashreghi & W. T. Ross Introduction to model spaces and their operators 149 C. Godsil & K. Meagher Erd˝os–Ko–Rado theorems: Algebraic approaches 150 P. Mattila Fourier analysis and Hausdorff dimension 151 M. Viana & K. Oliveira Foundations of ergodic theory 152 V. I. Paulsen & M. Raghupathi An introduction to the theory of reproducing kernel Hilbert spaces 153 R. Beals & R. Wong Special functions and orthogonal polynomials 154 V. Jurdjevic Optimal control and geometry: Integrable systems 155 G. Pisier Martingales in Banach spaces 156 C. T. C. Wall Differential topology 157 J. C. Robinson, J. L. Rodrigo & W. Sadowski The three-dimensional Navier–Stokes equations 158 D. Huybrechts Lectures on K3 surfaces 159 H. Matsumoto & S. Taniguchi Stochastic analysis 160 A. Borodin & G. Olshanski Representations of the infinite symmetric group 161 P. Webb Finite group representations for the pure mathematician 162 C. J. Bishop & Y. Peres Fractals in probability and analysis 163 A. Bovier Gaussian processes on trees 164 P. Schneider Galois representations and (ϕ , )-modules 165 P. Gille & T. Szamuely Central simple algebras and Galois cohomology (2nd Edition) 166 D. Li & H. Queffelec Introduction to Banach spaces, I 167 D. Li & H. Queffelec Introduction to Banach spaces, II 168 J. Carlson, S. Müller-Stach & C. Peters Period mappings and period domains (2nd Edition) 169 J. M. Landsberg Geometry and complexity theory 170 J. S. Milne Algebraic groups

Introduction to Banach Spaces: Analysis and Probability Volume 1 DA N I E L L I Université d’Artois, France

H E RV É Q U E F F É L E C Université de Lille I, France

Translated from the French by

DA N I È L E G I B B O N S a n d G R E G G I B B O N S

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 4843/24, 2nd Floor, Ansari Road, Daryaganj, Delhi – 110002, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107160514 DOI: 10.1017/9781316675762 Originally published in French as Introduction à l’étude des espaces de Banach by Société Mathématique de France, 2004 © Société Mathématique de France 2004 First published in English by Cambridge University Press 2018 English translation © Cambridge University Press 2018 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. Printed in the United States of America by Sheridan Books, Inc. A catalogue record for this publication is available from the British Library. ISBN – 2 Volume Set 978-1-107-16263-1 Hardback ISBN – Volume 1 978-1-107-16051-4 Hardback ISBN – Volume 2 978-1-107-16262-4 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Dedicated to the memory of Jean-Pierre Kahane

Contents

Volume 1 Contents of Volume 2 Preface Preliminary Chapter I II III IV

page x xiii

Weak and Weak∗ Topologies. Filters, Ultrafilters. Ordinals

Introduction Weak and Weak∗ Topologies Filters, Ultrafilters. Ordinals Exercises

1 1 1 7 12

1

Fundamental Notions of Probability I Introduction II Convergence III Series of Independent Random Variables IV Khintchine’s Inequalities V Martingales VI Comments VII Exercises

13 13 15 21 30 35 42 43

2

Bases in Banach Spaces I Introduction II Schauder Bases: Generalities III Bases and the Structure of Banach Spaces IV Comments V Exercises

46 46 46 59 74 76

3

Unconditional Convergence I Introduction II Unconditional Convergence

83 83 83

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Contents

III IV V VI VII VIII

Unconditional Bases The Canonical Basis of c0 The James Theorems The Gowers Dichotomy Theorem Comments Exercises

90 94 96 101 110 111

4

Banach Space Valued Random Variables I Introduction II Definitions. Convergence III The Paul Lévy Symmetry Principle and Applications IV The Contraction Principle V The Kahane Inequalities VI Comments VII Exercises

117 117 117 129 133 138 151 152

5

Type and Cotype of Banach Spaces. Factorization through a Hilbert Space I Introduction II Complements of Probability III Complements on Banach Spaces IV Type and Cotype of Banach Spaces V Factorization through a Hilbert Space and Kwapie´n’s Theorem VI Some Applications of the Notions of Type and Cotype VII Comments VIII Exercises

193 200 203 205

6

p-Summing Operators. Applications I Introduction II p-Summing Operators III Grothendieck’s Theorem IV Some Applications of p-Summing Operators V Sidon Sets VI Comments VII Exercises

210 210 211 217 227 231 258 260

7

Some Properties of Lp -Spaces I Introduction II The Space L1 III The Trigonometric System IV The Haar Basis in Lp

266 266 267 277 284

159 159 159 172 177

Contents

V VI VII 8

Another Proof of Grothendieck’s Theorem Comments Exercises

The Space 1 I Introduction II Rosenthal’s 1 Theorem III Further Results on Spaces Containing 1 IV Comments V Exercises

Annex I II III

Banach Algebras. Compact Abelian Groups Introduction Banach Algebras Compact Abelian Groups

ix

296 305 315 326 326 326 341 350 353 357 357 357 364

References Notation Index for Volume 1 Author Index for Volume 1 Subject Index for Volume 1

382 413 415 419

Notation Index for Volume 2 Author Index for Volume 2 Subject Index for Volume 2

425 426 429

Contents

Volume 2 Contents of Volume 1 Preface

page ix xiii

1

Euclidean Sections I Introduction II An Inequality of Concentration of Measure III Comparison of Gaussian Vectors IV Dvoretzky’s Theorem V The Lindenstrauss–Tzafriri Theorem VI Comments VII Exercises

1 1 1 8 18 40 45 46

2

Separable Banach Spaces without the Approximation Property I Introduction and Definitions II The Grothendieck Reductions III The Counterexamples of Enflo and Davie IV Comments V Exercises

51 51 53 59 68 70

3

Gaussian Processes I Introduction II Gaussian Processes III Brownian Motion IV Dudley’s Majoration Theorem V Fernique’s Minoration Theorem for Stationary Processes VI The Elton–Pajor Theorem VII Comments VIII Exercises x

72 72 72 76 79 85 95 122 123

Contents

xi

4

Reflexive Subspaces of L1 I Introduction II Structure of Reflexive Subspaces of L1 III Examples of Reflexive Subspaces of L1 IV Maurey’s Factorization Theorem and Rosenthal’s Theorem V Finite-Dimensional Subspaces of L1 VI Comments VII Exercises

127 127 128 142 150 157 176 180

5

The Method of Selectors. Examples of Its Use I Introduction II Extraction of Quasi-Independent Sets III Sums of Sines and Vectorial Hilbert Transforms IV Minoration of the K-Convexity Constant V Comments VI Exercises

193 193 193 217 223 228 230

6

The Pisier Space of Almost Surely Continuous Functions. Applications I Introduction II Complements on Banach-Valued Variables III The C as Space IV Applications of the Space C as V The Bourgain–Milman Theorem VI Comments VII Exercises

234 234 235 243 261 268 282 287

Appendix A

News in the Theory of Infinite-Dimensional Banach Spaces in the Past 20 Years

290

An Update on Some Problems in High-Dimensional Convex Geometry and Related Probabilistic Results

297

Appendix C

A Few Updates and Pointers

307

Appendix D

On the Mesh Condition for Sidon Sets

316

Appendix B

References Notation Index for Volume 2 Author Index for Volume 2 Subject Index for Volume 2

324 355 356 359

Notation Index for Volume 1 Author Index for Volume 1 Subject Index for Volume 1

363 365 369

Preface

This book is dedicated to the study of Banach spaces. While this is an introduction, because we trace this study back to its origins, it is indeed a “specialized course”,1 in the sense that we assume that the reader is familiar with the general notions of Functional Analysis, as taught in late undergraduate or graduate university programs. Essentially, we assume that the reader is familiar with, for example, the first ten chapters of Rudin’s book, Real and Complex Analysis (Rudin 2); Queffélec–Zuily would also suffice. It is also a “specialized course” because the subjects that we have chosen to study are treated in depth. Moreover, as this is a textbook, we have taken the position to completely prove all the results “from scratch” (i.e. without referring within the proof to a “well-known result” or admitting a difficult auxiliary result), by including proofs of theorems in Analysis, often classical, that are not usually taught in French universities (as, for example, the interpolation theorems and the Marcel Riesz theorem in Chapter 7 of Volume 1, or Rademacher’s theorem in Chapter 1 of Volume 2). The exceptions are a few results at the end of the chapters, which should be considered as complementary, and are not used in what follows. We have also included a relatively lengthy first chapter introducing the fundamental notions of Probability. As we have chosen to illustrate our subject with applications to “thin sets” coming from Harmonic Analysis, we have also included in Volume 1 an Annex devoted to compact Abelian groups. This makes for quite a thick book,2 but we hope that it can therefore be used without the reader having to constantly consult other texts. 1 The French version of this book appeared in the collection “Cours Spécialisés” of the Société

Mathématique de France.

2 However, divided into two parts in the English version.

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We have emphasized the aspects linked to Analysis and Probability; in particular, we have not addressed the geometric aspects at all; for these we refer, for example, to the classic Day, to Beauzamy or to more specialized books such as Benyamini–Lindenstrauss, Deville–Godefroy–Zizler or Pisier 2. We have hardly touched on the study of operators on Banach spaces, for which we refer to Tomczak-Jaegermann and to Pisier 2; Diestel– Jarchow–Tonge and Pietsch–Wenzel are also texts in which the part devoted to operators is more important. Dunford–Schwartz remains a very good reference. Even though Probability plays a large role here, this is not a text about Probability in Banach spaces, a subject perfectly covered in Ledoux–Talagrand. Probability and Banach spaces were quick to get on well together. Although the study of random variables with values in Banach spaces began as early as the 1950s (R. Fortet and E. Mourier; we also cite Beck [1962]), their contribution to the study of Banach spaces themselves only appeared later, for example, citing only a few, Bretagnolle, Dacunha-Castelle and Krivine [1966], and Rosenthal [1970] and [1973]. However, it was only with the introduction of the notions of type and cotype of Banach spaces (Hoffmann-Jørgensen [1973], Maurey [1972 b] and [1972 c], Maurey and Pisier [1973]) that they proved to be intimately linked with Banach spaces. Moreover, Probability also arises in Banach spaces by other aspects; notably it allows the derivation of the very important Dvoretzky’s theorem (Chapter 1 of Volume 2), thanks to the concentration of measure phenomenon, a subject still highly topical (see the recent book of M. Ledoux, The Concentration of Measure Phenomenon, Mathematical Surveys and Monographs 89, AMS, 2001), dating back to Paul Lévy, and whose importance for Banach spaces was seen by Milman at the beginning of the 1970s. We will also use Probability in a third manner, through the method of selectors, due to Erdös around 1955,3 and afterwards used heavily by Bourgain, which allows us to make random constructions. For all that, we do not limit ourselves to the probabilistic aspects; we also wish to show how the study of Banach spaces and of classical analysis interact (the construction by Davie, in Chapter 2 of Volume 2, of Banach spaces without the approximation property is typical in this regard); in particular we have concentrated on the application to thin sets in Harmonic Analysis. Even if we have privileged these two points of view, we have nonetheless tried to give a global view of Banach spaces (with the exception of the 3 Actually, this method traces back at least to Cramér [1935] and [1937].

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geometric aspect, as already mentioned), with the concepts and fundamental results up through the end of the 1990s. We point out that an interesting survey of what was known by the mid 1970s was given by Pełczy´nski and Bessaga [1979]. This book is divided into 14 chapters, preceded by a preliminary chapter and accompanied by an Annex. The first volume contains the first eight chapters, including the preliminary chapter and the Annex; the second volume contains the six remaining chapters. Moreover, it also contains three surveys, by G. Godefroy, O. Guédon and G. Pisier, on the major results and directions taken by Banach space theory since the publication of the French version of this book (2004), as well as an original paper of L. Rodríguez-Piazza on Sidon sets. Each chapter is divided into sections, numbered by Roman numerals in capital letters (I, II, III etc.), and each section into subsections, numbered by Arabic numerals (I.1 etc.). The theorems, propositions, corollaries, lemmas, definitions are numbered successively in the interior of each section; for example in Chapter 5 of Volume 1, Section IV they thus appear successively in the form: Proposition IV.1, Corollary IV.2, Definition IV.3, Theorem IV.4, Lemma IV.5, ignoring the subsections. If we need to refer to one chapter from another, the chapter containing the reference will be indicated. At the end of each chapter, we have added comments. Certain of these cite complementary results; others provide a few indications of the origin of the theorems in the chapter. We have been told that “this is a good occasion to antagonize a good many colleagues, those not cited or incorrectly cited.” We have done our best to correctly cite, in the proper chronological order, the authors of such and such result, of such and such proof. No doubt errors or omissions have been made; they are only due to the limits of our knowledge. When this is the case, we ask forgiveness in advance to the interested parties. We make no pretension to being exhaustive, nor to be working as historians. These indications should only be taken as incitements to the reader to refer back to the original articles and as complements to the contents of the course. The chapters end with exercises. Many of these propose proofs of recent, and often important, results. In any case, we have attempted to decompose the proofs into a number of questions (which we hope are sufficient) so that the reader can complete all the details; just to make sure, in most cases we have indicated where to find the corresponding article or book. The citations are presented in the following manner: if it concerns a book, the name of the author (or the authors) is given in small capitals, for example Banach, followed by a number if there are several books by this author: Rudin 3; if it concerns an article or contribution, it is cited by the name of

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the author or authors, followed in brackets by the year of publication, followed possibly by a lower-case letter: Salem and Zygmund [1954], James [1964 a]. We now come to a more precise description of what will be found in this book. In the Preliminary Chapter, we quickly present some useful properties concerning the weak topology w = σ (E, E∗ ) of a Banach space E and the weak∗ topology w∗ = σ (X ∗ , X) in a dual space X ∗ . Principally, we will prove the Eberlein–Šmulian theorem about weakly compact sets and the Krein– Milman theorem on extreme points. We then provide some information about filters and countable ordinals. Chapter 1 of Volume 1 is intended for readers who have never been exposed to Probability Theory. With the exception of Section V concerning martingales, which will not be used until Chapter 7, its contents are quite elementary and very classical; let us say that they provide “Probability for Analysts.” Moreover, in this book, we use little more than (but intensively!) Gaussian random variables (occasionally stable variables), and the Bernoulli or Rademacher random variables. The reader could refer to Barbe–Ledoux or to Revuz. Section III provides the theorems of Kolmogorov for the convergence of series of independent random variables, and the equivalence theorem of Paul Lévy. In Section IV, we show Khintchine’s inequalities, which, even if elementary, are of capital importance for Analysis. We also find here the majorant theorem (Theorem IV.5) which will be very useful throughout the book. Section V, a bit delicate for a novice reader of Probability, remains quite classical; we introduce martingales and prove Doob’s theorems about their convergence. In Chapter 2 (Volume 1) we begin the actual study of Banach spaces. We treat the Schauder bases, which provide a common and very practical tool. After having shown in Section II that the projections associated with a basis are continuous and given a few examples (canonical bases of c0 , p , Haar basis in Lp (0, 1), Schauder basis of C([0, 1])), we prove that the space C([0, 1]) is universal for the separable spaces, i.e. any separable Banach space is isometric to a subspace of C([0, 1]). In Section III, we see how the use of bases, or more generally of basic sequences, allows us to obtain structural results; notably, thanks to the Bessaga–Pełczy´nski selection theorem, to show that any Banach space contains a subspace with a basis. We next show a few properties of the spaces c0 and p . Finally, we see how the spaces possessing a basis behave with respect to duality; this leads to the notions of shrinking bases and boundedly complete bases and to the corresponding structure theorems of James.

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In Chapter 3 (Volume 1), we study the properties of unconditional convergence (i.e. commutative convergence) of series in Banach spaces. After having given different characterizations of this convergence (Proposition II.2) and showed the Orlicz–Pettis theorem (Theorem II.3) in Section II, we introduce in Section III the notion of unconditional basis, and show, in particular, that the sequences of centered independent random variables are basic and unconditional in the spaces Lp (P). In Section IV, we study in particular the canonical basis of c0 , and prove the theorems of Bessaga and Pełczy´nski which, on one hand, characterize the presence of c0 within a space by the existence of a scalarly summable sequence that is not summable, and, on the other hand, state that a dual space containing c0 must contain ∞ . In Section V, we describe the James structure theorems characterizing, among the spaces having an unconditional basis, those containing c0 , or 1 , or those that are reflexive. All of the above work was done before 1960 and is now very classical. In Section VI, we prove the Gowers dichotomy theorem, stating that every Banach space contains a subspace with an unconditional basis or a hereditarily indecomposable subspace (that is, none of its infinitedimensional closed subspaces can be decomposed as a direct sum of infinitedimensional closed subspaces). In addition, we provide a sketch of the proof of the homogeneous subspace theorem: every infinite-dimensional space that is isomorphic to all of its infinite-dimensional subspaces is isomorphic to 2 . In Chapter 4 (Volume 1), we study random variables with values in Banach spaces. Section II essentially states that the properties of convergence in probability, almost surely, and in distribution, seen in Chapter 1 in the scalar case can be generalized “as such” for the vector-valued case. Prokhorov’s theorem (Theorem II.9) characterizes the families of relatively compact probabilities on a Polish space. The conditional expectation, more delicate to define than in the scalar case, is introduced, as well as martingales; the vectorial version of Doob’s theorem (Theorem II.12) then easily follows from the scalar case. In Section III we describe the important symmetry principle, also known as the Paul Lévy maximal inequality, which allows us to obtain the equivalence theorem for series of independent Banach-valued random variables between convergence in distribution, almost sure and in probability. The contraction principle of Section IV will be of fundamental importance for all that follows; in its quantitative version, it essentially states that for a real (respectively complex) Banach space E, the sequences of independent centered

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random variables in Lp (E), 1  p < +∞, are unconditional basic sequences with constant 2 (respectively 4). In Section V, we generalize the scalar Khintchine inequalities to the vectorial case (Kahane inequalities); the proof is much more difficult than for the scalar case. These inequalities will turn out to be very important when we define the type and the cotype of Banach spaces (Chapter 5). The proof of the Kahane inequalities uses probabilistic arguments; in Subsection V.3, we will see how the use of the Walsh functions allowed Latała and Oleskiewicz, thanks to a hypercontractive property of certain operators (Proposition V.6), to obtain, in the case “L1 − L2 ,” the best constant for these inequalities (Theorem V.4). Chapter 5 (Volume 1) introduces the fundamental notions of type and cotype of Banach spaces. It is now common practice to define these using Rademacher variables, but it is often more interesting to use Gaussian variables, notably for their invariance under rotation. We thus begin, in Section II, by providing some complements of Probability; we first define Gaussian vectors, and show their invariance under rotation (Proposition II.8); we take advantage of this to present the vectorial version of the central limit theorem, which we will use in Chapter 4 of Volume 2. We next prove the existence of p-stable variables, also to be used in Chapter 4 of Volume 2, and present the classical theorems of Schönberg on the kernels of positive type, and of Bochner, which characterizes the Fourier transforms of measures. As notions of type and cotype are local, i.e. only involving the structure of finite dimensional subspaces, we give a few words in Section III to ultraproducts and to spaces finitely representable within another; we prove the local reflexivity theorem of Lindenstrauss and Rosenthal, stating, more or less, that the finite-dimensional subspaces of the bidual are almost isometric to subspaces of the space itself. In Section IV, we define the type and cotype, give a few examples (type and cotype of Lp spaces, cotype 2 of the dual of a C∗ -algebra), a few properties, and see how these notions behave with duality; this leads to the notion of K-convexity. We also show that in spaces having a non-trivial type, respectively cotype, we can, in the definition, replace the Rademacher variables by Gaussian variables (Theorem IV.8). In Section V, we prove Kwapie´n’s theorem, stating that a space is isomorphic to a Hilbert space if, and only if, it has at the same time type 2 and cotype 2; for this we first study the operators that factorize through a Hilbert space. In Section VI, we present a few applications, and in particular show how to obtain the classical theorems of Paley and Carleman (Theorem VI.2).

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In Chapter 6 (Volume 1), we will study a very important notion, that of a p-summing operator, brought out by Pietsch in 1967, and which soon afterward allowed Lindenstrauss and Pełczy´nski to highlight the importance of Grothendieck’s theorem, which, even though proven in the mid 1950s, had not until then been properly understood. We begin with an introduction showing that the 2-summing operators on a Hilbert space are the Hilbert–Schmidt operators. In Section II, after having given the definition and pointed out the ideal property possessed by the space of p-summing operators, we prove the Pietsch factorization theorem, stating that the p-summing operators T : X → Y are those that factorize by the canonical injection (or rather its restriction to a subspace) of a space C(K) in Lp (K, μ), where K is a compact (Hausdorff) space and μ a regular probability measure on K; in particular the 2-summing operators factorize through a Hilbert space. It easily follows that the psumming operators are weakly compact and are Dunford–Pettis operators. We next prove, thanks to Khintchine’s inequalities, a theorem of Pietsch and Pełczy´nski stating that the Hilbert–Schmidt operators on a Hilbert space are not only 2-summing, but even 1-summing. In Section III, we show Grothendieck’s inequality (Theorem III.3), stating that scalar matrix inequalities are preserved when we replace the scalars by elements of a Hilbert space, losing at most a constant factor KG , called the Grothendieck constant. We then prove Grothendieck’s theorem: every operator of a space L1 (μ) into a Hilbert space is 1-summing. The proof is “local,” meaning that it involves only the finite-dimensional subspaces; in passing we also show that the finite-dimensional subspaces of Lp spaces can be embedded, (1 + ε)-isomorphically, within spaces of sequences N p of finite dimension N. We then give the dual form of this theorem: every operator of a space L∞ (ν) into a space L1 (μ) is 2-summing. In Section IV, we present a number of results, originally proven in different ways, that can easily be obtained using the properties of p-summing operators (note that these do not depend on Grothendieck’s theorem, contrary to what might be suggested by the order of the presentation): the Dvoretzky–Rogers theorem (every infinite-dimensional space contains at least one sequence unconditionally convergent but not absolutely convergent), John’s theorem (the Banach–Mazur distance of every space of dimension n to the space n2 is √  n), and the Kadeˇc–Snobar theorem (in any Banach space, there exists, on √ every subspace of dimension n, a projection of norm  n). We then see that Grothendieck’s theorem allows us to show that every normalized unconditional basis of 1 or of c0 is equivalent to their canonical basis (this is also true for 2 , but this case is easy).

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Finally, Section V is devoted to Sidon sets (see Definition V.1). The fundamental example is that of Rademacher variables in the dual of the Cantor group  = {−1, +1}N ; another example is that of powers of 3 in Z. We prove a certain number of properties, functional, arithmetical and combinatorial, demonstrating the “smallness” of Sidon sets; we show in passing the classical inequality of Bernstein. Grothendieck’s theorem allows us to show that a set is Sidon if and only if the space C is isomorphic to 1 . We next present a theorem that is very important for the study of Sidon sets, Rider’s theorem (Theorem V.18), which involves, instead of the uniform norm of polynomials, another norm [[ . ]]R , obtained by taking the expectation of random polynomials constructed by multiplying the coefficients by independent Rademacher variables. This allows us to obtain Drury’s theorem (Theorem V.20), stating that the union of two Sidon sets is again a Sidon set, and the fact, due to Pisier, that is a Sidon set as soon as C is of cotype 2; for this last result, we need to replace, in the norm [[ . ]]R , the Rademacher variables by Gaussian variables, and are led to show a property of integrability of Gaussian vectors, due to Fernique (Theorem V.26), a Gaussian version of the Khintchine–Kahane inequalities, which will also be useful in Chapter 6 of Volume 2. In Chapter 7 (Volume 1), we present a few properties of the spaces Lp . In Section II, we study the space L1 . After having defined the notion of uniform integrability, we give a condition for a sequence of functions to be uniformly integrable (the Vitali–Hahn–Saks theorem), which allows us to deduce that the spaces L1 (m) are weakly sequentially complete. We then characterize the weakly compact subsets of L1 as being the weakly closed and uniformly integrable subsets (the Dunford–Pettis theorem). We conclude this section by showing that L1 is not a subspace of a space with an unconditional basis. We will continue the study of L1 in Chapter 4 of Volume 2; more specifically, we will examine the structure of its reflexive subspaces. In Section III, we will see that the trigonometric system forms a basis of Lp (0, 1) for p > 1. This is in fact an immediate consequence of the Marcel Riesz theorem, stating that the Riesz projection, or the Hilbert transform, is continuous on Lp for p > 1; most of Section III is hence devoted to the proof of this result. We have chosen not to prove it directly, but to reason by interpolation, allowing us to show in passing the Marcinkiewicz theorem, at the origin of real interpolation, as well as Kolmogorov’s theorem stating that the Riesz projection is of weak type (1, 1) (Theorem III.6). We conclude this section with a result of Orlicz (Corollary III.9) stating that the unconditional convergence of a series in Lp , for 1  p  2, implies the convergence of the sum of the squares of the norms, implying that the trigonometric system is unconditional only for L2 .

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In Section IV, we show, in contrast, that the Haar basis is unconditional in Lp (0, 1), for 1 < p < +∞. This unconditionality is linked to the facts that the Haar basis is a martingale difference and that martingale differences are unconditional in Lp , 1 < p < +∞ (Theorem IV.7). We also present some complements on martingales, notably on the behavior in Lp of the square function of a martingale (Theorem IV.6). The proof used here starts with the easy case, p = 2, and then passes successively, by doubling, to the cases p = 4, 8, 16, . . .; we finish by interpolation, using the Riesz–Thorin theorem, previously shown in Subsection IV.1. To conclude this section, we study a particular property of the Haar basis, in a way rendering it extremal; for this, we need Lyapounov’s theorem, stating that the image of vector measures with values in Rn is convex, and we prove this (Theorem IV.10). Finally, the aim of Section V is to present another proof of Grothendieck’s theorem, as a simple consequence of a theorem of Paley stating that +∞ 1  k 2 k=1 | f (2 )| < +∞ for every function f ∈ H (T). For this, we very succinctly develop the theory of the spaces H p , and prove the factorization theorem H 1 = H 2 H 2 (Theorem V.1) and the Frédéric and Marcel Riesz theorem. Grothendieck’s theorem then follows from the fact that the operator  k  f ∈ A(T) →  f (2 ) k1 ∈ 2 is 1-summing and surjective (Theorem V.6). In the Comments, we show that there is essentially only one space L1 (m), if we assume it separable and the measure m atomless. We also give an alternative proof of the F. and M. Riesz theorem, due to Godefroy, using the notions of nicely placed sets and Shapiro sets. Chapter 8 (Volume 1) is essentially devoted to Rosenthal’s 1 theorem, discovered in 1974. It provides a way to very easily detect when a Banach space contains 1 ; it is a very general dichotomy theorem: in any Banach space, from every bounded sequence, we can extract either a weakly Cauchy subsequence or a subsequence equivalent to the canonical basis of 1 . The majority of proofs currently given use a Ramsey-type theorem of infinite combinatorics, the Nash–Williams theorem; we proceed differently, by first showing, in Section II, by a method due to Debs in 1987, the Rosenthal–Bourgain–Fremlin– Talagrand theorem (Theorem II.3), which is also a dichotomy theorem for the extraction of subsequences, this time for the pointwise convergence of sequences of continuous functions on a Polish space. We then derive Rosenthal’s theorem for real Banach spaces. The complex case does not follow immediately; Dor was the first to show how to adapt the proof of the real case to show the complex case; we use here a method due to Pajor [1983] which uses combinatorial arguments to obtain the complex case from the real case. In Section III, we prove the Odell–Rosenthal theorem (Theorem III.2), stating that a separable Banach space X does not contain 1 if and only if every

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element of the unit ball BX ∗∗ of its bidual is the limit, for the weak∗ topology σ (X ∗∗ , X ∗ ), of a sequence of elements of the ball BX of X. We next show a result of Pełczy´nski (Theorem III.5), by a method due to Dilworth, Girardi and Hagler [2000], stating that a Banach space contains 1 if and only if its dual contains L1 (0, 1), or if and only if this dual contains the space of measures M([0, 1]) on [0, 1]. The Annex (Volume 1) serves especially to give a general framework to the elements of Harmonic Analysis that we use in this book, even though we essentially use those of the group T = R/Z and the Cantor group ∗  = {−1, +1}N (sometimes its finite version), as well as those of finite Abelian groups in Chapter 2 of Volume 2. In Section II, we present various notions on Banach algebras: invertible elements, maximal ideals, spectrum of an element, spectral radius; characters of a commutative algebra; involutive Banach algebras and their positive linear functionals (Theorem II.12); C∗ algebras. We show that every commutative C∗ -algebra is isometric to the algebra of continuous functions on a compact space (Theorem II.14). Section III concerns compact Abelian groups G, which we assume metrizable for simplicity. We begin by proving the existence, and uniqueness, of the Haar measure, thanks to the use of a strictly convex and lower semicontinuous function on the set of probabilities on G equipped with the weak∗ topology (this approach requires the metrizability). We then give some results on convolution. We next define the dual group  =  G as the set of characters of G and note that the metrizability of G implies that the dual is countable; we then determine the dual of the Cantor group (Proposition III.9), and show that  G separates the points of G (Theorem III.10; in fact shown in Theorem III.16), and hence that the set P(G) of trigonometric polynomials, i.e. finite linear combinations of characters, is dense in C(G) and in Lp (G) for 1  p < +∞; moreover  is an orthonormal basis of L2 (G). We next define the Fourier transform and show that it is injective. We conclude with results on approximate identities and on the Fejér and de la Vallée-Poussin kernels. We deduce that the norm of the convolution operator by a measure μ on L1 (G), and also on C(G), is equal to the norm of μ. The contents of Chapter 1 (Volume 2) are essentially of a local nature. We show a fundamental structure theorem concerning the finite-dimensional subspaces of Banach spaces, Dvoretzky’s theorem, which states that every n-dimensional space E contains, for any ε > 0, “large” subspaces (of dimension on the order of log n) which are (1 + ε)-isomorphic to Hilbert spaces. The proof is based on an argument of compactness, the Dvoretzky– Rogers lemma, and, in an essential manner, on a probabilistic argument linked to the concentration of measure phenomenon.

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We thus begin in Section II with some results from Probability; after reviewing the asymptotic behavior of Gaussian variables, we examine that of the associated maximal functions of independent Gaussian variables and their absolute value. We then prove the Maurey–Pisier deviation inequality (Theorem II.3), from which we can deduce their inequality of the concentration of measure (Theorem II.4). For this, we need Rademacher’s theorem (more or less classical, but rarely taught) on the almost everywhere differentiability of Lipschitz functions in RN . This inequality of concentration of measure allows us to prove Dvoretzky’s theorem in both real and complex spaces; nonetheless we also use another approach, due to Gordon, valid only for the real case, as it can easily be adapted to prove the isomorphic version of Milman and Schechtman (Subsection IV.5). In Section III, we prove a theorem concerning the comparison of Gaussian vectors, in a form due to Maurey (Theorem III.3). This allows us to easily obtain some important probabilistic results: Slepian’s lemma (Theorem III.5) and its variant, the Slepian–Sudakov lemma (Theorem III.4), to be used in the proof of Dvoretzky’s theorem for the real case, and Sudakov’s minoration (Theorem III.6); these three results will again serve, in an essential manner, in Chapters 3 and 6 (Volume 2). To prove Dvoretzky’s theorem, we need to be able to compare stochastically not only the max of Gaussian variables, but also their minimax; this is the purpose of Gordon’s theorem (Theorem III.7). The actual proof of Dvoretzky’s theorem is in Section IV. We in fact present two proofs; in both cases the principle is the same. First, we introduce the Gaussian dimension (Pisier calls it the concentration dimension) d(X) of a Gaussian vector X (Definition IV.9). Dvoretzky’s theorem is derived from what is known as the Gaussian version of Dvoretzky’s theorem (Theorem IV.10), stating that when a Banach space E contains a Gaussian vector X made up of m independent Gaussian variables, then E contains, for any ε > 0, a subspace (1 + ε)-Hilbertian of finite dimension controlled by the Gaussian dimension d(X) of X. The derivation from the Gaussian version is based on the Dvoretzky–Rogers lemma (Proposition IV.1), itself based on a compactness property in the spaces of operators between finite-dimensional spaces, given by Lewis’ lemma (Lemma IV.3). Next we prove Theorem IV.10. For this, we construct, out of independent copies of the Gaussian vector X, random operators on k2 with values in E, where k is an appropriate multiple, dependent on ε, of d(X). In the real case, the Slepian–Sudakov lemma allows us to limit from above the expectation of their norms, and Gordon’s theorem to limit them from below. In the second proof (for the complex case, but for the real case as well), the two estimations are obtained at the same time by the

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Maurey–Pisier concentration of measure inequality, by using the invariance of complex standard Gaussian vectors under the unitary group. In the rest of Section IV, we examine certain examples; we see for example that the theorem is optimal for E = n∞ . We also show that, with control of the cotype-2 constant of E, we can find, for any ε > 0, subspaces (1+ε)-Hilbertian of dimension proportional to that of E (Theorem IV.14). This will be useful in Chapter 6 (Volume 2). To conclude this section, we prove the isomorphic version (Theorem IV.15), due to Milman and Schechtman. This allows, in a real Banach space E of dimension n, to find, for any integer k  n, a subspace of dimension k, and whose distance to k2 is this time no longer arbitrarily close to 1, but is instead controlled by an explicit function of n and k. For this, we admit a delicate result, due to Bourgain and Szarek, that is an improvement of the Dvoretzky–Rogers lemma, and then apply Gordon’s theorem. Finally, we show in Section V the Lindenstrauss–Tzafriri theorem, for whose proof Dvoretzky’s theorem, associated with Kwapie´n’s theorem (Chapter 5 of Volume 1, Section V), is essential; it states that if in a Banach space all the closed subspaces are complemented, then this space is isomorphic to a Hilbert space. Chapter 2 (Volume 2), quite short, is dedicated to the construction by Davie of a separable Banach space without the approximation property. The problem of the existence of such a space was posed by Grothendieck in the mid 1950s; it generalized the old problem of the existence of a basis in every Banach space, which dates back to Banach himself, and was resolved in 1972 by Enflo. The construction given soon afterward by Davie is simpler than that of Enflo. It combines a probabilistic argument (method of selectors) with an argument from Harmonic Analysis concerning finite groups. It fits particularly well with the objectives of this book. In Section II, we give a certain number of equivalent formulations of the approximation property, and Section III contains the actual construction. We show that, for any p > 2, p contains a closed subspace without the approximation property. This is also the case for c0 and for p with 1  p < 2 (Szankowski), but the proof is more delicate; it can be found, for example, in Lindenstrauss–Tzafriri, Volume II, Theorem 1.g.4. In Chapter 3 (Volume 2), we study in more detail Gaussian vectors, as well as the more general notion of Gaussian processes. These are defined at the beginning of Section II. To each Gaussian process X = (Xt )t∈T we associate a (semi)-metric on T by setting dX (s, t) = Xs − Xt 2 ; we show, with the aid of Slepian’s lemma, that the condition dY  dX is sufficient to ensure that if X possesses a bounded version (respectively a continuous version), then so does Y (the Marcus–Shepp theorem).

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In Section III, we define Brownian motion as an example of a Gaussian process. Sections IV and V form the heart of this chapter. In Section IV, we define the entropy integral associated with a Gaussian  process: this is the integral, for ε ∈ [0, +∞[, of log N(ε), where N(ε) is the entropy associated with the metric dX of the process X, i.e. the minimum number of open dX -balls necessary to cover T. The Dudley majoration theorem gives an upper bound for the expectation of the supremum of the modulus (absolute value) of a process with the aid of this entropy integral; a process has continuous paths as soon as the entropy integral is finite (Theorem IV.3). We then give an example showing that this condition is not necessary. Next, in Section V, we see that, when the process is indexed by a compact metrizable Abelian group G and is stationary, i.e. its distribution does not change under translation, then the finiteness of the entropy integral J(d) becomes necessary to have continuous trajectories, and J(d) is, up   to a constant, equivalent to E supt∈G |Xt | : this is the Fernique minoration theorem (Theorem V.4). We conclude the section by giving an equivalent form of the entropy integral (Proposition V.5) that will be useful in Chapter 6 (Volume 2). Section VI returns to Banach spaces; we present the Elton–Pajor theorem (Theorem VI.12), which gives Elton’s theorem: in a real Banach space, if there are N vectors x1 , . . . , xN with norm  1 such that the average of  ± x1 + · · · ± xN  over all choices of signs is  δ N, then there is a subset of these vectors, of cardinality N  c(δ) N, which is equivalent to the canonical basis of N 1 , with constant β(δ) depending only on δ (Corollary VI.18). The proof uses probabilistic arguments: introduction of a Gaussian process and Dudley’s majoration theorem, combinatorial arguments, notably Sauer’s lemma (Proposition VI.3) and Chernov’s inequality (Proposition VI.4), and volume arguments: Urysohn’s inequality (Corollary VI.8), deduced from the Brunn–Minkowski inequality (Theorem VI.6), itself deduced from the Prékopa–Leindler inequality (Lemma VI.7). The complex version of Pajor requires several additional combinatorial lemmas (whose infinite-dimensional versions were used in Chapter 8 of Volume 1); it shows in particular that if a complex Banach space contains δ-isomorphically, as a real Banach space, the cN space N 1 , then it contains, in the complex sense, the complex space 1 , where c depends only on δ (Corollary VI.21). In Chapter 4 (Volume 2), we concentrate on the reflexive subspaces of L1 . In Section II, we first see that the reflexive subspaces of L1 are those for which the topology of the norm coincides with that of the convergence in measure (the Kadeˇc–Pełczy´nski theorem) and that, in consequence, any

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non-reflexive subspace contains a complemented subspace isomorphic to 1 (Corollary II.6). We then examine their local structure. Even though a priori, as L1 is weakly sequentially complete (Chapter 7 of Volume 1, Theorem II.6), its reflexive subspaces are those that do not contain 1 , by the Rosenthal 1 theorem (Chapter 8 of Volume 1), in fact we have much more: the reflexive subspaces of L1 are those not containing n1 ’s uniformly (Theorem II.7). We then show that the Banach spaces that do not contain n1 ’s uniformly are exactly those with a type p > 1 (Theorem II.8, of Pisier), so that the reflexive subspaces of L1 have a non-trivial type p > 1 (Corollary II.9). In Section III, we present some examples of reflexive subspaces. We first see that, for 1 < p  2, the sequences of independent p-stable variables generate isometrically p in the real L1 space (Theorem III.1). We then succinctly study the (q)-sets, which are the reflexive and translation-invariant subspaces of L1 (T). In particular we prove the Rudin transfer theorem, stating that the properties of Rademacher functions in the dual of the Cantor group are transferred to all the Sidon sets (Theorem III.10), so that, thanks to the Khintchine inequalities, every Sidon set is a (q)-set for any q < √ +∞, and, more precisely, f q  C S( ) q f 2 for every trigonometric polynomial f with spectrum in , where C is a numerical constant and S( ) is the Sidon constant of (Theorem III.11). The converse, due to Pisier, is shown in two different ways, first, in Chapter 5 (Volume 2), with a method of random extraction due to Bourgain, and then, in Chapter 6 (Volume 2), with the aid of Gaussian processes, which was the original proof of Pisier. Section IV is devoted to the deep theorem of Rosenthal showing that the reflexive subspaces of L1 embed in Lp , for some p > 1 (Theorem IV.1). We use in the proof the Maurey factorization theorem (Theorem IV.2), that Maurey isolated from the original proof of Rosenthal. We thus deduce that every (1)set is in fact (q) for some q > 1 (Corollary IV.3). In Section V, we study the finite-dimensional subspaces of L1 , and, more precisely, the dimension n of spaces n1 that they can contain (Theorem V.2, of Talagrand). To make this statement more precise, we first need to study the K-convexity constant of finite-dimensional spaces (Theorem V.3), and in particular of those of L1 (Theorem V.5). We also see that, up to a constant, nothing changes in the definition if the Rademacher variables are replaced by Gaussian variables (Theorem V.8). We need to prove an auxiliary result, due to Lewis (Theorem V.9). The proof of Talagrand’s theorem is then based on the method of selectors, as well as Pajor’s theorem from the preceding chapter to reduce to the real case.

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Chapter 5 (Volume 2) contains three results of Bourgain illustrating the method of selectors. This method was already used, in Chapters 2 and Chapter 4 of Volume 2; it involves selecting an independent sequence of Bernoulli variables ε1 , . . . , εn , taking on the values 0 and 1 with a certain probability, and then making constructions by randomly choosing the set (ω) of integers k  n for which εk (ω) takes the value 1. Section II treats the extraction of quasi-independent sets; these are particular Sidon sets, defined in an arithmetical manner, and whose Sidon constant is bounded by a fixed constant ( 8). We prove a theorem of Pisier stating that a set is Sidon if, and only if, there exists a constant δ such that every finite subset A of , not reduced to {0}, contains a quasi-independent subset B of cardinality |B|  δ |A| (Theorem II.3). In fact, we show that from every finite subset A, not reduced to {0}, we can extract a quasi-independent subset B of cardinality |B|  K (|A|/ψA )2 , where K is a numerical constant and ψA depends only on A (Theorem II.6). As an immediate consequence we have Drury’s theorem (Corollary II.4), and we easily obtain Pisier’s theorem (Theorem II.13), the converse of Rudin’s theorem seen in Chapter 4 of Volume 2, as well as Rider’s theorem (Theorem II.14). In Section III, we show that, for any N  1, there exists a subset ⊆ N∗ of  2/3 , where C is a numerical   cardinality N such that 0 k∈ sin kx ∞  C0 N constant (Theorem III.1). The interest in this result is linked to the vectorvalued Hilbert transform: if E is a Banach space of finite dimension N, John’s theorem immediately √  implies  that the Hilbert transform with values in E has a norm  N in L L2 (E) ; if E = N 1 , this norm is dominated by log N; the preceding result shows that for every N  1, we can find a Banach space E of dimension N so that this norm dominates N 1/3 . In Section IV, we show that the majoration K(X)  C log n for the K-convexity constant of spaces of dimension n seen in Chapter 4 (Volume 2) can essentially not be improved (Theorem IV.1). Chapter 6 (Volume 2) is for the most part devoted to Pisier’s space C as . In Section II, we prove two results that will be needed in the next section.  The first is the Itô–Nisio theorem, stating that, when n1 Xn is a series of independent symmetric random variables with values in C(K), where K is a metrizable compact space, such that, for every t ∈ K, the series +∞ n=1 Xn (·, t) converges almost surely to Xt , and in addition we assume that the process (Xt )t∈K has a continuous version, then the series is almost surely uniformly convergent (Theorem II.2). We then show a Tauberian theorem  (the Marcinkiewicz–Zygmund–Kahane theorem): if n1 Xn is a series of independent symmetric random variables with values in a Banach space E, then

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the fact that it is almost surely bounded (respectively almost surely convergent) according to a summation procedure implies that this holds in the usual sense (Theorem II.4). In Section III, the space C as is defined: let G be a compact metrizable Abelian group and  = {γn ; n  1} its dual group; let (Zn )n1 be a standard sequence of independent complex Gaussian variables; then C as (G) is the space of all the functions f ∈ L2 (G) for which, almost surely in ω, the sum of the  f (γn )γn is a continuous function f ω ∈ C(G). Theorem III.1 series n1 Zn (ω) gives several equivalent formulations (one of these being  Billard’s theorem).    Equipped with the norm defined by [[f ]] = supN1 E N n=1 Zn f (γn )γn ∞ , which is  f 2 , C as (G) is a Banach space for which the characters γn ∈  form a 1-unconditional basis (Theorem III.4). The Marcus–Pisier theorem (Theorem III.5) allows the Gaussian variables Zn in the definition to be replaced by Rademacher variables; the proof uses the Dudley majoration theorem and the Fernique minoration theorem. The fundamental result concerning C as is Theorem III.9. It establishes a duality between C as and the space of multipliers M2,2 from L2 (G) to L2 (G), where 2 is the Orlicz function 2 2 (x) = ex − 1, and shows that with this duality M2,2 can be identified, isomorphically, with the dual of C as . The first part of the theorem again uses the Fernique minoration theorem; the second part is more delicate, and in addition to the Marcus–Pisier theorem, requires several auxiliary results. Thanks to this duality, we easily establish a result of Salem and Zygmund that gives upper and lower bounds of the norm [[ . ]] of a sum of exponentials (Proposition III.13). In Section IV we present two more applications of C as . First we prove a theorem due to Pisier, a converse to Rudin’s theorem (Chapter 4 of Volume 2), √ that characterizes Sidon sets as those for which f q  C q f 2 for every trigonometric polynomial f with spectrum in (Theorem IV.1); note that this uses only the existence of a duality between C as and M2,2 , and not the fact that M2,2 is the dual of C as , and the Gaussian Rider theorem seen in Chapter 6 of Volume 1. Next, this space provides a response to the Katznelson dichotomy problem. Katznelson showed that only the real-analytic functions operate on the Wiener algebra A(T), while it is clear that all continuous functions operate on C(T); the problem was to know if, for every Banach algebra B possessing certain “nice” properties, and such that A(T) ⊆ B ⊆ C(T), either all continuous functions operate on B or only the analytic functions operate on B. Zafran found a counterexample to this conjecture; Theorem IV.2 (Pisier) reinforces the result of Zafran: P = C as (T) ∩ C(T), equipped with the norm f P = 8 f ∞ + [[f ]], is a Banach algebra possessing the required qualities, but in which all the Lipschitz functions operate.

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To conclude Chapter 5, we prove the Bourgain–Milman theorem (Theorem V.1): is a Sidon set as soon as C has a finite cotype (we have already seen in Chapter 6, Volume 1, that this is the case if the cotype is 2). The proof uses the notions of Banach diameter n(E) of a finite-dimensional Banach space E (Definition V.2) and arithmetic diameter for the finite subsets of the dual of a compact metrizable Abelian group G, where the latter is the entropy number NA (1/2) for a pseudo-metric dA on G, associated with the finite subset A of the dual of G (Definition V.3). Using Dvoretzky’s theorem for cotype-2 spaces (in fact for 1 ), the proof combines Theorem V.4 (of Maurey), which gives a lower bound for n(E) as a fonction of the cotype constant of E, and Theorem V.5 (of Pisier): if is a finite subset in the dual of G and if NA (δ)  eδ |A| for every A ⊆ , then the Sidon constant of is bounded above by a δ −b , where a, b > 0 are numerical constants. In the Comments, Section VI, as an application of random Fourier series, we prove two more results: one concerning functions of the Nevanlinna class (Theorem VI.1), and the other about random Dirichlet series (Theorem VI.2). For the reader who would like to dig a bit deeper, we refer to the works cited in the bibliography, and in particular to the recent Handbook of the Geometry of Banach Spaces, Vols. I and II. We kindly thank everyone that has assisted us in the preparation of this text; in particular Gabriel Li, who created the figures, and B. Calado, D. Choimet, M. Déchamps-Gondim, G. Godefroy, P. Lefèvre, F. Lust-Piquard, G. Pisier and Martine Queffélec, who proofread all or parts of the manuscript, for their comments and the improvements they have helped us to bring. We are especially thankful to G. Godefroy, J.-P. Kahane, B. Maurey and G. Pisier, from whom we have learned a great part of what is in this book. We also warmly thank the referees for their very precise and pertinent remarks, which we found extremely useful.

Acknowledgements for the English Edition We have not updated the French edition in this translation, but we have taken the opportunity to correct some mistakes and add some missing arguments. We warmly thank G. Godefroy, O. Guédon, G. Pisier and L. RodríguezPiazza, who were kind enough to write, especially for this English version, three surveys and an original paper (see Appendices A, B, C and D in Volume 2).

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Danièle and Greg Gibbons did a beautiful job with the translation of a very long, and at times highly specialized, mathematical text. Let them be warmly thanked for this achievement.

Conventions (1) In this book, the set N of natural numbers is N = {0, 1, 2, . . .}, and N∗ = {1, 2, . . .}. (2) Compact spaces are always assumed to be Hausdorff.

Preliminary Chapter Weak and Weak∗ Topologies. Filters, Ultrafilters. Ordinals

I Introduction The purpose of this chapter is to give a summary of results on the weak and weak∗ topologies (essentially to prove the Eberlein–Šmulian theorem), and on filters and ordinals. First recall that a Banach space is a real or complex vector space, equipped with a norm for which it is complete. The terminology Banach space appeared for the first time in 1928 in Fréchet (page 141). We assume that the reader is already familiar with the elementary properties of Banach spaces, as can be found, for example, in Rudin 2, Chapter 5; Rudin 3, Part I; or in the first chapters of more advanced texts, such as Fabian–Habala–Hájek–Montesinos; Santalucía–Pelant–Zizler, Chapters 1–2; Habala–Hájek–Zizler I, Chapters 1–4; or Megginson, Chapter 1.

II Weak and Weak∗ Topologies II.1 The Eberlein–Šmulian Theorem If X is a Banach space, its weak topology σ (X, X ∗ ), more simply denoted w, is the coarsest topology for which all the continuous (for the norm) linear functionals ϕ ∈ X ∗ remain continuous. By the Hahn–Banach theorem, the convex sets closed for the norm are also weakly closed. On the dual X ∗ , besides the topology of the norm  . X ∗ and the weak topology σ (X ∗ , X ∗∗ ), we can define the weak∗ topology, denoted w∗ ,

1

2

Weak and Weak∗ Topologies. Filters, Ultrafilters. Ordinals

or alternatively σ (X ∗ , X), which is the coarsest topology for which all linear maps ϕx :

X∗ x∗

−→ K −→ ϕx (x∗ ) = x∗ (x)

for x ∈ X are continuous. These topologies are locally convex, but they are never metrizable when X is infinite-dimensional. We nonetheless have the following result: Theorem II.1 (The Alaoglu–Bourbaki Theorem) ∗ 1) The unit ball BX ∗ of X ∗ equipped  with the weak topology is compact. ∗ 2) If X is separable, then BX ∗ , w is metrizable. The first part of this theorem is due to Banach when X is separable, and in the general case, independently, to Alaoglu [1940] and Bourbaki [1938]. Proof   (sketch) 1) BX ∗ , w∗ can be identified with a closed subspace of the product space  x∈X D(0, x), where D(0, r) = {u ∈ K ; |u|  r}, which is compact by Tychonov’s theorem. 2) If (xk )k1 is a dense sequence in BX , the distance defined by: d(x∗ , y∗ ) =

+∞ ∗ |x (xk ) − y∗ (xk )| k=1

2k

defines the topology σ (X ∗ , X) on BX ∗ . This theorem is closely linked to the following result, due to Goldstine [1938], which follows from the Hahn–Banach theorem applied to the weak∗ topology: Theorem II.2 (Goldstine’s Theorem) The unit ball BX of X is dense in the unit ball BX ∗∗ of X ∗∗ for the topology σ (X ∗∗ , X ∗ ). We now proceed to the main result of these preliminaries. Theorem II.3 (The Eberlein–Šmulian Theorem) A subset K of a Banach space X is relatively compact for the weak topology if and only if, from each sequence of elements of K, we can extract a weakly convergent subsequence. It should be noted that this holds even when K is not metrizable for the weak topology (this is so when X ∗ is separable, in which case the result is trivial). It is indeed a very useful and practical result as it allows us to only use sequences, instead of filters. The necessary condition is due to Šmulian [1940], and the sufficient condition to Eberlein [1947].

II Weak and Weak∗ Topologies

3

Proof 1) Necessary condition. By replacing K with its weak closure, we may assume that K is weakly compact. We first show: Lemma II.4 If X is a separable Banach space, there exists a norm on X defining on its unit ball a topology coarser than the weak topology. Proof If (xn )n1 is a sequence dense in BX , we can select linear functionals xn∗ ∈ X ∗ of norm 1 such that xn∗ (xn ) = xn . We set: ρ(x) =

+∞ 1 ∗ |x (x)| . 2n n n=1

ρ is indeed a norm because ρ(x) = 0 implies xn∗ (x) = 0 for any n  1, and thus x = 0: in fact, if ε > 0, there exists k  1 such that x − xk   ε, so that xk  = xk∗ (xk ) = xk∗ (xk − x)  xk − x  ε and hence x  2ε. The topology defined by this norm is coarser than the weak topology because, for any r > 0, the conditions x  1 and: |xk∗ (x)|  r/2

for 1  k  n,

imply ρ(x)  r as soon as 1/2n  r/2. Now let (xn )n1 be a sequence of elements of K. Consider the subspace X0 = span{xn ; n  1} generated by this sequence. It is obviously separable, w and it contains K0 = {xn ; n  1} . The norm ρ on X0 defined in the lemma induces the weak topology on K0 as K0 is weakly compact (it is weakly closed and contained within K; note that the weak topologies σ (X, X ∗ ) and σ (X0 , X0∗ ) coincide on X0 , thanks to the Hahn–Banach theorem). As the weak topology on K0 is metrizable, we can indeed extract from (xn )n1 a convergent subsequence. 2) Sufficient condition. We will use: Observation A bounded subset K of a Banach space X is relatively compact w∗ for the weak topology if and only if its weak∗ closure K in X ∗∗ is contained in X. In this statement, we have, as usual, identified X with a subspace of its bidual X ∗∗ .

4

Weak and Weak∗ Topologies. Filters, Ultrafilters. Ordinals

This observation is very useful, albeit evident: it is sufficient to recall that any weak∗ closed bounded subset in X ∗∗ is weak∗ compact, and to observe that the trace on X of the weak∗ topology σ (X ∗∗ , X ∗ ) of X ∗∗ is the weak topology σ (X, X ∗ ) of X. a) Note that if K satisfies the conditions of the theorem (i.e. that from any sequence of elements of K, a weakly convergent subsequence can be extracted), then K is a bounded subset of X; indeed, for any linear functional ϕ ∈ X ∗ , ϕ(K) is a bounded subset of K, since it is relatively compact, because any sequence of elements of ϕ(K) contains a convergent subsequence. b) We will now prove: Lemma II.5 If from any sequence of elements of K, some weakly convergent w∗ subsequence can be extracted, then every  ∈ K is in fact in X, and there exists a sequence of elements of K that converges weakly to . Corollary II.6 1) If from any sequence of elements of K we can extract a subsequence, weakly convergent in K, then K is weakly compact. w 2) If K ⊆ X is relatively compact for the weak topology, then every x ∈ K is the weak limit of a sequence of elements of K. We will use: Fact Let Y be a Banach space. For any finite-dimensional subspace F of Y ∗ , there exist a finite number of vectors y1 , . . . , yn ∈ Y, of norm 1, “norming” F in the following sense: for every y∗ ∈ F, we have: max |y∗ (yk )| 

1kn

1 ∗ y  . 2

Indeed, we can select a (1/4)-net {y∗1 , . . . , y∗n } in the unit sphere of F, which is compact, and then y1 , . . . , yn ∈ X of norm 1 such that |y∗k (yk )|  3/4. Then, if y∗ ∈ F and y∗  = 1, we can find some k  n such that y∗ − y∗k   1/4, and we have: |y∗ (yk )|  |y∗k (yk )| − y∗ − y∗k  

1 3 1 − = · 4 4 2

Proof of the lemma We will use the preceding “Fact” for Y = X ∗ . We start w∗ with an arbitrary element x1∗ in the unit sphere of X ∗ . As  ∈ K , there exists x1 ∈ K such that | x1∗ ,  − x1 |  1. According to the “Fact,” there exist x2∗ , . . . , xn∗2 ∈ X ∗ of norm 1 such that, for every  ∈ span{,  − x1 }, we have:

II Weak and Weak∗ Topologies

max |(xk∗ )| 

2kn2

5

1  . 2

But now, there exists x2 ∈ K such that, for 1  k  n2 , we have: | xk∗ ,  − x2 | 

1 · 2

Once again we use the “Fact” to obtain xn∗2 +1 , . . . , xn∗3 ∈ X ∗ of norm 1 such that: 1 max |(xk∗ )|   2 n2 +1kn3 for every  ∈ span{,  − x1 ,  − x2 }. Keeping the same method, we thus obtain a sequence (xk )k1 of elements of K, and xk∗ ∈ X ∗ of norm 1, satisfying: | xk∗ ,  − xp | 

1 p

for 1  k  np , and: max

np +1knp+1

|(xk∗ )| 

1  2

for every  ∈ span{,  − x1 , . . . ,  − xp }. By hypothesis, this sequence contains a weakly convergent subsequence. We may assume that the sequence (xn )n1 itself converges weakly, with weak limit x: x ∈ span{x1 , x2 , . . .}, and hence:  − x ∈ Z = span{ − x1 ,  − x2 , . . .} . However, by construction: sup |(xk∗ )| 

k1

1  2

for any  ∈ Z (the inequality is preserved when we pass to the closure in norm). This is true in particular for  =  − x. Hence it suffices to prove that  − x, xk∗ = 0 for any k  1 to obtain  = x, and thus the result. However: |  − x, xk∗ |  |  − xp , xk∗ | + | xk∗ , xp − x | 

1 + | xk∗ , xp − x | p

for k  p (because p  np implies k  np ). It remains to let p tend to infinity to obtain  − x, xk∗ = 0. This construction is known as Whitley’s construction (Whitley [1967]).

6

Weak and Weak∗ Topologies. Filters, Ultrafilters. Ordinals

II.2 The Krein–Milman Theorem We will state this in a more general context than that of the weak and weak∗ topologies. Note that it only concerns the real structure of the space, which we can thus assume real from the start. Theorem II.7 (The Krein–Milman Theorem) In any locally convex Hausdorff space E, any compact convex set K is the closed convex hull of the set of its extreme points. Proof If K is empty, there is nothing to prove. If not, we first show that K possesses extreme points. We will call an extreme face of K any nonempty closed convex subset F of K such that tx + (1 − t)y ∈ F, with x, y ∈ K and 0 < t < 1, implies x, y ∈ F. The set of extreme faces of K is inductive for “downwards” inclusion (the intersection of any totally ordered family of extreme faces is non-empty, since K is compact, and hence is again an extreme face); Zorn’s lemma ensures the existence of a minimal extreme face F0 . Suppose F0 contains at least two distinct points x and y: then we could find a continuous linear functional ϕ on E such that ϕ(x) = ϕ(y). With m = maxz∈F0 ϕ(z), it ensues that the convex set C = {z ∈ F0 ; ϕ(z) = m}, nonempty because F0 is compact, is an extreme face of K. First, with z1 , z2 ∈ K and z = tz1 + (1 − t)z2 ∈ C, with 0 < t < 1, we have z1 , z2 ∈ F0 , because F0 is an extreme face. Then m = ϕ(z) = tϕ(z1 ) + (1 − t)ϕ(z2 )  tm + (1 − t)m = m, so that ϕ(z1 ) = ϕ(z2 ) = m, and z1 , z2 ∈ C. But C is strictly contained in F0 , as it cannot contain x and y simultaneously: this contradicts the minimality of F0 . Hence F0 contains only a single point, which is of course an extreme point. We now denote by K0 the closed convex hull of the set of extremal points of K. If K0 were not equal to K, by the Hahn–Banach theorem there would exist a continuous linear functional ψ such that: max{ψ(y) ; y ∈ K0 } < max{ψ(x) ; x ∈ K} = M . However, as above, the set C = {x ∈ K ; ψ(x) = M} is an extreme face of K; as a non-empty compact convex set, it possesses extreme points, according to the preceding argument; moreover, as an extreme face, its extreme points are also extreme points of K. This contradicts the fact that all the extreme points of K are contained in K0 . Consequently, K = K0 , as was announced. This theorem is due to Krein and Milman [1940].

III Filters, Ultrafilters. Ordinals

7

III Filters, Ultrafilters. Ordinals III.1 Filters and Ultrafilters Let us recall some elementary properties of filters and ultrafilters. Not only will ultrafilters be useful to show convergence, but they will also be used in a different context in Chapter 8. Definition III.1 Let E be a non-empty set; a family of subsets F ⊆ P(E) is said to be a filter on E if: 1) it is not empty, and ∅ ∈ / F; 2) if A and B are in F, then A ∩ B ∈ F; 3) if A ∈ F and B ⊇ A, then B ∈ F. A filter F2 is said to be finer than a filter F1 if F2 ⊇ F1 . Another way to put it: F1 is coarser than F2 . A filter U is an ultrafilter if it is maximal, in the sense of inclusion, i.e. there is no strictly finer filter than U . Note that the whole set E is thence always contained in a filter. It is often practical to only work with some of the elements of a filter; if F is a filter, B ⊆ F is said to be a filter base of F if, for any A ∈ F, there exists B ∈ B such that B ⊆ A. For a non-empty family of subsets B ⊆ P(E) to be a filter base, it is necessary and sufficient that ∅ ∈ / B and that, for all A, B ∈ B, there exists C ∈ B such that C ⊆ A ∩ B; B is then a filter base of the filter FB made up of all subsets that contain at least one element B ∈ B; this is the smallest filter (the coarsest) that contains B. Examples a) On N, the set of subsets whose complement is finite is a filter, known as the Fréchet filter. b) More generally, let I be a set equipped with a filtering preorder: for any i, j ∈ I, there exists k ∈ I such that k  i and k  j; then, if we denote Ij = {i ∈ I ; i  j}, the set B = {Ij ; j ∈ I} is the base of a filter FI on I, called the section filter of I. Raymond Queneau (in his book, Bords: Mathématiciens, Précurseurs, Encyclopédistes, Hermann (1978), page 25) says that, in the first editions of Bourbaki, it was possible to have sets “flirtant (flirting!)” on the right or on the left; perhaps “philtres (love potions)” were used in those days . . . ! 1 1 However, we did not find this in the edition published in 1933; perhaps this appeared in

preliminary versions?!

8

Weak and Weak∗ Topologies. Filters, Ultrafilters. Ordinals

c) If E is a topological space, then, for every x ∈ E, the set V(x) of neighborhoods of x is a filter. d) For any set E, and any fixed a ∈ E, the set of subsets of E containing a is an ultrafilter Ua ; these ultrafilters are said to be trivial ultrafilters. With the exception of trivial ultrafilters, the existence of ultrafilters is assured only by Zorn’s lemma. Proposition III.2 than F.

For any filter F on a set E, there exists an ultrafilter finer

Proof It is clear that the union of any totally ordered family of filters is again a filter; consequently, the order given by fineness on the set of filters finer than F is inductive, and thus this set possesses maximal elements. An important property of ultrafilters is the following: Proposition III.3 A filter F on the set E is an ultrafilter if and only if, for all A ⊆ E, A ∈ F or Ac = E  A ∈ F. Proof First of all, assume that U is an ultrafilter. Then let A ⊆ E be such that / U ; we will show that A ∈ U . For this, we consider B = {B ∩ A ; B ∈ U }; Ac ∈ as ∅ ∈ / B (since ∅ ∈ B would mean that U contains a subset B ⊆ Ac , and hence that Ac ∈ U ), we can deduce that B is a filter base. The filter FB generated by B contains all the A ∩ B for B ∈ U , hence all the B ∈ U as B ⊇ B ∩ A. Thus FB is finer than U and so, by maximality, is equal to it. As, clearly, A ∈ FB , we have A ∈ U . Conversely, let F be a filter having the stated property. If it were not an ultrafilter, we could find a strictly finer filter F , and hence some A ∈ F  F. As A ∈ / F, the hypothesis would imply that Ac ∈ F, and hence that Ac ∈ F ; but then we would have ∅ = A ∩ Ac ∈ F , which is impossible. Thus F is indeed an ultrafilter. Corollary III.4 If U is an ultrafilter on E and if A1 ∪ . . . ∪ An ∈ U , then one of the sets Ak belongs to U . Proof Otherwise, we would have, by the preceding proposition, Ack ∈ U for any k  n; then ∅ = Ac1 ∩ . . . ∩ Acn ∩ (A1 ∪ . . . ∪ An ) ∈ U , which is not possible.

III.2 Limit along a Filter Definition III.5 Let X be a topological space; a filter F on X is said to converge to x0 ∈ X if it is finer than the neighborhood filter of x0 .

III Filters, Ultrafilters. Ordinals

9

This is the same as saying: “any neighborhood V of x0 is an element of the filter F.” If B is a base of the filter F, it converges to x0 if and only if any neighborhood V of x0 contains an element B of B. If X is Hausdorff, a filter can converge only to a single point x0 : then x0 is said to be the limit of the filter F. Clearly, for any subset A ⊆ X, x0 ∈ A if and only if there exists a filter F converging to x0 and such that A ∩ F = ∅ for every F ∈ F. As then FA = {A ∩ F ; F ∈ F} is a filter of elements of A, we see that x0 ∈ A if and only if there exists a filter of subsets of A converging to x0 . The notion of the cluster point of a sequence can be generalized in the following manner: a point x0 is said to be a cluster point of the filter F if x0 ∈ F for all F ∈ F. In other words, any neighborhood V of x0 meets all elements F of filter F: V ∩ F = ∅. This implies that x0 is a cluster point of the filter F if and only if there exists a filter F finer than F that converges to x0 . Thus any cluster point of an ultrafilter is a limit of this ultrafilter. Theorem III.6 If X is a Hausdorff topological space, X is compact if and only if all ultrafilters on X converge. Proof Suppose that X is compact, and let U be an ultrafilter on X; for all U1 , . . . , Un ∈ U , U 1 , . . . , U n ∈ U , hence U 1 ∩ . . . ∩ U n = ∅; compactness then

implies U∈U U = ∅, so that U possesses a cluster point, which is the limit of U (and the above intersection is reduced to this unique point). Conversely, let (Fα )α∈A be an arbitrary family of closed subsets of X such that Fα1 ∩ . . . ∩ Fαn = ∅ for every α1 , . . . , αn ; the set of finite intersections of the Fα is then a filter base on X. If U is an ultrafilter finer than this filter, by

hypothesis it has a cluster point. Thus, in particular, α∈A Fα = ∅, so that X is compact. As an application of this theorem, we have: Theorem III.7 (Tychonov’s Theorem) again compact.

Any product of compact spaces is

Proof First, the product is a Hausdorff space. Next, if U is an ultrafilter on this product, the images of U by the canonical projections are bases of ultrafilters on each of the factors. They are convergent because these factors are compact. Consequently U is convergent, and the product is compact. Definition III.8 Let X and Y be two topological spaces and f : X → Y; we say that f has a limit y0 ∈ Y along the filter F if the filter with base f (F) = { f (A) ; A ∈ F} converges to y0 .

10

Weak and Weak∗ Topologies. Filters, Ultrafilters. Ordinals

III.3 Filtering Families Instead of filters, filtering families, which are closer to the usual notion of sequences, are easier to “visualize.” If I is a filtering ordered set, and (xi )i∈I a family of elements of the topological space X indexed by I, we say that the filtering family (xi )i∈I converges to x0 ∈ X if, for any neighborhood V of x0 , there exists iV ∈ I such that xi ∈ V for all i  iV . With such a filtering family, we can associate the set: FI = {Y ⊆ X ; ∃ iY ∈ I, xi ∈ Y, ∀ i  iY } ; this is a filter, since, if Y, Z ∈ FI , there exists i0 ∈ I  iY and  iZ ; then xi ∈ Y ∩ Z for i  i0 , and hence Y ∩ Z ∈ FI (the other conditions being clearly satisfied), and it is finer than the neighborhood filter of x0 ; it thus converges to x0 . In fact, the image of the filter of sections of I by the mapping i → xi is a filter base FI . Conversely, if the filter F, or more generally a filter base B, converges to x0 , then for each F ∈ B, we can choose an element xF ∈ F. The set I = B is ordered by descending inclusion, and this order is filtering, because B is a filter base; moreover, it is clear that the filtering family (xF )F∈B converges to x0 . This point of view is attractive, but we must keep in mind a slight difference between the notion of filtering sub-family seen here and with sequences: the filtering family (yj )j∈J is said to be a filtering sub-family of (xi )i∈I if there exists a mapping ϕ : J → I with the following two properties: yj = xϕ( j) for any j ∈ J and for any i ∈ I, there exists a ji ∈ J such that j  ji implies ϕ( j)  i. The filter FJ as constructed above is then finer than the filter FI ; indeed, if Y ∈ FI , there exists iY ∈ I such that xi ∈ Y for i  iY ; moreover, there exists a jY ∈ J such that ϕ( j)  iY for j  jY ; then yj = xϕ( j) ∈ Y; consequently Y ∈ FJ . When B is the base of a filter F coarser than the filter F with base B , and when we have chosen yB ∈ B for all B ∈ B , it is a bit more subtle to state that (yB )B∈B is a filtering sub-family of (yB )B∈B : we take I = B and J = {(xB , B ) ; xB ∈ B , B ∈ B}, which we order by (xB1 , B 1 )  (xB2 , B 2 ) if B 2 ⊆ B 1 ; then we define ϕ : J → I by ϕ(xB , B ) = B; by setting yj = yB for j = (xB , B ) ∈ J, we indeed have yj = yB = yϕ(B) ; and, for all B ∈ I, we have ϕ( j) = B1 ⊆ B for j = (xB1 , B 1 )  jB = (xB , B).

III.4 Ordinals We will only give a very naive presentation and use of ordinals. For a rigorous introduction, Krivine, Chapitre II, is a good reference.

III Filters, Ultrafilters. Ordinals

11

As we will only be using countable ordinals, we limit ourselves to these, and admit that there exists a well-ordered uncountable set Ord1 , whose smallest element is denoted 0, and without a largest element, in which, for any α ∈ Ord1 , there exists an injection of the interval [0, α[ into N. Ord1 called the set of countable ordinals, and is generally denoted Ord1 = [0, ω1 [; ω1 is called the first uncountable ordinal. Also, by writing α < ω1 , we mean that α is a countable ordinal. If there exists an integer n ∈ N and a bijection of the interval [0, α] onto {0, 1, . . . , n} ⊆ N, the ordinal α is said to be finite, and we identify it with the integer n. Any ordinal smaller that a finite ordinal is again finite. The other ordinals are said to be infinite; the smallest infinite ordinal is denoted ω0 , or more simply ω. Any ordinal greater than ω is again infinite; N can thus be identified with the interval [0, ω0 [ of [0, ω1 [. Any ordinal α < ω1 possesses a successor, denoted α + 1, still < ω1 , which is, in [0, ω1 [, the smallest element of the set of ordinals β < ω1 which are > α. Whenever an ordinal β = α + 1 is the successor of the ordinal α, α is said to be the predecessor of β. An ordinal that does not have a predecessor is said to be a limit ordinal; for example, ω is a limit ordinal. If α is a limit ordinal, we have α = supβ 1  sup Re ϕ(2−(n+1) A + 2−(n+1) BX ). Show that ϕ  2n (note that BX ⊆ A), and that sup |ϕ(A)|  2n+1 − ϕ = α. b) Deduce that x ∈ / 2−n A (use (1/α)ϕ). −n c) Deduce that 2 A ⊆ 2−(n+1) A + 2−(n+1) BX . 2) a) Let xn ∈ 2−n A. Show that there exists a sequence of vectors xn+1 , xn+2 , . . . ∈ X such that, for m = n, n+1, . . . , we have xm+1 ∈ 2−(m+1) A and xm −xm+1   1/2m+1 . b) Deduce that 2−n A ⊆ N + 2−n BX . 3) a) Show that BM = {ϕ ∈ X ∗ ; |ϕ(x)|  1, ∀ x ∈ A} (use the fact that BM is weak∗ closed). b) Using the preceding questions, show that 2n BM ⊇ N ⊥ ∩ 2n BX ∗ , and conclude that M ⊇ N ⊥ . For a general version, with essentially the same proof, see Habala–Hájek– Zizler, Theorem 222. For a history of the Banach–Dieudonné theorem, refer to Alfsen, pages 8–9.

1 Fundamental Notions of Probability

I Introduction The aim of this chapter is two-fold: on one hand we want to introduce the quite elementary probabilistic tools that we will be using throughout the rest of this book; on the other hand, we will try to convince the reader who has unhappy memories of contacts with Probability in his early schooling that it is neither difficult nor boring, all while having very profound applications to Analysis. We recall the main notions. A probability space is a triple (, A, P) composed of a set  (called the sample space), a σ -algebra A of subsets of  (called the space of events) and a positive measure P : A → [0, 1] of total mass 1 (called the probability). We will always assume that A is complete for P, meaning if B is negligible (i.e. B ⊆ A for an A ∈ A such that P(A) = 0), then B ∈ A. A real random variable (in short, real r.v.) on (, A, P) is a mapping X :  → R that is measurable for the σ -algebra A and the σ -algebra Bor of Borel sets of R: X −1 (B) ∈ A for all B ∈ Bor. We can then define the probability law X, often called the probability distribution, as the image measure PX = X(P): this is the probability on (R, Bor) defined by: PX (B) = P[X −1 (B)] . When X is integrable, its integral is called the expectation of X, and is denoted E(X):   X(ω) dP(ω) = x dPX (x) . E(X) = R



The variable X is said to be centered when E(X) = 0. When X ∈ L2 (P), we define its variance by:  2 = E(X 2 ) − [E(X)]2 . V(X) = E X − E(X) 13

14

1 Fundamental Notions of Probability

The σ -algebra generated by X is denoted σ (X) = X −1 (Bor) ⊆ A. The concept that distinguishes Probability from the general theory of measure is that of independence. A family (Bi )i∈I of subsets of A is said to be independent (for P) if, for any choice of i1 , . . . , ip ∈ I (distinct) and any Ai1 ∈ Bi1 , . . . , Aip ∈ Bip , we have: P(Ai1 ∩ . . . ∩ Aip ) = P(Ai1 ) . . . P(Aip ) . We have the following useful result (see Neveu 1, pages 119–120), in which σ (B) denotes the σ -algebra generated by B: Lemma I.1 Let Bi be stable by intersection for each i ∈ I. Then, if the Bi ’s are independent, so are the generated σ -algebras σ (Bi ), i ∈ I. It then follows that the random variables X1 , . . . , Xn are independent if and only if the probability distribution of the n-tuple (random vector) (X1 , . . . , Xn ) is equal to the product of the distributions of X1 , . . . , Xn : P(X1 ,...,Xn ) = PX1 ⊗ · · · ⊗ PXn . In particular, if X and Y are two independent real r.v., we have E(XY) = E(X)E(Y), and V(X + Y) = V(X) + V(Y). Recall that, if (An )n1 is a sequence of events, its upper limit is:  lim An = Ak ; n→+∞

n1 kn

this is the set of points ω ∈  which belong to an infinite number of events Ak ; its lower limit is:  lim An = Ak ; n→+∞

n1 kn

this is the set of points which belong to all but a finite number of the events Ak . An often-useful elementary result is the following: Proposition I.2 (The Borel–Cantelli Lemma) Let (An )n1 be a sequence of events, and A its upper limit. Then:  a) If +∞ n=1 P(An ) < +∞, then P(A) = 0. +∞ b) If n=1 P(An ) = +∞, and if the events An , n  1, are independent, then P(A) = 1. This chapter is structured as follows: In Section II, we define the three fundamental types of convergence (in probability, almost surely, and in law/distribution) and recall the tools adapted to the study of these three types of convergence.

II Convergence

15

In Section III, we quickly prove the fundamental theorems on series of independent random variables. In Section IV, we prove Khintchine’s inequalities and give a few applications. In Section V, we present a study of the convergence of martingales, which is a first step towards the study of sums of dependent variables, and reveals itself a useful generalization in many cases. Section V can be omitted on a first reading.

II Convergence II.1 Convergence in Probability and Convergence Almost Sure We denote by L0 = L0 (, A, P; R) the vector space of equivalence classes a.s. (almost surely) of real random variables. A sequence (Xn )n1 of L0 is said to converge in probability to X ∈ L0 if: P(|Xn − X| > t) −−→ 0 n→+∞

P

for all t > 0. This is denoted Xn −−→ X. n→+∞

It is useful to equip L0 with a metric d associated with the convergence in probability. We set:   |X − Y| ; d(X, Y) = E 1 + |X − Y| and it is easy to see that d is a distance on L0 , translation-invariant, and that if P Xn , n  1, and X are real r.v.: d(Xn , X) −−→ 0 if and only if Xn −−→ X. n→+∞ n→+∞ Indeed, the convergence for d implies the convergence in probability, by the Bienaymé–Tchebychev inequality (or rather that of Markov), and the increasing nature of t → t/(1 + t). P Conversely, if Xn −−→ X, for a given ε > 0, select first t > 0 such that n→+∞ t/(1 + t)  ε/2, then n0  1 such that P(|Xn − X|  t)  ε/2 for n  n0 ; we then have, for n  n0 :   |Xn − X| |Xn − X| dP + dP d(Xn , X) = 1 + |X − X| 1 + |Xn − X| n |Xn −X| 1/p. In set-theoretic terms:  (Yn > 1/p) . A= p1 n1

P Suppose that Yn −−→ 0. Then, with Ap = n1 (Yn > 1/p) for each p  1, n→+∞   as the sequence of events (Yn > 1/p) n1 is non-increasing, we obtain: P(Ap ) = inf P(Yn > 1/p)  lim P(Yn > 1/p) = 0 ; n→+∞

n1

hence: P(A) 

+∞

P(Ap ) = 0 .

p=1

Thus (Xn )n1 is a.s. a Cauchy sequence, and hence converges a.s.

II Convergence

17

Conversely, if (Xn )n1 converges a.s., then, for any t > 0, Fatou’s inequality gives:   lim P(Yn > t)  P lim (Yn > t)  P(A) = 0 , n→+∞

n→+∞

since limn→+∞ (Yn > t) ⊆ A. 4) This can be shown similarly. Note that 1) is an immediate consequence of 4). Remark

We thus can easily deduce that the distance d is complete on L0 .

The preceding proposition shows that the almost sure convergence of (Xn )n1 is equivalent to the convergence in probability of a “maximal function” (Yn or Zn ) associated with (Xn )n1 , and explains the importance of the maximal inequalities (among others the symmetry principle of Paul Lévy) to be seen later. Along with the maximal inequalities and the Borel–Cantelli lemma, an important tool for the study of almost sure convergence is the following: Proposition II.2 (Kolmogorov’s zero–one law) Let (An )n1 be an indepenσ -algebra dent sequence of subsets of A. Let A∞ be the associated    asymptotic

. Then, for all B , where B = σ A of tail events: A∞ = n n k n1 kn A ∈ A∞ , we have P(A) = 0 or 1. The zero–one law acts like a generalization of the Borel–Cantelli lemma, with nonetheless one weak point: unlike this lemma, it cannot decide between P(A) = 0 and P(A) = 1. Proof

For any m < n, the two σ -algebras σ (A1 , . . . , Am )

and

Bn = σ (An , An+1 , . . .)

are independent; thus σ (A1 , . . . , Am ) and A∞ are independent for any m  1. Denote:  σ (A1 , . . . , Am ). C= m1

  C is stable by finite intersection (as the sequence σ (A1 , . . . , Am ) m1 is nondecreasing) and any element of C is independent of all elements of A∞ . By the criterion of independence seen in the Introduction, the σ -algebras σ (C) = σ (A1 , A2 , . . .) and A∞ are independent. As A∞ ⊆ σ (A1 , A2 , . . .), A∞ is independent of itself, and thus, for every A ∈ A∞ , we have: P(A) = P(A ∩ A) = P(A) × P(A) ; hence P(A) = 0 or P(A) = 1.

18

1 Fundamental Notions of Probability

Corollary II.3 If (Xn )n1 is a sequence of independent random variables, the variables limn→+∞ Xn and limn→+∞ Xn are a.s. constant. Proof Let An = σ (Xn ). Set Yn = infkn Xk . As the sequence (Yn )n1 is nondecreasing, we have, for any m  1, limn→+∞ Xn = supnm Yn ; the random variable limn→+∞ Xn is thus σ (Am , Am+1 , . . .)-measurable, for any m  1; hence it is A∞ -measurable. a.s. If P(X = −∞) = 1 (respectively P(X = +∞) = 1), then X = −∞ (respectively +∞). If not, as the σ -algebras A1 , A2 , . . . are independent, we have: P(X = −∞) = P(X = +∞) = 0 by the zero–one law. The r.v. X is thus a.s. with values in R. Moreover, for any x ∈ R: FX (x) = P(X  x) = 0 or 1 ; thus, because FX is right-continuous, there exists a ∈ R such that FX (x) = a.s. 1I[a,+∞[ (x), and hence X = a.

II.2 Convergence in Distribution in L0 This convergence is in a way the most natural: since a random variable X is known by its law PX , we would like to say that Xn tends to X if PXn somehow tends to PX . For an Analyst, the most natural definition of this convergence is the following: A sequence (Xn )n1 of random variables is said to converge in distribution (or converge in law) to the random variable X if, for any bounded continuous function f on R, we have:   f (x) dPXn (x) −−→ f (x) dPX (x) . n→+∞

R L

R

This is denoted Xn −−→ X. n→+∞

Equivalently, E[f (Xn )] −−→ E[f (X)], ∀ f ∈ Cb (R). n→+∞

In fact, convergence is only needed for all functions f ∈ C0 (R) (or even for continuous functions with compact support): indeed, if we start with a bounded function, it suffices to multiply it by a sequence of functions increasing to 1I and tending to 0 at infinity. It is therefore a matter of w∗ -convergence of probability laws in the dual space M(R) = [C0 (R)]∗ . Remark By approximation, we find that convergence in distribution is equivalent to the following condition: for all Borel sets A of R such that PX (∂A) = P(X ∈ ∂A) = 0, we have:

II Convergence

19

P(Xn ∈ A) −−→ P(X ∈ A) , n→+∞

where ∂A is the boundary of A: this is the most natural definition for Probabilists. A very useful criterion for convergence in distribution uses characteristic functions. The characteristic function of the random variable X is the function defined for all t ∈ R by:   X (t) = E eitX . In analysis terms, this is the Fourier transform of PX , up to the sign. Proposition II.4 The sequence of random variables (Xn )n1 converges in distribution to the random variable X if and only if the characteristic functions Xn (t) converge to X (t) for all t ∈ R. Proof The necessary condition is evident as x → eixt is continuous and bounded on R. Conversely, if Xn (t) −−→ X (t), for all t ∈ R, then, for any f ∈ L1 (R), we n→+∞ have, by using the Fubini and dominated convergence theorems:      −ixt  f (x) dPXn (x) = f (t)e dt dPXn (x) = f (t)Xn (−t) dt R R R   R f (x) dPX (x) . f (t)X (−t) dt =  −−→ n→+∞

By the density of g ∈ C0 (R):

FL1 (R)

R

R

= { f; f ∈

L1 (R)}



in C0 (R), we obtain, for every

 R

g(x) dPXn (x) −−→ n→+∞

R

g(x) dPX (x) .

Remark We can show then that the n ’s converge uniformly on all compact sets. Corollary II.5 distribution.

Convergence in probability implies convergence in

Proof

We have, for any ε > 0:    |E(eitXn ) − E(eitX )|  E eitXn − eitX  1I|Xn −X|ε    + E eitXn − eitX  1I|Xn −X|>ε  ε|t| + 2P(|Xn − X| > ε) ;

thus E(eitXn ) −−→ E(eitX ). n→+∞

20

1 Fundamental Notions of Probability

Recall that almost sure convergence implies convergence in probability. However, examples showing that the converse implications are false can easily be found. An important case where these three convergences coincide is that of series of independent real r.v.: this is a theorem of Paul Lévy, to be seen at the end of Section III. For the convergence in distribution, a sort of Cauchy criterion is at hand: the convergence can be tested without a priori knowledge of the limit. Theorem II.6 (The Paul Lévy Continuity Theorem) Let (Xn )n1 be a sequence of random variables. If the characteristic functions Xn , n  1, converge pointwise to a function , which is continuous at 0, then (Xn )n1 converges in distribution to a random variable X and  is the characteristic function of X. Proof Set μn = PXn . As the set of probabilities on R is a subset of the unit ball of the dual of the separable Banach space C0 (R), we can extract a subsequence (μnk )k1 , w∗ -converging to a measure μ of norm  1. Clearly, μ is a positive measure. It remains to show that μ is a probability, i.e. there is no loss of mass, or μ(R) = 1. This will complete the proof because (μnk )k1 will then converge in distribution to μ; the k = Xnk , k  1, will thus converge, for any real r.v. X of law PX = μ, pointwise to X , and thus we will have  = X . Next, as the sequence Xn , n  1, converges to  = X , the preceding proposition indeed shows that (Xn )n1 converges in distribution to X. For the conservation of mass, first we have (0) = 1 since Xn (0) = 1. Consider: n · Kn (x) = π(1 + n2 x2 ) As (Kn )n1 is an approximate identity, and as  is continuous at 0, we have:  R

Kn (x)(x) dx = (Kn ∗ )(0) −−→ (0) = 1. n→+∞

Besides, for any n  1, the dominated convergence theorem (since |k (x)|  1 and Kn ∈ L1 (R)) and Fubini’s theorem give:   Kn (x)(x) dx = lim Kn (x)k (x) dx k→+∞ R R     = lim Kn (t) dμnk (t) = Kn (t) dμ(t), k→+∞ R

R

III Series of Independent Random Variables

21

because  Kn ∈ C0 (R). Finally,  Kn (t) = e−|t|/n −−→ 1, so that: n→+∞    Kn (t) dμ(t) −−→ dμ = μ(R) . n→+∞

R

R

We thus indeed have μ(R) = 1. Remark We have only used the continuity of  at 0, but this implies continuity on R. Indeed, the inequality gives the inequality  Cauchy–Schwarz  |n (t + h) − n (t)|2  2 1 − Re n(h) , and hence, by passing to the limit, |(t + h) − (t)|2  2 1 − Re (h) .

III Series of Independent Random Variables For the rest of this section, (Xn )n1 will denote a sequence of independent random variables, and Sn = X1 + · · · + Xn will be the partial sum of the first n variables. By Kolmogorov’s zero–one law, the series converges or diverges a.s., since the convergence of the series does not depend on its first terms.

III.1 Inequalities of Kolmogorov and Paley–Zygmund The basic tool for the almost sure convergence of Sn is the following maximal inequality, a “maximal” version of the Bienaymé–Tchebychev inequality. Theorem III.1 (Kolmogorov’s Inequality) If the Xn , n  1, are independent, in L2 and centered (i.e. E(Xn2 ) < +∞ and E(Xn ) = 0), then, for any t > 0:   E(S2 ) N P sup |Sn | > t  · 2 t nN Remark As the Xn ’s are independent and centered, the inequality can as well be written: N   1 E(Xn2 ) · P sup |Sn | > t  2 t nN n=1

Proof

Let An = σ (X1 , . . . , Xn ), and T = inf{n  N ; |Sn | > t} ,

with the convention T = +∞ if maxnN |Sn |  t. This T is known as a stopping time adapted to the increasing sequence of sub-σ -algebras (An )n1 .

22

1 Fundamental Notions of Probability

The key is the following inequality, whose extension led Doob to the notion of submartingale (see Section V):   (1) If n  N and A ∈ An , we have Sn2 dP  SN2 dP. A

A

Indeed, Rn = Xn+1 + · · · + XN is independent of An , and we have: SN2 1IA = Sn2 1IA + 2Sn 1IA Rn + R2n 1IA  Sn2 1IA + 2(Sn 1IA )Rn , so that, by integration: E(SN2 1IA )  E(Sn2 1IA ) + 2E(Sn 1IA )E(Rn ) = E(Sn2 1IA ) , because E(Rn ) = 0. Next, let A be the event supnN |Sn | > t, and An the event (T = n). We have An ∈ An , and P(A) =

N

P(An ) 

n=1

N   S2 E 2n 1IAn t n=1

N   S2 1 1  E 2N 1IAn = 2 E(SN2 1IA )  2 E(SN2 ) , t t t n=1

by using (1). Kolmogorov’s inequality leads to sufficient conditions for the almost sure convergence of Sn : Theorem III.2 (Kolmogorov’s Theorem) Let X1 , X2 , . . . be independent real  2 r.v., centered and square integrable. If +∞ n=1 E(Xn ) < +∞, then the series +∞ n=1 Xn converges a.s. Proof

For N > n  1, let: Yn,N = sup |Sk − Sn | nn

By Kolmogorov’s inequality for the variables Xn+1 , Xn+2 , . . . we have: P(Yn,N > t) 

N +∞ 1 1 2 E(X )  E(Xk2 ) ; k t2 t2 k=n+1

k=n+1

then: +∞ 1 P(Yn > t)  2 E(Xk2 ) . t k=n+1

III Series of Independent Random Variables

23

+∞ P 2 As the series −→ 0. By k=1 E(Xk ) < +∞, this inequality implies Yn − n→+∞ Proposition II.1, (Sn )n1 converges a.s. The converse of Theorem III.2 holds in some cases. For such a study, the following inequality, due to Paley and Zygmund [1932], is a simple but efficient sort of backwards Tchebychev inequality: Proposition III.3 (Paley–Zygmund Inequality) in L2 . Then, for any λ ∈ ]0, 1[, we have:

Proof

Let X be a positive real r.v.

  [E(X)]2 · P X  λE(X)  (1 − λ)2 E(X 2 )   Let A be the event X  λE(X) . By the Cauchy–Schwarz inequality: E(X) = E(X1IAc ) + E(X1IA )  λE(X) + [E(X 2 )]1/2 [E(1IA )]1/2 = λE(X) + P(A)1/2 [E(X 2 )]1/2 ;

hence (1 − λ)E(X)  P(A)1/2 [E(X 2 )]1/2 , and the proof is complete. This inequality leads to: Theorem III.4 (Converse of Kolmogorov’s Theorem) Let (Xn )n1 be a sequence of independent random variables. We assume that they are uni formly bounded by a constant C. If the series +∞ n=1 Xn converges a.s., then +∞ V(X ) < +∞. n n=1 Proof

Only one fact will be used: the r.v. M = sup |Sn | = sup |X1 + · · · + Xn | n1

n1

is a.s. finite. First assume that the r.v. Xn are centered. Without loss of generality, we can N  2 2 assume that C = 1. Then N n=1 V(Xn ) = n=1 E(Xn ) = E(SN ). This value is 2 denoted σN . The Paley–Zygmund inequality gives: P(M 2  σN2 /2)  P(SN2  σN2 /2) 

1 σN4 · 4 E(SN4 )

We bound E(SN4 ). We have: SN4 =



4! X α1 . . . XNαN , α1 ! . . . αN ! 1

where the sum is taken over all N-tuples of integers such that α1 + · · · + αN = 4 .

24

1 Fundamental Notions of Probability

Thus: E(SN4 ) =



4! E(X1α1 ) . . . E(XNαN ) . α1 ! . . . αN !

Any αi equal to 1 gives a zero contribution because the Xi ’s are centered. Hence: 4! 2β 2β E(SN4 ) = E(X1 1 ) . . . E(XN N ) , (2β1 )! . . . (2βN )! where now the sum is taken over all N-tuples of integers such that β1 + · · · + βN = 2. As |Xn |  1, we obtain: E(SN4 )  6

N 

E(Xi4 ) +

6

E(Xi2 )E(Xj2 )



i=j

i=1 N 



E(Xi2 ) +

i=1



 E(Xi2 )E(Xj2 )  6 (σN2 + σN4 ) .

i=j

Back to the Paley–Zygmund inequality, we thus obtain: P(M 2  σN2 /2) 

σN4 1 · 24 σN2 + σN4

If we had σ 2 = limN→+∞ σN2 = +∞, a monotone passage to the limit would give: P(M 2 = +∞)  1/24, contradicting P(M < +∞) = 1. Thus indeed: +∞ 2 n=1 V(Xn ) = σ < +∞. When the real r.v. are not necessarily centered, we can often use a notion known as the symmetrization of the random variable. Let (Xn )n1 be a sequence of independent random variables, itself independent of the sequence (Xn )n1 , such that Xn has the same law as Xn for any n  1. The random variable Xns = Xn − Xn is called the symmetrization of Xn . One way to achieve this is to replace the probability space  by the product  × , and then to set Xns (ω, ω ) = Xn (ω) − Xn (ω ) (this is known as a decoupling of the variables). The symmetrization is centered: E(Xns ) = E(Xn ) − E(Xn ) = 0, and |Xns |   |Xn | + |Xn |  2 C. Moreover, if the series n1 Xn converges a.s., so does   s n1 Xn , and hence n1 Xn as well.  s2 By the beginning of the proof, we thus have: +∞ n=1 E(Xn ) < +∞. Hence E(Xns 2 ) = V(Xns ) = V(Xn − Xn ) = V(Xn ) + V(Xn ) = 2 V(Xn ), and the proof is complete.

III Series of Independent Random Variables

25

Remark The method of bounding E(SN4 ) will be used again, more systematically, in the proof of the Khintchine inequalities. Theorem III.4 provides a necessary and sufficient condition for the almost sure convergence of series of independent random variables, also due to Kolmogorov. Theorem III.5 (The Kolmogorov Three-Series Theorem) Let Xn , n  1, be  independent real r.v., and M > 0. Then the series n1 Xn converges a.s. if and only if the three following series of numbers converge: P(|Xn | > M), (b) V(Xn[M] ), (c) E(Xn[M] ), (a) n1

n1

n1

where Xn[M] is the variable Xn truncated at M: Xn[M] = Xn .1I{|Xn |M} . Note that if this is true for one M > 0, then it is true for all M > 0. Proof  1) Assume that n1 Xn converges a.s. Then:   a.s. (a) Xn −−→ 0; we thus have P limn→+∞ |Xn |  M = 0. But: n→+∞



 lim |Xn |  M ⊇ lim {|Xn | > M}.

n→+∞

n→+∞

As the events An = {|Xn | > M} are independent, the Borel–Cantelli  lemma gives +∞ n=1 P(|Xn | > M) < +∞. [M] (b) As the r.v. Xn are independent and bounded by M, we can use  Theorem III.4 as n1 Xn[M] converges a.s. (since Xn[M] (ω) = Xn (ω)  a.s. for n large enough), and we obtain n1 V(Xn[M] ) < +∞. [M] (c) Kolmogorov’s theorem applied to Xn[M] −E(X n ) then provides the a.s.   convergence of n1 Xn[M] − E(Xn[M] ) , and hence the convergence of   [M] [M] n1 E(Xn ), since n1 Xn converges a.s., by (b) .

2) Conversely, if the three series converge, then as above, condition (b) allows  us to apply Kolmogorov’s theorem to Xn[M] − E(Xn[M] ). Thus  [M] [M]  n1 Xn − E(Xn ) converges a.s. Condition (c) then provides  X [M] . But condition (a) implies that the a.s. convergence of  n1 n  P limn→+∞ {|Xn | > M} = 0, via the Borel–Cantelli lemma. As {|Xn | > M} = {Xn = Xn[M] }, we have Xn = Xn[M] a.s. for n large enough.  The a.s. convergence of n1 Xn then ensues.

26

1 Fundamental Notions of Probability

Remark If the Xn ’s are symmetric, condition (c) can be omitted as E(Xn[M] ) = 0. If the Xn ’s are positive, condition (b) can be omitted as it follows from (c) : V(Xn[M] )  ME(Xn[M] ). A sequence of random variables is said to be i.i.d. if these variables are independent and identically distributed (i.e. they all have the same distribution). Throughout, (gn )n1 will denote an i.i.d. standard Gaussian sequence, √ 2 with density (1/ 2π ) e−x /2 , and (εn )n1 will denote a Bernoulli sequence (or rather a Rademacher sequence), i.e. an i.i.d. sequence with law (1/2)(δ−1 + δ1 ), that is, the law of fair heads or tails: P(εn = −1) = P(εn = 1) = 1/2. The following is a corollary of the three-series theorem VI, often applied to gn or to εn : Corollary III.6 Let (an )n1 be a sequence of complex numbers, and (Xn )n1  an i.i.d. sequence centered and non-zero in L2 . Then the series +∞ n=1 an Xn  2 < +∞. converges a.s. if and only if +∞ |a | n=1 n Proof We may assume the an ’s real and also X1 2 = 1 (so that Xn 2 = 1 for any n  1). +∞ 2 Kolmogorov’s theorem immediately If the series n=1 |an | converges,  a X implies the a.s. convergence of +∞ n n , as Xn 2 = 1. n=1 Conversely, if the series converges a.s., the three-series theorem shows +∞ 2 +∞ [1/|an |] ) < +∞. that n=1 P(|X1 | > 1/|an |) < +∞ and n=1 an V(X1 In particular, the convergence of the first series implies that we have P(|X1 |  1/|an |) −−→ 1. Consequently, as V(X1 1IA ) −−→ V(X1 ), we have P(A)→1

n→+∞

[1/|a |] V(X1 n ) −−→ V(X1 ) +∞ 2 n→+∞ n=1 an < +∞.

=

E(X12 )

= 1; hence

[1/|a |] a2n V(X1 n )

∼ a2n , so that

III.2 The Paul Lévy Theorem Another important application of the three-series theorem is the fundamental theorem of Paul Lévy, whose proof by martingales is given in Section VI, and another in Chapter 4 (Volume 1). However, we begin with a “Fubinization principle” of independent interest, with numerous applications to be seen in this book. Proposition III.7 (Fubinization principle) Let (, A, P) be the product space of the two probability spaces (1 , A1 , P1 ) and (2 , A2 , P2 ). Let A ∈ A, and: Aω2 = {ω1 ; (ω1 , ω2 ) ∈ A}, the horizontal fiber of A with ordinate ω2 ; Aω1 = {ω2 ; (ω1 , ω2 ) ∈ A}, the vertical fiber of A with abscissa ω1 .

III Series of Independent Random Variables

27

If almost all horizontal fibers of A are of probability 1, then so are almost all vertical fibers, and vice versa. Proof

Fubini’s theorem and the hypothesis give:   P2 (Aω1 ) dP1 (ω1 ) = P1 (Aω2 ) dP2 (ω2 ) = 1 . P(A) = 1

2

Hence:



  1 − P2 (Aω1 dP1 (ω1 ) = 0 , 1

with 1−P2 (Aω1 )  0, which implies 1−P2 (Aω1 ) = 0 for P1 -almost all ω1 . Theorem III.8 (Paul Lévy) If the random variables Xn , n  1, are independent, the convergences almost sure, in probability and in distribution  of the series n1 Xn are equivalent. The a.s. convergence always implies the convergence in probability, and this in turn always implies the convergence in distribution. It thus suffices to show that the convergence in distribution for a series of independent real random variables implies the convergence a.s. For this, we will use the following inequalities: Lemma III.9 (Truncation Inequalities) For any random variable X, we have, for any t > 0:  1/t  3 x2 dPX (x)  2 1 − Re X (t) ; 1) t −1/t     7 t 1 − Re X (u) du. 2) P |X|  1/t)  t 0 Proof u2 u4 − , valid for any u ∈ R. This gives: 1) We use the inequality 1 − cos u  2 24       1 − Re X (t) = 1 − cos(tx) dPX (x)  1 − cos(tx) dPX (x) R

 

|x| t) dt . E[f (X)] = f (0) + 0

Proof

It suffices to take the expectation of:  +∞   f (t) 1I{X(ω)t} dt , f X(ω) = f (0) + 0

and apply the Fubini–Tonelli theorem. It will also be convenient to use the notion of Orlicz spaces, which generalize that of the classic Lebesgue spaces Lp . We call an Orlicz function any mapping ψ : R+ → R+ that is continuous, strictly increasing, convex and such that ψ(0) = 0 and ψ(∞) = ∞. The associated Orlicz space Lψ (, A, P), abbreviated Lψ , is the vector space of (classes variables X :  → C such that there exists a > 0   of) random  for which E ψ |X|/a < +∞. For X ∈ Lψ , we define:    |X|  1 . Xψ = inf a > 0 ; E ψ a When ψ(x) = xp (with p  1), this gives the classic notion of the space Lp , with the usual norm  . p . The general definition is less direct because of the lack of homogeneity of ψ. Nonetheless,  . ψ is the gauge of the balanced  ψ convex set {Y ∈ L ; E ψ(|Y|)  1}. It is an exercise to verify that  . ψ is a norm on Lψ and makes it a Banach space. This norm is called the Luxemburg norm. Remark Using the language of Orlicz spaces, we can give a shorter, somewhat less technical, proof of Khintchine’s inequalities. Indeed, let

IV Khintchine’s Inequalities

33

N N 2 X = n=1 εn un , where the un are real and n=1 un = 1. We will show that X has a “sub-Gaussian behavior,” meaning: P(|X| > t)  2 exp(−t2 /2)

for any t > 0.

In fact, for λ > 0, by independence: N N N  λ2 u2         2 n E eλX = = eλ /2 , E eλεn un = ch(λun )  exp 2 n=1

n=1

n=1

and Markov’s inequality gives     2 P(X > t) = P eλX > eλt  e−λt E eλX  e−λt+λ /2 . We optimize with λ = t, which gives P(X > t)  e−t /2 . By symmetry, we 2 obtain P(|X| > t)  2e−t /2 . 2 Now if ψ is the Orlicz function 2 (x) = ex − 1, the formula of integration √ by parts gives, for any a > 2:  |X|   +∞ 2t 2 2 = E 2 et /a P(|X| > t) dt a a2 0  2 +∞ t2 /a2 −t2 /2  2 2te e dt a 0  +∞ 1 2 1 2 , = 2 − 2 x dx = 2 exp − 2 a a 0 (a /2) − 1 √ √ so that X2  6 if X2 = 1. Hence, in general: X2  6 X2 . √ Then Exercise VII.7 shows that Xp  C p X2 , where C is a numerical constant. We thus obtain this additional factor C with respect to the first proof. 2

In Orlicz spaces, the following simple, but fundamental, inequality is very useful: Proposition IV.3 (The Orlicz–Jensen Inequality) random variables in an Orlicz space Lψ , and

Let X1 , . . . , XN be positive

M = sup(X1 , . . . , XN ) their maximal function. Then we have: E(M)  ψ −1 (N) sup Xn ψ . nN

Proof By homogeneity, we can assume supnN Xn ψ = 1. We have  ψ(M) = supnN ψ(Xn )  N n=1 ψ(Xn ). Hence E[ψ(M)] 

N n=1

E[ψ(Xn )]  N .

34

1 Fundamental Notions of Probability

Next, Jensen’s inequality shows that ψ[E(M)]  E[ψ(M)]  N, and the proof is complete. In this inequality, the more ψ grows, the more ψ −1 diminishes, and the more attenuated is the effect of the large number N of variables. But in compensation, Xn ψ grows . . . A compromise needs to be found, often 2 reached thanks to Khintchine’s inequalities: if ψ(x) = 2 (x) = ex − 1, and if Xn 2  CXn 2 , then the the Xn ’s have a sub-Gaussian behavior leading to  inequality of Proposition IV.3 reads as E(M)  C log(N + 1) supnN Xn 2 . For Gaussian variables, we can be even more precise: Corollary IV.4 Let g1 , . . . , gN be standard Gaussian variables, and M = max{|g1 |, . . . , |gN |} their maximal function. Then: E(M) 

8 log(N + 1) . 3 2

x Proof Indeed,  it suffices to take ψ(x) = 2 (x) = e − 1, so that −1 an elementary calculation, 2 (N) = log(N + 1); on the other hand, via√ 2, we have E2 (|g|/c) = with a standard Gaussian variable g and with c >   ! √ 2 1/ 1 − c2 − 1, which implies that g2 = 8/3.

A fundamental consequence of Khintchine’s inequalities, to be frequently used throughout this book, is the following variant of a result of Salem and Zygmund [1954]: Theorem IV.5 (The Boundedness Theorem) Let (ai,j ) 1iN be a rectangular 1jn

matrix of scalars, and (Xi )iN the associated “Bernoulli process”: Xi =

n

ai,j εj ,

j=1

where (εj )jn is a Bernoulli sequence. Then there exists a numerical constant C such that:    E sup |Xi |  C log(N + 1) sup Xi 2 . iN

Proof

iN 2

Let ψ(x) = 2 (x) = ex −1. The Orlicz–Jensen inequality shows that:    E sup |Xi |  2−1 (N) sup Xi 2 = log(N + 1) sup Xi 2 . iN

iN

iN

Since Xi 2  CXi 2 by Khintchine’s inequalities, as stated in the Remark, the proof is complete.

V Martingales

35

V Martingales V.1 Conditional Expectation Let B be a sub-σ -algebra of A. Starting with a random variable X, we wish to define another, which will “forget” everything that happens “outside B.” We begin by pointing out a slight difficulty. If Y is a B-measurable random variable and if Y is an A-measurable random variable which is a.s. equal to Y, i.e. P(Y = Y) = 0, then there is no reason for Y to be B-measurable, even if we did assume the σ -algebra A to be complete. In other words, the space of classes of B-measurable functions L0 (, B, P|B ) is not contained in the space of classes of A-measurable functions L0 (, A, P). Nevertheless, we have a natural injection from L0 (, B, P|B ) into L0 (, A, P), and this injection is an isometry of Lp (, B, P|B ) into Lp (, A, P) for any p such that 0 < p  +∞. Using this isometry, we can thus identify Lp (, B, P|B ) with a subspace of Lp (, A, P), which we will always do hereafter. Now let X ∈ L1 (, A, P). If we set, for every B ∈ B:  μ(B) = E(X1IB ) = X dP , B

we obtain a (real) measure on (, B), which is clearly absolutely continuous with respect to P|B . The Radon–Nikodým theorem thus ensures the existence of a B-measurable random variable Y ∈ L1 (, B, P|B ) such that μ = Y.P|B . Y is said to be the conditional expectation of X with respect to B, denoted E(X| B) or EB (X). By definition, EB (X) thus satisfies:   EB (X) dP = X dP B

B

  for every B ∈ B. In other words: E EB (X)1IB = E(X1IB ). We can then easily verify the following properties:   a) If C is a sub-σ -algebra of B, we have E(X| C) = E E(X| B)| C (nesting). b) If Z is B-measurable and bounded (or more generally if ZX ∈ L1 ), then EB (ZX) = ZEB (X) (ideal property). c) If X is independent of B, then EB (X) = E(X). If X is B-measurable, then EB (X) = X; in other words EB is a projection of L1 (, A, P) onto L1(, B, P).  Hereafter we will use the simpler notation P for P|B . d) E EB (X) = E(X) and EB (1I) = 1I. e) EB is a positive linear map that preserves constants, and we have Jensen’s inequality:     ϕ EB (X)  EB ϕ(X) , for any convex function ϕ.

36

1 Fundamental Notions of Probability

f) EB (X) ∈ Lp (, B, P) if X ∈ Lp (, A, P) for 1  p  +∞, and is a projection of norm 1. Actually, it is a projection of norm 1 even on any Orlicz space Lψ (, A, P). g) If p = 2, EB is the orthogonal projection of L2 (, A, P) onto L2 (, B, P).

V.2 Martingales Let A1 ⊆ A2 ⊆ . . . ⊆ An ⊆ . . . ⊆ A be an increasing sequence of sub-σ algebras of A (called a filtration). We often complete this filtration from below by adding the trivial σ -algebra A0 = {∅, } ⊆ A1 . A sequence (Mn )n1 of real r.v. is called a martingale adapted to the filtration (An )n1 if: a) Mn is integrable and An -measurable for any n  1; b) EAn (Mn+1 ) = Mn for all n  1. If we set M0 = 0 and dn = Mn − Mn−1 , we see that Mn = d1 + · · · + dn , and that EAn−1 (dn ) = 0. Thus Mn appears as the partial sum of order n of the series +∞ k=1 dk of conditionally centered random variables. In this sense, martingales are a generalization of series of independent variables. We note in fact that  the most immediate example of a martingale is exactly a series +∞ k=1 Xk of independent centered random variables, with Mn = Sn = X1 + · · · + Xn , and taking for An the σ -algebra σ (X1 , . . . , Xn ). The sequence (Mn )n1 is called a submartingale adapted to (An )n1 if, in the preceding conditions a) and b) , we replace b) by: b ) EAn (Mn+1 )  Mn for any n  1. Similarly, we can define supermartingales by replacing  by . This condition b ) is equivalent to:   Mn+1 dP  Mn dP for every A ∈ An , A

A

and indicates that a submartingale is a sequence that increases on average, while a martingale is a sequence that is constant on average. The submartingales possess remarkable properties of automatic convergence (reminiscent of the property “any bounded non-decreasing sequence of real numbers converges”) that provide useful generalizations of Kolmogorov’s theorems of the previous section. First, we present the following essential result:

V Martingales

37

Theorem V.1 (Doob’s Inequality) Let (M1 , . . . , MN ) be a submartingale. For t > 0, we have:  1  1   P sup Mn > t  E MN+ 1I" sup Mn >t#  E(|MN |) . nN t t nN Proof This is modeled on the proof of Kolmogorov’s inequality, with one exception: the inequality:   Mn dP  MN dP, A

A

Let T = with A ∈ An , is now part of the definition of submartingales.   inf{n  N ; Mn > t} or +∞, An = (T = n) ∈ An and A = supnN Mn > t . Then, we see that: P(A) =

N n=1

P(An ) 

N  1 E Mn .1IAn t n=1

N  1  1  E MN .1IAn = E MN .1IA t t n=1  1 1  E MN+ .1IA  E(|MN |) . t t

Remark We could have paid attention only to the bound (1/t)E(|MN |); but the intermediate bound (1/t)E(Mn+ 1IA ) will be needed in Chapter 7. Doob’s inequality and the following approximation property will lead to a theorem of almost sure convergence for “closed martingales,” in particular those bounded in L2 . The general case of those bounded in L1 will require further ingredients. Lemma V.2 Let B ⊆ A be a Boolean algebra, and let A ∈ σ (B) and ε > 0. Then there exists B ∈ B such that P(AB)  ε, where  denotes the symmetric difference. Proof It is convenient to note that P(AB) = 1IA − 1IB 1 . If A is the set of A ∈ σ (B) having the above approximation property for any ε > 0, we can easily verify that A is a σ -algebra. As it contains B, it is equal to σ (B). A martingale (Mn )n1 can be said to be closed if there exists M ∈ L1 such that Mn = EAn (M) for any n  1. The closed martingales must be considered as the “congenial” martingales; they are exactly the uniformly integrable martingales (Neveu 1), with limits easy to find, even in the vectorial case.

38

1 Fundamental Notions of Probability

Theorem V.3 (Doob’s Theorem) 1) Every closed martingale converges a.s. and in L1 -norm. 2) Every bounded martingale in L2 is closed, and hence converges a.s. and also in L2 -norm. Proof 1) Let (An )n1 be the filtration to which (Mn )n1 is adapted. By hypothesis, there exists M ∈ L1 (, A, P) such that Mn = EAn (M) for any n  1. Let  A∞ = σ (B∞ ), and M∞ = EA∞ (M). B∞ be the Boolean algebra +∞ n=1 An , For any ε > 0, we can find Y = Jj=1 aj 1IAj , with Aj ∈ A∞ , such that M∞ − Y1  ε. Thanks to Lemma V.2, we can find, for each j, a Bj ∈ B∞ such that: ε ; 1IAj − 1IBj 1  1 + |a1 | + · · · + |aJ |  hence Z = Jj=1 aj 1IBj is B∞ -measurable, and M∞ − Z1  M∞ − Y1 + Y − Z1  2ε . By the definition of B∞ , we can find n0 such that Z is An0 -measurable. We thus have, for n  n0 : Mn − M∞ = EAn (M) − M∞ = EAn (M∞ ) − M∞ = EAn (M∞ ) − EAn (Z) + Z − M∞ ; therefore: Mn − M∞ 1  EAn (M∞ − Z)1 + M∞ − Z1  2M∞ − Z1  4ε . L1

This shows that Mn −−→ M∞ . n→+∞ On the other hand, Doob’s maximal inequality shows that:   1 P sup |Mk − Mn | > t  MN − Mn 1 ; t n t  M∞ − Mn 1 . t k>n This inequality proves that the maximal function supk>n |Mk − Mn | converges in probability to 0. By Proposition II.1, (Mn )n1 converges a.s. As it converges to M∞ in the L1 -norm, the almost sure limit is necessarily M∞ .

V Martingales

39

2) Set M0 = 0 and dn = Mn − Mn−1 for n  1. These differences dn are orthogonal in L2 : indeed, for j < n, we have:     E(dj dn ) = E EAj (dj dn ) = E dj EAj (dn ) = E(dj .0) = 0 . n 2 Hence E(Mn2 ) = j=1 E(dj ). Therefore, as the martingale (Mn )n1 is  +∞ +∞ 2 assumed closed in L2 , j=1 E(dj ) < +∞. The series j=1 dj thus converges in L2 , and as Mn = d1 + · · · + dn , the martingale (Mn )n1 converges in L2 . Denote its limit by M. It only remains to show that EAn (M) = Mn : indeed, this would mean that (Mn )n1 is a closed martingale, and hence, by 1) , converges a.s. to M = M∞ . for N  n, we have To show this, each n  A ∈ An . Then,  for  1, fix  A  A n n E(Mn 1IA ) = E E (MN )1IA ) = E E (MN 1IA ) = E(MN 1IA ). As |E(MN 1IA ) − E(M1IA )|  MN − M1  MN − M2 , by letting N tend to infinity, we obtain E(Mn 1IA ) = E(M1IA ). The equality Mn = EAn (M) is therefore established. We could dispense with the hypothesis that (Mn )n1 is closed, but the proof is more delicate, and convergence in L1 -norm is not always guaranteed. The proof we will provide is essentially due to Chatterji [1968], and very much in the style of Analysts: notably, it avoids arguments involving stopping times and stopped martingales. Theorem V.4 (Doob)

Every martingale bounded in L1 converges a.s.

The proof is more difficult, “because of L1 ,” and not “because of martin2 by considering ϕ(M ), gales.” The idea is to bring the problem back to L√ n ). But this raises two where ϕ is a well-chosen function (for example difficulties: 1) for the composition with ϕ, we must already have limited the values of Mn , for example by supposing Mn  0; 2) the composition with ϕ leads us away from the domain of martingales, and lands us into that of sub- (or super-) martingales. These two difficulties will be overcome with the following two lemmas. Definition V.5 Let A1 ⊆ . . . ⊆ An ⊆ . . . be an increasing sequence of subσ -algebras of A. An increasing process adapted to (An )n1 is an increasing sequence 0 = A0  A1  . . .  An  . . .

40

1 Fundamental Notions of Probability

of random variables such that An+1 is An -measurable. We can also say that the sequence (An )n1 is predictable. Lemma V.6 (The Doob Decomposition) Let (Xn )n0 be a submartingale adapted to the sequence (An )n0 , where A0 is the trivial σ -algebra. Then, it can be uniquely decomposed in the form Xn = Mn + An , where (Mn )n0 is a martingale and (An )n0 is an increasing process, both adapted to (An )n0 . Moreover, if (Xn )n0 is bounded in L1 , then (An )n0 converges almost surely to a variable A∞ ∈ L1 , and if (Xn )n0 is positive and bounded in L2 , then (Mn )n0 is bounded in L2 . Proof We begin with the uniqueness. If such a decomposition exists, we must have Xn+1 − Xn = (Mn+1 − Mn ) + (An+1 − An ); hence: (1)

An+1 − An = EAn (Xn+1 − Xn ) = EAn (Xn+1 ) − Xn

and (2)

Mn+1 − Mn = (Xn+1 − Xn ) − (EAn (Xn+1 ) − Xn ) = Xn+1 − EAn (Xn+1 ).

As A0 = 0, and thus M0 = X0 , the formulas (1) and (2) imply the uniqueness of the decomposition. For the existence of the decomposition, it suffices to verify that, if we set A0 = 0 and M0 = X0 , then the formulas (1) and (2) provide the two desired sequences: this comes straight from (Xn )n0 being a submartingale. Now, if Xn 1  C, we have: E(An ) = E(Xn ) − E(Mn ) = E(Xn ) − E(M0 )  C − E(M0 ), and the monotone convergence theorem shows that if we set A∞ = supn0 An , then 0  E(A∞ )  C − E(X0 ); and hence A∞ ∈ L1 . In particular A∞ < +∞ a.s., and hence (An )n0 converges a.s. When Xn  0 and Xn 2  C, note that: (∗)

m > n ⇒ E(Xm Xn )  E(Xn2 ) .

Indeed, as Xn  0, we have:     E(Xm Xn ) = E EAn (Xm Xn ) = E Xn EAn (Xm )  E(Xn2 ). Moreover, by setting dn = Mn+1 − Mn and δn = An+1 − An , we also obtain: E(dn δn ) = 0.  A    In fact, E(dn δn ) = E E n (dn δn ) = E δn EAn (dn ) = E(δn .0) = 0. It ensues from (∗) and (∗∗) that:   2 ) − E(Xn2 )  E (Xn+1 − Xn )2 = E(dn2 ) + E(δn2 )  E(dn2 ) ; E(Xn+1 (∗∗)

V Martingales

41

hence, by taking the sum:   N−1 E (MN − M0 )2 = E(dn2 )  E(XN2 )  C2 . n=0

Remark We have the following variant: (∗) shows that E(Xm − Xn )2  2 )−E(X 2 ); hence (X ) 2 2 E(Xm n n1 is Cauchy in L , and thus converges in L , and a n 1 1 fortiori in L . Hence (Mn )n1 converges in L , thus is closed and converges a.s. Lemma V.7 (The Krickeberg Decomposition) Every bounded martingale (Mn )n1 in L1 is the difference of two non-negative martingales: Mn = Mn − Mn , where (Mn )n1 and (Mn )n1 are two non-negative martingales adapted to the same filtration as (Mn )n1 . Proof The submartingale (|Mn |)n1 is bounded in L1 . The Doob decomposition allows us to write |Mn | = Yn + An , where (Yn )n1 is a martingale and (An )n1 an increasing process, with A∞ ∈ L1 . We clearly have An = EAn (An )  EAn (A∞ ), so that |Mn |  Yn +EAn (A∞ ) = Mn . Then (Mn )n1 is a positive martingale, and it dominates (Mn )n1 . Thus, if we set Mn = Mn −Mn , (Mn )n1 is also a non-negative martingale, and Mn = Mn − Mn . Proof of Theorem V.4

We will proceed in three steps:

Step 1. We may assume, thanks to the Krickeberg decomposition, that the martingale (Mn )n1 is positive. Step 2. Any non-negative submartingale (Xn )n1 bounded in L2 converges a.s. Indeed, by the Doob decomposition, Xn = Mn + An , where (Mn )n1 is a martingale bounded in L2 , and (An )n1 is a process increasing to A∞ ∈ L1 . By Theorem V.3, (Mn )n1 , and consequently also (Xn )n1 , converge a.s. Step 3. All non-negative martingales converge a.s. We performour initial  program with ϕ(x) = e−x . As ϕ is convex, by Jensen’s inequality, e−Mn n1 is a submartingale, positive and bounded above by 1I, and hence a fortiori a.s. bounded in L2 . By Step 2, we have e−Mn −−→ L  0. Moreover, Doob’s n→+∞ inequality shows that:   E(M ) E(M1 ) , N P sup Mn > t  = t t nN   and thus P supn1 Mn > t  E(M1 )/t; hence (Mn )n1 is bounded a.s. It follows that L is a.s. strictly positive, and consequently we have a.s. Mn −−→ log(1/L). n→+∞

Remark This leads to an alternative proof of Paul Lévy’s theorem on the convergence of series of independent r.v. Indeed, assume that the series

42

1 Fundamental Notions of Probability

+∞

n=1 Xn converges in distribution. Set Sn = X1 + · · · + Xn . By hypothesis, n (t) = X1 (t) . . . Xn (t) = E eitSn converges pointwise to X (t). As X (0) = 1, there exists δ > 0 such that X (t) = 0 for |t|  δ. Thus we also have n (t) = 0 for any n  1 when |t|  δ. For these t, we set:

Mn (ω) =

eitSn (ω) · n (t)

Then (Mn )n1 is a (complex) martingale, bounded in L∞ a fortiori   , and hence in L2 . Consequently, it converges a.s. to a r.v. λ(t). Then eitSn (ω) n1 converges a.s. to λ(t)X (t). As in the first proof (see Exercise VII.11 for the details), we can conclude that (Sn )n1 converges a.s.

VI Comments A very thorough treatment of the asymptotic properties of sequences of real random variables can be found in the classic texts of Loève, Rényi and Billingsley, including the laws of large numbers that we will not need in this book. A more recent presentation can be found in the works of Shiryaev and Stroock, the latter notably containing a complete proof of the Hartman– Wintner Law of the Iterated Logarithm (for the case L2 equidistributed). The convergence of discrete martingales, or of sub- and supermartingales, is studied in detail in the two texts of Neveu (Neveu 1, Neveu 2): in particular the closed martingales are shown to be exactly the uniformly integrable martingales, or in other words those converging in L1 . Recently, extensions to the non-commutative context have been published (see for example Junge [2002] and Pisier and Xu [1997]). The Orlicz spaces are studied in detail in Neveu 2, as well as the duality of these spaces. The Fubinization principle, also called the reduction principle, is used systematically in Kahane 2. See Neveu 1 for its application to the theorem of Paul Lévy (see the Remark above). Khintchine’s inequalities are somewhat more popular with Analysts than with Probabilists! The classical proof given here can be found, for example, in Diestel. Analogous inequalities hold for Steinhaus sequences (Zn )n1 , which are sequences of i.i.d. random variables, uniformly distributed on the unit circle |z| = 1. Ullrich [1988 a], [1988 b] and [1988 c] brought to light an important difference from the Bernoulli sequences (εn )n1 : while the constant

VII Exercises

43

Ap tends to 0 as p tends to 0 for the Bernoulli sequences (consider ε1 + ε2 p ), it remains bounded below by a strictly positive constant for the Steinhaus sequences. Ball and Nazarov (unpublished) gave other results in this way. See also Montgomery–Smith [1990]. The boundedness theorem is implicit in the long article by Salem and Zygmund [1954] on the uniform norms of random polynomials. The Orlicz–Jensen inequality is found in Pisier [1983 c].

VII Exercises Exercise VII.1 Let (gn )n1 be a standard Gaussian sequence. |gn | Show that lim sup  = 1 a.s. (for θ > 1, apply the Borel–Cantelli 2 log  n n→+∞ lemma to the events {|gn |  2 θ log n }). Exercise VII.2 Let 0 < p  2 and let X be a symmetric p-stable random  p variable, i.e. whose characteristic function is E eitX = e−|t| . The existence of such variables will be shown in Chapter 5, Subsection II.2. For p = 2, these are the Gaussian variables. Let (Xn )n1 be an i.i.d. sequence of variables with the same distribution as X. Let (an )n1 be a sequence of scalars. +∞ Show that the series n=1 an Xn converges a.s. if and only if +∞ p < +∞. |a | n=1 n Exercise VII.3 Let (Xn )n1 be as in Exercise VII.2. We admit (see Chapter 5, Subsection II.2) that, for 0 < p < 2, the variables Xn have a continuous density f with f (x) ∼ cp |x|−p−1 when |x| → +∞ (with cp > 0). Show that: +∞ +∞ p 1) If 0 < p < 1: n=1 |an Xn | < +∞ a.s. ⇐⇒ n=1 |an | < +∞. 2) If p = 1: +∞

 1  < +∞ . |an | log 1 + |an | n=1 +∞ +∞ n=1 |an Xn | < +∞ a.s. ⇐⇒ n=1 |an | < +∞.

|an Xn | < +∞ a.s. ⇐⇒

n=1

3) If 1 < p  2:

+∞

Exercise VII.4 1) Let (un )n1 be a sequence of real numbers > 0 increasing to +∞, and  xn is let (xn )n1 be a sequence of real numbers such that the series +∞ n=1 un convergent. 1 n Show that xk −−→ 0. un k=1 n→+∞

44

1 Fundamental Notions of Probability

2) Deduce that, if (Xn )n1 is an i.i.d. sequence in L2 , then 1 a.s. Xk −−→ E(X1 ) . n→+∞ n n

k=1

3) Show that this remains true if the i.i.d. variables Xn are in L1 : the strong law of large numbers (use the truncated variables Yn = Xn 1I{|Xn |n} , and apply the Borel–Cantelli lemma to get limn→+∞ (Xn − Yn ) = 0 a.s.). Exercise VII.5 Let 1 < p < 2 and X ∈ Lp be a centered r.v. Let (Xn )n1 be an i.i.d. sequence of r.v. each having the same distribution as X.  −1/p X converges a.s. Show that +∞ n n=1 n X1 + · · · + Xn Conclude that − −→ 0 almost surely and in Lp (this version of n→+∞ n1/p the strong law of large numbers for X is an improvement over L1 ). Exercise VII.6 1) Show that there exists a sequence (In )n1 of segments of [0, 1] whose length tends to 0 and such that all points  of [0, 1] belong to an infinity of In . 2) Show that any subsequence of 1IIn n1 contains another subsequence that converges almost everywhere to 0. 3) Show that the notion of almost sure convergence is not associated with a topology. q

Exercise VII.7 Let q be the Orlicz function ex − 1, where q  1. Show that, for X ∈ Lq , we have: Xr Xr  Xq  C sup 1/q , 1/q r1 r r1 r

C−1 sup

where C is a constant depending only on q (use Stirling’s formula). Exercise VII.8 Let Z1 , . . . , Zn n be complex i.i.d. random variables uniformly distributed on the unit circle of C (Steinhaus variables). Let A be a fixed integer  2.   √ 1) Show that E sup1knA |Z1k + · · · + Znk |  CA n log n. 2) Show that we can find complex numbers z1 , . . . , zn ∈ C, all of modulus 1, √ such that |zk1 + · · · + zkn |  CA n log n if k = 1, 2, . . . , nA . It should be noted that, even for A = 2, we do not know if the logarithmic factor can be removed. Furthermore, it can be shown (Turán, page 80) that √ we always have max1k2n |zk1 + · · · + zkn |  δ n, where δ is a numerical constant (see also Biró [2000]).

VII Exercises

45

Exercise VII.9 Let (Xn )n1 be an i.i.d. sequence of positive r.v. such that E(X1 ) = 1 and P(X1 = 1) > 0. 1) We set Mn = X1 . . . Xn . Show that (Mn )n1 is a martingale. √  2) Show that +∞ n=1 Mn < +∞ a.s. 3) Show that (Mn )n1 converges a.s. to zero, but does not converge in L1 (hence (Mn )n1 is not uniformly integrable; see Chapter 7). Exercise VII.10 Let (Xn )n1 be an i.i.d. sequence of centered r.v. each in the unit ball of L2 .  2 Let (an )n1 be a sequence of real numbers such that +∞ n=1 an < +∞. +∞ Show that the series n=1 an X1 . . . Xn converges a.s., and in L2 . Exercise VII.11 (Two alternative proofs of Theorem III.8) 1) Let (n )n1 be a sequence of characteristic functions converging to 1 on a segment [−δ, δ]. Show that then n (t) −−→ 1 for any t ∈ R (the inequality n→+∞   |n (t + h) − n (t)|2  2 1 − Re n (h) may be used). L

P

2) If Xn −−→ 0, show that Xn −−→ 0 (Let t > 0 and f : R → R+ be continuous n→+∞ n→+∞ such that f (x) = 1 if |x|  t and f (0) = 0: an upside-down triangle function; pass to the limit in the inequality P(|Xn | > t)  E[f (Xn )])). 3) Let (Xn )n1 be a sequence of independent variables such that Sn = X1 +· · ·+Xn converges in distribution. Let (nk )k1 be an increasing sequence of integers.   a) Show that there exists δ > 0 such that E eit(Snk+1 −Snk ) −−→ 1 for all k→+∞

|t|  δ. P b) Using 1) and 2) , show that Snk+1 − Snk −−→ 0, and deduce that Sn k→+∞

converges in probability. 4) Let (xn )n1 be a sequence of real numbers, and G the set of t ∈ R such that (eitxn )n1 converges. Show that G is a measurable subgroup of R and deduce that G = R as soon as G has positive measure. We now assume that eitxn −−→ 1 for all t ∈ R. Show that xn −−→ 0 n→+∞ n→+∞ $ +∞ (consider 0 e−t e−itxn dt). Suppose now that (eitxn )n1 converges for t ∈ A, a Borel subset of R with positive measure. Show that (xn )n1 converges (show that xnk+1 −xnk −−→ 0 k→+∞ for any increasing sequence of integers (nk )k1 ). 5) Use 4) (and the Fubinization principle) to fill out the proof by martingales of Theorem III.7 outlined in the remark at the end of Section V.

2 Bases in Banach Spaces

I Introduction The notion of bases in Banach spaces involves convergence of series (so it is not the notion of bases used in linear algebra); it generalizes the notion of orthonormal bases in Hilbert spaces. These bases are known as Schauder bases. The existence of bases is used to easily verify the presence of certain Banach spaces within others. In particular this will provide techniques to study the structure of Banach spaces. This chapter is divided into two sections. In Section II, we first see that any Schauder basis can be associated with a sequence of continuous projections, and then we provide a few examples. Then we show that the space C([0, 1]) is universal for the separable spaces, i.e. that any separable Banach space is isometric to a subspace of C([0, 1]). This, combined with the existence of a basis for C([0, 1]), leads us to conclude that every Banach space contains subspaces each with a basis. In Section III, we see how to use the existence of bases to obtain isomorphisms between Banach spaces; in particular we obtain a few structure theorems for the spaces p . Finally, we examine the behavior of bases with respect to duality.

II Schauder Bases: Generalities II.1 Definition. Associated Projections Definition II.1 Let X be a Banach space (real or complex). A sequence (en )n1 of non-null elements of X is said to be a (Schauder) basis for X if, for every x ∈ X, there exists a unique sequence of numbers (an )n1 = (an (x))n1 , real or complex, such that: +∞ an (x)en . x= n=1

46

II Schauder Bases: Generalities

47

The convergence is taken in the sense of the norm:   N    lim x − an (x)en   = 0. N→+∞  n=1

With the existence of a basis, the space X is necessarily separable; the definition could be adapted to non-separable spaces, but it is less interesting. If (en )n1 is a basis of X, and if λn , n  1, are non-zero scalars, then (λn en )n1 is also a basis of X. The basis is said to be normalized if en  = 1 for any n  1. By replacing (en )n1 with (en /en )n1 , a basis can always be normalized. Immediate Examples 1) Any orthonormal basis of a (separable) Hilbert space H is a basis of H. 2) For 1  p < +∞, denote:  p = x = (an )n1 ;

+∞

% |an | < +∞ p

n=1

and " c0 = x = (an )n1 ;

# lim an = 0 .

n→+∞

Define a norm on p (resp. c0 ) by setting: xp =

 +∞

1/p |an |p

(resp. x∞ = sup |an | )

n=1

n1

which makes it a Banach space. With: ⎧ e = (1, 0, 0, 0, . . .) ⎪ ⎪ ⎪ 1 ⎪ ⎨ e = (0, 1, 0, 0, . . .) 2 ⎪ e3 = (0, 0, 1, 0, . . .) ⎪ ⎪ ⎪ ⎩ .................. the sequence (en )n1 is a basis of p (resp. c0 ): every x = (an )n1 ∈ p  (resp. c0 ) can be written x = +∞ n=1 an en , since:

48

2 Bases in Banach Spaces   N    x − a e n n  = (0, 0, . . . , aN+1 , aN+2 , . . .)p  p

n=1

=

 +∞

1/p |an |p

n=N+1

−−→ 0 , N→+∞

respectively:   N   x − an en    n=1

∞

= sup |an | −−→ 0. N→+∞

nN+1

This basis is called the canonical (or natural) basis of p (respectively c0 ). We will see further examples. Whether every Banach space has a basis is a question that long remained open. It was solved in the negative by Enflo [1973]: he showed that there exist reflexive separable Banach spaces without the approximation property, a weaker property than the existence of a basis. We will give a proof, due to Davie, in Chapter 2 of Volume 2. In fact, every space p (for p = 2) contains subspaces without the approximation property (see Lindenstrauss–Tzafriri I, pages 87–90). The following property is fundamental: Theorem II.2 (Banach) A sequence (en )n1 of non-null elements of a Banach space X is a basis if and only if: (a) X = span {en ; n  1}; (b) There exists a constant K > 0 such that: m n         ak ek   K  ak ek   k=1

k=1

for any n  m, and any scalars a1 , . . . , an . In particular: Theorem II.3 defined by:

If (en )n1 is a basis of X, the projections Pm : X → X,

x=

+∞

ak ek −→ Pm (x) =

k=1

are continuous, and Pm   K for any m  1.

m k=1

ak ek ,

II Schauder Bases: Generalities

49

In particular, the linear functionals ϕm = e∗m : X → K = R or C, defined by x=

+∞

ak ek −→ ϕm (x) = am ,

k=1

are continuous. These are called the coordinate linear functionals. The number K0 = supm1 Pm  (< +∞ by the theorem) is called the constant of the basis (en )n1 . The basis is said to be monotone if K0 = 1, i.e. for any n  m:   n   m        ak ek    ak ek  .  k=1

k=1

Proof of the theorem 1) First we verify that the conditions are necessary: (a) is clear. Consider (b) : (i) We know that any x ∈ X can be written in a unique manner as x = +∞ k=1 ak (x)ek . Set:  m     |||x||| = sup  ak (x)ek  . m1

k=1

   As limm→+∞  m k=1 ak (x)ek = x, we have x  |||x||| < +∞. Moreover, we can easily see that ||| . ||| is a norm on X. (ii) We will now show that X is complete for this new norm. Let (x(r) )r1 be a Cauchy sequence in (X, ||| . |||). Each x(r) can be written: x(r) =

+∞

(r)

ak ek .

k=1

For any ε > 0, there exists rε  1 such that:   m   (r) (r )    r, r  rε ⇒  ak − ak ek    ε,

∀ m  1.

k=1

In particular, using the triangle inequality, we have, for all k  1:  (r )  r, r  rε ⇒ a(r)  2ε/ek  ; k − ak  (r)  the sequence ak r1 is thus Cauchy in K for all k  1. Let: ak = lim a(r) k . r→+∞

50

2 Bases in Banach Spaces  We will show that the series +∞ k=1 ak ek converges in X for the norm  . . Indeed, by the triangle inequality:    n  (r) (r )    a e − a r, r  rε ⇒  k   2ε , ∀ n  m. k k  k=m

Letting

r

tend to infinity, this gives:   n   (r)    a e − a r  rε ⇒  k k   2ε , ∀ n  m. k  k=m

Moreover, as the series exists mε  1 such that:

+∞ k=1

(r )

ak ε ek converges in (X,  . ), there

  n  (rε )   ⇒  ak ek    ε.

n  m  mε

k=m

We thus obtain, for n  m  mε :   n   n   n  (rε )      (rε )        ak − ak ek  +  ak ek    ak ek    2ε + ε = 3ε .  k=m

k=m

k=m

+∞

The partial sums of the series k=1 ak ek are hence Cauchy in (X,  . ), and consequently converge in (X,  . ).  (r) in Set x = +∞ k=1 ak ek . It remains to show that x = limr→+∞ x (X, ||| . |||). However, as:   m   (r) (r )    ak − ak ek  r, r  rε ⇒    ε , ∀ m  1, k=1

we obtain, by letting r tend to infinity:    m  (r)    ak − ak ek  r  rε ⇒    ε , ∀ m  1; k=1

hence: r  rε ⇒ |||x(r) − x|||  ε . (iii) Now, as X is complete for both norms and ||| . ||| is finer than  . , Banach’s isomorphism theorem says that they are equivalent; thus there exists a constant K > 0 such that: |||x|||  K x

II Schauder Bases: Generalities

51

for every x ∈ X; in other words:      +∞  m      ak ek   K  ak ek    k=1

k=1

for any m  1. The projections Pm are thus continuous, and Pm   K for any m  1.  Then condition (b) ensues, with x = nk=1 ak ek , and n  m. 2) We now examine the sufficiency of these conditions.  +∞ (i) For the uniqueness, let x = +∞ k=1 ak ek =  nk=1 bk ek . Fix m  1. For any ε > 0, there exists n  m such that  k=1 (ak − bk )ek   ε. By the triangle inequality, the condition (b) then implies:   n    |am − bm | em   2K  (a − b )e k k k   2Kε ,  k=1

which gives am = bm , because ε > 0 was arbitrary. (ii) For the existence of the development, note that the condition (b) means that the projections Pm are defined and continuous on X0 = span {ek ; k  1}, and that Pm L(X0 )  K. By density (which is the condition (a) ), they can be extended to continuous linear functionals on the whole of X while keeping the same norms. In particular, the coordinate linear functionals defined by:  e∗1 (x)e1 = P1 (x) e∗k (x)ek = Pk (x) − Pk−1 (x)

k2

are continuous on X, and moreover, for every x ∈ X, we have: Pn (x) =

n

e∗k (x)ek .

k=1

+∞

It remains only to see x = k=1 e∗k (x)ek , i.e. x = limn→+∞ Pn (x).  Indeed, for any ε > 0, there exists xm = m k=1 ck ek ∈ X0 such that x − xm   ε; for n  m, as Pn (xm ) = xm , we have then: x − Pn x  x − xm  + Pn xm − Pn x  (1 + Pn ) x − xm   (1 + K) ε , which completes the proof. Remark In the proof, we have seen that e∗k  ek  infk1 e∗k  = 0 if supk1 ek  = +∞.



2K: hence

52

2 Bases in Banach Spaces

Examples 1) The Haar basis in Lp (0, 1), 1  p < +∞ Let us define the Haar system. Set h1 (t) = 1 for any t, and for k  0 and 1  l  2k : ⎧ 2l − 2 2l − 1 ⎪ 1 if k+1 < t < k+1 , ⎪ ⎪ ⎪ 2 2 ⎨ 2l 2l − 1 h2k +l (t) = −1 if < t < k+1 , ⎪ k+1 ⎪ 2 2 ⎪ ⎪ ⎩ 0 otherwise . We show the graph of these functions in Figure 2.1 for k  2. h1

h2

1

h4

h3 1

1

1

1/2 1

–1

1/2

1

–1

–1 h5 1

1/4

1/2

1

–1 h7

h6

1

h8

1

1

1/2 1

–1

1

1

–1

1/2

–1

3/4

1

1

–1

Figure 2.1

Proposition II.4 (Schauder) Lp (0, 1) for 1  p < +∞. Remark

The Haar basis (hn )n1 is a monotone basis of

  1/p  It is not a normalized basis, as h2k +l p = 1/2k .

Proof We first verify the conditions (a) and (b) of Theorem II.2. (a) is satisfied since span {hn ; n  1} contains all the indicator functions  1  h1 + h2 of the dyadic intervals. For example: 1I]1/4,1/2[ = − h3 , almost 2 2 everywhere; more generally, for 1  l  2k we have:

II Schauder Bases: Generalities

1I 2l−2 , 2l−1  = 2k+1 2k+1

and 1I 2l−1 , 2k+1

53

 1 h2k +l + 1I l−1 , l  2 2k 2k

 1    h = , − 1 I k 2l l−1 , l 2 2 +l 2k+1 2k 2k

and then the assertion is proved by induction. (b) Let a1 , . . . , an+1 be scalars; let: f =

n

ak hk

g=

and

k=1

n+1

ak hk .

k=1

f and g differ only on a dyadic interval where f is equal to a certain constant value b, and where g is equal to b + an+1 on the first half, and b − an+1 on the second half. As |b + an+1 |p + |b − an+1 |p  2|b|p for 1  p < +∞, we obtain f p  gp . We then finish by induction. Remark We will see in Chapter 7 that the trigonometric system is also a basis of Lp (0, 1) for 1 < p < +∞; this is a consequence of a theorem of Marcel Riesz. This is no longer the case for p = 1; in other words, there are functions in L1 whose Fourier series does not converge in the L1 -norm. It was this question of Steinhaus, resolved by the young Banach, that gave rise to the Banach–Steinhaus theorem. We see this in Proposition II.5. Proposition II.5 The trigonometric system is not a basis of L1 (0, 1). Proof

We need to show that, if f ∈ L1 (0, 1), and if we set: (Pn f )(t) =

n

 f (k) e2π ikt ,

k=−n

then supn1 Pn  = +∞. But Pn is the convolution operator by the Dirichlet kernel of order n: Dn (t) =

n

e2π ikt =

k=−n

sin π(2n + 1)t , sin π t

and hence Pn  = Dn 1 (see the Annex, Corollary III.24). It is thus sufficient to show: Lemma II.6 For the Dirichlet kernel Dn of order n, we have: Dn 1 

4 log n . π2

54

2 Bases in Banach Spaces

Proof

We write:  Dn 1 =

1  sin(2n + 1)π t 

0

 

sin π t

  dt = 2



1/2  sin(2n + 1)π t 

  dt sin π t  1/2    sin(2n + 1)π t  2  dt .  πt 0 0

 

Set u = (2n + 1)t; we obtain:   2 n  sin π u  2 n+1/2  sin π u  Dn 1   du   du   π 0 u π 0 u n−1  n−1  2 k+1  sin π s  2 1 sin π s = ds   ds = π s π s+k k=0 k k=0 0  n 2 1 1 4  sin π s ds  2 log n . π k 0 π k=1 4 Note that in fact the computation gives Dn 1 = 2 log n + O(1). π 2) The Schauder basis of C([0, 1]) We obtain the Schauder basis by integrating the Haar system. More precisely, we set ϕ0 (t) = 1 and ϕ1 (t) = t for all t ∈ [0, 1], and for k  0 and 1  l  2k :  t k+1 h2k +l (u) du. ϕ2k +l (t) = 2 0

Explicitly:

ϕ2k +l (t) =

⎧ ⎪ 2k+1 t − (2l − 2) ⎪ ⎪ ⎪ ⎨ −2 ⎪ ⎪ ⎪ ⎪ ⎩ 0

k+1 t

+ 2l

2l − 2 t 2k+1 2l − 1 if k+1  t  2 otherwise if

2l − 1 2k+1 2l k+1 2

(see Figure 2.2). Proposition II.7 (Schauder)

The Schauder system is a basis of C([0, 1]).

Remark As in the case of L1 (0, 1), the norm Dn 1 of the Dirichlet kernel tends to infinity: we deduce that the trigonometric system is not a basis of C([0, 1]); hence there exist continuous functions whose Fourier series do not converge uniformly. Proof Let f ∈ C([0, 1]). We recursively define a sequence (pn )n0 of continuous functions interpolating f at dyadic points; more precisely, pn is

II Schauder Bases: Generalities j1

j2

1

j3

1

1/2

1/2

1

1

1

j8

1

1/4 1/2

1/2

1

j7

1

1/4

1

1

j6

1

j4

1

1 j5

55

1

1/2 1/4

1

1/4

1

Figure 2.2

a linear combination of ϕ0 , . . . , ϕn , and interpolates f at the dyadic points of order n; the uniform continuity of f then implies f − pn ∞ −−→ 0. Set: n→+∞

p0 = f (0) ϕ0 ,   p1 = p0 + f (1) − p0 (1) ϕ1 ,   p2 = p1 + f (1/2) − p1 (1/2) ϕ2 ,   p3 = p2 + f (1/4) − p2 (1/4) ϕ3 ,   p4 = p3 + f (3/4) − p3 (3/4) ϕ4 ,   p5 = p4 + f (1/8) − p4 (1/8) ϕ5 ,   p6 = p5 + f (3/8) − p5 (3/8) ϕ6 ,   p7 = p6 + f (5/8) − p6 (5/8) ϕ7 ,   p8 = p7 + f (7/8) − p7 (7/8) ϕ8 , and, in general, for k  0 and 1  l  2k :    2l − 1  2l − 1  − p p2k +l = p2k +l−1 + f ϕ2k +l . k 2 +l−1 2k+1 2k+1 n As pn+1 = pn + αn+1 ϕn+1 , we have pn = k=0 αk ϕk . Moreover, pn interpolates f at the dyadic points of order n. Indeed, we see that p0 (0) = f (0), as ϕ0 = 1I; next, as ϕ1 (0) = 0 and ϕ1 (1) = 1, we have p1 (0) = f (0) and p1 (1) = f (1); then p2 (0) = f (0), p2 (1) = f (1) and p2 (1/2) = f (1/2) since ϕ2 (0) = ϕ2 (1) = 0 and ϕ2 (1/2) = 1; continuing by induction, we thus obtain p2k +l ( j/2k+1 ) = f ( j/2k+1 ) for 0  j  l  2k . Hence, as said previously,  (pn )n0 converges uniformly to f , and f = +∞ k=0 αk ϕk .

56

2 Bases in Banach Spaces

Only the uniqueness of this representation remains to be shown. Indeed, if  +∞ ever +∞ k=0 αk ϕk = k=0 βk ϕk , by successively calculating the values at 0, 1, and then at (2l − 1)/2k+1 for l = 1, . . . , 2k and k = 0, 1, 2, . . ., we obtain β0 = α0 , then β1 = α1 , β2 = α2 , etc.

II.2 Universality of C([0, 1]) for Separable Spaces The fact that C([0, 1]) possesses a basis is particularly interesting because it is universal among the separable Banach spaces: Theorem II.8 (The Banach–Mazur Theorem) Every separable Banach space is isometric to a subspace of C([0, 1]). In fact, we will not work directly with [0, 1], but with other compact spaces. For this, we will need a few topological lemmas. Lemma II.9 Let K be the Cantor set and  = {0, 1}N . The mapping 

−→

(an )n0

−→

ϕ:

K +∞ 2 an 3n+1 n=0

is a homeomorphism from  onto K. The proof is left as an exercise. Remember only that the Cantor set is the compact set obtained by removing the (open) middle third of each remaining interval, starting with [0, 1] (see Figure 2.3). 1/9

2/9

1/3

2/3

0

7/9

8/9 1

Figure 2.3

Lemma II.10

The mapping σ :



−→

(an )n0

−→

[0, 1] +∞ an n=0

2n+1

is surjective and continuous. Lemma II.11 There exists a surjective continuous mapping ψ :  → [0, 1]N .

II Schauder Bases: Generalities

57

Proof We start with the continuous surjection σ :  → [0, 1] given in Lemma II.10, and define: * σ :

N (xn )n0

N

−→ −→

 [0, 1] σ (xn ) n0 .

We check that * σ is surjective and continuous, which provides ψ as  and N are homeomorphic, since N × N has the same cardinality as N: N  N = {0, 1}N = {0, 1}N×N ≈ {0, 1}N =  . Lemma II.12 For every compact metrizable space L, there exists a continuous surjection θ :  → L. This ensues that any compact metrizable space L is a continuous image of the Cantor set K. Proof 1) First we show that L is homeomorphic to a closed subset of [0, 1]N . Indeed, we can find a sequence of continuous functions fn : L → [0, 1] that separates the points of L (for example fn (x) = δ(x, xn ), where (xn )n0 is a dense sequence in L and δ is a metric ( 1) defining the topology of L). Then the mapping: −→  [0, 1]N −→ fn (x) n0

L x

provides a homeomorphism from L onto a closed subset F of [0, 1]N . 2) Now we use the continuous surjection ψ of Lemma II.11 to construct a continuous mapping ϕ from  onto G = ψ −1 (F), in order to obtain:  = {0, 1}N

ψ

/ [0, 1]N 

ϕ

 G = ψ −1 (F)

ψ|G

/ F ≈ L.

Define: +∞   |an − bn | d (an )n0 , (bn )n0 = · 3n+1 n=0

Then d is a metric on , defining the topology (product topology) of . By compactness, for every x ∈  there exists y ∈ G such that d(x, y) = d(x, G). Moreover this y is unique: d(x, y) = d(x, z) implies y = z; indeed:

58

2 Bases in Banach Spaces +∞ |xn − yn | n=0

3n+1

=

+∞ |xn − zn | n=0

3n+1

,

and as xn , yn , zn = 0 or 1, we also have |xn − yn |, |xn − zn | = 0 or 1; the equality of the two sums is only possible if |xn − yn | = |xn − zn | for any n  0, which implies xn − yn = xn − zn for any n, because xn , yn , zn = 0 or 1, and hence yn = zn for all n. Then we define ϕ by setting ϕ(x) = y. It remains to show that the mapping ϕ :  → G thus defined is continuous (it is clearly surjective because   ϕ(x) = x if x ∈ G). Let xk −−→ x in , and let y be a cluster point of ϕ(xk ) k0 in G. k→+∞

Modulo an extraction, we can assume that ϕ(xk ) −−→ y . If z ∈ G, we have k→+∞   d xk , ϕ(xk )  d(xk , z); hence, by passing to the limit, d(x, y )  d(x, z); shown above means that y = ϕ(x). Hence, the sequence  the uniqueness ϕ(xk ) k0 has only a single cluster point ϕ(x), and it thus converges to ϕ(x). Consequently, ϕ is continuous. Proposition II.13 of C().

Every separable Banach space is isometric to a subspace

First, we present a very easy, albeit useful, lemma (use the mapping J : X → C(L), x → x˜ = ϕx ) : Lemma II.14 Every Banach space  X is∗ isometric to a subspace of C(L), where L is the compact space L = BX ∗ , w . In general this compact space is very large, without a nice representation. However, when X is separable, Lemma II.12 can be used. Proof of Proposition II.13 As L is compact and metrizable, Lemma II.12 provides a continuous surjection θ :  → L, that induces a mapping: T:

C(L) f

−→ C() −→ f ◦ θ .

This mapping is an isometry since θ is surjective. The composition T ◦ J is then an isometry from X onto C(). We are now ready to complete the proof: Proof of the Banach–Mazur theorem As the Cantor set K is homeomorphic to , Proposition II.13 provides an isometry U : X → C(K). It thus suffices to prove the existence of an isometry V : C(K) → C([0, 1]). First [0, 1]  K is a union of disjoint open intervals In , n  1. If f : K → K is continuous, it can be extended to a continuous function f˜ : [0, 1] → K, affine on each interval In . The mapping V : f → f˜ is indeed an isometry from C(K) onto C([0, 1]).

III Bases and the Structure of Banach Spaces

59

Corollary II.15 Every separable Banach space is isometric to a subspace of ∞ = ∞ (N). Proof It suffices to select a sequence (tn )n0 dense in   [0, 1]; we obtain an isometry W : C([0, 1]) → ∞ by setting W( f ) = f (tn ) n0 . We could in fact directly obtain an isometry of the Banach space X onto ∞ by choosing a sequence (xn∗ )n0 w∗ -dense in the unit ball of X ∗ (which is and metrizable for the weak∗ topology) and associating the sequence  compact xn∗ (x) n0 to x ∈ X. Note that this isometry into ∞ is much less interesting than the isometry into C([0, 1]), since ∞ is not separable. Moreover, the existence of a basis in C([0, 1]) allows us to deduce the existence of “basic sequences” in all Banach spaces: see the following section (Corollary III.9). Definition II.16 A sequence (ek )k1 in a Banach space is said to be a basic sequence if it is a basis of the (closed) vector space span {en ; n  1} that it generates.

III Bases and the Structure of Banach Spaces III.1 Bases and Isomorphisms Definition III.1 Let (un )n1 be a basis of a Banach space X, and let (vn )n1 be a basis of another Banach space Y. They are said to be equivalent if, for  any sequence (an )n1 of scalars, the convergence of +∞ n=1 an un is equivalent +∞ to that of n=1 an vn . Examples 1) Two orthonormal bases in (separable, infinite-dimensional) Hilbert spaces are always equivalent. 2) Let X = c be the space of convergent sequences a = (an )n1 , equipped with the uniform norm, (en )n1 the canonical basis of c0 , and e0 = 1I. Then  (en )n0 is a basis of c, and we have a = le0 + +∞ −→ l. n=1 (an − l)en if an − n→+∞ Set s0 = e0 = 1I, and for n  1: sn = (0, 0, . . . , 1, 1, . . .) = e0 − (e1 + · · · + en ) . + ,- . n times

The sequence (sn )n0 is a basis of c, called the summing basis, because: n

bk sk = (b0 , b0 +b1 , . . . , b0 +· · ·+bn−1 , b0 +· · ·+bn , b0 +· · ·+bn , . . .) .

k=0

The bases (en )n0 and (sn )n0 are not equivalent.

60

2 Bases in Banach Spaces

This notion allows us to easily obtain isomorphisms between Banach spaces. Proposition III.2 Let (un )n1 be a basis of X and (vn )n1 a basis of Y. The following three properties are equivalent: 1) (un )n1 and (vn )n1 are equivalent; 2) there exists an isomorphism T : X → Y such that T(un ) = vn for any n  1; 3) there exists a constant C > 0 such that:       n  n   n   1         C a u a v a u k k k k k k    C k=1

k=1

k=1

for any n  1 and any scalars a1 , . . . , an ∈ K. Note the following implication: the normalization of a basis provides an equivalent basis only if infn1 un  > 0 and supn1 un  < +∞. Proof Clearly that 2) ⇔ 3) ⇒ 1) . Let us show: 1) ⇒ 2) . As the bases are equivalent, we can define T : X → Y by:   +∞ +∞ an un = an vn . T n=1

n=1

T is bijective; for a proof of its continuity (and of T −1 ), we prove that the graph of T is closed. We need to show that if x( j) −−→ x and Tx( j) −−→ y, then y = Tx. Consider j→+∞

j→+∞

the coordinate linear functionals u∗n and v∗n , n  1, associated with the bases (un )n1 and (vn )n1 in X and Y respectively. They are known to be continuous; hence, for any n  1:     u∗n x( j) −−→ u∗n (x) and v∗n Tx( j) −−→ v∗n (y) . j→+∞

j→+∞

    However, by the definition of T, v∗n Tx( j) = u∗n x( j) ; thus u∗n (x) = v∗n (y). As this is true for any n  1, again by the definition of T, it ensues that y = Tx. A very convenient method to verify the equivalence of bases is given now: Theorem III.3 (The Bessaga–Pełczy´nski Equivalence Theorem) Let (un )n1 be a basis of X, and (u∗n )n1 the sequence of coordinate linear functionals. Let (vn )n1 be a sequence of (non-null) vectors of X. If: (1)

+∞

u∗n  un − vn  < 1 ,

n=1

then (vn )n1 is also a basis of X, and it is equivalent to (un )n1 .

III Bases and the Structure of Banach Spaces

61

If (un )n1 is simply a basic sequence of X, (vn )n1 will also be a basic sequence of X, equivalent to (un )n1 . Moreover, if there exists a projection P : X → span {un ; n  1}, such that +∞

(2)

u∗n  un − vn 
0, for any k  1. Denote by K the constant of the basis (en )n1 , and set p1 = 1. As limk→+∞ e∗1 (yk ) = 0, there exists k1  1 such that: α · e∗1 (yk1 ) e1   2 × 8K  ∗ As the series yk1 = +∞ n=1 en (yk1 ) en converges, there exists p2 > p1 such that:  +∞   ∗  α   e (y ) e · n k1 n    2 × 8K n=p2 +1

64

2 Bases in Banach Spaces

Now, for n = 1, 2, . . . , p2 , we have limk→+∞ e∗n (yk ) = 0; thus there exists k2 > k1 such that:  p2   ∗  α   · e (y ) e n k2 n   2  2 × 8K n=1

As above, there exists p3 > p2 such that:   +∞   ∗ α   · e (y ) e n k2 n   2  2 × 8K n=p3 +1

Continuing this process, we construct two strictly increasing sequences of integers: 1 = p1 < p2 < · · · and 1  k1 < k2 < · · · such that:  pj  +∞    ∗    α α ∗    and  en (ykj ) en   j en (ykj ) en     2j × 8K · 2 × 8K n=pj+1 +1

n=1

This is illustrated in Figure 2.4.

pj

pj+ 1

Figure 2.4

Set: zj =

pj+1

e∗n (ykj ) en .

n=pj +1

 0; indeed: Then (zj )j1 is a block basis of the basis (en )n1 as zj =     p   +∞   j ∗ ∗    − zj   ykj  −  e (y ) e e (y ) e n kj n  n kj n    n=1

α−

n=pj+1 +1



α α 1  7 − = 1− α  α, 2 × 8K 2 × 8K 8K 8

since K = supm1 Pm   1 as the relation Pm = P2m implies Pm   1. The equivalence of (zj )j1 and (ykj )j1 remains to be proved.

III Bases and the Structure of Banach Spaces

65

With z∗j , j  1, the coordinate linear functionals, as the constant of the basis (zj )j1 is  K, we know that: z∗j  zj   2K ; hence: z∗j  

16K 2K  · zj  7α

Then we calculate: +∞ j=1

z∗j  zj − ykj    pj +∞    16K  ∗   en (ykj ) en    + 7α j=1



16K 7α

n=1

 +∞ j=1

2j

α α + j × 8K 2 × 8K

+∞ n=pj+1 +1



=

 

e∗n (ykj ) en  

4 < 1. 7

We complete the proof using the Bessaga–Pełczy´nski equivalence theorem: (ykj )j1 is a basic sequence equivalent to (zj )j1 . Remark Corollary III.8 cannot be inferred from Corollary III.7; indeed, even though in infinite dimensions the weak topology is always strictly coarser than the norm topology, there are nonetheless infinite-dimensional spaces in which every weakly convergent sequence converges automatically in norm. Such spaces are said to possess the Schur property. The classical example is 1 ; we will obtain this result as a consequence of the Bessaga–Pełczy´nski selection theorem, even though the same idea of a “gliding hump” could be used again for a direct proof. Theorem III.10 (Schur’s Theorem) Every weakly convergent sequence in 1 converges in norm. Proof Suppose there is a sequence converging weakly to 0 without converging in norm: then by the Bessaga–Pełczy´nski selection theorem, we can extract a subsequence (ykj )j1 equivalent to a block basis of the canonical basis (en )n1 of 1 . As (ykj )j1 does not converge in norm to 0, we can assume that infj1 ykj  > 0. Moreover, supj1 ykj  < +∞, since any sequence that converges weakly is bounded. However, any such block basis of (en )n1 is equivalent to (en )n1 (see also Proposition III.12 below). This implies that (en )n1 , as well as (ykj )j1 , converge weakly to 0. But this cannot be the case: when λ = (λn )n1 ∈ ∞ , in general λ, en = λn does not have a limit.

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2 Bases in Banach Spaces

Then, by the Eberlein–Šmulian theorem, we obtain: Corollary III.11 In 1 , the weakly compact subsets are compact in norm. Therefore, 1 does not contain any infinite-dimensional reflexive subspaces. But in fact we can go further. Proposition III.12 If X is one of the spaces c0 or p with 1  p < +∞, and if (en )n1 is its canonical basis, every normalized block basis (un )n1 of (en )n1 is equivalent to (en )n1 and the space U = span{un ; n  1} is isometric to X. Moreover, there exists a projection P : X → U of norm 1. Proof

We treat p ; the case of c0 requires only a change of notation. We write:

qn+1

un = As un p = 1, we have

ck ek .

k=qn +1

qn+1

k=qn +1 |ck |

p

= 1; thus:

  N 1/p  qn+1 N   p p  =  a u |a | |c | n n n k   p

n=1

k=qn +1

n=1

=

 N

1/p |an |p

    N  = a e n n . 

n=1

n=1

p

The equivalence of the bases and the isometry between U and p ensue. To construct the projection, select v∗n ∈ ∗p = q such that: (i) v∗n ∈ span {e∗qn +1 , . . . , e∗qn+1 }, (ii) v∗n q = v∗n (un ) = 1, and set, for every x ∈ p : P(x) =

+∞

v∗n (x) un .

n=1

Then P is a projection from p onto U, and P = 1: indeed, with qn+1  x = +∞ k=1 ak ek and xn = k=qn +1 ak ek , we have: P(x)pp

=

+∞ n=1

|v∗n (x)|p

=

+∞

|v∗n (xn )|p

n=1

 +∞ +∞  q n+1  ∗ p vn q xn p =  |ak |p = xpp . n=1

n=1

k=qn +1

III Bases and the Structure of Banach Spaces

67

Thanks to Corollary III.8 and the Bessaga–Pełczy´nski selection theorem, we now infer the following result: Theorem III.13 Let X be one of the spaces c0 or p , with 1  p < +∞. Then any infinite-dimensional subspace Y of X contains a subspace Z, isomorphic to X and complemented in X (and hence also in Y). In particular, we again see that 1 does not have any infinite-dimensional reflexive subspace. Corollary III.14 If X is one of the spaces c0 , or p , with 1  p < +∞, no subspace of X is isomorphic to one of these spaces, other than X. Proof Let Y ⊆ X. We split this into two cases. First, assume that Y ≈ Y0 = c0 or p with 1 < p < +∞. As the canonical basis of Y0 converges weakly to 0, so does its image (un )n1 in Y. We can then extract a subsequence equivalent to a block basis of the canonical basis of X. But this subsequence remains equivalent to the canonical basis of Y0 . It follows that the canonical basis of Y0 is equivalent to a block basis of the canonical basis of X, and, by the preceding proposition, is hence equivalent to this latter canonical basis; this is only possible if X = Y0 . If Y ≈ Y0 = 1 , by the theorem, there exists a subspace Z of Y isomorphic to X. This is only possible when X = 1 , otherwise Z would contain a sequence (the image of the canonical basis of c0 or p with 1 < p < +∞) converging weakly to 0, but not in norm: this cannot be possible since Y has the Schur property. As we have previously mentioned, each of these spaces contains subspaces without bases. These subspaces cannot be complemented. Indeed: Theorem III.15 (Pełczy´nski) If X is one of the spaces c0 , or p , with 1  p < +∞, any infinite-dimensional complemented subspace of X is isomorphic to X. Proof

Let Y be a complemented subspace of X: X = Y ⊕ X1 .

By the preceding theorem, Y contains a subspace Z complemented in X, hence in Y, and isomorphic to X. We thus have: Y = Z ⊕ Y1 . ∼

Then (≈ meaning “isomorphic” and = “isometric”): X ⊕ Y ≈ X ⊕ (Z ⊕ Y1 ) ≈ (X ⊕ Z) ⊕ Y1 ≈ (Z ⊕ Z) ⊕ Y1 ≈ Z ⊕ Y1 = Y .

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2 Bases in Banach Spaces ∼

But, on the other hand, as we have X = (X ⊕ X ⊕ . . .)p if X = p , and ∼ X = (X ⊕ X ⊕ . . .)c0 if X = c0 , this gives:   ∼ X ⊕ Y = (X ⊕ X ⊕ . . .)X ⊕ Y ≈ (Y ⊕ X1 ) ⊕ (Y ⊕ X1 ) ⊕ . . . X ⊕ Y ≈ (X1 ⊕ X1 ⊕ . . .)X ⊕ (Y ⊕ Y ⊕ . . .)X ⊕ Y ≈ (X1 ⊕ X1 ⊕ . . .)X ⊕ (Y ⊕ Y ⊕ . . .)X  ∼ ≈ (X1 ⊕ Y) ⊕ (X1 ⊕ Y) ⊕ . . .)X = X . In conclusion, X ≈ Y. This is known as the Pełczy´nski decomposition method; it can be used in many different contexts. Remark The block basis property of c0 and p is characteristic of these spaces; in fact, Zippin [1966] showed the following result: Theorem Any Banach space X that possesses a basis equivalent to all its normalized basic blocks is isomorphic to c0 or to p with 1  p < +∞.

III.3 Bases and Duality Proposition III.16 Let X be a Banach space with a basis (en )n1 , of constant K, and let (e∗n )n1 be the sequence of coordinate linear functionals. Then: 1) e∗n   2K/en , for any n  1; 2) (e∗n )n1 is a basic sequence in X ∗ , of constant  K. Moreover, the projections associated with this basic sequence are the adjoints of the projections associated with (en )n1 . Proof We have already seen 1) . For 2) , let Pn , n  1, be the projections associated with the basis (en )n1 : Pn (x) =

n

e∗k (x) ek .

k=1

We have: Pn   K for any n  1. Let P∗n be the adjoint operator of Pn . It is a projection, and for m  n, we have:  0 / m m n /  0 P∗n ak e∗k , x = ak e∗k , Pn (x) = ak e∗k (x) ; k=1

k=1

hence: P∗n

 m k=1

ak e∗k

k=1

 =

n k=1

ak e∗k .

III Bases and the Structure of Banach Spaces

69

Since P∗n  = Pn   K, we have proved that (e∗k )k1 is a basis of the space span {e∗k ; k  1}, of constant  K, and that the P∗n ’s, n  1, are the associated projections. Now let us study the following problem: when do (e∗k )k1 form a basis of X ∗ ? This cannot always be the case, as this would require X ∗ to be separable. For example, it cannot be the case when X = 1 or X = C([0, 1]). In fact, there exist spaces with bases and with separable dual such that these dual spaces do not even have the approximation property (Lindenstrauss [1971]; see Lindenstrauss–Tzafriri I, Theorem 1.e.7). However, if X possesses a basis and if X ∗ is separable and has the approximation property, then X ∗ has a basis (Johnson, Rosenthal and Zippin [1971]). According to the preceding proposition, (e∗k )k1 is a basis of X ∗ if and only if span {e∗k ; k  1} = X ∗ . Indeed, we always have a “weak” version (or rather weak∗ !): Proposition III.17 Let (en )n1 be a basis of X. Then for every x∗ ∈ X ∗ , we have: x∗ = w∗ -

+∞

x∗ (ek )e∗k .

k=1

In other words, the series converges to x∗ in the sense of the weak∗ topology w∗ = σ (X ∗ , X). Proof We need to show that x∗ = w∗ - limn→+∞ P∗n x∗ , and this is easy: for every x ∈ X: | x∗ − P∗n x∗ , x | = | x∗ , x − Pn x |  x∗  x − Pn x −−→ 0 . n→+∞

The following criterion deals with convergence in norm in Proposition III.17. Proposition III.18 Let (en )n1 be a basis of X. The coordinate linear functionals e∗n , n  1, form a basis of X ∗ if and only if we have, for every x∗ ∈ X ∗ :     ∗ x  span{e ; kn}  −−→ 0 . n→+∞ k Definition III.19

A basis satisfying this property is said to be shrinking.

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2 Bases in Banach Spaces

Example The canonical basis of c0 is shrinking as (c0 )∗ = 1 , and with x∗ = (λn )n1 ∈ 1 , we indeed have:   ∗ x  span {e

k

+∞   |λk | −−→ 0 . = ; kn} n→+∞

k=n

We can also verify directly that the coordinate linear functionals are the vectors of the canonical basis of 1 . Proof of Proposition III.18 1) If (e∗n )n1 is a basis of X ∗ , then for any x∗ ∈ X ∗ , we have: P∗n−1 x∗ − x∗  −−→ 0 . n→+∞

  As (P∗n−1 x∗ ) span {e ; kn} = 0, we get x∗  span {e ; kn}  −−→ 0. n→+∞ k k   2) Conversely, assume x∗  span {e ; kn}  −−→ 0. Then, for any x ∈ X, we n→+∞

k

have: | x∗ − P∗n x∗ , x | = | x∗ , (I − Pn )x |  x∗  span {e  x∗  span {e

k

; kn+1}

k

; kn+1}

 (I − Pn )x

 (K + 1)x ;

thus: x∗ − P∗n x∗   (K + 1) x∗  span {e

k

; kn+1}

 −−→ 0 . n→+∞

The “dual” notion is: Definition III.20 A basis (εn )n1 of a Banach space is said to be boundedly  an εn converges as soon as complete if the series n1

 n     sup  a ε k k  < +∞ . 

n1

Example

k=1

The canonical basis of 1 is boundedly complete because:   +∞ n  n   = sup sup  a ε |a | = |ak | < +∞ . k k k  n1

k=1

n1 k=1

k=1

However, the canonical basis of c0 is not:    n   sup  e k  = sup (1, 1, . . . , 1, 0, 0, . . .)∞ = 1 .  n1 n1 + ,- . ∞ k=1

n times

III Bases and the Structure of Banach Spaces

71

Theorem III.21 1) Let (en )n1 be a shrinking basis of a Banach space X. Then (e∗n )n1 is a boundedly complete basis of X ∗ . 2) Let Y be a Banach space possessing a boundedly complete basis (εn )n1 . Then Y is isomorphic to a dual space. More precisely, Y ≈ X ∗ , where X = span {εn∗ ; n  1}. Moreover, the en = εn∗ , n  1, form a shrinking basis of X, and e∗n = Jεn , where J : Y → X ∗ is the mapping defined by (Jy)(x) = x(y) for y ∈ Y and x ∈ X. The result 2) is due to Alaoglu [1940]. Proof 1) We have already seen that (e∗n )n1 is a basis of X ∗ . It remains to see that it is boundedlycomplete. Thus let an , n  1, be scalars such that  n ∗ supn1  nk=1 ak e∗k  < +∞. Set xn∗ = k=1 ak ek . By hypothesis, the ∗ sequence (xn )n1 is bounded. We will use the fact that (closed) balls of X ∗ are w∗ -compact. Let x∗ be a w∗ -cluster point of the sequence (xn∗ )n1 . As (e∗n )n1 is a basis of X ∗ , we know that: x∗ =

+∞

x∗ (ek ) e∗k

k=1

(for the convergence in norm). On the other hand, by definition of the ∗ ∗ topology  σ (X , X), for any k  1, x (ek ) is a cluster point of the sequence  ∗ xn (ek ) n1 . Since:  a if k  n = k 0 if k > n ,  we obtain x∗ (ek ) = ak . The series k1 ak e∗k is thus indeed convergent (in norm). Moreover, x∗ is the only w∗ -cluster point of (xn∗ )n1 , and hence its limit. 2) Let X = span {εn∗ ; n  1} ⊆ Y ∗ and let J : Y → X ∗ be the mapping defined by (Jy)(x) = x(y). We will see that J is an isomorphism. Of course  Jy  y. We now seek a minoration. Let y = nk=1 ak εk . There exists y∗ ∈ Y ∗ such that y∗  = 1 and y∗ (y) = y. Let Qk , k  1, be the projections associated with the basis (εk )k1 . For any z ∈ Y, we have: xn∗ (ek )

Q∗n (y∗ )(z) = y∗ , Qn z = y∗ , z =

n k=1

εk∗ (z) y∗ (εk ) ,

72

2 Bases in Banach Spaces as Qn (z) =

n

∗ k=1 εk (z) εk ;

hence:

Q∗n y∗ =

n

y∗ (εk ) εk∗ ,

k=1

and therefore

Q∗n y∗

∈ X. Since:

(Jy)(Q∗n y∗ ) = (Q∗n y∗ )(y) = y∗ , y = y , we obtain: 1 y , K because Q∗n y∗   Q∗n  y∗   K, K being the constant of the basis 1 (εn )n1 . By density, we thus have Jy  y for every y ∈ Y. K It remains to show that J is onto. First, note that (εn∗ )n1 is a basis of X of constant  K: indeed, it is a basic sequence in Y ∗ and by definition X = span {εn∗ ; n  1}. Also, note that the associated coordinate linear functionals are the Jεn , n  1. Now let x∗ ∈ X ∗ . We have, for any x ∈ X:  n    n  ∗ ∗    ∗  ∗   x (εk ) (Jεk )(x)  x   (Jεk )(x) εk   x∗  Kx ;  Jy 

k=1

hence:

k=1

   n ∗ ∗  ∗  x (εk ) εk    Kx  .  k=1

As the basis (εn )n1 is boundedly complete, the series converges in Y.  ∗ ∗ ∗ Denote y = +∞ k=1 x (εk ) εk . Clearly Jy = x (since they coincide on the ∗ elements of the basis (εn )n1 ): the surjectivity of J is thus proved. To conclude, the shrinking nature of the basis (εn∗ )n1 follows from the fact that the coordinate linear functionals e∗n = Jεn , n  1, form a basis of X ∗ = JY. We can now state James’ theorem: Theorem III.22 (James) Let X be a Banach space with a basis. Then X is reflexive if and only if this basis is at the same time shrinking and boundedly complete. Proof

Denote this basis by (en )n1 .

1) First, assume that X is reflexive. Then: (a) We have seen that, for any x∗ ∈ X ∗ , we always have: x∗ = w∗ -

+∞ n=1

x∗ (en ) e∗n .

III Bases and the Structure of Banach Spaces

73

As X is reflexive here, the weak∗ topology w∗ = σ (X ∗ , X) coincides with the weak topology w = σ (X ∗ , X ∗∗ ). It then follows that X ∗ is equal to the weak closure of the space span {e∗n ; n  1}. But this closure is the same as closure in norm. Thus (e∗n )n1 is a basis of X ∗ , and consequently (en )n1 is shrinking. (b) We have just seen that (e∗n )n1 is a basis of X ∗ . As X ∗ is also reflexive, (e∗n )n1 is a shrinking basis of X ∗ , by a) . By 1) of Theorem III.21, the associated coordinate linear functionals form a boundedly complete basis of X ∗∗ = X. These coordinate linear functionals are precisely the en , n  1, so (en )n1 is boundedly complete. 2) Conversely, assume that the basis (en )n1 is shrinking and boundedly complete. The conclusion will ensue from the next proposition: indeed, by hypothesis  the basis is also boundedly complete, hence k1 x∗∗ (e∗k ) ek will converge (in norm) in X; thus x∗∗ will be in X, and consequently X ∗∗ = X. Proposition III.23 Let X be a Banach space with a shrinking basis (en )n1 . Then, for any x∗∗ ∈ X ∗∗ , we have:   a) supn1  nk=1 x∗∗ (e∗k ) ek  < +∞ ; +∞ b) x∗∗ = w∗ x∗∗ (e∗k ) ek . k=1

Proof of Proposition III.23 As (en )n1 is shrinking, (e∗n )n1 is a basis of X ∗ . When the Pn , n  1, are the projections associated with the basis (en )n1 , the P∗n , n  1, have been seen to be the projections associated with the basis (e∗n )n1 . Thus, if x∗∗ ∈ X ∗∗ , for any x∗ ∈ X ∗ , we have: 1 2 n ∗∗ ∗∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ Pn x , x = x , Pn x = x , x (ek )ek ; k=1

hence: ∗∗ a) P∗∗ n x =

n

k=1 x

∗∗ (e∗ ) e , k k

and we indeed have:

n     ∗∗ ∗∗ ∗∗ sup  x∗∗ (e∗k ) ek  = sup P∗∗ n x   sup Pn  x 

n1

k=1

n1

n1

= sup Pn  x∗∗  = Kx∗∗  < +∞; n1

since P∗n x∗ − x∗  −−→ 0 . n→+∞  n  ∗   Remark Conversely, k=1 ak ek < +∞, any w -cluster point  ifn supn1 ∗ ∗∗ ∗∗ x of the sequence k=1 ak ek n1 will satisfy x , ej = aj for any j  1,

b)

∗∗ ∗ P∗∗ −→ x∗∗ , x∗ , n x , x − n→+∞

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2 Bases in Banach Spaces

 n  ∗ ∗ as e∗j k=1 ak ek = aj as soon as n  j. Since X = span {en ; n  1}, there  is only a single cluster point, and x∗∗ = w∗ - +∞ k=1 ak ek . To conclude this chapter, we state the previously mentioned results: Theorem A (Lindenstrauss) There exist Banach spaces X possessing bases, and with separable dual X ∗ , but such that X ∗ have no basis (and not even the approximation property). Theorem B (Johnson–Rosenthal–Zippin) If X is a Banach space such that X ∗ has a basis, then X possesses a shrinking basis. The proof of this last theorem is delicate, and makes use of the structure of the finite-dimensional subspaces of X.

IV Comments 1) The notion of basis was introduced by Schauder [1927], who also proved Proposition II.7; Proposition II.4 is found in Schauder [1928] (see Banach, pages 110–112 and page 238). The problem of the existence of bases appeared in 1932 in Banach, page 111: “On ne sait pas si tout espace de type (B) séparable admet une base” (in English: “We do not know if every separable space of type (B) possesses a basis”); it was then presented as problem 153 in the Scottish Book on November 6, 1936, by Mazur, who offered a live goose for its resolution. Enflo’s solution won him a goose, awarded to him in person by Mazur at the time of his presentation in Warsaw in 1972 (see Roman Kalu˙za, The Life of Stefan Banach, Birkhäuser, 1996).1 The textbook of Singer, even if a bit old, is still an invaluable reference about bases. For a treatment of bases in function spaces, see Figiel and Wojtaszczyk [2001]; see also Chapter 7. 2) The Banach–Mazur theorem can be found in Banach, Chapter XI § 8. Easily, if K and H are two homeomorphic compact sets, then C(K) is isometric to C(H). The Banach–Stone theorem states that the converse is true (see Exercise V.7); more generally, it remains true when dist[C(K), C(H)] < 2 (Amir [1965] and Cambern [1967]; see Benyamini [1981] for an extension to the case of non-surjective isometries). Without precision on the distance, this is no longer the case: for any uncountable compact metric 1 W. Zelazko ˙ reported in a conference at the Université d’Artois in Lens (France) in 2014 that

P. Wojtaszczyk, then a student of Mazur, had to kill the goose, and that Mrs. Mazur then cooked it.

IV Comments

3)

4)

5) 6)

75

space K, C(K) is isomorphic to C([0, 1]) (Milyutin [1966], but proved as early as 1951 in his thesis). As for the countable compact spaces, they are not all isomorphic; the classification was done by Bessaga and Pełczy´nski [1960]: if α and β are two countable ordinals, and α  β, then C([0, α]) is isomorphic to C([0, β]) if and only if β < α ω . In particular, the space of convergent sequences c = C([0, ω]) is not isomorphic to C([0, ωω ]) (see Chapter 3, Exercise VIII.6). For more information about spaces of continuous functions, one may turn to Lacey, Chapters 2 and 3, or to Semadeni. The results of Subsection III.1 as well as Theorem III.6 and its corollaries are due to Bessaga and Pełczy´nski [1958 a]; Corollary III.9 is announced in Banach, page 238: “Remarquons toutefois que tout espace de type (B) à une infinité de dimensions renferme un ensemble linéaire fermé à une infinité de dimensions qui admet une base” (in English: “We remark nonetheless that any infinite-dimensional space of type (B) contains an infinite-dimensional closed linear set that possesses a basis”). However, the first proofs were not published until 1958 (Bessaga and Pełczy´nski [1958 a], Gelbaum [1958] and Day); one could refer to Megginson, page 361, for further comments on this subject. The notion of a boundedly complete basis was introduced by Dunford and Morse [1936]. Theorem III.15 is in Pełczy´nski [1960]. The Pełczy´nski decomposition method is not applicable to every Banach space: Gowers [1996 a] constructed a Banach space X isomorphic to X ⊕ X ⊕ X, but not to X ⊕ X, so that X and X ⊕ X are each isomorphic to a complemented subspace of the other, but are not isomorphic to each other. Another construction can be found in Gowers and Maurey [1997]. For 1  p < +∞, p = 2, there exist subspaces of p isomorphic to p , but not complemented (Rosenthal [1970], Bennett, Dor, Goodman, Johnson and Newman [1977] and Bourgain [1981 b]). The theorems of Subsection III.3 are essentially due to James [1950]. Theorem A is due to Lindenstrauss [1971], and can be found in Lindenstrauss–Tzafriri I, Theorem 1.e.7; Theorem B is due to Johnson, Rosenthal and Zippin [1971]. In connection with this result, Zippin [1988] showed that any Banach space having a separable dual could be isomorphically embedded in a space with a shrinking basis (see Lindenstrauss–Tzafriri I, Problem 1.b.16; a generalization, with an alternative proof, can be found in Ghoussoub, Maurey and Schachermayer [1992]). Zippin also showed that any reflexive separable space could be embedded in a reflexive space possessing a basis.

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2 Bases in Banach Spaces

7) Between the notions of basis and of approximation property, numerous intermediate notions have been introduced. The closest is that of FDD (finite-dimensional decomposition): a sequence of finite-dimensional subspaces Xn of the Banach space X is said to be an FDD if any x ∈ X  can be uniquely written as x = +∞ n=1 xn , with xn ∈ Xn for each n  1. The notion of a basis thus corresponds to the case where each Xn is of dimension 1. As in the case of bases, the projections x → xn are continuous and with uniformly bounded norms. When each Xn possesses a basis, with a uniformly bounded constant for n  1, then clearly X possesses a basis. Szarek [1987] constructed a Banach space with FDD, but without a basis. A particularly interesting result is the following: for any separable infinite-dimensional Banach space X, there exists a subspace Y such that Y and X/Y both possess an FDD (Johnson and Rosenthal [1972]; see also Lindenstrauss–Tzafriri, Theorem 1.g.2). We refer to Casazza [1986] or Casazza [2001] for an exhaustive development of this subject. A separable Banach space X is said to have the bounded approximation property (BAP) if there exists a sequence of operators of finite rank converging pointwise to the identity. Pełczy´nski [1971] and Johnson and Rosenthal [1972] independently showed that a separable Banach space has the BAP if and only if it is isomorphic to a complemented subspace of a space possessing a basis (see Lindenstrauss–Tzafriri I, Theorem 1.e.13; see also Pełczy´nski and Wojtaszczyk [1971] with a very simple proof that X has the BAP if and only if it is complemented in a space having an FDD). There exist Banach spaces having the BAP, but without an FDD (Read, unpublished work; see Casazza and Kalton [1990]). For other comments, we refer to Chapters 3 (Volume 1) and 2 (Volume 2).

V Exercises Exercise V.1 1) Show that any space possessing a basis (en )n1 can be re-normed (i.e. has an equivalent norm) so that the new basis constant is 1 (show that |||x||| = supn1 Pn x is satisfactory). 2) Show that the unit ball * BX ∗ of X ∗ for this new norm ||| . ||| is equal to BX ∗ . Deduce then that N = the weak∗ closure of span {e∗n ; n  1} ∩ * ∗ span {en ; n  1} norms X: there exists a > 0 such that: sup ϕ∈N,ϕ1

for every x ∈ X.

|ϕ(x)|  a x

V Exercises

77

Exercise V.2 Let (An )n1 be a measurable partition of [0, 1] in sets of positive   1 1IAn measure; show that, in Lp (0, 1), 1  p < +∞, the sequence n1 λ(An ) is isometrically equivalent to the canonical basis of p . Exercise V.3 Show that, if X possesses an FDD, the associated projections are continuous and their norms are uniformly bounded. Exercise V.4 Let T : X → c0 be a continuous linear mapping. We denote by (en )n1 the canonical basis of 1 . 1) Show that T is compact if and only if limn→+∞ T ∗ (en ) = 0. 2) Show that T is weakly compact if and only if w∗ -limn→+∞ T ∗ (en ) = 0. Exercise V.5

Let 1  p < +∞ and p = 2.

1) Show that, if x, y ∈ p and x + yp = x − yp = xp + yp , then (supp x) ∩ (supp y) = ∅ (see also Chapter 7, Exercise VII.7). 2) Deduce that, if T : p → p is a surjective isometry and if (en )n1 is the canonical basis of p , then (supp Tek ) ∩ (supp Tel ) = ∅ for k = l. 3) Deduce that there exist a permutation π of integers  and signs εn , real or  complex: |εn | = 1 such that T (xn )n1 = εn xπ(n) n1 . 4) Use the same method to show that the result holds for p = +∞. Deduce that the same holds for any surjective isometry T : c0 → c0 . 5) Is it true for 2 ? Exercise V.6 Show that if K is a countable compact space, then C(K)∗ is isometric to 1 . Exercise V.7 1) Show that if H and K are two homeomorphic compact spaces, then C(H) and C(K) are isometric. 2) a) For any compact space K, determine the set of extreme points of M(K) = C(K)∗ . b) Show that  K = {δt ; t ∈ K}, equipped with the weak∗ topology, is homeomorphic to K. 3) Let T : C(K) → C(H) be an isometric isomorphism. a) Show that, for every x ∈ H, there exist, in a unique manner, σ (x) ∈ K and a scalar α(x) such that |α(x)| = 1 and T ∗ (δx ) = α(x) δσ (x) . b) Show that the mapping σ : H → K appearing in a) is bijective. c) Show that the mapping α is continuous (show that α = T1I), and deduce that f ◦ σ ∈ C(H) for every f ∈ C(K), and then that σ is continuous. d) Show that H and K are homeomorphic.

78

2 Bases in Banach Spaces

Exercise V.8 1) Let X be a Banach space such that (X ⊕ R)∞ is isometric to a subspace of X. By induction, construct a subspace Y of X isometric to c0 . 2) Deduce that there does not exist any reflexive and separable Banach space such that every reflexive and separable Banach spaces is isometric to a subspace of X (Lindenstrauss [1966 b]). Szlenk [1968] showed that this holds even when “isometric” is replaced by “isomorphic.” Exercise V.9 Show that, for 1  p < r < +∞, every operator T : r → p is compact (Pitt’s theorem, Pitt [1936]): show that, if T were not compact, the Bessaga–Pełczy´nski selection theorem and Proposition III.12 would imply the existence of a sequence (xn )n1 in r equivalent to the canonical basis of r and such that (Txn )n1 is equivalent to the canonical basis of p , which is not possible. Show similarly that every operator T : c0 → p is compact. Exercise V.10 An operator T : X → Y is said to be strictly singular if there does not exist any infinite-dimensional (closed) subspace Z of X such that T is an isomorphism between Z and T(Z). 1) Show that every compact operator is strictly singular. 2) Show that the canonical injection j : 1 → 2 is strictly singular, but not compact (use Pitt’s theorem). 3) Show that every strictly singular operator T : p → p (1  p < +∞) is compact (adapt the proof of Pitt’s theorem). 4) a) Show that, for every separable Banach space X, there exists a surjective operator T : 1 → X. Show that, for any surjective operator T : Y → X, the adjoint operator T ∗ : X ∗ → Y ∗ is an isomorphism between X ∗ and T ∗ (X ∗ ). b) Show that if T : 1 → 2 is surjective, then T is strictly singular. Is T ∗ strictly singular? 5) Let I be a closed bilateral ideal of L(p ), 1  p < +∞, not reduced to {0}. a) Show that I contains the ideal K(p ) of compact operators on p . b) We assume that I = K(p ). Show that there exist an operator T ∈ I and a subspace M ⊆ p such that T(M) is isomorphic to p and complemented in p , and such that T is an isomorphism from M onto T(M). (Hint: use 3) and Theorem III.13.) c) Then deduce that I contains an invertible operator * T ∈ L(p ), and hence that I = L(p ).

V Exercises

79

Exercise V.11 Show that if (en )n1 is a norm-bounded shrinking basis in a Banach space, then the sequence (en )n1 converges weakly to 0 (Pełczy´nski and Szlenk [1965] constructed a space X with a basis converging weakly to 0, even though it is not shrinking). Exercise V.12 We denote by J the set of all real sequences x = (xn )n1 ∈ c0 such that:  1 x = √ sup (xp1 − xp2 )2 + (xp2 − xp3 )2 + · · · + 2 1/2 < +∞ , + (xpk−1 − xpk )2 + (xpk − xp1 )2 where the upper bound is taken over all choices of p1 < p2 < · · · < pk and all k  2. 1) Show that  .  is a norm on J, which makes it a Banach space. J is called the James space; it was constructed by James [1950], in fact with the second norm ||| . ||| to be seen below in 4). 2) Show that the usual canonical basis (en )n1 is a monotone basis for J. 3) Show that e1 + · · · + en  = 1 for any n  1, but no weakly convergent subsequence can be extracted from (e1 + · · · + en )n1 . Deduce that J is not reflexive. 4) Show that: 1/2  |||x||| = sup (xp1 − xp2 )2 + (xp2 − xp3 )2 + · · · + (xpk−1 − xpk )2 is an equivalent norm on J. 5) Let (uk )k1 be a normalized block basis of (en )n1 . Show that:  k2    k2  1  √ 1 1/2      u 6 k  k  k2 qk+1 −1

6) 7) 8) 9)

k=k1

k=k1

(if uk = n=qk ak ek , split the indices pj for which pj and pj+1 are both in the same block {qk , . . . , qk+1 − 1} and those for which pj and pj+1 are in different blocks). Deduce that the basis (en )n1 is shrinking. Describe J ∗∗ (use Proposition III.23), and deduce that J ∗∗ = J ⊕ R1I. Also deduce that J is not isomorphic to J ⊕ J. Show that J contains neither c0 , nor 1 . Can J have an unconditional basis (see Chapter 3)? For x∗∗ ∈ J ∗∗ , denote a = limn→+∞ x∗∗ (e∗n ). Show that the mapping U : J ∗∗ → J, defined by: Ux∗∗ = (−a, x∗∗ (e∗1 ) − a, x∗∗ (e∗2 ) − a, . . .), is a surjective isometry (for the norm  . , James [1951]).

80

2 Bases in Banach Spaces

10) Show that J2 = {x = (xn )n1 ; x2k = 0 ∀ k  1} is isomorphic to 2 . 11) Show that (e∗n )n1 is a basis of J ∗ that is not shrinking, even though the dual J ∗∗ of J ∗ is separable. 12) Set εn = en − en+1 for any n  1. Show that (εn )n1 is a boundedly complete basis of J. Deduce that J is isomorphic to the dual of a Banach space, denoted J∗ . Describe it. Show that if g ∈ J ∗ is defined by: +∞ +∞   an εn = an , g n=1

n=1

J∗

then = J∗ ⊕ Rg. Pisier [1988] showed that J ∗ is of cotype 2 (see Chapter 5). Exercise V.13 1) Let Y be a separable Banach space containing c0 . Denote by (e∗n )n1 the canonical basis of 1 = c∗0 , and extend each e∗n to an element y∗n ∈ Y ∗ of ∗ ∗ norm 1. Let F = BY ∗ ∩ c⊥ 0 , and let d be a metric on BY defining the weak topology. a) Show that there exists z∗n ∈ F such that d(y∗n , z∗n ) −−→ 0. n→+∞  b) Show that the mapping P : Y → c0 defined by P(y) = y∗n (y) − z∗n (y) n1 is a projection of norm  2 (Sobczyk’s theorem, Sobczyk [1941]; this proof is due to Veech [1971]; conversely, a profound and difficult result of Zippin [1977] states that any infinite-dimensional separable space, complemented in every separable space that contains it, is isomorphic to c0 ). 2) a) Show that if (Bn )n1 is a sequence of finite-dimensional spaces, then the  3+∞  ∼ space n=1 Bn c0 is isometric to a subspace of (c0 ⊕ c0 ⊕ · · · )c0 = c0 . b) Let X be a Banach space and Y a subspace of X isomorphic to a subspace of c0 . Let T : Y → c0 be such an embedding. Show that there exists an extension  T : X → ∞ (use the Hahn–Banach theorem, coordinate by coordinate); by using the separability of  T X and Sobczyk’s theorem, T X, c0 ) → c0 ; then deduce show that there exists a projection P : span ( that S = P T : X → c0 extends T. c) Furthermore, suppose that X/Y is also isomorphic to a subspace of c0 . If U : X/Y → c0 is such an embedding, show that if Q : X → X/Y ∼ is the quotient mapping, then R : X → c0 ⊕ c0 = c0 , defined by R(x) = (Sx, UQx), is an embedding, and hence that X is isomorphic to a subspace of c0 .

V Exercises

81

Remark From these results, we can show that any quotient of c0 is isomorphic to a subspace of c0 (Johnson and Zippin [1974]). A partial converse was given by Godefroy, Kalton and Li [1996]: If X is a subspace of c0 with the metric approximation property, then X is isomorphic to a quotient of c0 if and only if X ∗ is isomorphic to a subspace of L1 . Exercise V.14 1) Show that there exists an uncountable family (Ai )i∈I of infinite subsets of N such that Ai ∩ Aj is finite for i = j. Indication. We could replace N by the countable set Q ∩ [0, 1]; for any i ∈ I = [0, 1]  Q ∩ [0, 1], select for Ai a sequence of elements of Q ∩ [0, 1] converging to i; another possibility is to take for I the " # set of infinite subsets  α(i,k) ; l  1 , where (pk )k1 is the of N, and, for i ∈ I, Ai = n = lk=1 pk sequence of prime numbers, and where α(i, k) = 1Ii (k). 2) Let T : ∞ → ∞ a be a continuous operator such that ker T ⊇ c0 . / Ai . a) For any i ∈ I, let xi ∈ ∞ , of norm 1, such that xi (k) = 0 when k ∈ Let J be a finite subset of I. Show that, for every i ∈ J, there exists yi , zi ∈ ∞ of norm  1, such that xi = yi + zi , and such that the yi ’s have finite supports,   and the zi ’s have disjoint supports. b) Deduce that  i∈J Txi ∞  T. c) Show that the set Bn = {i ∈ I ; (Txi )(n) = 0} is countable, and that  / I  n∈N Bn . Txi = 0 if i ∈ d) Conclude that there exists an infinite subset A ⊆ N for which ker T ⊇ ∞ (A). 3) Show that there does not exist any continuous injective operator U : ∞ /c0 → ∞ , and there does not exist any continuous projection P : ∞ → c0 (Phillips [1940], page 539; Lindenstrauss [1967] showed that any infinite-dimensional complemented subspace of ∞ is isomorphic to ∞ ). Exercise V.15 (James [1964 b]) 1) Let ||| . ||| be an equivalent norm on 1 , and α > 0 a positive number such that α |||x|||  x1  |||x||| for any x = (xn )n1 ∈ 1 . Set sn = sup{x1 ; |||x||| = 1 and x1 = · · · = xn = 0}. a) Show that the sequence (sn )n1 converges. Denote its limit by s. Take ε > 0. Let N be such that sN < s (1 + ε). b) Show that there exists a block basis (uk )k1 of the canonical basis of 1 such that, for any k  1, we have |||uk ||| = 1, uk 1 > s/(1 + ε) and

82

2 Bases in Banach Spaces PN (uk ) = 0, PN being the canonical projection onto the space generated by the N first vectors of the basis. c) Show that:   +∞   +∞ 1 s     ak uk   |ak | .  sN 1 + ε k=1

k=1

d) Conclude that the sequence (uk )k1 is (1 + ε)2 -equivalent to the canonical basis of 1 (see also Tomczak-Jaegermann, Proposition 30.5). 2) Let ||| . ||| be an equivalent norm on c0 . Take ε > 0, and let δ > 0 be such that (1 + δ)2 /(1 − 2δ − δ 2 )  1 + ε. Set σn = inf{x∞ ; |||x||| = 1 and x1 = · · · = xn = 0}, and σ = limn→+∞ σn . Show that there exists a block basis (vk )k1 of the canonical basisof c0 such that  |||vk ||| = 1, vk ∞ < σ (1 + δ) for any k  1, and such   +∞ that  k=1 ak vk   (1 + δ)2 a∞ for every a = (ak )k1 ∈ c0 . With |aK | = a∞ , show that:    +∞  +∞     2          |||2a a v v ||| − a v − 2a v k k  K K k k K K   (1 − 2δ − δ ) a∞ ,   k=1

k=1

and conclude that (vk )k1 is (1 + ε)-equivalent to the canonical basis of c0 . Remark A Banach space X is said to be distortable if there exists an equivalent norm ||| . ||| on X and a λ > 1 such that, for any infinite-dimensional subspace Y of X, we have: sup{|||y|||/|||x||| ; x = y = 1, x ∈ X, y ∈ Y} > λ . The results of Exercise V.15 show that 1 and c0 are not distortable. Lindenstrauss and Pełczy´nski [1971] showed that C([0, 1]) and Lp (0, 1), for 1  p < +∞ and p = 2, have a weak form of distortability. Odell and Schlumprecht [1994] proved that p , for 1  p < +∞, is distortable, and, more generally, that any space containing neither 1 nor c0 contains a distortable subspace. In fact their paper presents a proof, due to Maurey, that 2 is arbitrarily distortable (i.e. for any λ > 1); see also Maurey [1995 b]. Maurey [1995 a] next showed that any space having an unconditional basis, but not containing n1 ’s uniformly, contains an arbitrarily distortable subspace. Further results on the subject can be found in Milman and Tomczak-Jaegermann [1993] as well as the survey in Odell [2002].

3 Unconditional Convergence

I Introduction This chapter is devoted to the properties of unconditional convergence of series in Banach spaces. In Section II, we give different characterizations of this convergence (Proposition II.2) and show the Orlicz–Pettis theorem (Theorem II.3). In Section III, we introduce the notion of unconditional basis, and, in particular, show that the sequences of centered independent random variables are basic and unconditional in the spaces Lp (P). In Section IV, we focus on the canonical basis of c0 , and prove the Bessaga– Pełczy´nski theorems: on one hand, they characterize the presence of c0 within a space by the existence of a scalarly summable but not summable sequence; on the other hand, they assert that a dual space containing c0 must contain ∞ . In Section V, we prove the James structure theorems: these theorems characterize, among the spaces possessing unconditional bases, those containing c0 or 1 , and those being reflexive. In Section VI, we prove the Gowers dichotomy theorem: every Banach space contains a subspace with an unconditional basis or a hereditarily indecomposable subspace (i.e. none of its infinite-dimensional subspaces can be decomposed as a direct sum of infinite-dimensional subspaces). In addition, we provide an outline of the proof of the homogeneous subspace theorem: every infinite-dimensional space, isomorphic to all its infinite-dimensional subspaces, is isomorphic to 2 .

II Unconditional Convergence  Definition II.1 A series +∞ n=1 xn in a Banach space is said to be uncondition+∞ ally convergent if n=1 xπ(n) converges for every permutation π of integers. 83

84

3 Unconditional Convergence

In French, the term used is: commutativement convergente (commutatively convergent). Remark As the sum of an unconditionally convergent series of scalars (which is equivalent to absolutely convergent in this case) is unchanged under modification of the order of the terms, we have, for every x∗ ∈ X ∗ : /

x∗ ,

+∞ n=1

+∞ +∞ +∞ 0 0 /   xπ(n) = x∗ xπ(n) = x∗ (xn ) = x∗ , xn ; n=1

n=1

+∞

n=1

hence, if n=1 xn is unconditionally convergent, we have +∞ n=1 xn for every permutation π of integers.

+∞

n=1 xπ(n)

=

We have the following characterization: Proposition II.2 (Orlicz) Let (xn )n1 be a sequence in a Banach space X. The following properties are equivalent:  xn converges unconditionally; 1) the series +∞ n=1 +∞ 2) the series k=1 xnk converges for every choice of n1 < n2 < · · · (such a series is said to be subseries convergent);  θ x converges for every choice of signs θn = ±1; 3) the series +∞ n=1 n n 4) the family n1 xn is summable.  ∈ X such Recall that the family n1 xn is summable if there is some  that,  x for every ε > 0, there exists an integer N  1 such that x − n∈σ xn   ε for any finite subset σ ⊇ [1, N]. For this, satisfying the Cauchy summability criterion issufficient: for any ε > 0, there exists an integer N  1 such that    x n∈σ n  ε for any finite subset σ ⊆ [N, +∞[. Remark We know (Riemann) that in finite dimension, unconditional con vergence is equivalent to absolute convergence, i.e. +∞ n=1 xn  < +∞. This no longer holds in infinite dimension: in that case, Dvoretzky and Rogers [1950] proved the existence of unconditionally convergent but not absolutely convergent series. More precisely, for every sequence (λn )n1 ∈ 2 , there  exists an unconditionally convergent series +∞ n=1 xn with xn  = λn for any n  1. This theorem will be proved in Chapter 5 of this volume, and further detailed in Chapter 1 of Volume 2. Example

If X = c0 , with xn =

  1 1 en = 0, . . . , 0, , 0, . . . , + ,. n n n−1 times

II Unconditional Convergence

85

 1 1 1, , , . . . converges unconditionally (for 2 3 example, we can use the criteria 2) or 3) of Proposition II.2); however, +∞ +∞ 1 n=1 xn  = n=1 n = +∞.

then the series



+∞

n=1 xn =

Proof of Proposition II.2

We are going to show:

2) ⇐⇒ 3) ,

2) ⇐⇒ 4) ,

1) ⇐⇒ 4) .

2) ⇒ 3) . Given a sequence of signs (θn )n1 , we write: {n ; θn = 1} = {n1 < n2 < · · · } and {n ; θn = −1} = {n 1 < n 2 < · · · } .  +∞ +∞ By hypothesis +∞ k=1 xnk and k=1 xn k converge, hence n=1 θn xn = +∞ +∞ k=1 xnk − k=1 xnk as well. 3) ⇒ 2) . Given n1 < n2 < · · · , set θj = 1 if j is one of the nk ’s, and θj = −1 otherwise; then the series:   +∞ +∞ +∞ 1 xkj = xj + θj xj 2 j=1

j=1

j=1

converges. 4) ⇒ 2) . Select a strictly increasing sequence n1 < n2 < · · ·. Let ε > 0. By  hypothesis, there exists an integer N  1 such that  n∈σ xn   ε for every finite subset σ ⊆ [N, +∞[. Select a J  1 such that n  N; then, for j  J, by taking σ = {nJ , . . . , nj } we have  Jj   k=J xnk  ε. 2) ⇒ 4) . Suppose that 4) does not hold. Then, there exist an ε > 0 and finite subsets σn ⊆ N, n  1, such that, for any n  1:     xk   ε ,    k∈σn

with qn = max σn < min σn+1 = pn+1 (as illustrated by Figure 3.1). pn

qn

pn+ 1

sn

qn + 1 sn+ 1

Figure 3.1

 Then, if we set σ = n1 σn = {n1 < n2 < · · · }, the series +∞ j=1 xnj does not converge. 1) ⇒ 4) . We proceed as above, and write σn = { j1,n < · · · < jln ,n }. Select a permutation πn : [pn , qn ] → [pn , qn ] such that πn ( jk,n ) =

86

3 Unconditional Convergence pn + (k − 1). We thus obtain a permutation π : N∗ → N∗ by setting π( j) = πn ( j) if j ∈ [pn , qn ], and π( j) = j otherwise; we have:     pn     +ln −1    x xk  −1 π ( j)  =    ε.  j=pn

+∞

k∈σn

Hence the series j=1 xπ −1 ( j) does not converge. 4) ⇒ 1) . Let π be a permutation of N∗ , and let ε > 0. There exists  an integer N  1 such that x − n∈σ xn   ε for every finite subset σ ⊇ [1, N]. Let k0 = max π −1 ([1,N]); then, for k  k0 , σ =  {π(1), . . . , π(k)} ⊇ [1, N], and we obtain  kj=1 xπ( j) −x  ε.  Remark Let +∞ n=1 xn be an unconditionally convergent series, and x ∈ X its sum. For any continuous linear functional x∗ ∈ X ∗ , the scalar series +∞ ∗ x (xn ) is unconditionally convergent, hence absolutely convergent: +∞ ∗ n=1 +∞ ∗ |x (x )| < +∞, and we have x∗ (x) = n=1 n=1 x (xn ). This means +∞ n that n=1 xn converges unconditionally to x in the weak topology σ (X, X ∗ ). This last condition does not imply the unconditional convergence in norm (take X = c0 and x2n−1 = en , x2n = −en , n  1, where (en )n1 is the canonical basis of c0 ). However, with a slightly stronger hypothesis, we have the following remarkable result:  Theorem II.3 (The Orlicz–Pettis Theorem) If +∞ n=1 xn is subseries conver∗ gent (in X) in the weak topology σ (X, X ), then it converges unconditionally (in norm).1 The proof of this theorem is facilitated by the following notion, weaker than unconditional convergence for the weak topology:  Definition II.4 A series n1 xn is said to be weakly unconditionally Cauchy (abbreviated w.u.C.) when: +∞

|x∗ (xn )| < +∞

n=1

for every

x∗

∈

X∗.

Remark These series are often referred to as weakly unconditionally convergent, but in general such a series does not converge weakly. Nevertheless, it converges weak∗ in X ∗∗ : there exists an x∗∗ ∈ X ∗∗ such that x∗∗ = +∞ xn . σ (X ∗∗ , X ∗ ) n=1 1 In the French version of this book, this theorem was wrongly stated. We thank E. Oja and L.

Rodríguez-Piazza who pointed that out to us.

II Unconditional Convergence

87

Proposition II.5 The following properties are equivalent:  1) the series n1 xn is w.u.C.;  ∗ ∗ 2) there exists some C > 0 such that +∞ n=1 |x (xn )|  Cx  for every ∗ ∗ x ∈X ;   3) there exists some C > 0 such that supn1  nk=1 λk xk   Cλ∞ for every λ = (λn )n1 ∈ ∞ ;  4) for any t = (tn )n1 ∈ c0 , the series +∞ n=1 tn xn converges;   5) there exists some C > 0 such that supθn =±1  n∈σ θn xn   C for every finite subset σ of N∗ .  Remark Condition 3) implies: if the series n1 xn converges unconditionally, then there exists a constant C > 0 such that:    +∞   λn xn     Cλ∞ n=1

   +∞   for any λ ∈ ∞ . In particular: sup  θn xn    C. θn =±1

n=1

Proof 1) ⇒ 2) We define T : X ∗ → 1 by Tx∗ = (x∗ (xn ))n1 . The closed graph theorem shows that T is continuous. 2) ⇒ 3) Indeed:    n  n     ∗      = sup x λ x λ x k k k k    ∗ k=1

x 1

 sup

x∗ 1



k=1

sup |λk | k1

+∞ 

 |x∗ (xk )|  Cλ∞ .

k=1

3) ⇒ 4) When (tk )k1 ∈ c0 , for 1  m  n and m  k  n, set λk =  tk , and set λk = 0 otherwise; condition 3) gives  nk=m tk xk   C supmkn |tk | −−−−→ 0. n>m→+∞ 3) ⇒ 5) is evident. 5) ⇒ 1) Let x∗ ∈ X ∗ and N  1. Set θn = 1 if Re x∗ (xn )  0, and θn = −1 if Re x∗ (xn ) < 0; similarly set θn = 1 if Im x∗ (xn )  0, and θn = −1 if Im x∗ (xn ) < 0. Then:

88

3 Unconditional Convergence N

|x∗ (xn )| 

n=1

=

N n=1 N

| Re x∗ (xn )| +

N

| Im x∗ (xn )|

n=1 N

θn Re x∗ (xn ) +

n=1

θn Im x∗ (xn )

n=1

      N ∗   N θn x∗ (xn ) +  θn x (xn )  2 C x∗  .   n=1

n=1

4) ⇒ 1) We will use the following result: for every sequence (αn )n1 of  real numbers such that +∞ n=1 |αn | = +∞, we can find a strictly increasing sequence of integers (pk )k1 , with p1 = 1, such that pk+1 −1 2 n=pk |αn |  k. If we set tn = (sgn αn )/k for pk  n  pk+1 − 1, +∞ then (tn )n1 ∈ c0 , while n=1 tn αn = +∞. This classical result is known as the du Bois–Reymond lemma. Here, taking αn = Re x∗ (xn ) or Im x∗ (xn ), we obtain the diver ∗ gence of the series +∞ n=1 tn x (xn ), and thus, a fortiori, that of the +∞ series n=1 tn xn . In the proof of the Orlicz–Pettis theorem, the essential point is the following:  Proposition II.6 If the series n1 xn is subseries convergent (in X) for the weak topology, the operators: U:

c0 (λn )n1

−→ X +∞ −→ n=1 λn xn

and

T : X∗ x∗

−→ −→

1 (x∗ (xn ))n1

are compact. Proof First, the operator U is well defined thanks to condition 4) of Proposition II.5. Then, by condition 3) of this same theorem, it is a bounded operator. Next note that T = U ∗ . According to Schauder’s theorem, it hence suffices to prove the compactness of T. Therefore we must show that, given any sequence (xk∗ )k1 in BX ∗ , we can extract, from (Txk∗ )k1 , a subsequence convergent in norm in 1 . However, by Schur’s theorem (Chapter 2, Theorem III.10), it suffices to extract a weakly convergent subsequence. Replacing if necessary X by X0 = span{xn , n  1}, we may, and do, assume that X is separable. Then the compact space (BX ∗ , w∗ ) is metrizable. Thus, from (xk∗ )k1 , we can extract a subsequence (xk∗j )j1 w∗ -convergent to an element x∗ ∈ BX ∗ . +∞ σ (1 ,∞ ) Let us show that T(xk∗j ) −− → T(x∗ ). Indeed, as the series n=1 xn is j→+∞

subseries convergent for the weak topology, for every σ ⊆ N∗ , there exists

II Unconditional Convergence

89

 an xσ ∈ X such that xσ = w - n∈σ xn (the elements of σ being numbered in increasing order). We have: T(xk∗j )(n) = xk∗j (xn ) 1Iσ , T(xk∗j ) = n∈σ

=

xk∗j ,



n∈σ

xn

since the series is w-convergent

n∈σ

= xk∗j , xσ −−→ x∗ , xσ = x∗ , j→+∞



xn = 1Iσ , Tx∗ .

n∈σ

As span{1Iσ ; σ ⊆ N} = ∞ (density of the simple functions), and as the sequence (T(xk∗j )j1 is bounded, we obtain: λ, T(xk∗j ) −−→ λ, Tx∗ j→+∞

for any λ ∈ ∞ , i.e. T(xn∗k ) −−→ T(x∗ ). w

k→+∞

For every σ ⊆ N, there exists xσ ∈ X

Proof of the Orlicz–Pettis theorem such that:

x∗ , xσ =



x∗ (xn )

n∈σ

for every

x∗

∈

X∗.

Consider the projections:

Qn = (I − Pn ) :

1 α = (αk )k1

−→ −→

1 Qn (α) = (0, . . . , 0, αn+1 , . . .) . + ,- . n times

We have Qn   1, and Qn (α) −−→ 0 for every α ∈ 1 . Since the operator T n→+∞

is compact by Proposition II.6, the set K = T(BX ∗ ) is compact, and hence the Qn , n  1, converge uniformly to 0 in K, by Ascoli’s theorem. Then:         ∗     xσ − = sup x x x − x |x∗ (xk )| k σ k   sup   ∗ ∗ k∈σ kn

x ∈BX ∗

 sup



x∗ ∈BX ∗ kn+1

k∈σ kn

x ∈BX ∗

k∈σ kn+1

|x∗ (xk )| = sup Qn T(x∗ )1 −−→ 0 , x∗ ∈BX ∗

n→+∞

which proves the unconditional convergence in norm of the series. +∞ Theorem II.7 If n=1 xn is unconditionally convergent, then the series +∞ λ x converges for every sequence (λn )n1 ∈ ∞ . n n n=1 Note that the converse is obviously true, by using 2) or 3) of Proposition II.2. Moreover, the convergence will be unconditional, since the series +∞ n=1 θn λn xn will also be convergent for every choice of signs θn = ±1.

90

3 Unconditional Convergence

Proof

We use the same notation as above. We have:  q    q q     ∗      sup = sup x λ x λ x |λn | |x∗ (xn )| n n n n    n=p

x∗ ∈BX ∗

 λ∞ sup

n=p q

x∗ ∈BX ∗ n=p

x∗ ∈BX ∗ n=p

|x∗ (xn )|

= λ∞ sup (Qq − Qp−1 )(Tx∗ )1 −−−−→ 0 . x∗ ∈BX ∗

q>p→+∞

III Unconditional Bases Definition III.1 A basis (en )n1 is said to be unconditional if, for every  ∗ x ∈ X, the series +∞ n=1 en (x)en converges unconditionally. Examples 1) The canonical bases of c0 and of p , 1  p < +∞, are unconditional. However, the summing basis (sn )n1 of c, defined by s0 = (0, 0, 1,  . . .), . . . is not, because we (1, 1, 1, . . .), s1 = (0, 1, 1, . . .), s2 =   have  nk=0 (−1)k sk ∞ = 1, while  nk=0 sk ∞ = n + 1. 2) Every orthogonal basis of a Hilbert space is unconditional. We will see in Chapter 7: 3) The Haar basis of Lp (0, 1) is unconditional for 1 < p < +∞ (Chapter 7, Theorem IV.8). However, the trigonometric system is not unconditional except for p = 2 (Chapter 7, Proposition III.10). 4) L1 (0, 1) is not contained in any space possessing an unconditional basis (Chapter 7, Theorem II.10). Note that Maurey [1980 a] showed that the space H 1 (T) = { f ∈ L1 (T) ;  f (n) = 0, ∀ n < 0} possesses an unconditional basis. Maurey’s proof is extremely difficult, and indirect. Soon afterwards, Carleson [1980], and then Wojtaszczyk [1982], gave explicit unconditional bases of H 1 . Proposition II.5 leads to the following: Proposition III.2 Let (en )n1 be a basis of X. It is unconditional if and only if one of the following equivalent conditions is satisfied:

III Unconditional Bases

91

1) for any subset σ of N∗ (the elements of σ being numbered in increasing   order), the convergence of +∞ k=1 ak ek implies that of∗ k∈σ ak ek ; 2) for any choices of signs θ = (θk )k1 ∈ {−1, 1}N , the convergence of +∞ +∞ k=1 ak ek implies that of k=1 θk ak ek . More precisely, we have: Proposition III.3 Let (en )n1 be a total sequence in X; it is an unconditional basis if and only if one of the following conditions is satisfied:  1) There exists a constant K > 0 such that, with x = nk=1 ak ek , we have, for any finite subset σ of N∗ :      ak ek    Kx .  k∈σ kn

∞  In other words, the projections Pσ : x = k∈σ ak ek are k=1 ak ek → continuous, and supσ Pσ   K.  2) There exists a constant K > 0 such that, with x = nk=1 ak ek , we have, for ∗ any choice of signs θ = (θk )k1 ∈ {−1, 1}N :    n   θk ak ek     K x . k=1

In other words, the symmetries Mθ : x = continuous, and supθ Mθ   K .

∞

k=1 ak ek

→

∞

k=1 θk ak ek

are

Proof Indeed, with σ = {1, 2, . . . , m} for m  n, condition 1) shows that (en )n1 is a basis of X. The rest follows easily from the properties of unconditionally convergent series. Note that the two conditions are equivalent: if σ = {k ; θk = 1}, we have Pσ = (Id +Mθ )/2, and sup Pσ   sup Mθ   2 sup Pσ  . σ

θ

σ

Definition III.4 The number Ku = supθ Mθ  is called the unconditional constant of the basis. If Kb denotes the basis constant, then Kb  Ku . For the canonical bases of c0 and p , 1  p < +∞, we have Ku = 1, and hence Ku = Kb . However, in general Ku > Kb . For example, Burkholder [1984] showed that the unconditional constant of the Haar basis of Lp (0, 1), for 1 < p < +∞, is max(p − 1, q − 1) = max(p − 1, 1/(p − 1)), where q is the conjugate exponent of p. We will see this in Chapter 7.

92

3 Unconditional Convergence

Remark This unconditional constant is in fact the real unconditional constant, most commonly used. However, at times we need the complex version: * Ku = supθ Mθ , where this time the θ ’s are complex “signs,” i.e. complex numbers of modulus 1. In the following proposition, when the space X is Ku . complex, the term 2Ku can be replaced by * Proposition III.5 Let (en )n1 be an unconditional basis of X, with uncondi tional constant Ku . Then, for any x = +∞ k=1 ak ek ∈ X and any λ = (λk )k1 ∈ ∞ , we have:  +∞     Ku λ∞ x if the space X is real,  λk ak ek     2 K λ x if the space X is complex. u ∞ k=1 Proof

Select x∗ ∈ X ∗ such that x∗  = 1 and  4+∞ 5  +∞         ∗ λk ak ek  =  λk ak ek  x .   k=1

k=1

By separately summing the real and imaginary parts of ak x∗ (ek ), we can assume that these values are all real. Set θk = 1 if ak x∗ (ek )  0, and θk = −1 if ak x∗ (ek ) < 0. Then we have:   +∞  +∞  +∞ ∗   λ a e |λ | |a x (e )| = |λk | θk ak x∗ (ek ) k k k k k k   k=1

k=1

k=1

 sup |λk | x∗ (Mθ x)  sup |λk | . Ku x . k1

k1

The following theorem gives a particularly interesting example of an unconditional basic sequence: Theorem III.6 Any sequence X1 , X2 , . . . of independent centered random variables in Lp (, P) (1  p < +∞) is basic and unconditional in Lp (, P), with unconditional constant  2. Moreover, if the variables are symmetric, the unconditional constant is 1. In this last case, the property given in Proposition III.5 is often called the contraction principle. We will see in Chapter 4 that Theorem III.6 holds for Banach-valued random variables, with the same proof. Recall that a random variable    X is symmetric if (−X) has the same distribution as X; we thus have  nk=1 θk ak Xk p =  n    k=1 ak Xk p for every choice of signs θk = ±1; then clearly we have unconditionality with constant 1. For the case of centered variables, we will begin with a simple lemma:

III Unconditional Bases

93

Lemma III.7 Let X and Y be two independent random variables in Lp (, P), with E(Y) = 0. Then: X + Yp  Xp . Proof have:

Let PX and PY be the distributions of X and Y. By independence, we   X + Ypp = 

R R

|x + y|p dPX (x) dPY (y) |x + y| dPY (y)

p

dPX (x) R   R p     (x + y) dPY (y) dPX (x) R R = |x|p dPX (x) = Xpp . R

Proof of Theorem III.6 Given scalars a1 , . . . , an and a choice of signs θk = ±1, 1  k  n, we set: I = {k ∈ {1, . . . , n} ; θk = 1} then: X=



ak Xk =

k∈I



θk ak Xk

and

J = {k ∈ {1, . . . , n} ; θk = −1} ;

and

Y=

k∈I



ak Xk = −

k∈J



θk ak Xk .

k∈J

As X and Y are centered, Lemma III.7 gives:  n     θk ak Xk    = X − Yp  Xp + Yp k=1

p

 X + Yp + Y + Xp   n    = 2 X + Yp = 2  ak Xk   , k=1

p

hence the result by Proposition III.3. To conclude this section, we present the following easy but useful proposition: Proposition III.8 If (en )n1 is an unconditional basis of a Banach space X, then every block basis of (en )n1 is an unconditional basic sequence, whose unconditional constant is less than or equal to that of (en )n1 . The coordinate linear functionals e∗n , n  1, form an unconditional basic sequence in X ∗ , with the same unconditional constant as (en )n1 .

94

3 Unconditional Convergence

Proof For the first assertion, with θk = ±1:   kN+1   N  k   k n+1   N+1           =  K θ a c e θ a c e a c e n n k k  u k k k k k k k   k=kn +1

n=1

k=1

n=1

θk

k=1

 N  k  n+1    = Ku  an ck ek  , k=kn +1

a k

by setting = θn and = an for kn + 1  k  kn+1 . For the second assertion, it suffices to note that Mθ∗  = Mθ .

IV The Canonical Basis of c0 The classification of Banach spaces is done by comparing them to each other (with isomorphisms, isometries or, more recently, their completely bounded versions, or with Lipschitz or uniform homeomorphisms), and in particular by comparing them to some reference spaces. The so-called classical spaces: the sequence spaces c0 , p , 1  p  +∞, the continuous function spaces C(K) and the Lebesgue spaces Lp are most frequently used. Recall that a Banach space X is said to contain another Banach space Y if Y is isomorphic to a (closed) subspace of X. A criterion is given here to detect the presence of c0 within a Banach space. Throughout this section, (en )n1 denotes the canonical basis of c0 . Proposition IV.1 Any basic sequence (zn )n1 such that: a) infn1 zn  > 0; +∞ b) n=1 zn is w.u.C. is equivalent to the canonical basis of c0 . +∞ Proof If −→ 0, and hence tn − −→ 0 by n=1 tn zn converges, then tn zn − n→+∞ n→+∞ +∞ a) . In other words: n=1 tn en converges. Conversely, condition b) , with the characterization of series w.u.C. (Proposition II.5), states that the series +∞ n=1 tn en converges for every (tn )n1 ∈ c0 . This leads to: Theorem IV.2 (The Bessaga–Pełczy´nski c0 Theorem) A Banach space X contains c0 if and only if X contains a w.u.C. series that does not converge (in X). More precisely, we have:

IV The Canonical Basis of c0

Theorem IV.3 equivalent:

95

For any Banach space X, the following properties are

1) X contains a w.u.C. series that is not unconditionally convergent;  2) X contains a w.u.C. series +∞ n=1 yn such that infn1 yn  > 0; 3) X contains c0 . Proof  convergent, there exists a permu1) ⇒ 2) . If +∞ n=1 xn is not unconditionally +∞ tation π of integers such that n=1 xπ(n) does not converge. Thus we can find integers q1 < q2 < · · · such that:  qn+1     inf  x π( j)  = a > 0 .  n1

j=qn +1

Then, if we set:

qn+1

yn =

xπ( j) ,

j=qn +1

we indeed have infn1 yn  = a > 0. Moreover, for every x∗ ∈ X ∗ : +∞ n=1

|x∗ (yn )|  +∞

+∞

|x∗ (xπ( j) )| =

j=1

+∞

|x∗ (xk )| < +∞ ,

k=1

in other words n=1 yn is w.u.C. 2) ⇒ 3) . Condition 2) implies, in particular, that the sequence (yn )n1 converges weakly to 0; but it does not converge in norm. Thanks to the Bessaga–Pełczy´nski selection theorem, we can extract a basic subsequence (zn )n1 such that infn1 zn  > 0. This subsequence hence satisfies the conditions of Proposition IV.1 above; it is thus equivalent to the canonical basis of c0 . 3) ⇒ 1) . It suffices to note that the canonical basis (en )n1 of c0 is w.u.C.;  however, as en ∞ = 1, the series +∞ n=1 en does not converge. Theorem IV.4 (Bessaga–Pełczy´nski) If the dual X ∗ of a Banach space X contains c0 , then X contains a complemented subspace isomorphic to 1 . As a result, X ∗ contains ∞ . Corollary IV.5

A separable dual space never contains c0 .

Proof Let T : c0 → X ∗ be a bounded linear mapping providing an isomorphism between c0 and T(c0 ). Let T ∗ : X ∗∗ → 1 be the adjoint operator, and S = T∗ : X −→ 1 X

96

3 Unconditional Convergence

its restriction to X. We have:   Sx = (Te1 )x, (Te2 )x, . . . . As T is an isomorphism, T ∗ is onto. As the canonical basis of 1 is the coordinate linear functionals (e∗n )n1 , there exists a K > 0 so that, for any n  1, we can find a xn∗∗ ∈ X ∗∗ such that: xn∗∗   K

T ∗ (xn∗∗ ) = e∗n .

and

In particular: xn∗∗ , Tek = T ∗ xn∗∗ , ek = e∗n , ek =

 1 0

if n = k, if n =  k.

As BX is w∗ -dense in BX ∗∗ , we can find xn ∈ X, n  1, so that xn   K and: |(Ten )(xn )|  1/2

and

|(Tek )(xn )|  1/n for k  n − 1 .

Thus Sxn 1  |(Ten )(xn )|  1/2, and Sxn , ek = (Tek )(xn ) −−→ 0, n→+∞

for any k  1. By the Bessaga–Pełczy´nski selection theorem, there exists a subsequence (xnk )k1 such that (Sxnk )k1 is equivalent to a block basis (yk )k1 of the canonical basis (e∗n )n1 of 1 . However Y = span {yk ; k  1} is isometric to 1 and there exists a projection Q : 1 → Y ⊆ 1 of norm 1; hence Z = span {Sxnk ; k  1} is also isomorphic to 1 and there exists an M > 0 such that:   +∞ +∞    |ak |  M  a Sx k nk  ,  k=1

k=1

and a projection P : 1 → Z (see Chapter 2, Theorem III.3). Then, as:  +∞     +∞  +∞ +∞            ak xnk   K |ak |  KM  ak Sxnk   KM S  ak xnk  ,  k=1

k=1

k=1

k=1

S is an isomorphism between X0 = span {xnk ; k  1} and Z. Thus X0 is isomorphic to 1 , and S−1 PS : X → X0 is a projection of X onto X0 .

V The James Theorems Spaces with unconditional bases deserve special attention because of their interesting theorems of structure: the James theorems.

V The James Theorems

97

Theorem V.1 (James) Let X be a Banach space possessing an unconditional basis (en )n1 . Then (en )n1 is shrinking if and only if X does not contain 1 . This is the case if and only if X ∗ is separable; then X ∗ also possesses an unconditional basis. Remark In Chapter 2 we indicated that a Banach space with a basis could have a separable dual, without the dual satisfying the approximation property. We see here that, for a space with an unconditional basis, the separability of the dual automatically implies the existence, within this dual, of a basis (which is moreover unconditional). By the Johnson–Rosenthal–Zippin theorem, also seen in Chapter 2, the existence of a basis for X ∗ implies the same for X. However, X ∗ may possess an unconditional basis without X having one. For example, if K is a countable compact metric space with an infinite number of limit points, we can show that C(K) does not possess an unconditional basis (one reason is that C(K) does not satisfy Pełczy´nski’s property (u), Pełczy´nski [1958]), even though C(K)∗ is isometric to 1 . Prior to the second theorem, the notion of a weakly sequentially complete space needs to be introduced. Definition V.2 A sequence (xn )n1 is said to be weakly  (abbreviated  Cauchy w-Cauchy) if it is Cauchy for the weak topology, i.e. if x∗ (xn ) n1 converges for every x∗ ∈ X ∗ . Such a sequence defines an element x∗∗ ∈ X ∗∗ of the bidual of X: x∗∗ , x∗ = lim x∗ (xn ) . n→+∞

Then x∗∗ = w∗ - limn→+∞ x∗ (xn ) for the topology w∗ = σ (X ∗∗ , X ∗ ) of X ∗∗ . Definition V.3 A Banach space X is said to be weakly sequentially complete (w.s.c.) if every w-Cauchy sequence converges weakly (in X). In other words, X is w.s.c. if the above element x∗∗ of X ∗∗ is in fact in X. Examples 1) Every reflexive space is w.s.c., by the weak compactness of the unit ball, and the Eberlein–Šmulian theorem. 2) 1 is w.s.c., by Schur’s theorem (Chapter 2, Theorem III.10). 3) Every space L1 (S, T , m) is w.s.c.; this will be seen in Chapter 7. 4) Every subspace of a w.s.c. space is w.s.c. 5) c0 is not w.s.c.: sn = e1 + · · · + en , n  1, is w-Cauchy, but does not converge weakly. Hence, if X is w.s.c., it does not contain c0 .

98

3 Unconditional Convergence

Theorem V.4 (James) Let X be a Banach space possessing an unconditional basis (en )n1 . The following assertions are equivalent: 1) (en )n1 is boundedly complete; 2) X is weakly sequentially complete; 3) X does not contain c0 . In particular, a weakly sequentially complete space possessing an unconditional basis is isomorphic to a dual space. Since a space with a basis is reflexive if and only if this basis is simultaneously shrinking and boundedly complete, it ensues that: Corollary V.5 (James) When a Banach space X possesses an unconditional basis, it is reflexive if and only if it contains neither c0 nor 1 . In particular, this occurs when X ∗∗ is separable and possesses an unconditional basis. Proof of Theorem V.1 1) If X contains (a subspace isomorphic to) 1 , let J : 1 −→ X0 ⊆ X be such an isomorphism. Then J ∗ : X ∗ → ∞ is onto, and as ∞ is not separable, neither is X ∗ . Thus X ∗ cannot possess a basis, and hence no basis of X is shrinking. 2) Conversely, if (en )n1 is not shrinking, there exists x∗ ∈ X ∗ such that:    ∗  x  span (e ,e ,...)  −− /→ 0 . n+1 n+2

n→+∞

We can thus find an α > 0 and a normalized block basis (uk )k1 of (en )n1 such that x∗ (uk )  α for any k  1. Then, for a1 , . . . , am  0, we have:  m    m m   ∗    x  α a u a u ak . k k k k  k=1

k=1

k=1

But as the basic sequence (uk )k1 is unconditional, with constant K, then, for a1 , . . . , am ∈ C arbitrary complex numbers, we obtain:  m   m    m m   iθ    1  α k       ak uk  =  e |ak |uk   |ak |uk   |ak | .  2K  2K k=1

k=1

k=1

k=1

When the scalar field is R, we can replace α/(2K) by α/K.   m  As we obviously have  m k=1 ak uk  k=1 |ak |, the sequence (uk )k1 is equivalent to the canonical basis of 1 , so that span {uk ; k  1} is isomorphic to 1 .

V The James Theorems

99

The last assertion follows: indeed, on one hand, every space having a shrinking basis has a separable dual (Chapter 2, Proposition III.18), and, on the other hand, if X contains a subspace isomorphic to 1 , then ∞ is isomorphic to a quotient of X ∗ ; thus the latter cannot be separable. Proof of Theorem V.4 2) ⇒ 3) . This is clear because c0 is not w.s.c. 3) ⇒ 1) . Suppose that (en )n1 is not boundedly complete. We can thus find scalars a1 , a2 , . . . such that:  n      a e k k  1  k=1

 for any n  1, but without convergence of the series +∞ k=1 ak ek . Thus the partial sums do not form a Cauchy sequence, so we can find an α > 0 and integers: p1 < q1 < p2 < q2 < · · · such that, with: un =

qn

ak ek ,

k=pn

we have un   α for any n  1. Then (un )n1 is a block basis of (en )n1 and is hence an unconditional basic sequence. Thus there exists a constant K > 0 (the unconditional constant of (un )n1 if the scalar field is R, or twice this constant if it is C), such that, for any scalars λ1 , . . . , λm :     m  m  m qk            λk uk   K sup |λk |  uk   K sup |λk |  aj ej    1km

k=1

k=1

1km

k=1 j=pk

    qm 2   K sup |λk | K  aj ej    K sup |λk | . 1km

j=1

1km

Since uj  u∗j   K, we also have:    m m  u   uj   ∗   α j     u  |λj |  |λj | λ u λ u k k k k =  K  j K K k=1

k=1

for any j  1; hence  m    α  λk uk     K sup |λj | . 1jm k=1

100

3 Unconditional Convergence

Therefore (uk )k1 is equivalent to the canonical basis of c0 , and hence X contains c0 . 1) ⇒ 2) . Assume that (en )n1 is boundedly complete. To show that X is w.s.c., we need the following lemma: Lemma V.6 Let X be a space possessing an unconditional basis (en )n1 , and let (yj )j1 be a bounded sequence in X. If both the following conditions hold: a) limj→+∞ x∗ (yj ) exists for every x∗ ∈ X ∗ (i.e. (yj )j1 is w-Cauchy); b) limj→+∞ e∗n (yj ) = 0 for any n  1, then (yj )j1 converges weakly to 0. Indeed, once this lemma is established, we consider a w-Cauchy sequence (xj )j1 in X. In particular the limit an = limj→+∞ e∗n (xj ) exists for any n  1. As   m   m   ∗      ak ek  = lim  ek (xj )ek   = lim Pm (xj )  K sup xj  ,  k=1

j→+∞

+∞

k=1

j→+∞

j1

the series k=1 ak ek converges, since the basis (en )n1 is boundedly complete. +∞ Denote x = k=1 ak ek ; Lemma V.6, applied to yj = xj − x, shows that w w yj −−→ 0, i.e. xj −−→ x. j→+∞

j→+∞

Proof of Lemma V.6 If the conclusion were false, by extracting a subsequence if necessary, we would be able to assume that there exists an α > 0 and an x∗ ∈ X ∗ such that |x∗ (yj )|  α for any j  1. As e∗n (yj ) −−→ 0 for any n  1, by the Bessaga–Pełczy´nski selection theorem j→+∞

(or more precisely its proof), we can find a block basis (uk )k1 of (en )n1 and a subsequence (yjk )k1 such that: α uk − yjk   k ∗ · 2 x  We then have |x∗ (uk )|  α/2, and hence, as in the proof of Theorem V.1, (uk )k1 is equivalent to the canonical basis of 1 . But then (yjk )k1 is also equivalent to this canonical basis of 1 . In particular there exists a y∗ ∈ X ∗ such that y∗ (yjk ) = (−1)k for any k  1. Hence (yjk )k1 is not w-Cauchy. This contradicts the fact that (yj )j1 is w-Cauchy. We conclude with a final result complementing Theorem V.1. Theorem V.7 Let X be a space possessing an unconditional basis (en )n1 . If its dual X ∗ is weakly sequentially complete, or, more generally, if X ∗ does not contain c0 , then the basis (en )n1 is shrinking. In particular, X ∗ is separable.

VI The Gowers Dichotomy Theorem

101

We will use the following result, which deserves to be set apart:  ∗ ∗ Proposition V.8 Any series n1 xn in a dual space X such that +∞ ∗ n=1 |xn (x)| < +∞ for all x ∈ X is weakly unconditionally Cauchy.   Proof We can define a linear mapping T : X → 1 by Tx = xn∗ (x) n1 , which is continuous by the closed graph theorem. Its adjoint operator T ∗ : ∞ → X ∗ satisfies: T ∗ (λ), x =

+∞

λn xn∗ (x)

n=1

for every λ = (λn )n1 ∈ ∞ . Hence:  N  N       ∗  sup  λn xn  = sup sup  λn xn∗ (x)  T λ∞ . N1

n=1

N1 x1 n=1

 ∗ ∗ ∗ Proof of Theorem V.7 We have +∞ n=1 |en (x) x (en )|  Ku x x , for every x ∈ X and x∗ ∈ X ∗ , since (en )n1 is unconditional. By the preceding lemma,  the series n1 x∗ (en ) e∗n is hence weakly unconditionally Cauchy in X ∗ . As we have assumed that X ∗ does not contain c0 , it converges unconditionally. Proposition III.17 of Chapter 2 ensures that the limit is x∗ . The sequence (e∗n )n1 is thus an unconditional basis of X ∗ , which, by Proposition III.18 of Chapter 2, translates to the basis (en )n1 being shrinking.

VI The Gowers Dichotomy Theorem In Chapter 2, the existence of a basis for the universal space (for separable spaces) C([0, 1]) enabled us to deduce the existence of basic sequences for every Banach space. But C([0, 1]) does not possess an unconditional basis, and is not even a subspace of a space with an unconditional basis, as we will see in Chapter 7; the same reasoning is thus not possible. The following question is hence natural: Question 1. Does every Banach space contain an unconditional basic sequence? According to the James theorems, a positive response to this first question would also lead to a positive answer to the following question: Question 2. Does every Banach space contain one of: a reflexive subspace, 1 , or c0 ? This would be a gem of a theorem of structure! These questions remained open for a long time. They were resolved in the negative. For Question 1,

102

3 Unconditional Convergence

Gowers and Maurey [1993], working independently during the summer of 1991, constructed a counter-example. They did not answer the second question, as their space GM was reflexive. Nonetheless, somewhat later (1992), Gowers [1994 a] used the same techniques to construct a Banach space containing neither a reflexive subspace, nor 1 , nor c0 . These constructions are extremely complicated, and we refer to the article of Gowers and Maurey [1993] (we point out that this contains a small error, detected and corrected by Kalton [1995]), or to the article of Ferenczi [1997 a] (and [1995]), who constructed a space of the same type, but also uniformly convex. We can nonetheless explain the starting idea. First, note that if X possesses an unconditional basis (en )n1 , then X can be decomposed into a direct (topological) sum of two infinite-dimensional (closed) subspaces Y and Z, with, for example: Y = span (e1 , e3 , e5 , . . .) and Z = span (e2 , e4 , e6 , . . .). The space GM constructed by Gowers and Maurey satisfies the following property: whenever Y and Z are two infinite-dimensional (closed) subspaces of X, their sum Y + Z is not closed. Definition VI.1 A Banach space is said to be hereditarily indecomposable (HI) if it is infinite-dimensional and if, whenever Y and Z are two infinitedimensional (closed) subspaces of X, their sum Y + Z is not closed. An HI space cannot contain an (infinite-dimensional) subspace with an unconditional basis. The space GM has its origins in the Tsirelson space (Tsirelson [1974]; see Exercise VIII.8): it is a reflexive space T (but with an unconditional basis), not containing any of the spaces p for 1  p < +∞ (nor c0 , as it is reflexive). In spite of its unconditional basis, T carries the seeds of the space GM. The important point is the way to iteratively construct the norm. In early 1991, a decisive step was the construction by Schlumprecht [1991] of a space S, of the same type as Tsirelson’s; Gowers and Maurey showed that, given any constant C  1, there exists a renorming for which S does not possess an unconditional basic sequence of constant C. These hereditarily indecomposable spaces could appear as oddities, especially after Gowers and Maurey [1997] had constructed Banach spaces without unconditional basic sequences, but not HI (and truly not HI , since GM⊕GM is not, but in an artificial way); however, Gowers discovered that Banach spaces are full of HI subspaces, even though we could not detect them earlier: Theorem VI.2 (The Gowers Dichotomy Theorem) Every Banach space contains either an infinite-dimensional subspace with an unconditional basis, or a hereditarily indecomposable subspace.

VI The Gowers Dichotomy Theorem

103

The proof of Gowers (Gowers [1994 b], [1996 b] and [2002]) in fact provides a more general result on the construction of basic blocks; it uses infinite game theory (roughly: one player attempts to construct an unconditional basic sequence, and the opponent an HI space). We will give hereafter an elementary (i.e. not requiring any prerequisites) proof by Maurey [1998]. First note that Gowers was thus able to resolve positively an old problem of Banach about homogeneous spaces: Theorem (Gowers) Any infinite-dimensional homogeneous Banach space, i.e. isomorphic to all its infinite-dimensional subspaces, is isomorphic to 2 . We attribute this theorem to Gowers, as he contributed the final touch, but we have to mention Komorowski and Tomczak-Jaegermann, who provided a difficult result essential to the proof (see Theorem B below), and whose names should be associated with Gowers’. We simply give a brief outline of this proof: 1) First, as every Banach space possesses an infinite-dimensional subspace with a basis, every homogeneous space itself possesses a basis. It ensues that all its subspaces also have bases. 2) Next, we use (see Chapter 5 of this volume for the notions of type and cotype): Theorem A (Szankowski [1978]) If X is a Banach space with every subspace satisfying the approximation property (in particular possessing a basis), then X is of type 2 − ε and of cotype 2 + ε, for any ε > 0. This result must be compared with Kwapie´n’s theorem (to be seen in Chapter 5): ´ Theorem A Banach space is isomorphic to a Hilbert space if Kwapien’s and only if it is simultaneously of type 2 and of cotype 2. A homogeneous Banach space thus appears “very close” to being a Hilbert space. This “very close” turns out in fact to be quite misleading, as Johnson showed: Theorem (Johnson [1980]) The “convexified Tsirelson space” * T does not T possesses a contain 2 , even though every quotient of every subspace of * basis. 3) The most delicate point is then the following result of Komorowski and Tomczak-Jaegermann [1995] and [1998]: Theorem B (Komorowski and Tomczak-Jaegermann) If an (infinitedimensional) Banach space is of finite cotype and if each of its subspaces possesses an unconditional basis, then it hereditarily contains 2 .

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3 Unconditional Convergence

Therefore, as a homogeneous space is of cotype 2 + ε for any ε > 0, either it contains 2 and hence is isomorphic to it because of its homogeneity, or it contains a subspace without an unconditional basis and hence, by homogeneity, contains no unconditional basic sequence. In the latter case, by the Gowers dichotomy theorem, it must contain an HI subspace; but then it must itself be HI , which is not possible since the HI spaces contain very few operators; in particular: Theorem C (Gowers–Maurey [1993]) A hereditarily indecomposable space is not isomorphic to any of its strict subspaces. Consequently, only the former case is possible: the space is isomorphic to 2 ! We now proceed to the proof of the dichotomy theorem. Proof of the Gowers dichotomy theorem (Maurey) We present the proof given in Maurey [1998]; for a slightly different proof, with a remarkably clear presentation of the subject, we refer to Maurey [1994]. It is based on the notion of the angle between two subspaces. Given two subspaces L and M of X, the angle between L and M is defined as the number: α(L, M) = inf{y − z ; y ∈ SL and z ∈ SM } , where SL and SM are the unit spheres of L and M. For a space X, to be HI means that the angle between any two infinite-dimensional subspaces Y and Z of X is null. Moreover, if (en )n1 is an unconditional basic sequence in X and if K is its unconditional constant, denote, for any set of integers I: EI = span {en , n ∈ I} and E = EN∗ . Consider the projection PI : E → EI , associating the vector +∞  n∈I an en to n=1 an en : it is of norm  K; thus, when I and J are two disjoint sets of integers, we have x and y  K x − y for every x ∈ EI and every y ∈ EJ . Hence, the angle between EI and EJ is  1/K. Rather than directly constructing HI spaces, we first use particular intermediate spaces. Given ε > 0, the (infinite-dimensional) Banach space X is said to be an HI (ε) space if α(Y, Z)  ε for every infinite-dimensional subspace Y and Z of X. Clearly a space is HI if and only if it is HI (ε) for any ε > 0. Then, to prove the theorem, it suffices to show the following: Proposition VI.3 Let X be an infinite-dimensional Banach space. Then, for any ε > 0, X contains either an (infinite) unconditional basic sequence of constant  4/ε, or an HI (ε) subspace.

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105

We deduce the theorem from this proposition by using a diagonalization procedure. Indeed, if X does not possess any infinite unconditional subsequence, then every infinite-dimensional subspace Y of X contains, for every ε > 0, a subspace Z that is HI (ε). Using this iteratively for ε = 1/2n , we thereby construct a decreasing sequence of subspaces Zn , each being HI (1/2n ) for any n. Next, for each n, select an element zn ∈ Zn of norm 1. Then Z = span {zn ; n  1} is HI , as it is contained in Zn + span {z1 , . . . , zn−1 }, and hence is HI (1/2n ) for any n  1. Proof of Proposition VI.3 Of course, X can be assumed separable. For any couple (E, F) of finite-dimensional subspaces of X and any infinitedimensional subspace Z of X, set: A(E, F; Z) = sup α(E + U, F + V) , U,V⊆Z

where the upper bound is taken over all the infinite-dimensional subspaces U and V of Z. Note that we always have (1)

A(E, F; Z)  α(E, F),

because α(E + U, F + V)  α(E, F). With the number ε > 0 being fixed once and for all, the couple (E, F) is said to accept Z if A(E, F; Z) < ε . In a more geometric language, we could say that (E, F) “parallelizes” Z. The strict inequality is required for approximation reasons, to be seen later. Note that the expression “the couple ({0}, {0}) accepts Z” means that Z is HI (ε ) for a certain ε < ε, hence a fortiori HI (ε). The relation of acceptance is clearly symmetric: (E, F) accepts Z if and only if (F, E) accepts Z, and if (E, F) accepts Z, it also accepts every Z ⊆ Z. Moreover, if (E, F) accepts Z, then it also accepts Z + G for every finitedimensional subspace G of X; in fact, if U and V are two arbitrary infinitedimensional subspaces of Z + G, the subspaces U = U ∩ Z and V = V ∩ Z of Z are also infinite-dimensional, and thus, as (E, F) accepts Z, we have: α(E + U, F + V)  α(E + U , F + V )  A(E, F; Z) < ε . Hence A(E, F; Z) = A(E, F; Z + G); the function Z → A(E, F; Z) does not change when Z is perturbed by a finite-dimensional space. This point is essential for the proof. Note that if (E, F) does not accept Z, then (1) implies α(E, F)  ε. The couple (E, F) is said to reject Z if no subspace Z ⊆ Z is accepted by (E, F).

106

3 Unconditional Convergence

In more vivid language, we could say that (E, F) makes Z “angular.” This notion of rejection is the principal tool to be used in the inductive construction of subspaces forming angles uniformly bounded from below. Observe that this relation is clearly symmetric in E and F, and that if the couple (E, F) rejects Z, it rejects all subspaces Z of Z. It then also rejects every super-space Z+G when G is finite-dimensional (since otherwise it would accept a subspace Z ⊆ Z+G, and hence also the infinite-dimensional subspace Z = Z ∩ Z of Z). It ensues that if Z is accepted or rejected by (E, F), so is every subspace Z of a finite-dimensional perturbation Z + G of Z (dim G < +∞). This remark is the starting point of the construction. Construction Step 1. Construction of a subspace. First we construct a subspace Z0 of X within which we can construct an unconditional sequence if Z0 is not HI (ε). Lemma VI.4 There exists an infinite-dimensional subspace Z0 of X such that, for every couple (E, F) of finite-dimensional subspaces of X, either (E, F) rejects Z0 or (E, F) accepts Z0 . We first prove this lemma for a dense family of couples (E, F): for this, we need a metric on the set F of finite-dimensional subspaces of X. With E and F ∈ F, we define:   δ(E, F) = max sup d(x, SF ), sup d(y, SE ) . x∈SE

y∈SF

This is the Hausdorff distance between the spheres SE and SF . As we have assumed X separable, we can select a countable family E ⊆ F which is dense in F for the distance δ. Sub-Lemma VI.5 There exists an infinite-dimensional subspace Z0 of X such that, for every couple (E, F) ∈ E × E, and any rational a ∈ ]0, ε[, we have: • either A(E, F; Z0 ) < a • or A(E, F; Z )  a for every infinite-dimensional subspace Z of Z0 . Proof Denote by (En , Fn , an )n1 the countable family of all triples (E, F, a), where E, F ∈ E and a ∈ Q ∩ ]0, ε[. We construct a decreasing sequence (Xn )n0 of infinite-dimensional subspaces of X by setting X0 = X, and then, by iterating the following procedure: • if A(En+1 , Fn+1; Z )  an+1 for every subspace Z of Xn , we take Xn+1 = Xn ; • if this is not the case, there exists a subspace of Xn , that we select for Xn+1 , such that A(En+1 , Fn+1 ; Xn+1 ) < an+1 .

VI The Gowers Dichotomy Theorem

107

Using a diagonal procedure, we then construct the infinite-dimensional subspace Z0 of X, choosing by induction zn+1 ∈ Xn+1 , of norm 1, such that zn+1 ∈ span {z1 , . . . , zn }. We call Z0 the closed subspace generated by the zn , n  1. Then, as Z0 ⊆ Xn + Gn , where Gn = span {z1 , . . . , zn−1 } is finitedimensional, we have: • either A(En , Fn ; Z ) = A(En , Fn ; Z ∩ Xn )  an • or A(En , Fn ; Z0 )  A(En , Fn ; Xn + Gn ) = A(En , Fn ; Xn ) < an . We would now like to extend this dichotomy from dense subsets to all finitedimensional subspaces. For this, we need the following approximation result: Lemma VI.6 Let E, E , M and Z be subspaces of X, with E and E finitedimensional, and Z infinite-dimensional. We have: sup α(E + U, M)  sup α(E + U, M) + 2δ(E , E) , U⊆Z

U⊆Z

where the upper bound is taken over all the infinite-dimensional subspaces U of Z. Proof

Set δ = δ(E, E ), and let: s > sup α(E + V, M) , V⊆Z

t > 1 and U be an arbitrary infinite-dimensional subspace of Z. As E is finitedimensional, its unit ball is compact, and we can thus find a finite number of linear functionals ϕ1 , . . . , ϕk of norm 1 such that: t sup{|ϕ1 (e)|, . . . , ϕk (e)|}  e for every e ∈ E. Set U = U∩ker ϕ1 ∩. . .∩ker ϕk . We have t u +e  e for every u ∈ U and every e ∈ E. Moreover, as U is an infinite-dimensional subspace of Z, we have α(E + U , M) < s. We can thus find e + u ∈ SE+U and y ∈ SM such that (e + u ) − y < s. However e + u  = 1 implies e  t; hence there exists e ∈ E such that e − e  tδ. Then 1 − tδ  e + u   1 + tδ, and thus there exists an x ∈ SE +U such that x − (e + u )  tδ. Finally, this gives: α(E + U, M)  α(E + U , M)  x − y < s + 2tδ , which completes the proof of the lemma. We can then deduce: Proof of Lemma VI.4 In fact, if (E, F) does not reject Z0 , it accepts a subspace Z of Z0 , and we can chose a rational a ∈ Q ∩ ]0, ε[ such that A(E, F; Z ) < a. Let b be a rational such that 0 < b < (ε − a)/8, and let

108

3 Unconditional Convergence

E , F ∈ E such that δ(E, E ) < b and δ(F, F ) < b. Lemma VI.6 then states that A(E , F ; Z ) < a + 4b < ε, and hence, by Sub-lemma VI.5, we have A(E , F ; Z0 ) < a + 4b. An additional application of Lemma VI.6 finally gives us A(E, F; Z0 ) < a + 4b + 4b < ε, so that (E, F) indeed accepts Z0 . Step 2. Construction of an unconditional sequence. The construction now continues within the subspace Z0 , and here the dichotomy starts. Recall that if ({0}, {0}) accepts Z0 , then Z0 is HI (ε). Thanks to Lemma VI.4, we can assume that ({0}, {0}) rejects Z0 . The idea is to construct an unconditional sequence (ek )k1 , with an unconditional constant K  4/ε. For this, we choose a sequence of non-null vectors so that, for any n  1, and for any disjoint subsets I, J ⊆ {1, . . . , n}, the couple (EI , EJ ) rejects Z0 , where EI = span {ek ; k ∈ I}. The construction of en+1 based on e1 , . . . , en is possible thanks to the following two lemmas. Lemma VI.7 If (E, F) rejects Z0 , then, for every infinite-dimensional subspace Z of Z0 , there exists another infinite-dimensional subspace U , contained in Z , such that, for every finite-dimensional subspace E of U , the couple (E + E , F) rejects Z0 . Proof If the conclusion does not hold, there exists Z ⊆ Z0 so that, for every U ⊆ Z , there would be E ⊆ U such that (E + E , F) does not reject Z0 . But then, by Lemma VI.4, (E + E , F) accepts Z0 . We thus have, for any subspace V ⊆ Z , since E + U = E + E + U : α(E + U , F + V ) = α(E + E + U , F + V )  A(E + E , F; Z0 ) < ε . This means that (E, F) accepts Z . Hence (E, F) does not reject Z0 , and consequently, again by Lemma VI.4, accepts it. Next, we deduce: Lemma VI.8 Suppose that (El , Fl )l∈L is a finite family of couples of finitedimensional subspaces all rejecting Z0 . Then, for any infinite-dimensional subspace Z of Z0 , there exists another infinite-dimensional subspace U ⊆ Z such that, for every finite-dimensional subspace E of U , the couple (El + E , Fl ) rejects Z0 , for any l ∈ L. Proof Denote L = {l1 , . . . , lp }. Let Z = Z0 be a subspace of Z0 . By Lemma VI.7, there exists U = Z1 ⊆ Z0 such that, for all E ⊆ Z1 , the couple (El1 + E , Fl1 ) rejects Z0 . We again apply Lemma VI.7, but this time to the couple (El2 , Fl2 ), and with Z = Z1 ; next, we only repeat the procedure until ⊆ Z1 ⊆ Z0 = Z . we reach U = Zp ⊆ Zp−1

VI The Gowers Dichotomy Theorem

109

We can now finish the proof. In fact, we use Lemma VI.8 in the following weakened form, where [z] denotes the subspace generated by z: Lemma V.8 Let (El , Fl )l∈L be a finite family of couples, each rejecting Z0 : then, for every infinite-dimensional subspace Z of Z0 , there exists a non-null vector z ∈ Z such that every couple (El + [z], Fl ), for l ∈ L, again rejects Z0 . Recall the assumption that ({0}, {0}) rejects Z0 , and our wish to construct by induction a sequence of non-null vectors ek , k  1, such that, for every partition I ∪ J = {1, . . . , n}, the couple (EI , EJ ) rejects Z0 , with EI = span {ek ; k ∈ I}. Lemma V.8 allows the construction of e1 starting from the couple ({0}, {0}). Suppose now that e1 , . . . , en have already been constructed. A partition of length n is any couple of the form (EI , EJ ), where (I, J) is a partition of {1, . . . , n}. Denote by {(El , Fl ) ; l ∈ Ln } the finite set of all partitions of length n. By our induction hypothesis, all the couples (El , Fl ), for l ∈ Ln , reject Z0 . Let Z be an infinite-dimensional subspace of Z0 such that Z ∩ span {e1 , . . . , en } = {0}. By Lemma V.8 , there exists a non-null vector z ∈ Z such that every couple (El + [z], Fl ), l ∈ Ln , rejects Z0 . Note that (Fl , El ) = (El , Fl ) for a certain l ∈ Ln , so that the (Fl + [z], El ), l ∈ Ln , also reject Z0 . Hence if we set en+1 = z, then (EI , EJ ) rejects Z0 , now for all partitions of length n + 1. It only remains to show: Lemma VI.9 If α(E, F)  ε, then x + y 

4 x − y ε

for every x ∈ E and every y ∈ F. Proof By symmetry, we can assume y  x. Next, by homogeneity, we can also assume x = 1. We thus have x + y  2, and hence it suffices to show that x − y  ε/2. Indeed: • either y  1 − ε/2, and then  ε ε = ; x − y  x − y  1 − 1 − 2 2 • or 1 − ε/2 < y  1, and then:   y  y  ε     ε  x−   x−y+ y−  = x−y+(1−y)  x−y+ , y y 2 so that also x − y  ε/2.

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3 Unconditional Convergence

We now finish: given scalars a1 , . . . , an and choices of signs θ1 , . . . θn ∈ {−1, 1}, denote I = {k ; θk = 1} and J = {k ; θk = −1}. Then x =   k∈I ak ek ∈ EI and y = − k∈J ak ek ∈ EJ ; hence: n n     4     θk ak ek  = x + y  x − y =  ak ek  ,  ε k=1

k=1

which shows that the sequence (ek )k1 is unconditional with constant  4/ε. This completes the proof of Proposition VI.3 and hence the Gowers dichotomy theorem.

VII Comments 1) The Orlicz theorem and the Orlicz–Pettis theorem were proved by Orlicz [1929]; the latter was rediscovered by Pettis [1938 a]. Other proofs can be found in Diestel, pages 27–28, 46 and 85–86, as well as in Talagrand, pages 28–29. 2) A notion weaker than unconditional basis is that of unconditional FDD. They are related as follows: any space having an unconditional FDD is a subspace of a space with an unconditional basis (see Lindenstrauss– Tzafriri I, Theorem 1.g.5). The unconditionality of independent random variables can be found in Rosenthal [1970], but was undoubtedly known earlier. 3) The results of Section IV are due to Bessaga and Pełczy´nski [1958 a]. A characterization of the spaces containing c0 was given more recently by Rosenthal: a basic sequence (xn )n1is said to be strongly summing  if it is weakly Cauchy and if supn1  nk=1 ak xk  < +∞ implies +∞ the convergence of the scalar series k=1 ak ; the typical example is the summing basis of c0 , given by sn = e1 + · · · + en = (1, . . . , 1, 0, 0, . . .). + ,- . n times

This leads to the following theorem: Theorem (Rosenthal [1994]) not weakly convergent, then:

If (xn )n1 is a weakly Cauchy sequence, but

– either (xn )n1 has a strongly summing basic subsequence; – or there is a sequence of blocks of (xn )n1 which is equivalent to the summing basis of c0 . It follows that a separable Banach space X has the property (u) of Pełczy´nski (see Exercise VIII.6) and does not contain 1 if and only if, for every subspace Y of X, Y ∗ is weakly sequentially complete (Rosenthal [1994]).

VIII Exercises

111

There are many other characterizations of c0 (see, for example, Argyros and Gasparis [2001] and Farmaki [2002]). We single out that of Kwapie´n [1974]: X does not contain c0 if and only if, for every sequence (xn )n1 in  X, the series n1 εn xn converges in norm for almost all choices of signs εn = ±1 as soon as its partial sums are bounded for almost all choices of signs: see Chapter 4, Exercise VII.6. Also, Odell and Schlumprecht [1994] showed that if X is an infinitedimensional Banach space, then every infinite-dimensional subspace of X contains c0 (i.e. X contains c0 hereditarily) if and only if every Lipschitz function on the unit sphere of X possesses, for any infinite-dimensional subspace Y of X, an arbitrarily small oscillation on an infinite-dimensional subspace of Y. 4) The James theorems were extended, first to the subspaces of spaces with unconditional bases (Bessaga and Pełczy´nski [1958 b]), then to the subspaces of spaces more general than those with unconditional bases, known as cyclic (Tzafriri [1969]), and finally to the subspaces of Banach lattices (Tzafriri [1972]). Note that, as the measure space M([0, 1]) on [0, 1] is weakly sequentially complete (see Chapter 7, Theorem II.7), and as C([0, 1]) contains 1 , it follows from Theorem IV.7 that C([0, 1]) does not have an unconditional basis, and from the results of Bessaga and Pełczy´nski cited above it is not even a subspace of such a space. We will see another proof in Chapter 7, Subsection II.2. 5) For Section VI, we refer to Maurey [2003 b]. Pelczar [2001] gives a generalization of Theorem VI.2. Argyros and Felouzis [2000] present a general method to construct HI spaces; they also show that every (infinitedimensional) Banach space contains 1 or a quotient of an HI space. For the dichotomy theorem, see also Gowers [2002]. Another proof was recently found by Figiel, Frankiewicz, Komorowski and Ryll–Nardzewski (see Odell [2002], pages 414–416 for an outline).

VIII Exercises Exercise VIII.1 An operator T : X → Y between two Banach spaces is said to be unconditionally convergent if it transforms every weakly unconditional Cauchy series in X into an unconditionally convergent series in Y. 1) Show that T is not unconditionally convergent if, and only if, there exists a subspace X0 of X, isomorphic to c0 , such that T realizes an isomorphism between X0 and TX0 . 2) Show that any weakly compact operator is unconditionally convergent.

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3 Unconditional Convergence

Exercise VIII.2 By using the parallelogram identity, generalized to n vectors, show that, for any normalized unconditional basis (en )n1 in a Hilbert +∞  2 space, the convergence of +∞ n=1 an en implies that of n=1 |an | (prove that the latter series is Cauchy), and hence this basis is equivalent to an orthonormal basis. Exercise VIII.3 Show that any space with an unconditional basis can be renormed in such a way that the unconditional constant of this basis is equal to 1. Exercise VIII.4 Show that the James space (Chapter 2, Exercise V.12) does not possess any unconditional basis. Exercise VIII.5 1) Show that if X is a Banach space whose dual is separable, then every bounded sequence in X possesses a weakly Cauchy subsequence (use the fact that the unit ball of X ∗∗ is w∗ -metrizable and compact). 2) Show that if X is a space with an unconditional basis and if X does not contain 1 , then every bounded sequence possesses a weakly Cauchy subsequence (in Chapter 8, the hypothesis of the existence of an unconditional basis will be proved to be superfluous, but this is a much more difficult result). Exercise VIII.6 A Banach space is said to satisfy the Pełczy´nski property (u) (Pełczy´nski [1958]; see also Singer, pages 445–448, or Lindenstrauss– Tzafriri II, pages 31–33) if, for every weakly Cauchy sequence (xn )n1 ,  there exists a weakly unconditionally Cauchy series n1 un such that n w xn − k=1 uk −−→ 0. n→+∞

1) Show that every weakly sequentially complete space has the property (u). 2) Show that c0 satisfies the property (u). 3) Show that if a Banach space satisfies the property (u), then it is weakly sequentially complete if, and only if, it does not contain c0 . Conclude that the James space J does not satisfy the property (u). 4) Let X be a Banach space with an unconditional basis (en )n1 . Let (xk )k1 be a weakly Cauchy sequence in X, and let  = w∗ - lim xk ∈ X ∗∗ . k→+∞

We set uk =

, e∗k ek .

 uk is weakly unconditionally Cauchy. k1 k  ∗ b) Show that limk→+∞ en xk − j=1 uj = 0 for any n  1. a) Show that the series

VIII Exercises c) Deduce that xk −

k

w −→ 0 j=1 uj − k→+∞

113

(use Lemma V.6), and that X satisfies

the property (u). 5) Let X be a Banach space satisfying the property (u), and let Y be a subspace of X. Let (yn )n1 be a weakly Cauchy sequence in Y, and  un be a weakly unconditionally Cauchy series in X such that zn = n1 w yn − nk=1 uk −−→ 0. Show that there exist p1 < p2 < · · · and numbers n→+∞  pn+1 −1  pn+1 −1    1/2n . Let λk  0 such that k=p λ = 1 and λ z k k k k=p n n pn+1 −1 λk yk ∈ Y; we set v1 = z1 and vn = zn − zn−1 for n  2. zn = k=p n  Show that the series n1 vn is weakly unconditionally Cauchy. Conclude that Y satisfies the property (u). 6) Deduce that C([0, 1]) is not a subspace of a space with an unconditional basis (use the fact that the James space does not satisfy the property (u)). 7) Show that C(ωω ) does not satisfy the property (u) (if K = [0, ωω ], we denote by K = K (0) , K = K (1) , K (2) , . . . the successive derived sets of K, i.e. each one is the set of limit points of its predecessor; verify that the function f equal to 1 on K (2n)  K (2n+1) and −1 on K (2n+1)  K (2n+2) , for any n ∈ N, is the limit of a weakly Cauchy sequence in C(ωω ), but is not the sum of a weakly unconditionally Cauchy series). 8) A Banach space X is said to satisfy the Pełczy´nski property (V) (see Pełczy´nski [1962]) if every unconditionally convergent operator T : X → Y (see Exercise VIII.1) is weakly compact. a) Show that if X satisfies the property (u) and if from every bounded sequence of X we can extract a weakly Cauchy subsequence (which is equivalent to the fact that X does not contain 1 , to be seen in Chapter 8), then X satisfies the property (V). b) Show that if X satisfies the property (V), then X ∗ is weakly sequentially complete (use Schur’s theorem, Chapter 2, Theorem III.10). 9) Let K be a compact space, and (μn )n1 be a sequence of measures in M(K) = [C(K)]∗ . Show that if there exist δ > 0 and disjoint measurable subsets En ⊆ K such that μn (En ) > δ for any n  1, then there exist a subsequence (νk )k1 of (μn )n1 and disjoint open sets G1 , G2 , . . . such that νk (Gk ) > δ/2. Then deduce that C(K) satisfies the property (V). Examples of spaces satisfying the property (u) can be found in Godefroy and Li [1989], Godefroy, Kalton and Saphar [1993] and Knaust and Odell [1989], where it is shown that the Hagler space (Hagler [1977]) has the property (u). Every C∗ -algebra satisfies the property (V) (Pfitzner [1994]). The dual of this notion is the property (V ∗ ): a Banach space Y satisfies (V ∗ ) if, for every non-weakly compact operator T : Z → Y, there exists a subspace Z0 of Z

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3 Unconditional Convergence

isomorphic to 1 for which T realizes an isomorphism, and such that T(Z0 ) is complemented in Y. The dual of any space possessing the property (V) has the property (V ∗ ); every space having the property (V ∗ ) is weakly sequentially complete; every space L1 (μ) has the property (V ∗ ); a more general class of spaces having the property (V ∗ ) was given by Pfitzner [1993]. Exercise VIII.7 Let X be an HI space, and GX be the set of infinitedimensional subspaces of X. If Y, Z ∈ GX , we say that Y Z if there exists an operator T : Y → X of the form T = IY + S, where IY : Y → X is the canonical injection, and where S : Y → X is strictly singular (see Chapter 2, Exercise V.10), and T realizes an isomorphism between Y and T(Y) ⊆ Z. Show that we thereby define a pre-order relation on GX . We now wish to show that this pre-order is filtering (Ferenczi [1997 b]), i.e. if Y, Z ∈ GX , there exists W ∈ GX such that W Y and W Z. 1) Show that every infinite-dimensional Banach space contains, for any ε > 0, an infinite-dimensional subspace possessing a basis with a basis constant  1 + ε (go back to the proof of Theorem III.6 and Corollary III.9 of Chapter 2, and note that the Schauder basis of C([0, 1]) is monotone). 2) Show that the spaces Y and Z can thus be assumed to possess bases, with constants  2. Whenever a Banach space possesses a basis, then for every vector x in this space, the support of x is by definition the set of indices for which the component of x in this basis is not null. 3) Using the quality HI of X, show that there exists y0 ∈ Y and z0 ∈ Z, of norm 1, and with finite support, such that y0 − z0   1/16. 4) By induction, construct two sequences (yn )n0 in Y and (zn )n0 in Z, of norm 1, with finite support, such that max(supp yn ) < min(supp yn+1 ) ,

max(supp zn ) < min(supp zn+1 ) ,

(1/16) 2−n

and yn − zn   for any n  0. 5) Show that the subspace W of Y generated by the yn ’s works (if we set T(yn ) = zn , this defines an operator from W onto the subspace of Z generated by the zn ’s, such that T = IY + K where K is compact, and T − IY   1/2). Exercise VIII.8 Let c00 be the space of all sequences that are eventually null. Its canonical basis is denoted (en )n1 , and, for x = (xn )n1 ∈ c00 , x0 = maxn1 |xn | is the uniform norm. By induction on m  0, we define:   p  k   j+1  1  xm+1 = max xm , max xn en    , 2 m j=1

n=pj +1

VIII Exercises

115

where the maximum between the brackets is taken over all choices of p1 < p2 < · · · < pk+1 such that p1  k. 1) Show that xT = limm→+∞ xm exists for all x ∈ c00 and defines a norm on c00 . 2) The completion of c00 for this norm is called the Tsirelson space and is denoted T. In fact it is the dual of the space constructed by Tsirelson [1974]; the construction presented here is due to Figiel and Johnson [1974] (see Lindenstrauss–Tzafriri I, 2.e1). Show that (en )n1 is a normalized unconditional basis of T. 3) Show that:  p   k   j+1  1  xn en  xT = max max |xn |, max   ; 2 n1 T j=1 n=pj +1  k  p1 < p2 < · · · < pk+1 , k  1  for every x = +∞ n=1 xn en ∈ T. 4) Deduce that, for any k  1, and every sequence of k normalized blocks pj+1 a e , 1  j  k, in which k  p1 < p2 < · · · < pk+1 , we uj = n=p j +1 n n have:   k k k   1  |cj |   c e  |cj | , j j  2 j=1

j=1

T

j=1

for all scalars c1 , . . . , ck . Conclude that T contains neither c0 , nor p for 1 < p < +∞ (use Theorem III.6–Bessaga–Pełczy´nski selection theorem– and Theorem III.12 of Chapter 2). 5) We suppose that T contains a subspace isomorphic to 1 . a) Show that there exists a normalized block basis (vj )j0 of (en )n1 such that:   +∞ +∞   +∞ 8  |bj |   b v  |bj | j j  9 j=0

j=0

T

j=0

(use Exercise V.15 of Chapter 2), and, in particular:  v1 + · · · + vr  16   v0 +   T r 9 for r = 1, 2, . . . Let k  p1 < p2 < · · · < pk+1 , and let Pσ1 , . . . , Pσk be the projections associated with the basis (en )n1 , where σj = {pj + 1, . . . , pj+1 }. Let n0 = max(supp v0 ).

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3 Unconditional Convergence

b) Show that, if k  n0 , we have: k   v1 + · · · + vr      2. Pσj v0 + T r j=1

c) For k < n0 , we set:  = {i  0 ; Pσj vi = 0 for at least two values of j},  = {i  0 ; Pσj vi = 0 for at most one value of j} . By noting that  has at most (k − 1) elements, show that: k   v1 + · · · + vr     Pσj v0 + T r j=1   1 2  2 v0 T + vi T + vi T r i∈

i∈

n0 − 1 · 3+ r d) Then show that, if r  2n0 , we have:  v1 + · · · + vr  7   v0 +   · T r 4 e) By using a) , conclude that actually T does not contain 1 . 6) Show that T is reflexive (use James’ theorems). For more information about the Tsirelson space, we refer to Casazza– Shura, Beauzamy–Lapresté, Casazza, Johnson and Tzafriri [1984] and Figiel and Johnson [1974].

4 Banach Space Valued Random Variables

I Introduction Let (, A, P) be a probability space, whose σ -algebra A is always assumed complete, and let E be a Banach space (most often separable and infinitedimensional). In this chapter, we study Banach space valued random variables, i.e. measurable functions on  with values in E; initially, this generalization does not give rise to any major difficulties, all the while proving extremely useful, notably in the study of Fourier series and Taylor series.

II Definitions. Convergence II.1 The Space L0 (E) Definition II.1 Let E be a Banach space, and B its Borel σ -algebra. A random variable with values in E is a mapping X :  → E satisfying the following two properties: (a) X is (A - B)-measurable, i.e. X −1 (B) ⊆ A; (b) there exists a separable Banach subspace E0 of E such that X takes a.s. its values in E0 : P(X ∈ E0 ) = 1. The set (of a.s. equivalence classes) of these random variables, which is clearly a vector space, is denoted L0 (, A, P; E), or simply L0 (E). As in the scalar case, fundamental elements of this space are the simple random variables, those taking on only a finite number of values: X=

n

1IAi xi ,

i=1

117

118

4 Banach Space Valued Random Variables

with Ai ∈ A and xi ∈ E. Clearly X satisfies (a) and (b) ; we also write X = n i=1 1IAi ⊗ xi . Condition (b) permits an approximation of every random variable by simple random variables, as is shown in the following simple proposition: Proposition II.2 1) Let X ∈ L0 (E) be a random variable. Then: (a) X is tight, i.e. for any ε > 0, there exists a compact K ⊆ E such that: P(X ∈ K)  1 − ε . (b) X is the almost sure limit of a sequence (Xn )n1 of simple variables such that: Xn (ω)  2 X(ω) for any n  1, and almost all ω ∈ . 2) If a sequence (Xn )n1 of L0 (E) converges almost surely to Y :  → E, then Y ∈ L0 (E). Proof 1) (a) If necessary replacing E by a separable Banach subspace in which X takes almost surely its values, without loss of generality we can assume E separable. Let D = {x1 , . . . , xn , . . .} be a dense countable subset of E with x1 = 0, and let q be an integer  1. As E is the union, for n = 1, 2, . . . , of the closed balls B(xn , 1/q), we can find an integer Nq B(xn , 1/q), Nq  1 such that PX (Aq )  1 − 2−q ε, where Aq = n=1 and where PX denotes the distribution of X, i.e. the probability image of

P by X: PX (A) = P(X ∈ A), for A ∈ B. Let K = ∞ q=1 Aq . Then K is compact as it is totally bounded, by construction, and complete. On the other hand, we have: PX (K c ) 

∞

PX (Acq ) 

q=1

∞

2−q ε = ε ,

q=1

which proves 1) (a). (b) Let " # Tn (ω) = inf k  n ; X(ω) − xk  = min X(ω) − xl  . 1ln

  For i, j ∈ we have X − xi  < X − xj  ∈ A, since X ∈ L0 (E); n hence Tn ∈ L0 (E), and Xn = xT = k=1 1I(Tn =k) xk is a simple n   variable. By definition X(ω)−Xn (ω) = dist X(ω), {x1 , . . . , xn } ; thus   X(ω) − Xn (ω) −−→ dist X(ω), D = 0 . N∗ ,

n→+∞

II Definitions. Convergence

119

Also: X(ω) − Xn (ω)  X(ω) − x1  = X(ω), so that Xn (ω)  2 X(ω) . 2) If Xn almost surely takes its values in the separable Banach subspace En ⊆ E, Y takes almost surely its values in the separable Banach space generated by the En ’s; as on the other hand the σ -algebra A is complete, without loss of generality we can assume that E is separable and that Xn (ω) −−→ Y(ω) n→+∞ for all ω ∈ . Now let G be an open subset of E, and k an integer  1; we set: Gk = {x ∈ E ; dist(x, Gc ) > 1/k} , and we verify that: (Y ∈ G) =

∞  

lim n→+∞

k=1

  Xn ∈ Gk .

  Indeed, if Y(ω) ∈ G, we can find k  1 such that dist Y(ω), Gc > 1/k, since Gc is closed; as Xn (ω) −−→ Y(ω), we have Xn (ω) ∈ Gk for n n→+∞ large enough; therefore the inclusion from left to right is proved. The reverse inclusion can similarly be shown. This equality shows that the set (Y ∈ G) = Y −1 (G) belongs to A. As the open subsets of E generate B, we obtain that Y is (A - B)-measurable and finally that Y ∈ L0 (E).

II.2 Convergence in Probability in L0 (E) This is a trivial generalization of the scalar case; therefore we do not give a detailed proof of the proposition that follows. A sequence (Xn )n1 of L0 (E) is said to converge in probability to X ∈ L0 (E) if, for any t > 0: P(Xn − X > t) −−→ 0 . n→+∞

We then denote: P

Xn −−→ X . n→+∞

For X, Y ∈ L0 (E), we set: dP (X, Y) = E Then:



X − Y 1 + X − Y

 ·

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4 Banach Space Valued Random Variables

Proposition II.3 1) For Xn , X in L0 (E), n  1: P

Xn −−→ X n→+∞

if and only if dP (Xn , X) −−→ 0 . n→+∞

2) dP is a metric on L0 (E), which is translation-invariant and complete.

II.3 Almost Sure Convergence in L0 (E) A sequence (Xn )n1 in L0 (E) is said to converge almost surely to X if there exists 0 ∈ A, of probability 1, such that Xn (ω) − X(ω) −−→ 0 for every n→+∞

ω ∈ 0 (Proposition II.2 then shows that X ∈ L0 (E)). We denote a.s.

Xn −−→ X . n→+∞

Here again, the following proposition is an immediate generalization of the scalar case. Proposition II.4 Let Xn , n  1, and X be variables in L0 (E). Then: a.s.

P

n→+∞

n→+∞

1) Xn −−→ X implies Xn −−→ X; P

2) if Xn −−→ X, there exists a subsequence of integers (nk )k1 for which n→+∞ a.s.

Xnk −−→ X; n→+∞ 3) (Xn )n1 converges almost surely if and only if P

Yn = sup Xk − Xn  −−→ 0 ; n→+∞

kn

4) (Xn )n converges almost surely to X if and only if P

Zn = sup Xk − X −−→ 0. n→+∞

kn

II.4 Convergence in Distribution in L0 (E) Let Cb (E) be the space of bounded continuous functions from E into its scalar field, and let X, X1 , . . . , Xn , . . . ∈ L0 (E). Analogously to the scalar case, the definition of “(Xn )n1 converges in distribution to X,” denoted: L

Xn −−→ X, n→+∞

would be E[ f (Xn )] −−→ E[ f (X)] n→+∞

II Definitions. Convergence

121

for every f ∈ Cb (E); or, with a more probabilistic style, P(Xn ∈ A) −−→ P(X ∈ A)

(1)

n→+∞

whenever P(X ∈ ∂A) = 0, A being a Borel set of E and ∂A its topological boundary. A few additional difficulties appear, due to the loss of local compactness of E. To avoid bogging down the presentation, we limit ourselves to the case where E is separable. Moreover, as the convergence in distribution plays a relatively minor role in this book, we will only outline the proofs, and refer to Borkar or Ledoux–Talagrand for more details. The proper framework is the following: (S, d) is a Polish space (i.e. a metrizable, complete and separable topological space), equipped with a metric d  1 (we can always reduce to this case); B is the σ -algebra of Borel subsets of S; P(S) is the set of probability measures on (S, B); Cb (S) denotes the vector space of bounded continuous functions f from S into C, normed by f ∞ = supx∈S |f (x)|. Finally, if d is a metric on S topologically equivalent to d, U Cb (S, d ) denotes the subspace of Cb (S) formed by all functions uniformly continuous for d . Lemma II.5 There exists a sequence ( fn )n1 in the unit ball of Cb (S), “separating” in the following sense:   If λ, μ ∈ P(S) and fn dλ = fn dμ for any n  1, then λ = μ. S

S ∗

N Sketch of the proof We embed S in the compact hypercube  1] with  K = [0, the homeomorphism h : S → h(S) defined by h(x) = d (x, sj ) j1 , where (sj )j1 is a dense sequence in S. As K is a compact metric space, the unit ball of C(K) contains a dense sequence (gn )n1 ; if we set fn = gn ◦ h, the sequence ( fn )n1 answers the question.

We can then equip P(S) with a metric ρ by setting    ∞  1  . ρ (λ, μ) = f dλ − f dμ n n   2n S S n=1

The following proposition clarifies formula (1) and connects it with the scalar case seen in Chapter 1. Proposition II.6 Let μ, μ1 , . . . , μn , . . . ∈ P(S). The following assertions are equivalent: 1) ρ(μn , μ) −−→ 0; n→+∞   2) f dμn −−→ f dμ, for every f ∈ Cb (S); S

n→+∞

S

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4 Banach Space Valued Random Variables

3) there exists a metric d on S equivalent to d such that   f dμn −−→ f dμ n→+∞

S

S

(S, d );

for every f ∈ U Cb 4) limn→+∞ μn (F)  μ(F) for every closed set F of S; 5) limn→+∞ μn (G)  μ(G) for every open set G of S; 6) limn→+∞ μn (A) = μ(A) for every A ∈ B such that μ(∂A) = 0. Then, (μn )n1 is said to converge in distribution to μ, abbreviated L μn −−→ μ. n→+∞

Sketch of the proof

We proceed in the order

2) ⇒ 1) ⇒ 3) ⇒ 4) ⇒ 5) ⇒ 6) ⇒ 2) . 2) ⇒ 1) is trivial. ∗ For 1) ⇒ 3) ,let d1 be a metric on K = [0, 1]N defining its topology, and let d (x, y) = d1 h(x), h(y) , with h as in Lemma II.5. d is equivalent to d. Let  μ = h(μ), for μ ∈ P(S). Keeping the notation of Lemma II.5, we have, for any k  1:     gk d6 μn = fk dμn −−→ fk dμ = gk d μ; K

n→+∞

S

hence:



S

K

 K

g d6 μn −−→ n→+∞

g d μ K

for any g ∈ C(K). If now f ∈ U Cb (S, d ), f ◦ h−1 : h(S) → C is uniformly continuous for the metric d1 , and can be extended to g ∈ C(K); we then have:     f dμn = g d6 μn −−→ g d μ = f dμ . S

K

n→+∞

3) ⇒ 4) . For ε > 0, let Oε = {x ∈ S ; ϕε (x) =

K

d (x, F)

S

< ε}, and

d (x, Ocε ) d (x, F) + d (x, Ocε )

;

then ϕε ∈ $ U Cb (S, d ),$ and ϕε = 1 on F; hence limn→+∞ μn (F)  limn→+∞ S ϕε dμn = S ϕε dμ, and limn→+∞ μn (F)  μ(F), on letting ε tend to 0. 4) and 5) are clearly equivalent by passing to the complement.  ◦ 4) and 5) ⇒ 6) : By hypothesis, μ(A) = μ A = μ(A); we then apply 4) and ◦

5) to A and A.

II Definitions. Convergence

123

6) ⇒ 2) : Let f ∈ Cb (S) be a real function, and ε > 0. We can find a0 < . . . < aN in R such that: a0 = −f  − 1, aN = f  + 1, μ( f = ai ) = 0 if 0  i  N, and ai − ai−1  ε

if 1  i  N.  Let Bi = {x ; ai−1  f (x) < $ai } and ϕ = $N i=1 ai 1IBi ; the Bi ’s form a partition of S, and μ(∂Bi ) = 0; hence S ϕ dμn −−→ S ϕ dμ. Moreover, f − ϕ∞  ε; $ $  n→+∞  $ $  f dμn − f dμ  2ε +  ϕ dμn − ϕ dμ, so that we thus obtain S S S S  $ $ $ $ limn→+∞  S f dμn − S f dμ  2ε, and limn→+∞ S f dμn = S f dμ. In the special case (S, d) = (E,  . ), convergence in distribution of Xn to X actually means convergence in distribution of PXn to PX (also known as weak convergence), where, in general, PY ∈ P(E) is the distribution of Y ∈ L0 (E): PY (A) = P(Y ∈ A) for every A ∈ B. An immediate corollary of Proposition II.6 is the following: Corollary II.7 Convergence in probability implies convergence in distribuL P tion: Xn −−→ X implies Xn −−→ X for X, X1 , . . . , Xn , . . . ∈ L0 (E). n→+∞ n→+∞ Proof We test condition 3) of Proposition II.6 with d =  . : let f ∈ U Cb (E) and δ its modulus of uniform continuity: δ(ε) = sup{f (x) − f (y) ; x − y  ε} . We have:       f dPX − f dPX  = |Ef (Xn ) − Ef (X)| n   E

E

 δ(ε) + 2 f ∞ P(Xn − X > ε) . The hypothesis implies

      lim  f dPXn − f dPX   δ(ε) n→+∞ E E $ $ for any ε > 0, so that limn→∞ E f dPXn = E f dPX .

  It can be useful to describe the relatively compact subsets of P(S), ρ , where ρ is the metric of Proposition II.6. A criterion due to Prokhorov provides such a description: it is analogous to the result of Ascoli for the spaces of continuous functions, where the notion of equicontinuity is replaced by that of uniform tightness.

124

4 Banach Space Valued Random Variables

Definition II.8 A subset T of P(S) is said to be uniformly tight if, for any ε > 0, the elements of T are supported by a single compact set, up to ε, i.e. for every ε > 0 there exists a compact subset K of S such that: μ(K)  1 − ε ,

∀μ ∈ T .

Note that a singleton is uniformly tight according to Proposition II.2 (valid for every Polish space). Theorem II.9   (Prokhorov’s Theorem) Let S be a Polish space and T a subset of P(S), ρ . Then T is relatively compact if and only if T is uniformly tight. Proof We first observe that, if G is an open subset of S, the function μ → μ(G) is lower semi-continuous on (P(S), ρ) by 5) of Proposition II.6. Now, fix ε > 0. Let (sj ) be a dense sequence in S and, for each pair of integers k, n  1, set: n  Gk,n = B(sj , 1/k). j=1

Fix k. The functions fn , defined by fn (μ) = μ(Gk,n ), are lower semi-continuous by the previous observation, and converge pointwise to μ(S) = 1. By the Dini theorem, they converge uniformly to 1 on the compact set T. Therefore, an nk B(sj , 1/k), we have: integer nk can be found such that, setting Ok = j=1 μ(Ok )  1 − ε 2−k for each μ ∈ T.

Now, if K = k1 Ok , the set K is totally bounded and closed, hence compact. Additionally if μ ∈ T, then:   ∞ Ok  1 − ε 2−k = 1 − ε. μ(K)  μ k1

k=1

This completes the proof of the necessary condition. For the sufficient condition, we keep the notation of Lemma II.5: let (μn )n1 be a sequence in T, and νn = h(μn ): νn ∈ P(K). As K is a compact metric space, by using the Banach–Alaoglu theorem and if necessary extracting a subsequence, we can assume that there exists ν ∈ P(K) such that $ $ f dν f dν, for every f ∈ C(K); in other words, νn tends weakly − − → n K n→+∞ K to ν in the sense of Proposition II.6. Moreover, for every integer m  1, the hypothesis provides a compact subset Km of S such that: μn (Km )  1 −

1 , m

II Definitions. Convergence

125

  with n = 1, 2, . . . We thus also have νn h(Km ) = μn (Km )  1 − 1/m, and Proposition II.6, 4) shows that     1 ν h(Km )  lim νn h(Km )  1 − ; n→+∞ m hence

    ν h(S)  sup ν h(Km ) = 1 , 



m

whereby ν h(S) = 1, i.e. that ν is supported by h(S), rather than the whole of h(S). This allows us to write ν = h(μ), where μ = h−1 (ν) ∈ P(S). Now let F be a closed subset of S; h(F) is a closed subset of h(S) since h is a homeomorphism. Then, by another   application   of Proposition II.6, as limn→+∞ μn (F) = limn→+∞ νn h(F)  ν h(F) = μ(F), μn tends weakly to μ: ρ (μn , μ) −−→ 0. Hence, every sequence of T contains a n→+∞ convergent subsequence, and T is relatively compact in the complete metric   space P(S), ρ . Now we study how two theorems of Paul Lévy from Chapter 1 can be generalized in the Banach space valued context. If E is a Banach space, and Er∗ its real dual (the space of continuous R-linear functionals of E into R), the characteristic function X of X ∈ L0 (E) is defined by the formula:   X ( f ) = E eif (X) , ∀ f ∈ Er∗ . Theorem II.10 Let (Xn )n1 be a sequence in L0 (E) and X ∈ L0 (E). Then Xn converges in distribution to X if and only if: 1) Xn converges pointwise to X on Er∗ 2) and moreover (Xn )n1 is uniformly tight. Here “(Xn )n1 is uniformly tight” means “(PXn )n1 is uniformly tight.” Proof Necessary condition: The sequence of distributions (PXn )n1 is uniformly tight since it is relatively compact (easy half of Prokhorov’s theorem). Moreover, if f ∈ Er∗ , the function g = eif is bounded by 1 and (uniformly) continuous; hence, by Proposition II.6: Eg(Xn ) −−→ Eg(X), i.e. n→+∞

Xn ( f ) −−→ X ( f ). n→+∞ Sufficient condition: By hypothesis the sequence (Xn )n1 is uniformly tight, so by Prokhorov’s theorem (PXn )n1 is relatively compact. It thus suffices to see that PX is its unique cluster point. Let μ be a cluster point of μn = PXn in P(E). We fix f ∈ Er∗ , and consider f (μ) and f (PX ), the probability images by f of μ and PX . For t ∈ R, we have:

126

4 Banach Space Valued Random Variables 

 R

eitx df (μ)(x) = E

eitf (y) dμ(y) = lim Xn (tf ) = X (tf ) = f (X) (t) , n∈A

where A is an infinite subsequence of integers. As t ∈ R is arbitrary, it follows from Chapter 1 that f (μ) = Pf (X) . But Pf (X) = f (PX ). By letting f vary, we obtain μ = PX . This theorem will be used in Chapter 5 with E = Rd . Note that the convergence in distribution requires the additional hypothesis of tightness of (Xn )n1 , which is an automatic consequence of Xn −−→ X when dim E < n→+∞ ∞ (the proof in Chapter 1 can be immediately generalized). The generalization of the second theorem of Paul Lévy is seen in the next section.

II.5 Spaces Lp (, A, P; E) For 0 < p < +∞, Lp (, A, P; E), abbreviated Lp (E) if there is no risk of confusion, denotes the set of X ∈ L0 (E) such that:  Xp dP < +∞ . E(Xp ) = 

Lp (E) is a vector space and the Lp (E) are non-increasing: Lq (E) ⊆ Lp (E) if q  p. For X ∈ Lp (E), p  1, we wish to define a vectorial expectation E(X) ∈ E. This is possible thanks to the following theorem: Theorem II.11 There exists a unique linear mapping E : L1 (E) → E, called the expectation, such that:   1) E(X) = ni=1 P(Ai ) xi if X = ni=1 1IAi xi ; 2) E(X)  E(X), for every X ∈ L1 (E). Proof Let L00 (E) be the subspace of L1 (E) formed by the simple variables.  Even when the Ai ’s are not disjoint, we can verify that ni=1 P(Ai ) xi does n not depend on the decomposition chosen for X = i=1 1IAi xi , and that, with  E(X) = ni=1 P(Ai ) xi , X → E(X) is a linear functional on L00 (E). This linear  functional is contracting, since E(X)  ni=1 xi P(Ai ) = E(X) (the last equality is obtained with the Ai ’s chosen disjoint). Now let X ∈ L1 (E); Proposition II.2 provides a sequence (Xn )n1 of L00 (E) such that Xn (ω) −−→ X(ω) for almost all ω and such that Xn (ω)  2 X(ω). n→+∞

Since for almost all ω, Xp (ω) − Xq (ω) −−−→ 0 and Xp (ω) − Xq (ω)  p,q→+∞

4 X(ω), we have EXp − Xq  −−−→ 0, according to the Lebesgue dominated p,q→+∞

convergence theorem. A fortiori, EXp − EXq  −−−→ 0: indeed, since Xp − Xq p,q→+∞

are in L00 (E), we have EXp − EXq  = E(Xp − Xq )  EXp − Xq . Thus E(Xn ) converges to an element l ∈ E. We then perform the usual verifications:

II Definitions. Convergence

127

l does not depend on the choice of the sequence (Xn )n1 , and we can thereby set l = E(X). The map E thus defined is a linear operator. Moreover, a passage to the limit in E(Xn )  E(Xn ) gives E(X)  E(X). The expectation E has the properties that we are entitled to expect (. . . !)  from this extension of the scalar case. For example, with X = N j=1 θj xj , xj ∈ E, and with θj real integrable random variables, we can easily verify that the N p given definition leads to E(X) = j=1 E(θj )xj . For 1  p < +∞, L (E) p 1/p is a Banach space for the norm Xp = XLp (E) = (EX ) ; the proof is not harder than in the scalar case: similarly we show that every absolutely convergent series in Lp (E) is convergent. Finally, the notion of independence in L0 (E) is the same as in the scalar case: X, Y ∈ L0 (E) = L0 (, A, P; E) are said to be independent if: P(X ∈ A1 , Y ∈ A2 ) = P(X ∈ A1 ) P(Y ∈ A2 ), for all Borel sets A1 , A2 ∈ B. Sequences of independent variables in L0 (E) etc. can be similarly defined. Such sequences turn out, as we will see, to be basic and unconditional in Lp (E) if additionally they are centered, i.e. with expectation zero. The next proposition (Proposition II.13) will play an important role for this. But first let us define the conditional expectation in the vectorial case. The Radon–Nikodým theorem is no longer true in general for measures with values in E (when it holds, E is said to have the Radon–Nikodým property, abbreviated “E has RNP”; for example, the space E = L1 [0, 1] does not have RNP: see Exercise VII.7). Nevertheless, the conditional expectation of X ∈ L1 (E) knowing A0 (A0 sub-σ -algebra of A) can be defined as follows:  If Y = ni=1 1IAi xi is a simple variable, set: E(Y | A0 ) = EA0 (Y) =

n

E(1IAi | A0 ) xi ,

i=1

with the scalar conditional expectation appearing in the expression on the right. We can verify that the result does not depend on the decomposition of Y, and contracts the L1 -norm. We can then extend the definition to L1 (E) by density, since the simple variables are dense “from below” in L1 (E) (see Proposition II.2). Clearly, the operation E( . | A0 ) inherits many of the properties of the scalar case. For example, it is linear, idempotent and contracts every Lp -norm; moreover:   E E(X | A0 ) = E(X) ; E(X | A0 ) = X

if X is A0 -measurable;

E(X | A0 ) = E(X)

if X is independent of A0 .

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4 Banach Space Valued Random Variables

If now A1 ⊆ . . . ⊆ An ⊆ . . . is an increasing sequence of sub-σ -algebras of A, the notion of martingale can be defined as in the scalar case: Mn = E(Mn+1 | An ) . The martingale is said to be closed if there exists M ∈ L1 (E) such that Mn = E(M | An ) for any n. A part of Doob’s theorem continues to hold in the vectorial case. Theorem II.12 (Doob’s Vectorial Theorem) Let (Mn )n1 be a closed martingale of L1 (E): Mn = E(M | An ). Then, with A∞ the σ -algebra generated by the union of the An , n  1, and with M∞ = E(M | A∞ ), the martingale (Mn )n1 converges in L1 and almost surely to M∞ . Proof The proof given in the scalar case can be reproduced to show that Mn −−→ M∞ in L1 (E); next, we apply Doob’s maximal inequality to the n→+∞ scalar submartingale Mn+1 − Mn , Mn+2 − Mn , . . . (it is a submartingale for the filtration An+1 , An+2 , . . ., where n is fixed), and we obtain P Yn = supk>n Mk − Mn  −−→ 0. Finally, by Proposition II.4, Mn converges n→+∞ a.s. (necessarily to M∞ ). The following proposition will be needed (see also Chapter 3, Lemma III.7, for a slight variant of the proof): Proposition II.13 Let 1  p < +∞ and X, Y ∈ Lp (E), independent with Y centered: E(Y) = 0. Then we have the inequality: Xp  X + Yp . Proof In terms of conditional expectation, the result is obvious: indeed, X = E(X+Y | X), and the conditional expectation contracts the Lp -norms. However we give an alternative direct proof, based on the following remark: If Z :  ×  → E is defined by Z(ω, ω ) = X(ω) + Y(ω ), Z has the same distribution as X + Y :  → E, which is an immediate consequence of the independence of X and Y. It ensues that Zp = X + Yp . 

This being so, we can write X(ω) = [X(ω)+Y(ω )] dP(ω ), as E(Y) = 0;   hence X(ω)p  X(ω) + Y(ω )p dP(ω ), by Jensen’s inequality. Then, 

by Fubini’s theorem,   X(ω)p dP(ω)  

X(ω) + Y(ω )p dP(ω) dP(ω );

×

hence Xp  Zp = X + Yp , which completes the proof.

III The Paul Lévy Symmetry Principle and Applications

129

III The Paul Lévy Symmetry Principle and Applications III.1 Symmetry Principle We begin with a fundamental definition: Definition III.1 A random variable X ∈ L0 (E) is said to be symmetric if (−X) has the same distribution as X: P(−X) = PX , or P(X ∈ A) = P(−X ∈ A) for every Borel set A ∈ B. A sequence of random variables X1 , . . . , Xn , . . . ∈ L0 (E) is said to be symmetric if the sequence (ε1 X1 , . . . , εn Xn , . . .) has the same distribution as  (X1 , . . . , Xn , . . .) for every choice of signs εi = ±1 (abbreviated εn Xn ) ∼ (Xn )). If the Xn ’s are independent and symmetric, (X1 , . . . , Xn , . . .) is clearly symmetric; the centered Gaussian and centered Bernoulli variables are symmetric. Such variables possess remarkable properties, as is shown in the following theorem, also known under the name of the Paul Lévy maximal inequality: Theorem III.2 (The Paul Lévy Symmetry Principle) Let X1 , . . . , XN ∈ L0 (E) be independent and symmetric random variables and Sn = X1 + · · · + Xn , for 1  n  N . Then, for every t > 0, we have:   P max Sn  > t  2 P(SN  > t) ; (1) nN   P max Xn  > t  2 P(SN  > t) . (2) nN

Proof The symmetry principle is illustrated by the following diagram (Figure 4.1) and maxim: As soon as we pass to the other side of the mirror (Sn  > t), then SN , or its reflection 2Sn − SN with respect to Sn , is also on the other side of the mirror, and this with the same probability. In mathematical terms, we set: T = inf{k  N ; Sk  > t} or T = ∞ if no Sk  exceeds t; this can be written: (3)

(T = n) ⊆ (SN  > t , T = n) ∪ (2Sn − SN  > t , T = n) ;

(4)

P(SN  > t , T = n) = P(2Sn − SN  > t , T = n) .

The inclusion (3) can be seen as follows: if T = n and SN   t, then: 2Sn − SN   2 Sn  − SN  > 2t − t = t .

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4 Banach Space Valued Random Variables

2Sn – SN

The other side of the mirror: || || > t

Sn

Mirror

One side of the mirror: || || £ t

SN

Figure 4.1

For (4), we set ε1 = · · · = εn = 1, εn+1 = · · · = εN = −1 and j Sj = i=1 εi Xi ; hence Sj = Sj if j  n, and 2Sn − SN = SN . The sequences (S1 , . . . , SN ) and (S1 , . . . , SN ) are equidistributed; hence: P(SN  > t , T = n) = P(S1   t , . . . , Sn−1   t , Sn  > t , SN  > t)   t , Sn  > t , SN  > t) = P(S1   t , . . . , Sn−1

= P(2Sn − SN  > t , T = n) . It ensues that P(T = n)  2 P(SN  > t , T = n), and then: N N   P(T = n)  2 P(SN  > t , T = n) P max Sn  > t = nN

n=1

n=1

= 2 P(SN  > t , T  N) = 2 P(SN  > t) . The proof of (2) is analogous: we set T = inf{k  N ; Xk  > t} or T = ∞, and if T = n, we reflect SN with respect to Xn , replacing it by 2Xn − SN if SN   t. Remarkably in (1), there is no assumption about the integrability of the Xn ’s, in contrast to most other maximal inequalities (Kolmogorov, Doob etc.).

III.2 Applications Here is an important application, already mentioned in Section II: Theorem III.3 (The Paul Lévy Equivalence Theorem) Let (Xn )n1 be an arbitrary sequence of independent random variables in L0 (E), and Sn = X1 + · · · + Xn the sequence of its partial sums. Then, the following assertions are equivalent:

III The Paul Lévy Symmetry Principle and Applications

131

1) Sn converges in distribution (to S); 2) Sn converges in probability; 3) Sn converges almost surely. Proof The scalar case was already seen in Chapter 1. The vectorial case is more delicate, as we can no longer rely on the three-series theorem. 1) ⇒ 2) : The key point is the following: P

Lemma III.4 If Sn = X1 +· · ·+Xn converges in distribution, then Xn −−→ 0 . n→+∞

Proof The space E can be assumed separable. Its dual E∗ thus contains a sequence (ϕj )j1 that separates the points of E: ϕj (x) = 0 ∀ j  1 ⇒ x = 0 .     By Theorem II.10, we have E eitϕ(Sn ) −−→ E eitϕ(S) for any ϕ ∈ E∗ and n→+∞

L

every t ∈ R. Hence ϕ(Sn ) = ϕ(X1 ) + · · · + ϕ(Xn ) −−→ ϕ(S). As the variables n→+∞  are now scalar, the series n1 ϕ(Xn ) converges a.s. (Chapter 1, Theorem a.s. III.8). In particular: ϕ(Xn ) −−→ 0. Thus there exists 0 ⊆  of probability 1 n→+∞ such that: lim ϕj [Xn (ω)] = 0

n→+∞

for any j  1 and every ω ∈ 0 . Moreover, as Xn = Sn − Sn−1 , the sequence (Xn )n1 is uniformly tight, and hence, for any ε > 0, we can find a compact set Kε ⊆ E such that supn1 P(Xn ∈ Kε )  ε. Then, for any t > 0 and with A(ε) = limn→+∞ {Xn  > t, Xn ∈ Kε }, we obtain:     lim P(Xn  > t)  ε + P A(ε) = ε + P A(ε) ∩ 0 , n→+∞

by Fatou’s lemma, applied to the inequality P(Xn  > t)  P(Xn  > t, Xn ∈ Kε ) + P(Xn ∈ / Kε ) . We now show that A(ε) ∩ 0 = ∅. If not, let ω ∈ A(ε) ∩ 0 ; we can find an increasing sequence of integers (nk )k1 such that:  Xnk (ω) ∈ Kε and Xnk (ω) > t. Let xω ∈ Kε be a cluster point of Xnk (ω) k1 ( for the norm); then, on one hand, xω   t, and on the other, ϕj (xω ) = 0, for any j  1 (since ω ∈ 0 ); however, the sequence (ϕj )j1 is separating, and thus xω = 0, which is a contradiction. This proves the lemma. The rest of the proof of 1) ⇒ 2) in the theorem is standard: for every increasing sequence of positive integers (nk )k0 , with n0 = 0, we set S0 = 0

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4 Banach Space Valued Random Variables

and Tk = Snk − Snk−1 . Then T1 + · · · + Tk = Snk converges in distribution to P S, and the Tk ’s are independent. Then, by the lemma, Tk −−→ 0. As this is true k→+∞

for every sequence (nk )k0 , (Sn )n1 is thus a Cauchy sequence in probability, and hence converges in probability. 2) ⇒ 3) : First we assume the Xn symmetric, and set: Yn,N = sup Sk − Sn  , nn

Let ε > 0 and t > 0. The hypothesis and Lévy’s inequality (1) give: P(Yn,N > t)  2 P(SN − Sn  > t)  ε, when n  n0 = n0 (ε, t). Then, by letting N tend to +∞, P(Yn > t)  ε when P n  n0 , which shows that Yn −−→ 0, and hence that Sn converges almost surely, n→+∞ by Proposition II.4, 3) . In the general case, we can define the symmetrization X of a variable X by setting X(ω, ω ) = X(ω) − X(ω ); then X is symmetric (on the product  P P space  × ), and if Sn −−→ S, we have Sn = nj=1 X j −−→ S. The first part n→+∞ n→+∞ a.s.

of the reasoning shows that Sn −−→ S. Then, by Fubini’s theorem, for almost n→+∞

all ω , Sn (ω, ω ) converges for almost all ω. In particular, there exists ω0 such that Sn (ω) − Sn (ω0 ) −−→ L(ω) for almost all ω. Set xn = Sn (ω0 ): we have n→+∞

P

P

P

n→+∞

n→+∞

Sn −−→ S, and Sn − xn −−→ L, thus xn −−→ S − L. But xn is a constant variable, n→+∞

a.s.

hence xn −−→ x ∈ E, and consequently Sn −−→ L + x. n→+∞

n→+∞

Here is another application of the symmetry principle (a more general assertion will be seen in Chapter 6 of Volume 2): Theorem III.5 Let (Xn )n1 be a sequence of independent symmetric variables, and Sn = X1 +· · ·+Xn be its sequence of partial sums. Assume that there a.s. exists a subsequence of integers p1 < · · · < pn < · · · for which Spj −−→ S. j→+∞

Then Sn tends to S almost surely. Proof First note the following variant of Theorem III.2: If the Yn ’s are independent and symmetric, if is an infinite subset of N∗ and if Tn = Y1 + · · · + Yn , M = supn1 Tn , and M = supn∈ Tn , then, for any t > 0, we have: (∗)

P(M > t)  2 P(M > t) .

IV The Contraction Principle

133

Indeed, if T = inf{n  1 ; Tn  > t} or +∞ if there is no such n, we see  that P(M > t) = ∞ n=1 P(T = n), and, when λn ∈ with λn > n, the proof of Theorem III.2 shows that: P(T = n)  2 P(Tλn  > t , T = n) ; hence, by summing: P(M > t)  2 P(M > t , T < ∞) = 2 P(M > t) . Then, set Yj = supk>j Spk − Spj  and Zn = supm>n Sm − Sn . For n > p1 , r = r(n) denotes the integer such that pr < n  pr+1 . When m > n, we have: Sm − Sn   Sm − Spr  + Sn − Spr  ; hence:

  P(Zn > t)  P sup Sl − Spr  > t/2  2 P(Yr > t/2) , l>pr

the last inequality resulting from (∗) applied to the variables Xn of index n > pr and to = {pr+1 , pr+2 , . . .}. To conclude, we apply Proposition II.4 twice: P(Yr > t/2) −−→ 0; hence, as r→+∞

P

r → +∞ with n, Zn −−→ 0, and so finally Sn converges almost surely. n→+∞

IV The Contraction Principle The contraction principle is a basic tool in the geometry of Banach spaces, even essential in the study of series of vectorial random variables, as we now see.

IV.1 Qualitative Version An almost surely convergent series of symmetric random variables is in general neither unconditionally convergent nor summable; nonetheless, with independence, it inherits one of the properties of summable families, as shown in Theorem IV.1: Theorem IV.1 (Qualitative Contraction Principle) Let (Xn )n1 be a sequence  almost of independent and symmetric variables such that ∞ n=1 Xn converges ∞ surely. Then, for every bounded sequence (λn )n1 of scalars, n=1 λn Xn converges almost surely as well. Proof We can assume λn ∈ R and −1  λn  1. We use the following lemma:

134

4 Banach Space Valued Random Variables

Lemma IV.2 Set Sn = X1 + · · · + Xn , Tn = λ1 X1 + · · · + λn Xn , for n  1, and S0 = T0 = 0. Then, for every t > 0 and any N  1: P(TN  > t)  2 P(SN  > t) . Proof of the lemma We can assume the λn  0, since the sequences (λn Xn )n1 and (|λn |Xn )n1 have the same distribution, thanks to the symmetry of the Xn ’s. First assume λ1  · · ·  λN  0. An Abel summation by parts gives: TN =

N

λn (Sn − Sn−1 ) =

n=1

N−1

(λn − λn+1 )Sn + λN SN .

n=1

Then, setting M = supnN Sn , we have: TN  

N−1

(λn − λn+1 )Sn  + λN SN 

n=1

M

N−1

(λn − λn+1 ) + λN = Mλ1  M .

n=1

Next, the Lévy maximal inequality gives: P(TN  > t)  P(M > t)  2 P(SN  > t) . In the general case, we rearrange the λn ’s in non-increasing order, with a permutation σ of {1, . . . , N} such that λ∗1  . . .  λ∗N , where λ∗n = λσ (n) , N ∗ and set Xn∗ = Xσ (n) . It suffices then to note that SN = k=1 Xk and TN = N ∗ ∗ k=1 λk Xk (the summation from 1 to N lands us back on our feet): the first part of the reasoning shows that P(TN  > t)  2 P(SN  > t), and this completes the proof of the lemma. It is now easy to finish: Lemma IV.2 implies P(Tq − Tp  > t)  2 P(Sq − Sp  > t) for p < q: hence (Tn )n1 is a Cauchy sequence in probability. It is thus convergent in probability, and almost surely convergent according to Theorem III.3. Remarks 1) The contraction principle is obviously false with deterministic series  n n n1 xn , as can be seen with the example xn = (−1) /n, λn = (−1) . 2) It is also false for independent variables that are only centered (i.e. EXn = 0), as shown by the following example, where the Xn ’s (scalars) are “large with small probability”:  (−1)n /n with probability 1 − 2−n Xn = αn /n with probability 2−n ,

IV The Contraction Principle

135

where αn = (−1)n (1 − 2n ) is adjusted so that EXn = 0. By the Borel– Cantelli lemma, almost surely Xn is equal to (−1)n /n for n large enough; ∞ ∞ n thus n=1 Xn converges almost surely, while n=1 (−1) Xn is always divergent. 3) An almost surely convergent series of symmetric independent variables is not always almost surely summable, even in the scalar case; indeed, let Xn = an εn , where (εn )n1 is a Bernoulli sequence and the an ’s positive +∞ 2 +∞ then real numbers such that n=1 an = +∞ and n=1 an < +∞; +∞ +∞ +∞ |X | = a = +∞, and by Kolmogorov’s theorem n n n=1 n=1 n=1 Xn converges almost surely. The fact is that, in Theorem IV.1, the almost sure set of convergence depends on the multipliers (λn )n1 in an uncountable manner. 4) In the scalar case, or even for Hilbert spaces, we can give a more direct proof of the contraction principle, thanks to the equivalence: ∞

Xn converges a.s.

⇐⇒

n=1

∞

Xn 2 < +∞ a.s.

n=1

which is a symmetrization of the “Bernoulli” case: ∞

εn xn converges a.s.

⇐⇒

n=1

∞

xn 2 < +∞

n=1

(see Exercise VII.4). 5) If the Xn ’s are independent and symmetric, and if σ is an injection from N∗ into itself, then: ∞

Xn converges a.s.

⇒

n=1

∞

Xσ (n) converges a.s. (ω ∈ / Nσ ) .

n=1

Indeed, if u < v, set p = minunv σ (n) and q = maxunv σ (n); then unv

Xσ (n) =



λk Xk ,

pkq

with λk = 1 if k ∈ σ (N∗ ), and λk = 0 otherwise. Lemma IV.2 shows that:               Xσ (n)  > t  2 P  Xk  > t ; P  unv

pkq

 as p, q → +∞ when u, v → +∞, the sequence N n=1 Xσ (n) is Cauchy in probability, and hence, by Theorem III.3, converges almost surely.

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4 Banach Space Valued Random Variables

IV.2 Quantitative Version The sequences of symmetric or centered independent random variables are good examples of unconditional basic sequences (see Chapter 3), as is shown by the following theorem, generalizing the one already seen in the scalar case: Theorem IV.3 (Quantitative Contraction Principle) Let E be a real Banach space and X1 , . . . , XN ∈ Lp (E) be independent variables, with 1  p < ∞. Then: 1) If the Xn ’s are symmetric, they are unconditional with constant 1:      N    N     λ a X  sup |λ | a X n n n n  n n  p p n L (E)

n=1

n=1

L (E)

for any an , λn ∈ R. 2) If the Xn ’s are centered, they are unconditional with constant 2:  N    N        λn an Xn   2 sup |λn |  an Xn    n

Lp (E)

n=1

n=1

Lp (E)

for any an , λn ∈ R. If the Banach space E is complex, the constants 1 and 2 have to be replaced by 2 and 4, respectively. Proof Recall that the unconditional constant of a sequence x1 , . . . , xN in a real Banach space (here Lp (E)) is the best constant M such that      N  N        M θ a x a x n n n n n   n=1

n=1

for θn = ±1 and an ∈ R. As the sequence (an Xn )n1 and the sequence (Xn )n1 share the same properties, we can assume an = 1 to prove the theorem. Set  . p as an abbreviation for  . Lp (E) . With θn = ±1, we need to show the following inequality:        N  N    θn Xn    Xn   ,  n=1

p

n=1

p

but in fact this is trivial since the two sides of the inequality are equal when the Xn ’s are symmetric. When the Xn ’s are only centered, with θn = ±1, we need to show:    N  N        θn Xn   2  Xn   .  n=1

p

n=1

p

IV The Contraction Principle

For this we set X=



Xn

and

θn =1

Y=



137

Xn ,

θn =−1

and apply Proposition II.13, noting that X and Y are both centered; this gives:      N    N    θn Xn  = X − Yp  Xp + Yp  2 X + Yp = 2  Xn    . n=1

p

n=1

p

The passage to the complex case is evident, with the decomposition into real and imaginary parts, and the proof is then complete. Remark 1 In the case of a complex space, with the Xn assumed symmetric in the complex sense, i.e. θ Xn ∼ Xn whenever |θ | = 1, then clearly the sequence (Xn )nN is unconditional with constant 1, in the complex sense (meaning that we replace θn = ±1 by θn ∈ C with |θn | = 1 in the definition), and thus a fortiori unconditional with constant 1, in the usual real sense. Remark 2 Up to a change of variables, the inequalities of Theorem IV.3 can also be stated in the following frequently used form:    N  N         λ a X  inf |λ | a X n n n n  n n ,  n p

n=1

n=1

p

     N    N 1     inf λ a X  |λ | a X n n n n  n n .  2 n p p n=1

n=1

It is often useful to consider random multipliers λn , and we obtain the following result: Theorem IV.4 Let (εn )n1 be a Bernoulli sequence, (ξn )n1 a real symmetric sequence and (un )n1 a sequence of vectors of the Banach space E. For 1  p < ∞, we have the following two inequalities:       N   N     ξ u  2 sup ξ  ε u n n n ∞ n n ;  n=1  N

  

n=1

p

nN

n=1

p

   N    1   inf ξn un   E|ξ | ε u n  n n .  2 nN p p n=1

Proof As the sequence (εn )n1 is symmetric, the inequality in Remark 2, with λn = sgn ξn (ω), provides:

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4 Banach Space Valued Random Variables

p    N    ξ (ω)ε (ω )u n n n  dP(ω )  

n=1



p N  1 p       |ξ (ω)| ε (ω )u n n n  dP(ω ) .  2 

n=1

Integrating with respect to ω and using Fubini’s theorem, we obtain, via the symmetry of (ξn )n1 : p     N    dP(ω) = ξ (ω)u n n   

n=1



 p  N   ξn (ω)εn (ω )un    dP(ω) dP(ω ) n=1

p  N  1 p         dP(ω) dP(ω ) |ξ (ω)| ε (ω )u  n n n   2 



n=1

p   N  1 p       dP(ω ) |ξ (ω)| ε (ω )u dP(ω)  n n n   2 

 n=1

(via the convexity of the integral)  p  N  1 p     = E|ξ | ε (ω )u n n n  dP(ω ) ;  2  n=1

hence:  N     N  N      1 1       inf ξ u  E|ξ | ε u  E|ξ | ε u n n n n n n  n n ,   2 2 n p p p n=1

n=1

n=1

again thanks to Remark 2. This proves the second inequality; the proof of the first is similar, actually even simpler. Remark It ensues from Remark 1 that, if the ξn ’s are symmetric in the complex sense, we can replace 1/2 by 1 in the right-hand side of the second inequality.

V The Kahane Inequalities V.1 Differences with the Scalar Case If we attempt to extend the Khintchine inequalities to the case of X = N n=1 εn un , where the un ’s are vectors in a Banach space E and the εn ’s are independent Bernoulli variables, we encounter several difficulties:

V The Kahane Inequalities

139

  N 2 1/2 when 1) In general, X2 no longer has anything to do with n=1 un  E is not a Hilbert space. 2) To estimate E|X|p , we used the fact that the modulus can be ignored ( for X real) when p is an even integer: |X|p = X p ; we can do nothing of the sort with Xp , and worse, X p does not even make sense: E is a vector space, not an algebra. For the same reason, the proof of the Remark of Chapter 1, Subsection IV.2, with its introduction of an exponential of X, no longer works.

V.2 The Kahane Inequalities The fact that Khintchine’s inequalities still hold in the vectorial case is thus a non-trivial and unexpected property. We have the following theorem: Theorem V.1 (The Kahane Inequalities) For every p ∈ ] 0, +∞[, there exist constants Ap and Bp > 0 such that, for any Banach space E and any random  0 variable of the form X = N n=1 εn un ∈ L (, A, P; E), where un ∈ E and (εn )n1 is a Bernoulli sequence, we have: Ap X2  Xp  Bp X2 . Moreover, for p  2, Bp can be chosen so that: Bp = O(p) . Proof (Kahane)

First we show a result that holds for an infinite sum:  Theorem V.2 Let ∞ n=1 εn xn be a Bernoulli series, with xn ∈ E, almost surely bounded in the sense that M = supn Sn  < +∞ almost surely, with  Sn = nj=1 εj xj . Then, for any s, t > 0, we have: (1)

P(M > s + t)  4 P(M > s) P(M > t) .

In particular we have M ∈ Lp for every p < +∞, and, if 2  p < +∞: Mp  Bp M2 . The inequality quantifies the following phenomenon: if the probability of M being large is small, the probability of M being really large is very small. Proof

We introduce the events A = {M > s + t} ;

An =



 sup Sp − Sn−1  > s , pn

and the stopping time T = inf{n  1 ; Sn  > t} .

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4 Banach Space Valued Random Variables

The key observation is: the events (T = n) and An are independent. Indeed, if p  n, Sp − Sn−1  = εn xn + εn+1 xn+1 + · · · + εp xp  = xn + εn εn+1 xn+1 + · · · + εn εp xp  ; hence An ∈ σ (εn εn+1 , . . . , εn εn+j , . . .), while (T = n) ∈ σ (ε1 , . . . , εn ). However the two σ -algebras σ (ε1 , . . . , εn ) and σ (εn εn+1 , . . . , εn εn+j , . . .) are independent, since, for any N  1 and every choice of α1 , . . . , αN and β1 , . . . , βN = ±1, we have:   P (ε1 , . . . , εn ) = (α1 , . . . , αn ), (εn εn+1 = β1 , . . . , εn εn+N = βN ) = P(ε1 = α1 , . . . , εn = αn , εn+1 = αn β1 , . . . , εn+N = αn βN ) = 2−(n+N) = 2−n × 2−N . Moreover, we have the inclusion: A ∩ (T = n) ⊆ An . Indeed, if M(ω) > s + t and T(ω) = n, there exists p  n such that Sp (ω) > s + t; hence: Sp (ω) − Sn−1 (ω)  Sp (ω) − Sn−1 (ω) > s + t − t = s , and therefore ω ∈ An . It ensues that: P(A) =

∞ ∞     P A ∩ (T = n)  P An ∩ (T = n) n=1

=

∞ n=1

n=1

P(An ) P(T = n)  sup P(An ) n

∞

P(T = n)

n=1

= sup P(An ) P(T < ∞) = sup P(An ) P(M > t) . n

n

To bound P(An ), we use the contraction principle of Lemma IV.2 with λ1 = . . . = λn−1 = 0 and λk = 1 if k  n, and the symmetry principle. For q  n:   P sup Sp − Sn−1  > s  2 P(Sq − Sn−1  > s) npq

 4 P(Sq  > s)  4 P(M > s) ; hence: P(An )  4 P(M > s) by letting q tend to +∞. We have thus obtained: P(M > s + t)  4 P(M > s) P(M > t) .

V The Kahane Inequalities

141

It is easy to finish: q(t) = P(M > t) −−→ 0, by hypothesis. Let s0 > 0 such t→+∞

that q(s0 )  1/16, and λ > 0 such that eλs0  2, and set:  (n+1)s0 In = eλt q(t) dt . ns0

We have:  In+1 =

(n+1)s0

eλ(s0 +t) q(s0 + t) dt

ns0



 4eλs0 q(s0 )

(n+1)s0

eλt q(t) dt 

ns0

1 In ; 2

thus: 

+∞ 0

eλt q(t) dt =

∞

In n=0 ∞

 I0

2−n = 2 I0 = 2



s0

eλt q(t) dt  4 s0 .

0

n=0

An integration by parts leads to:  λM E(e ) = 1 +

+∞

λeλt q(t) dt  1 + 4λs0 .

0

Hence, eλM ∈ L1 , and a fortiori M ∈ Lp for every p < +∞. In particular, M ∈ L1 and Markov’s inequality shows that we can take s0 = 16 M1 to obtain, via a change of λ to 1/a:   es0 /a  2 ⇒ E eM/a  1 + (4 s0 /a) . In terms of the Orlicz function 1 (x) = ex − 1, we thus obtain: M  4 s 0 es0 /a  2 ⇒ E1  · a a The choice a = 4s0 is valid and thereby: M1  a = 64 M1 . However: Mp , p p2

M1 ≈ sup

by Stirling’s formula, and the result follows, with Bp = O(p).

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4 Banach Space Valued Random Variables

∞ Corollary V.3 Let n=1 εn xn be an almost surely convergent Bernoulli ∞ p ε x . series S = n=1 n n Then, S ∈ L (E) for any p < +∞, and moreover, for p  2: Sp  Bp S2 , with Bp = O(p).

  Proof Let M = supn1  nj=1 εj xj . We have S  M and M < +∞ a.s. By the Paul Lévy maximal inequality:  +∞  +∞ M22 = 2 t P(M > t) dt  4 t P(S > t) dt = 2 S22 . 0

0

Hence, for p  2, we obtain: Sp  Mp  O(p) M2 

√ 2 O(p) S2 .

Clearly Corollary V.3 restricted to finite sums implies the Kahane inequalities, and that finishes the proof of Theorem V.1.  Remark 1 Corollary V.3, applied to a Bernoulli series ∞ n=1 εn xn , implies the equivalence of the following assertions: 1) 2) 3) 4)

the series converges in probability; the series converges in Lp for some p ∈ ]0, +∞[; the series converges in Lp for every p ∈ ]0, +∞[; the series converges almost surely.

When E is the scalar field, or even a Hilbert space, we again find the equivalence: +∞ n=1

εn xn

converges a.s.

⇐⇒

+∞

|xn |2 < +∞ ,

n=1

since this last condition is equivalent to the convergence of the series in L2 . Remark 2 The proof of Corollary V.3 only uses the almost sure boundedness  of the partial sums nj=1 εj xj , and thus shows (see Chapter 1, Theorem III.4) the equivalence of the following assertions: 1) 2) 3) 4)

the series is bounded in probability: supn P(Sn  > t) −−→ 0; t→+∞ Sn is bounded in Lp for some p ∈ ]0, +∞[; Sn is bounded in Lp for every p ∈ ]0, +∞[; Sn is almost surely bounded.

V The Kahane Inequalities

143

Remark 3 For Bp , the proof of Theorem V.2, based on the concentration inequality (1), provides a weaker dependency on p than in the scalar case: Bp = √ O(p) instead of Bp = O( p). But an argument of “independent copies” allows √ a self-reinforcement of Theorem V.1 and leads to Bp = O( p) in the vectorial ∞ case. In fact, let S = n=1 εn xn be an almost surely convergent Bernoulli  ε series and let Sj = ∞ n=1 n,j xn (j  1) be independent copies of S. For k fixed, define Xn by:  +∞  ∞ εn,1 + · · · + εn,k S1 + · · · + Sk = xn = Xn xn . √ √ k k n=1 n=1 By Khintchine’s inequalities (Theorem IV.1 of Chapter 1), we obtain E|Xn |  A1 . For p  1, an integer, the contraction principle in the form given in Theorem IV.4 leads to:    S1 + · · · + Sk p   (2A1 )−p ESp ;  E √  k hence:

     S1 + · · · + Sk p S1 + · · · + Sk p p −p/2 p   ES  C E  λ p! E √  C k λp p! k   S1 + · · · + Sk   Cp k−p/2 λp p! E exp λ k    S  Cp k−p/2 λp p! E exp , λ p

p

where C = 2A1 and λ > 0. Let us assume S1 = 1. The end of the proof of Theorem V.1 shows that  we have E exp(S/64)  2. The choices k = p and λ = 64 then give: ESp  (128 C)p p−p/2 p!  (128 C)p pp/2 ; hence: Sp  128 C

√

p,

and the result ensues.

V.3 Walsh Functions and Hypercontractivity Kahane’s proof of Theorem V.1 is very “probabilistic.” The use of the Walsh functions and of the phenomenon of hypercontractivity leads to “analytic” proofs of the Khintchine–Kahane inequalities; these proofs have the merit of extending these inequalities to the “Bernoulli chaos” and of providing very

144

4 Banach Space Valued Random Variables

√ precise values for the constants Ap and Bp : we can take Bp = p − 1 and √ A1 = 1/ 2. We recall the definition of the Walsh functions: let (εn )n1 be a Bernoulli sequence on (, A, P); restricting A if necessary, we can assume that A = σ (ε1 , . . . , εn , . . .). For every finite subset A of N∗ , we set:  εn , wA = n∈A

with the convention w∅ = 1I. The wA ’s are called the Walsh functions associated with (εn )n1 ; they are indexed by the finite subsets of N∗ and form an orthonormal basis of L2 (, A, P) (see the Annex, the wA ’s being the characters of the Cantor group ∗ {−1, 1}N ). A first use of these functions is the determination of the best A1 . Theorem V.4 (Latała–Oleszkiewicz) Let E be a Banach space. For x1 , . . . , xn ∈ E, and ε1 , . . . , εn independent Bernoulli variables, we have,  with S = ni=1 εi xi : 1 S1  √ S2 , 2 √ and the constant 1/ 2 is optimal. √ Proof In the scalar case, for S = ε1 + ε2 , we have S1 = 1 and S2 = 2; the best √ constant A1 in the Khintchine–Kahane inequalities is thus necessarily  1/ 2. √ To show that A1 = 1/ 2 is the best constant, we use (εi )1in and (wA )A⊆{1,...,n} in the following model: let e1 , . . . , en be the canonical basis of Rn , let  = {−1, 1}n be the set of the 2n extreme points of the cube [−1, 1]n , i.e.: ω∈

⇐⇒

ω=

n

εi (ω) ei ,

with εi (ω) = ±1 ,

i=1

let A = P() be the set of all subsets of , and finally let P be the normalized counting measure. We also equip  with the Hamming metric d: 1 |εi (α) − εi (β)| . 2 n

d(α, β) = #{i  n ; εi (α) = εi (β)} =

i=1

An element β of  is said to be a neighbor of α if it differs from α by a single coordinate; that is, d(α, β) = 1. We set f (ω) = S(ω). For A ⊆ {1, . . . , n}, the wA ’s form an orthonormal basis of L2 (, A, P) (see the Annex), so, for any g ∈ L2 (P), we have:

V The Kahane Inequalities

g=

145

 g(wA ) wA ,

A⊆{1,...,n}

with

  g(wA ) =

g wA dP . 

The key to the proof is the introduction of a linear operator L on L2 (P) such that:  (wA ) = |A| g(wA ), ∀ g ∈ L2 (P), a) Lg b) Lf  f . f (wA ). Note that a) and b) lead to the result: In what follows, we set cA =  indeed, cA = 0 for |A| = 1 as f is even: f (−ω) = f (ω), and b) gives f 2  f Lf ; thus, by integrating and using Parseval’s identity:     (wA ) f (wA ) Lf f 2 dP  f Lf dP = 



A⊆{1,...,n}



=

|A| cA2

A⊆{1,...,n}

2



cA2









f dP − 2

 

− c∅2

2 f dP ;

2

2

 cA2

A⊆{1,...,n}





hence:

|A| cA2

|A|2

=2

|A|2

=2

=



2  f dP  f 2 dP , 

which proves the theorem. It remains to construct L : L2 (P) → L2 (P) satisfying a) and b) . For L, we take the discrete Laplacian:   1 n 1 g(α) − [g(α) − g(β)] = g(β) , Lg(α) = 2 2 n β

β

where the sum is taken over the β neighbors of α (there are n of them). To verify a) , it suffices to see that L(wA ) = k wA when |A| = k. Take, for example, wA = ε1 . . . εk ; among the neighbors β of α, n − k of them coincide with α on 1, . . . , k (and then wA (β) = wA (α)), and k of them coincide with α on k + 1, . . . , n (and then wA (β) = −wA (α)); this gives:

146

4 Banach Space Valued Random Variables

  n k (n − k) wA (α) + wA (α) − wA (α) 2 n n n 2k wA (α) = k wA (α) . = 2 n To verify b) , we note the two following points: 1 β; 1) α = n−2 LwA (α) =

d(α,β)=1

Indeed, a) gives Lεi = εi for 1  i  n; thus, by using the definition of L and the representation of the elements of , we have: α=

n

εi (α) ei =

i=1

=

1 2

n

{β ; d(α,β)=1}

1 (α − β) , = 2 

Lεi (α) ei

i=1  n

εi (α) ei −

i=1

n

 εi (β) ei

i=1

β

and 2α = nα − β β. R, convex 2) f is the restriction to  of a function F : Rn →  and homogeneous  n n   of order 1; indeed it suffices to take F(t) = i=1 ti xi , for t = i=1 ti ei . Now, 1) and 2) imply: f (α) = F(α) = F 

1   n 1  n F β = β n−2 n n−2 n 1

n n−2 n

β

F(β) =

β

1 n−2



β

f (β) ,

β

    so n f (α)−2 f (α)  β f (β), or β f (α)−f (β)  2 f (α), which proves b) , given the definition of L. A second use of the Walsh functions is provided by the following theorem: ! Theorem V.5 (C. Borell) Let 1 < p < q < +∞ and λ0 = p−1 q−1 · Then, for 0  λ  λ0 , and for every Banach space E, we have:      |A|       λ wA xA    wA xA    , A⊆{1,...,n}

q

A⊆{1,...,n}

p

for any n  1 and xA ∈ E with A ⊆ {1, . . . , n} . The key of the proof is the following harmless-looking inequality:

V The Kahane Inequalities

147

Proposition V.6 If x, y ∈ R and 0  λ  λ0 , we have:     |x + λy|q + |x − λy|q 1/q |x + y|p + |x − y|p 1/p  . 2 2 Proof (∗)

The change of variables x + y = u, x − y = v reduces the inequality to:    p  |u| + |v|p 1/p |au + bv|q + |cu + dv|q 1/q  , 2 2

1−λ where we have set a = d = 1+λ 2 , b = c = 2 . We introduce the symmetric bistochastic matrix ⎞ ⎛ (1 + λ)/2 (1 − λ)/2 ⎠, Tλ = ⎝

(1 − λ)/2 (1 + λ)/2 and the probability space  = {1, 2} with the normalized counting measure. Then (∗) expresses a phenomenon of hypercontractivity: if 0  λ  λ0 , Tλ “improves the integrability,” i.e. it sends p () into q () with norm  1 (the same notation refers to the matrix and the operator it represents). To prove (∗), we perform some reductions: 1) The semi-group property: Tλ Tμ = Tμ Tλ = Tλμ is valid with 0  λ, μ  1. This is foreseeable given Theorem V.5, and also immediate to verify. 2) It suffices to prove (∗) for 1 < p < q  2. = q∗ = q/(q − 1) and q1 = p∗ = p/(p − 1). In fact,!if 2 < p < !q, set p1 ! ∗

q −1 p−1 We have pq11 −1 −1 = p∗ −1 = q−1 ; hence if λ  λ0 , Tλ sends p1 into q1 with norm  1 and, by transposition, Tλ∗ = Tλ sends p into q with norm  1. If now  λ0 , we can write λ = μν, with μ  ! p < 2 < q and λ ! p−1 2−1 and ν  ν0 = Tμ T p −−−→ 2 −−−ν→ q shows that:

μ0 =

2−1 q−1 ;

then μ0 ν0 = λ0 , and the diagram

Tλ p →q = Tν Tμ p →q  Tν 2 →q Tμ p →2  1 . Thus we need to prove (∗) when 1 < p < q  2. It is in fact more convenient to go back now to the equivalent original form of Theorem V.5. We can assume x = 0 and by homogeneity x = 1.

148

4 Banach Space Valued Random Variables

Suppose the case |y|  1 has already been dealt with. If |y| > 1, we have |1 ± λy|  |y ± λ|, so that:   |1 + λy|q + |1 − λy|q 1/q 2   1/q  q q |y + λ| + |y − λ| |1 + λy−1 |q + |1 − λy−1 |q 1/q  = |y| 2 2  1/p   −1 p −1 p p |1 + y | + |1 − y | |1 + y| + |1 − y|p 1/p  |y| = . 2 2 It thus remains to show that, for |y|  1 and 0  λ  λ0 :   (1 + y)p + (1 − y)p q/p (1 + λy)q + (1 − λy)q ∗  (∗) . 2 2 The left-hand side is equal to: q(q − 1) · · · (q − 2k + 1) 1+ λ2k y2k . (2k)! k1

Moreover, using the convexity inequality (1 + u)q/p  1 + (q/p) u when |u|  1, we see that the right-hand side of ( ∗ ∗ ) is greater than or equal (p − 1) · · · (p − 2k + 1) 2k y . It thus suffices to show that, if to 1 + q (2k)! k1

k = 1, 2, . . .:  ∗  (q − 1) · · · (q − 2k + 1) λ2k  (p − 1) · · · (p − 2k + 1) . ∗∗ For k = 1, the inequality can be read (q − 1) λ2  p − 1, which is true because λ  λ0 . For k  2, by switching an even number 2(k − 1) of signs, we must show that: (q − 1)(2 − q) · · · (2k − 1 − q) λ2k  (p − 1)(2 − p) · · · (2k − 1 − p) , which can be written: λ2k 

p−1 q−1

 2j2k−1

j−p · j−q

Hence it suffices to treat the case λ = λ0 , which reads:  j−p ;  λ2k−2 0 j−q 2j2k−1

this inequality is evident,  as the left-hand side is  1, and the right-hand side  1. This proves ∗∗∗ and completes the proof of Proposition V.6.

V The Kahane Inequalities

149

Proof of Theorem V.5 Note that the inequality   (∗) transfers automatically to vectors u, v ∈ E, since the matrix Tλ = ac db has positive coefficients (note that λ0 < 1). With u, v ∈ E, we thus have:   au + bvq + cu + dvq 1/q 2   (au + bv)q + (cu + dv)q 1/q  2 1/p  p p u + v  . 2 The equivalent inequality of Proposition V.6, thus “vectorialized,” corresponds to the case n = 1 of Theorem V.5. The general case ensues by induction on n, thanks to the following form of Minkowski’s inequality: Let (X, dμ) and (Y, dν) be two σ -finite measure spaces. Then, for every measurable function f : X × Y → R+ and for 0 < α  1: 1/α   α 1/α   f α (x, y) dμ(x) dν(y)  f (x, y) dν(y) dμ(x) . Y

X

X

Y

Let us go from n to n + 1 in Theorem V.5. We must show that Xq  Yp , with: λ|A| wA xA and Y = wA xA . X= A⊆{1,...,n+1}

A⊆{1,...,n+1}

Distinguishing the cases n + 1 ∈ A and n + 1 ∈ / A, we write instead: X= λ|A| wA (xA + λ εn+1 xA ) and Y = wA (xA + εn+1 xA ) , A⊆{1,...,n}

xA

A⊆{1,...,n}

xA , xA

with = εn+1 , we write:

= xA∪{n+1} . To highlight the independence of the wA ’s and

⎧   ⎪ X(x, y) = λ|A| wA (x) xA + εn+1 (y)xA ⎪ ⎨ A   ⎪ ⎪ wA (x) xA + εn+1 (y)xA . ⎩ Y(x, y) = A

The induction hypothesis first gives:  X(x, y)q dP(x) 



     



 q/p p  wA (x) xA + λ εn+1 (y)xA  dP(x) . 

{A⊆{1,...,n}

150

4 Banach Space Valued Random Variables

Denoting the sum within the integral by Z(x, y), integrating with respect to y and using the generalized Minkowski inequality with f = Zq and α = p/q (so that f α = Zp ), we obtain:   p/q q/p Z(x, y)q dP(y) dP(x) . Xqq  



According to the case n = 1 already proved, we have:  q/p  q p Z(x, y) dP(y)  Y(x, y) dP(y) ; 



hence:

  Xqq

q/p



Y(x, y) dP(y) dP(x) p

,

×

i.e. Xq  Yp , which completes the proof of Theorem V.5. By “specializing” Theorem V.5, we obtain Kahane’s inequalities for the k-th “Bernoulli chaos”: Theorem V.7 Let E be a Banach space, let A ⊆ N∗ and let xA ∈ E be vectors, almost all null. Let k be an integer  1, and 2  p < +∞. Then:         k/2     w x  (p − 1) w x A A A A .   p

|A|=k

In particular:

2

|A|=k

            εn xn   p − 1  εn xn    , n1

p

n1

2

for xn ∈ E, almost all null. The second inequality shows that we can take Bp = Khintchine–Kahane inequalities.

√ p − 1 in the

Proof We apply Theorem V.5, where 2 and p play the roles of p and q respectively; hence λ0 = (p − 1)−1/2 . Taking λ = λ0 and xA = 0 if |A| = k, we obtain:        k   λ  wA xA    wA xA   ,  0 |A|=k

p

|A|=k

2

which proves the first inequality. The other one corresponds to k = 1. ! p−1 Remark Borell’s theorem states that, for λ  , the convolution q−1 semi-group formed by the Riesz products Rλ = n1 (1 + λ εn ) sends

VI Comments

151

contractively Lp (E) into Lq (E) (and not only into Lp (E)), which explains the term hypercontractivity: indeed, with: wA (ω)xA , Y(ω) = A⊆N∗

we have: X(ω) =

A⊆N∗

|A|



Y(ωα −1 ) d(Rλ .P)(α) = (Y ∗ Rλ )(ω) .

λ wA (ω)xA = 

VI Comments Banach space valued random variables are exhaustively studied in the article of Hoffmann-Jørgensen [1974] (see also [1973]) and in the book of Ledoux– Talagrand. A complete study of probability and random variables in metric spaces (mostly Polish spaces) is presented in the books of Borkar and Parthasaraty. The Paul Lévy symmetry principle was first shown for real variables (not necessarily symmetric; this then requires the introduction of the median of a random variable): see Loève or Stroock; the extension to the vectorial case as treated here, is due to Kahane (see Kahane 2), and the symmetric case is sufficient for us. Further applications of the principle of symmetry, which are generalizations of Theorems III.3 and III.5, will be seen in Chapter 5 of Volume 2. The contraction principle was first highlighted by Billard [1965] before being systematically used for vectorial series (Kahane 2) and unconditional sequences in Banach spaces (Lindenstrauss–Tzafriri). The Banach aspect was seen in Chapter 3; the “random” form of this contraction principle can be found in Maurey [1973 a], Lemma 1 (see also Pisier [1974 a], Proposition 1). The proof of Theorem V.2 (the Kahane inequality) is due to Kahane (Kahane 2); the quantitative version (Theorem V.2) is due to HoffmannJørgensen [1973] and [1974], who gave a more general formulation: if the Xn ’s are independent and if Sn = X1 + · · · + Xn , then, with N = supn Xn , and M = supn Sn , as soon as Sn is bounded in probability, we have: N ∈ Lp ⇒ M ∈ Lp , for 0 < p < +∞; see also Ledoux–Talagrand. In fact, Kahane was focused on showing the “good” integrability of sums  of convergent series ∞ n=1 εn xn . The formulation of Theorem V.1 in terms of Khintchine’s inequalities was popularized by Maurey and Pisier (see, for example, Pisier [1973 a] and [1973 c]).

152

4 Banach Space Valued Random Variables

The possibility of self-improvement of the equality Bp = O(p) to Bp = √ O( p) was shown by Kwapie´n [1976]; see also Kahane 2. The best constant A1 in the scalar case was determined by Szarek [1976] (see also Tomaszewski [1987]); Haagerup [1978] and [1982] gave a simpler proof. Another proof was given by Nazarov and Podkorytov √ [2000]. For a long time, the “record” in Latała and the vectorial case was A1  1/ 3 (Tomaszewski [1982]) before √ Oleszkiewicz [1994] won the case, with a proof of A1 = 1/ 2, based on the use of Walsh functions. C. Borell [1979] also used these functions to prove the generalization (Theorem V.5) of the Khintchine–Kahane inequalities (see also Maurey [1991]). The scalar case, in connection with some problems of the (p) sets in Harmonic Analysis, was studied by A. Bonami [1970], the first to prove the inequality of Proposition V.6 and to emphasize its importance. For the hypercontractivity of other semi-groups (Hermite, Poisson etc.), see Gross [1975] and Weissler [1980]. For the Steinhaus variables, in the vectorial case, Ullrich [1988 c] showed the same phenomenon as in the scalar case: the > constant Ap of the inequality of Theorem V.1 stays bounded below when p − →0 (Ball and Nazarov [1994], unpublished work; Nazarov [1996], unpublished work); p can even descend below 0 (!) (Favorov [1998]), but must remain > −1. See also Guédon [1999].

VII Exercises In what follows, (εn )n1 denotes a Bernoulli sequence and (gn )n1 a (real) standard Gaussian sequence. Exercise VII.1 Assume that the space E contains n∞ ’s uniformly (i.e. there exists λ  1 so that, for every n, E contains a subspace En such that d(En , n∞ )  λ, with d the Banach–Mazur distance). Show that there exists a  sequence (xn )n1 of E such that ∞ n=1 εn xn converges almost surely, whereas ∞ g x diverges almost surely. n n n=1 Maurey and Pisier [1973] proved the converse: if E does not contain n∞ ’s  uniformly, then the almost sure convergence of ∞ n=1 εn xn implies that of ∞ n=1 gn xn . Exercise VII.2 Show that, for every Banach space E and any sequence  (xn )n1 of E, if the series ∞ n=1 gn xn converges almost surely, then so does ∞ the n=1 εn xn . Hint. Let σn = sign (gn ) and A0 = σ (σ1 , . . . , σn , . . .); show that:   N N 2 A0 E gn xn = σn xn , π n=1

n=1

VII Exercises

153

and from this deduce the inequality:   N    E εn xn   n=1

 N   π   E gn xn  . 2  n=1

Exercise VII.3 (vectorial form of Prokhorov’s theorem, following Ledoux–Talagrand) 1) Let K be a subset of a Banach space E; show the equivalence of the following two assertions: (i) K is relatively compact; (ii) K is bounded and flat, i.e. for every ε > 0, there is a finite-dimensional subspace F of E such that K ⊆ Fε = {x ∈ E ; dist(x, F)  ε}. (Turpin [1973] showed that this holds for E = L0 (0, 1), even though L0 (0, 1) is not locally convex; he used the terminology mince (thin) in (ii) ). 2) Let E be a separable Banach space and T a subset of (P(E), ρ), where ρ is the metric defined in Section II. Show the equivalence of the following three assertions: (i) T is relatively compact in P(E) (ii) For every ϕ ∈ E∗ , the dual of E, ϕ(T) is a relatively compact subset of P(K) (K = R or C), and, for any ε > 0, there exists a finitedimensional subspace F of E such that μ(Fε )  1 − ε for any μ ∈ T. (iii) For any ε > 0, there exist a bounded subset L of E and a finitedimensional subspace F of E such that μ(L)  1−ε and μ(Fε )  1−ε for all μ ∈ T.

Hint. To show (ii) ⇒ (iii) , possible steps are: (∗) There exist ϕ1 , . . . , ϕn ∈ E∗ , of norm 1, such that x  2 supi |ϕi (x)| for every x ∈ F. (∗∗) For any a > 0, supi |ϕi (x)| > a if x ∈ [(2a + 3 ε) BE ]c ∩ Fε . (∗ ∗ ∗) There exists a > 0 such that μ {x ∈ E ; supi |ϕi (x)| > a}  ε for every μ ∈ T. Then take L = (2a + 3 ε) BE . For (iii) ⇒ (i) , select for each 0 < δ  ε a finite-dimensional subspace F such that μ(Fδ )  1 − δ for μ ∈ T, and set K(δ) = L ∩ Fδ and K =

∞ −q c q=1 K(ε2 ); then show that K is totally bounded and that μ(K )  ε for every μ ∈ T.

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4 Banach Space Valued Random Variables

Exercise VII.4 Let H be a Hilbert space. Show that the Kolmogorov maximal inequality (Chapter 1) can be extended to independent random variables Xn with values in H. Deduce that, with xn ∈   2 surely if and only if +∞ H, +∞ n=1 εn xn converges almost n=1 xn  < +∞, ∞ and that, with symmetric Xn , n=1 Xn converges almost surely if and only if ∞ 2 n=1 Xn  < +∞ almost surely. Give a direct proof of the contraction principle in the Hilbertian case. Exercise VII.5 Let H be a Hilbert space, and (xn )n1 a sequence of elements of H; consider Sn = ε1 x1 + · · · + εn xn .    1) Show that P Sn 2  12 nj=1 xj 2  1/24.  2 2) Assume (Sn )n1 is almost surely bounded. Show that ∞ n=1 xn  < +∞ and that (Sn )n1 is almost surely convergent. Exercise VII.6 In an answer to a question of Hoffmann-Jørgensen [1974], Kwapie´n [1974] showed the equivalence between the two following assertions: (i) The Banach space E does not contain c0 ; ∞ (ii) n=1 εn xn is almost surely convergent as soon as its partial sums are almost surely bounded, for every sequence (xn )n1 of E. This exercise is devoted to proving this result. 1) Show that (ii) implies (i) . For the following, denote:   # " ∗ B(E) = (xn )n1 ∈ EN ; M = supn1  nk=1 εk xk  < +∞ a.s. " #  ∗ C(E) = (xn )n1 ∈ EN ; +∞ n=1 εn xn converges a.s. . 2) Let (xn )n1 ∈ B(E)  C(E). a) Show that there exist a > 0 and a sequence of integers n1 < n2 < · · · <  nk < · · · such that, with Xk = nk 0 almost surely. d) Show that, if B(E) = C(E), then there exists a sequence (yn )n1 in B(E) such that infn1 yn  = α > 0.

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3) Let (xn )n1 be a sequence in B(E) such that xn   α > 0 for any n  1, and let (εn )n1 be a Bernoulli sequence in the space (, A, P). Denote by B0 the σ -algebra generated by this Bernoulli sequence. a) Show that, for every R ∈ B0 and every a = ±1,   1 P R ∩ (εn = a) −−→ P(R). n→+∞ 2   b) Let M(ω) = supn1  nk=1 εk (ω)xk . Show that there exists λ > 0 such that, with A = (M  λ), we have P(A) > 1/2. With such a fixed λ, show that we can construct a sequence n1 < · · · < nk < · · · so that, for every choice of signs a1 , . . . , ak = ±1, we have   P A ∩ (εn1 = a1 , . . . , εnk = ak ) > 1/2k+1 . c) Set εj = εj if j ∈ {n1 , n2 , . . .}, and εj = −εj otherwise. Let n       εk (ω)xk   λ , A = ω ; sup  n1

(k) and a

=

k

j=1 (εnj

k=1

= aj ), for a fixed choice of signs a = (a1 , . . . , ak ).

Show that A ∩ A ∩ a = ∅ (bound its probability from below). (k) d) Show that, if ω0 ∈ A ∩ A ∩ a , we have: (k)

k j=1

aj xnj =

1 2

 nk j=1

εj (ω0 )xj +

nk

 εj (ω0 )xj .

j=1

e) Show that the sequence (xnk )k1 is equivalent to the canonical basis of c0 . Exercise VII.7 A Banach space E is said to have the Radon–Nikodým property (RNP) if every operator T : L1 ([0, 1]) → E is representable, i.e. if there exists g ∈ L∞ ([0, 1]; E) such that:  1 ϕ(t) g(t) dt Tϕ = 0

for any ϕ ∈ L1 ([0, 1]). 1) Show that, for every $g ∈ L1 (, A, P; E), the map Tg : L∞ (, P) → E defined by Tg (ϕ) =  ϕ g dP is a compact operator (approximate g by simple functions to show that Tg is the limit of finite rank operators). Then deduce that the identity operator of L1 ([0, 1]) is not representable, and hence that L1 ([0, 1]) does not have RNP.

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  2) Let T : L1 ([0, 1]) → c0 be the operator defined by Tϕ =  ϕ (n) n1 . Show that T is not representable and hence that c0 does not have RNP. 3) $Show that the operator T : L1 ([0, 1]) → C([0, 1]) defined by (Tϕ)(x) = x 0 ϕ(t) dt is not representable, and hence that C([0, 1]) does not have RNP. Exercise VII.8 Let E be a Banach space. 1) Let (, A, P) be a probability space, and let π be a finite partition of A into measurable almost surely non-empty subsets. If Aπ is the algebra generated by this partition, show that 1 1IA E( f ) E Aπ ( f ) = P(A) A∈π

for every f ∈ L1 (, P; E). 2) Let E1 be a subspace of E, and let g ∈ L∞ ([0, 1]; E). Show that if the operator Tg : L1 ([0, 1]) → E takes its values in E1 , then g is almost surely with values in E1 : if πn is the partition of dyadic intervals of order n, the martingale EAπn (g) n1 converges in L1 (E) to g; hence there is a subsequence converging almost surely to g (in fact, the martingale itself converges almost surely to g, by Theorem II.12); then use 1) to obtain that EAπn (g) ∈ E1 . 3) Conclude that any subspace of a space with RNP also has RNP. Exercise VII.9 Let (, A, P) be a probability space. 1) Let f :  → E be a mapping with values in a separable Banach space E (more generally, we can always assume that f almost surely takes its values in a separable subspace of an arbitrary Banach space E). Assume that all the functions x∗ ◦ f :  → R or C, for x∗ ∈ E∗ , are measurable. Let (xk∗ )k1 be a sequence of unit vectors of E∗ , norming E: x = supk1 | xk∗ , x | for every x ∈ E. a) Show that the mappings ω ∈  → f (ω) and ω ∈  → f (ω) − x, x ∈ E, are measurable. b) Let (xn )n1 be a dense sequence in E, and let ε > 0. Set An = {ω ∈  ; f (ω) − xn   ε} ,  Bn = An  j 0. The degenerate case σ = 0 corresponds to X = m, and by convention a constant variable is Gaussian. In abbreviation, we write X ∼ N (m, σ 2 ) and we have E(X) = m, σ 2 t2

V(X) = σ 2 and X (t) = E(eitX ) = eimt− 2 . The distribution of X is thus entirely determined by its expectation and its variance. When m = 0 and σ = 1, X is said to be a standard Gaussian variable. If g is a standard Gaussian variable, X = σ g + m has distribution N (m, σ 2 ). In this section, all variables are centered. Definition II.1 Let X1 , . . . , Xn :  → R be real random variables; the random vector X = (X1 , . . . , Xn ) :  → Rn is said to be a Gaussian vector if its image by every linear functional ϕ : Rn → R is a real Gaussian variable. In other words, this says that every linear combination a1 X1 + · + an Xn , with a1 , . . . , an ∈ R, is a Gaussian variable. Under this form, since the sum of two independent Gaussian variables is a Gaussian variable (as can immediately be seen from its characteristic function), the following proposition easily ensues: Proposition II.2 If X1 , . . . , Xn are independent real random variables, then the vector X = (X1 , . . . , Xn ) is Gaussian. Without independence, this no longer holds in general. The following proposition also immediately follows from the definition: Proposition II.3 The image of any Gaussian vector X :  → Rn by a linear mapping  : Rn → Rm is a Gaussian vector, in Rm . An equivalent expression for “the vector X is Gaussian” is: the scalar product X, a is a Gaussian variable for any a = (a1 , . . . , an ) ∈ Rn . For such a variable X, a , assumed centered, its distribution is determined by its variance  a a E(Xi Xj ); everything thus relies on the matrix σ 2 = E( X, a 2 ) = i,j  X  i j    C = ci,j i,j = ci,j i,j = E(Xi Xj ) i,j , known as the covariance matrix of the vector X, and satisfying the following properties: Proposition II.4 a) The matrix C is positive and, conversely, every positive matrix is the covariance vector.  matrix  of a Gaussian   b) E ei t,X = exp − 12 Ct, t for any t ∈ Rn . c) Two (centered) Gaussian vectors X and Y with the same covariance have the same distribution. d) If C is invertible, then the density f of X is given by, for x ∈ Rn :   1 1 f (x) = exp − C−1 x, x . √ 2 (2π )n/2 det C

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Proof  2 a) For a ∈ Rn , we have Ca, a = j,k aj ak E(Xj Xk ) = E( X, a )  0. 2 Conversely,   if C ∈ Mn (R) is positive, it can be written C = B , with B = bj,k j,k symmetric. Let g1 , . . . , gn be independent standard Gaussian variables; the vector G = (g1 , . . . , gn ) is Gaussian by Proposition II.2, as well as the vector X = BG, by Proposition II.3. n As X = (X1 , . . . , Xn ), with Xj = k=1 bj,k gk , and since E(Xj Xk ) = n b b = c , the matrix C is indeed the covariance matrix of X. j,l k,l j,k l=1 b) Let Y = t, X ; then Y has distribution N (0, σ 2 ), with σ 2 = E(Y 2 ) = Ct, t . According to the reminders at the start of this section, we have:   1     2 E ei t,X = E eiY = e−σ /2 = exp − Ct, t . 2 c) If X and Y have the same covariance matrix C, b) shows that   1     E ei t,X = exp − Ct, t = E ei t,Y 2 for any t ∈ Rn . Thus X and Y have the same characteristic function, and consequently the same distribution.   d) When the characteristic function X (t) = E ei t,X =   C1 is invertible, exp − 2 Ct, t of X is integrable; X thus possesses a density, given by the Fourier inversion formula:   1 1 1 −i t,x e ϕ (t) dt = e−i t,x e− 2 Ct,t dt f (x) = X n n (2π ) Rn (2π ) Rn  1 1 2 = e−i t,x e− 2 Bt dt , (2π )n Rn where B is symmetric positive and B2 = C. With the change of variables Bt = u, then t = B−1 u, and as det (B−1 ) = (det C)−1/2 , we obtain:  1 1 1 −1 2 f (x) = e−i u,B x e− 2 u du . n 1/2 n (2π ) (det C) R Setting y = B−1 x and separating the variables, we get: n   2 1 1 e−iuj yj e−uj /2 duj f (x) = (2π )n (det C)1/2 j=1 R   1 1 1 2 y , = exp − 2 (2π )n/2 (det C)1/2 which gives the result, since y2 = B−1 x, B−1 x = B−2 x, x = C−1 x, x .

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5 Type and Cotype of Banach Spaces

Note that an alternative presentation of this reasoning can be given as follows: consider the probability space formed by Rn equipped with its Borel σ -algebra and with the probability:  1    1  1 1 2 y exp − y2j dyj , dy = exp − √ 2 2 (2π )n/2 2π n

j=1

where y = (y1 , . . . , yn ). The coordinate linear functionals Y1 , . . . , Yn : Rn −→ R form a sequence of independent standard Gaussian variables; the vector Y = (Y1 , . . . , Yn ) (which is none other than the identity mapping of Rn onto itself) is thus Gaussian. The covariance matrix C of X can be written C = B2 with B symmetric positive. Then, as the covariance matrix of the Gaussian vector BY is equal to B2 = C, this vector hasthe same distribution  as X, i.e. the image by B of the probability (2π1)n/2 exp − 12 y2 dy on Rn . Since C is invertible, so is B, and the formula of the change of variables provides the rest. Later (Chapter 4 of Volume 2), we will need the following version of the central limit theorem: Theorem II.5 (Vectorial Central Limit Theorem) Let V = (U1 , . . . , Ud ) be a centered random vector in L2 . Let (Vn )n1 a sample of V, and Sn = V1 + · · · + Vn for n = 1, 2, . . . Then: Sn a) √ converges in distribution to a Gaussian vector X with the same n   covariance matrix C = E(Ui Uj ) i,j as V. b) Moreover, if V is a bounded random variable and if f is a continuous function on Rd with moderate growth (i.e. |f (x)|  a ebx ), then:    Sn E f √ −−→ E[f (X)] . n n→+∞ Proof   a) Let (t) = E ei t,V . As V is centered and in L2 ,  is C 2 , with ∂j (0) = 0 and ∂j,k (0) = −E(Uj Uk ). We thus have:  1 n  i t, √Sn    t n  1 n =  √ = 1− tj tk cj,k + o E e 2n n n j,k       1 1 exp − tj tk cj,k = exp − Ct, t = E ei t,X . − − → n→+∞ 2 2 j,k

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We have implicitly used the following fact:  n if zn −−→ z, then lim 1 + (zn /n) = ez . n→+∞

n→+∞

 n  (n) k Let us prove it: ez − 1 + (z/n) = +∞ k=0 ak z , with positive coefficients (n) ak ; therefore  n  n   z  e − 1 + z   e|z| − 1 + |z|   n n  n for every z ∈ C, and hence 1 + (z/n) −−→ ez uniformly on all compact n→+∞ subsets of C. b) The argument is based on the following lemma: Lemma II.6 Let μ, μ1 , . . . , μn , . . . be probabilities on Rd such that (μn )n1 converges in distribution to μ. Let f , g : Rd → R be continuous functions such that: g(x) > 0 for any x ∈ Rd , g(x) −−−→ + ∞, and x→+∞   f$(x) = o g(x) when x → +∞. Moreover, assume that the integrals Rd g dμn are uniformly bounded above by a constant C , independent of n. Then:   Rd

f dμn −−→

n→+∞

Rd

f dμ .

First, let us explain why this completes the proof of Theorem II.5. As V is centered and bounded, by the proof the Khintchine inequalities (see Chapter 1), we have, for p  0: ESn p  α p pp/2 np/2 , with α > 0 not depending on n (but depending on d), and with the convention 00 = 1. Then, for 0 < δ < 1/(2 e α 2 ): 2  +∞    +∞ p δp   Sn 2p δ  √Snn  δ 2p   = E α (2p)p E e √ p!  n  p! p=0



+∞

p=0

δ p α 2p 2p ep = C < +∞ .

p=0

√ 2 Now we apply the lemma, with g(x) = eδx , μn the distribution of Sn / n and μ that of the Gaussian vector X, and we obtain b) of Theorem II.5. It remains to prove the lemma. Denote by C0 (Rd ) and K(Rd ) the spaces of continuous functions on Rd tending to 0 at infinity and with compact support, respectively. By hypothesis, we can write f = ϕ g, with ϕ ∈ C0 (Rd ).

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5 Type and Cotype of Banach Spaces

$ First note that Rd g dμ  C : indeed, if ψ ∈ K(Rd ) and 0  ψ  1, we have ψg ∈ K(Rd ), and the convergence in distribution of μn to μ implies:   ψg dμ = lim ψg dμn  C ; n→+∞ Rd

Rd

it suffices to let ψ increase to 1I to obtain the stated upper bound. Then let ε > 0 and ψ ∈ K(Rd ) such that ϕ − ψ∞  ε; we have:                d f dμn − d f dμ   d ψg dμn − d ψg dμ R R R R   + |ϕ − ψ| g dμn + |ϕ − ψ| g dμ d Rd  R     ψg dμn − ψg dμ + 2 ε C ;   Rd

hence

  lim  n→+∞

Rd

 Rd

f dμn −

R

  f dμ  2 ε C , d

which proves the lemma, as ε > 0 is arbitrary. Remark 1 A theorem of Skorohod (see Billingsley, page 325) states that we can find a probability space (, A, P) and random variables X :  → Rd and X1 , . . . , Xn , . . . :  → Rd , with respective distributions μ, μ1 , . . . , μn , . . ., such that (Xn )n1 converges to X almost surely. The hypothesis implies that the variables f (Xn ) are uniformly integrable, and hence E[f (Xn )] −−→ E[f (X)], by Vitali’s theorem (see Chapter 7); in other n→+∞ $ $ words Rd f dμn −−→ Rd f dμ, which provides a clearer proof of Lemma II.6, n→+∞ if we admit the theorem of Skorohod stated above. Remark 2 Gaussian vectors can be defined in a more general framework: if E is a real Banach space, and X :  → E a random variable, X is said to be Gaussian if ϕ(X) is a Gaussian for every continuous linear functional ϕ ∈ E∗ (see Chapter 4, Exercise VII.11). Definition II.7 the form:

A complex standard Gaussian variable is a variable of Z=

g1 + i g2 , √ 2

where g1 , g2 are independent real standard Gaussians. We have E(Z) = 0 and E(|Z|2 ) = 1. The real Gaussians g1 , g2 are symmetric: gj ∼ (−gj ), but the complex Gaussian Z is even better: we have eiα Z ∼ Z

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for every α ∈ R; in other words, Z is invariant under the rotations of the plane. This is special case of the following proposition: Proposition II.8 (Rotation Invariance of Gaussian Vectors) Let X = (X1 , . . . , Xn ) be a Gaussian vector, with covariance CX . Let A ∈ Mn (R),  and Y = AX = (Y1 , . . . , Yn ), with Yj = nk=1 aj,k Xk . Then: a) Y is a Gaussian vector with covariance A CX A∗ . b) If X is standard (i.e. CX = In ), and if A ∈ O(n), then Y = AX is also standard: this is the property of rotation invariance of standard Gaussian vectors. c) If X is standard, if A ∈ Mn (R) is a contraction (A  1 for the operator norm of the Euclidean space Rn into itself ) and if h : Rn → R is convex and of moderate growth, as in Theorem II.5, then we have the inequality: E[h(AX)]  E[h(X)] . Remark Similarly, if Z1 , . . . , Zn are independent complex standard Gaussian variables, then, for any unitary matrix A ∈ U(n), the complex Gaussian vectors Z = (Z1 , . . . , Zn ) and AZ have the same distribution. Proof a) The first assertion is Proposition II.3. For the second, set ck,l = E(Xk Xl ) and a∗j,k = ak,j . We have: ai,k aj,l ck,l = ai,k ck,l a∗l,j = (A CX A∗ )i,j . E(Yi Yj ) = k,l

k,l

A A∗

= In , since A is orthogonal. b) By a) , CY = c) The hypothesis means that |h(x)|  a ebx , where a and b are positive constants. We equip L(Rn ) with the operator norm derived from the Euclidean norm on Rn , and denote by B = BL(Rn ) the closed unit ball of this space. Let Extr(B) be the set of its extreme points. The functional φ : B → R, defined by φ(A) = E[h(AX)], is convex and continuous on the compact set B; it thus attains its maximum on Extr(B), which we identify thanks to the following lemma: Lemma II.9 The set of extreme points of the unit ball B of L(Rn ) is the orthogonal group O(n) of order n. Proof of lemma The inclusion O(n) ⊆ Extr(B) is immediate, as every A ∈ O(n) preserves the Euclidean unit sphere of Rn , whose all the points are extreme points; thus, with A = (A + A )/2 and with x = 1, as A x and A x  1 and as Ax is an extreme point, we have A x = A x = Ax.

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5 Type and Cotype of Banach Spaces

For the reverse inclusion, let A ∈ Extr(B); we use a polar decomposition A = U S, with U ∈ O(n) and S self-adjoint with positive eigenvalues. We have S = A  1; thus there exists an orthonormal basis (ej )jn of Rn such that S(ej ) = λj ej , with 0  λj  1. If one of the λj ’s, for example λ1 , is < 1, we take ε > 0 small enough so that −1  λ1 − ε < λ1 + ε  1, and we define S1 , S2 ∈ B by: S1 (e1 ) = (λ1 − ε)e1 , S2 (e1 ) = (λ1 + ε)e1 and S1 (ej ) = S2 (ej ) = λj ej for j  2. We have A = (US1 + US2 )/2, and US1 , US2 ∈ B, whereas US1 = US2 ; this contradicts the extremality of A. Hence, all the λj ’s are equal to 1, so S = In and A = U ∈ O(n), which completes the proof of the lemma. Remark The same proof shows that the set U(n) of unitary matrices forms the set of extreme points of the unit ball of L(Cn ). Indeed, and this is even U1 + U2 , with U1 , U2 ∈ U(n) easier, if A ∈ L(Cn ) and A  1, we have A = 2 (see Halmos, problem 136). This being so, the convex functional φ attains its maximum at A0 ∈ O(n). Since A0 X has the same distribution as X, by the invariance under rotation of b) , we obtain: E[h(AX)] = φ(A)  φ(A0 ) = E[h(A0 X)] = E[h(X)] , which completes the proof.

II.2 p-Stable Variables A Gaussian variable N (0, 2) has e−t for characteristic function. Paul Lévy p discovered that e−|t| is still a characteristic function for 0 < p  2. The following theorem proves this assertion and lists the properties needed for the corresponding random variables: 2

Theorem II.10 Let p be a real number with 0 < p  2. Then: a) There exists a real random variable X, called a p-stable variable, such that:   p E eitX = e−|t| for every t ∈ R. b) If X1 , . . . , Xn are independent p-stable variables, and if a1 , . . . , an ∈ R, we have:  1/p n n p aj Xj ∼ |aj | X1 . j=1

j=1

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c) For 0 < p < 2, X has a continuous density f such that f (x) ∼ cp |x|−p−1 as |x| → +∞, where cp is a positive constant. In particular, E|X|q < +∞ if q < p, and E|X|p = +∞. d) For p = 2, X is a Gaussian variable; it has the continuous density 2 1 √ e−t /4 and thus has moments of all orders. 2 π Remarks p 1) Analogous statements hold for (t) = e−c|t| (c > 0); the case p = 2, c = 1/2 corresponds to the standard Gaussian. 2) The result no longer holds for p > 2: see Exercise VIII.4. 3) The adjective “stable” is justified by property b) : a linear combination of p-stable variables is again such a variable, up to a multiplicative constant. In truth, the variables introduced here are the symmetric p-stable variables; we have omitted this word as we use only such variables. The proof of the theorem requires several auxiliary results and notions. Definition II.11

Let S be a set, and G an Abelian group.

1) The mapping ϕ : S × S → C is said to be a kernel of positive type,  abbreviated ϕ ! 0, if i,j ϕ(xi , xj ) ci cj  0 for every x1 , . . . , xn ∈ S and every c1 , . . . , cn ∈ C. The function  : G → C is said to be of positive type if the kernel ϕ defined by ϕ(x, y) = (x − y) is of positive type. 2) The mapping ψ : S × S → R is said to be a kernel of negative type, and we write ψ " 0, if: (a) ψ(x, x) = 0 for any x ∈ S; (b) ψ(x, y) = ψ(y, x) for any x, y ∈ S;  (c) i,j ψ(xi , xj ) ci cj  0 for every x1 , . . . , xn ∈ S and c1 , . . . , cn ∈ C, with sum c1 + · · · + cn = 0. Note that a kernel of negative type is not the opposite of a kernel of positive type; this holds only conditionally to c1 + · · · + cn = 0. First we give a criterion determining whether a kernel is of negative type, avoiding this conditioning; next we establish the link between the two types of kernels thanks to a theorem of Schönberg (Theorem II.13), part a) of which can also be used in certain proofs of Bochner’s theorem. Proposition II.12 Let ψ : S × S → R be a mapping such that ψ(x, x) = 0 and ψ(y, x) = ψ(x, y). Then ψ " 0 if and only if for every x0 , x1 , . . . , xN ∈ S and every c1 , . . . , cN ∈ C, we have: ci cj [ψ(xi , xj ) − ψ(xi , x0 ) − ψ(xj , x0 )]  0 . 1i,jN

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5 Type and Cotype of Banach Spaces

Proof Assume that ψ " 0, and let c0 = −(c1 + · · · + cN ). By hypothesis,  since c0 + c1 + · · · + cN = 0, we have 0i,jN ci cj ψ(xi , xj )  0; hence, as ψ is symmetric and ψ(x, x) = 0: c0

N

cj ψ(x0 , xj ) + c0

j=1

N

ci ψ(xi , x0 ) +

i=1

=





ci cj ψ(xi , xj )

1i,jN

ci cj [ψ(xi , xj ) − ψ(xi , x0 ) − ψ(xj , x0 )]  0 .

1i,jN

The converse implication can be shown similarly. Theorem II.13 (Schönberg’s Theorem) a) The product of two kernels of positive type remains of positive type. b) If ϕ ! 0, then F(ϕ) ! 0 for every entire function F with positive coefficients. c) Let ψ be a real symmetric kernel, null on the diagonal of S ×S; then ψ " 0 if and only if e−tψ ! 0 for any t > 0. Proof     a) We need to show that if A = ai,j i,j and B = bi,j i,j ∈ Mn (C) are positive   Hermitian matrices, then so is their Hadamard product ai,j bi,j i,j . Since the   matrix tB = bj,i i,j is also a positive Hermitian matrix, it can be written = D2 , where D is positive Hermitian. With (ei )in the canonical basis of Cn , we have: ai,j bi,j = tr(A tB) = tr(A D2 ) = tr(D A D)

tB

i,j

=

n n D A D ei , ei = A D ei , D ei  0 , i=1

i=1

since the operator A is positive. b) More generally,  replacing A by Az , where Az is the positive Hermitian  matrix ai,j zi zj i,j , we have

ai,j bi,j zi zj = tr(Az tB)  0 .

i,j

It follows immediately by induction that if ϕ ! 0 and if F(z) = +∞ k k=0 ck z is an entire function with coefficients ck  0, then F(ϕ) ! 0. c) First we assume ψ " 0. Fix x0 ∈ S, and set: ϕ(x, y) = ψ(x, x0 ) + ψ(y, x0 ) − ψ(x, y) .

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By Proposition II.12, we have ϕ ! 0; by b) , we thus obtain etϕ ! 0. Hence: ci cj etψ(xi ,x0 ) etψ(xj ,x0 ) e−tψ(xi ,xj ) = ci cj etϕ(xi ,xj )  0 . i,j

i,j

The replacement of ci by ci e−tψ(xi ,x0 ) then gives e−tψ ! 0. Conversely, if e−tψ ! 0, select c1 , . . . , cN ∈ C with sum zero. By hypothesis, we have:   1 − e−tψ(xi ,xj ) 1 =− ci cj ci cj e−tψ(xi ,xj )  0 t t i,j

i,j

for t > 0. By letting t tend to zero, we obtain ψ " 0.



ci cj ψ(xi , xj )  0, i.e.

i,j

Theorem II.14 (Bochner’s Theorem) Every continuous function of positive type on R is the Fourier transform of a positive bounded measure. Note that the converse is true and easy: if μ is a positive bounded measure, then:    n 2   cj ck  μ(xj − xk ) = cj ck eit(xj −xk ) dμ(t) = cj eitxj  dμ(t)  j,k

R j,k

R

j=1

(here the probabilistic definition of the Fourier transform is used). The proof given here follows Revuz; for a proof based on the Bochner– Weil–Raikov theorem (Annex, Theorem II.12), see Rudin 1, page 303; another proof is in Katznelson. Proof First note that if  is of positive type, then (0)  0. Moreover, if in the definition we take c1 = 1, c2 = c ∈ C, x1 = 0 and x2 = x ∈ R  {0}, we obtain: (1 + |c|2 )(0) + c (x) + c (−x)  0 ; this is not possible for every c ∈ C unless (−x) = (x)

and

|(x)|  (0) .

In particular it ensues that, if  is not null, we may assume in what follows that (0) = 1. We now denote* g(x) = g(−x). For every continuous and integrable complex function g, we have:  S= (x) (g ∗ * g)(x) dx  0 ; R

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5 Type and Cotype of Banach Spaces

indeed, S can also be written:  R2

(x − y) g(x) g(y) dxdy ,

and hence, when the support of g is in [−a, a], S is the limit of Riemann sums: n (2a)2  aj ak   aj   ak  − g ;  g n n n n (2n)2 j,k=−n

thus, in this case, S  0. By density this holds for every continuous and integrable function g. Then, if the density of the distribution N (0, σ 2 ) is denoted: fσ (x) = we set: hσ = 2σ

  1 √ exp − x2 /2σ 2 , σ 2π

√ √ π  · ( fσ ∗ fσ ) = 2σ π  · fσ √2 .

This function is integrable, because  is bounded (by (0) = 1) and because fσ is integrable. Its Fourier transform can be written:  hσ (y) = (x) (g ∗ * g)(x) dx , R

with g(x) = fσ (x) eixy ; it is hence positive, by the above. As hσ ∈ Cb (R), we deduce that hσ is integrable from the following lemma: Lemma II.15 If h ∈ L1 (R) ∩ Cb (R) and if its Fourier transform  h is positive, this Fourier transform is integrable. From this lemma and the Fourier inversion theorem, it follows that:  1 hσ (y) e−ixy dy hσ (x) = 2π R 1  hσ (y) dy (it is indeed a is the Fourier transform of the probability measure 2π probability density, as we can see by taking x = 0 in the preceding formula, given that hσ (0) = 1). To obtain the desired result, it then only remains to apply the Paul Lévy continuity theorem, since hσ −−→  and  is continuous. σ →+∞

Proof of the lemma theorem gives:

Thanks to the positivity of  h, the monotone convergence  R

 h(y) dy = lim



τ #0 R

2 2  h(y) e−τ y /2 dy .

II Complements of Probability

171

Now, for each τ , the function within the integral on the right is the Fourier transform of h ∗ fτ , and consequently:    h(y) dy = lim 2π h(−x) fτ (x) dx R

τ →0

R

= lim 2π (h ∗ fτ )(0) = 2π h(0) < +∞ τ →0

(for the last equality we have used the dominated convergence theorem, given that h is bounded and continuous). Proof of Theorem II.10 a) We can assume p < 2. According to the theorems of Schönberg and Bochner, we need the kernel ψ(x, y) = |x − y|p to be of negative type. For this, we use the integral representation:  +∞ 1 − cos tx p dt , |x| = γ tp+1 0 where γ = γp > 0 is some constant (this is immediate with the change of variables tx = s). If then c1 , . . . , cN ∈ C are of sum null, we have:  +∞  j,k cj ck cos t(xj − xk ) p cj ck |xj − xk | = −γ dt tp+1 0 j,k   +∞  N itxj 2 j=1 cj e dt  0 . = −γ tp+1 0  n  p 1/p ; we have: b) We set a = j=1 |aj | n n    n   ita X    it j=1 aj Xj j j = = E e exp − |t|p |aj |p E e j=1



= exp − a |t| p

p



j=1

  = E eitaX1 ,

thus the result. p c) Since 0 (t) = e−|t| = X (t) is integrable, the Fourier inversion formula states that X possesses a density f , and that:   1 +∞ 1 p p e−|t| e−itx dt = cos(tx) e−t dt . f (x) = 2π R π 0 By parity, we can limit ourselves to the case x > 0. If 0 < p < 1, an integration by parts and a change of variables show that:  +∞ p p sin(tx) tp−1 e−t dt f (x) = πx 0  +∞ p p p = sin u up−1 e−u /x du , π xp+1 0

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and by additional integration by parts, or the second mean value formula, this last integral tends to:  +∞  +∞ up−1 sin u du = (1 − cos u)(1 − p) up−2 du > 0 . 0

0

The case p = 1 corresponds to a Cauchy variable, for which we have 1 1 ∼ · explicitly f (x) = π(1 + x2 ) π x2 Finally, for 1 < p < 2, three integrations by parts show that:    p(p − 1) +∞ p cos(tx) tp−2 e−t dt + O x−3 f (x) = π x2 0    p(p − 1) +∞ p p = cos u up−2 e−u /x + O x−3 π xp+1 0  p(p − 1) +∞ ∼ cos u up−2 du π xp+1 0  +∞ p(p − 1) = (2 − p)(3 − p) (1 − cos u) up−4 du π xp+1 0 = Cp x−p−1 , with Cp > 0. d) This is the Gaussian case already seen.

III Complements on Banach Spaces III.1 Local Reflexivity Recall that the Banach–Mazur distance between two Banach spaces X and Y is defined as: d(X, Y) = inf{T T −1  ; T : X −→ Y isomorphism} , with the convention d(X, Y) = +∞ if X and Y are not isomorphic. We have d(X, Y)  1. Note that d(X, Y) = 1 does not mean that X and Y are necessarily isometric (see Exercise VIII.7); in this case they are said to be quasi-isometric. Moreover, clearly d(X, Z)  d(X, Y) d(Y, Z), so in fact it is log d which is a (semi-) metric. This notion is particularly useful when we work with finite-dimensional spaces, as two spaces of the same dimension are always isomorphic.

III Complements on Banach Spaces

173

The local theory of Banach spaces is the study of properties of Banach spaces that only depend on their finite-dimensional subspaces. In this framework, the following notion is natural: Definition III.1 A Banach space Y is said to be crudely finitely representable in X if there exists a constant C  1 such that, for every finite-dimensional subspace E of Y, and for any ε > 0, we can find a finite-dimensional subspace F of X satisfying d(E, F)  C(1 + ε). For C = 1, we simply say finitely representable. In particular, Dvoretzky’s theorem (Chapter 1 of Volume 2) states that 2 is finitely representable in every Banach space. In Chapter 6, we will see that every space Lp (μ) is finitely representable in p . The following result is important: Theorem III.2 (Principle of Local Reflexivity) Let X be an arbitrary Banach space. For every finite-dimensional subspace E of X ∗∗ , every finite-dimensional subspace F of X ∗ and any ε > 0, there exists an isomorphism T from E onto a subspace T(E) of X such that: 1) T T −1   1 + ε; 2) T|E∩X = Id|E∩X ; 3) ϕ, T  = , ϕ for every  ∈ E and every ϕ ∈ F. In particular, X ∗∗ is finitely representable in X. The proof given here is due to C. Stegall [1980], and is based on the theory of operators. The key point is the following: Lemma III.3 Let X and Y be two Banach spaces, and u : X → Y an operator with a closed range. Let x∗∗ ∈ X ∗∗ such that u∗∗ x∗∗ = y ∈ Y. Then, for any δ > 0, there exists x ∈ X such that ux = y and with x  (1 + δ)x∗∗ . Proof First, y ∈ im u; in fact, as im u is closed, by the Hahn–Banach theorem, it suffices to show that if y∗ ∈ Y ∗ vanishes on im u, then it vanishes on y: for such a y∗ , we have u∗ y∗ = 0, and hence: y∗ , y = y∗ , u∗∗ x∗∗ = x∗∗ , u∗ y∗ = x∗∗ , 0 = 0 . Thus there exists x0 ∈ X such that y = ux0 . Then x∗∗ − x0 ∈ ker u∗∗ , and hence: x0 X/ ker u = x0 (X/ ker u)∗∗ = x0 X ∗∗ /(ker u)⊥⊥ = x∗∗ X ∗∗ /(ker u)⊥⊥  x∗∗  ; so some x ∈ x0 + ker u can be found such that x  (1 + δ)x∗∗ .

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5 Type and Cotype of Banach Spaces

At this point, the following observation is useful: Observation III.4 If u : X → Y has a closed range and v : X → Z is of finite rank, then w = u ⊕ v : X → Y ⊕ Z also has a closed range. In fact: w∗ (y∗ , z∗ )(x) = (y∗ , z∗ )(ux, vx) = y∗ (ux) + z∗ (vx) = u∗ y∗ (x) + v∗ z∗ (x) , so that w∗ (y∗ , z∗ ) = u∗ y∗ + v∗ z∗ . Hence im w∗ = im u∗ + im v∗ . But im u∗ is closed, because im u is closed, and im v∗ is finite-dimensional, because v, and hence v∗ , are of finite rank; it follows that im w∗ is closed, and thus im w as well. Proof of Theorem III.2 Let E, F and ε > 0 be as in the statement of the theorem, and let δ > 0 to be adjusted. We select a δ-net 1 , . . . , n of SE , the unit sphere of E, such that (1 , . . . , p ) is a basis of E ∩ X, and (1 , . . . , q ) is a basis of E (p  q  n). Next let ϕ1 , . . . , ϕm ∈ SX ∗ , the unit sphere of X ∗ , with: (1)

| k , ϕk | 

1 1+δ

for k = 1, 2, . . . , n ,

and with m  n chosen large enough so that (ϕ1 , . . . , ϕm ) contains a basis of F. Setting r = n − q, we can write, for l = 1, . . . , r: (2)

q+l =

q

tj,l j .

j=1

Then we define A0 : X n → X p × X r by the formula:   q q A0 (x1 , . . . , xn ) = x1 , . . . , xp , tj,1 xj − xq+1 , . . . , tj,r xj − xq+r . j=1

j=1

The mapping A0 is surjective because, if (y1 , . . . , yp , z1 , . . . , zr ) ∈ X p × X r , we have (y1 , . . . , yp , z1 , . . . , zr ) = A0 (x1 , . . . , xn ), with: p tj,l yj − zl , x1 = y1 , . . . , xp = yp ; xp+1 = · · · = xq = 0 ; xq+l = j=1

for 1  l  r. In particular, A0 has a closed range. p Now we let Y = X p × X r × Kmn = ∞ (X) ⊕∞ r∞ (X) ⊕∞ mn ∞ (K), and n define A : X → Y by the formula:     A(x1 , . . . , xn ) = A0 (x1 , . . . , xn ), ϕk (xj ) 1jn , 1km .

III Complements on Banach Spaces

175

It follows from Observation III.4 that A has a closed range. Moreover, as: q  ∗∗ A∗∗ (x1∗∗ , . . . , xn∗∗ ) = x1∗∗ , . . . , xp∗∗ , tj,1 xj∗∗ − xq+1 ,..., j=1 q

   ∗∗ tj,r xj∗∗ − xq+r , xj∗∗ (ϕk ) 1jn , 1km ,

j=1

by (2), the definition of A0 being tailor-made for this, we have:     A∗∗ (1 , . . . , n ) = 1 , . . . , p , 0, . . . , 0, j (ϕk ) 1jn , 1km , and hence A∗∗ (1 , . . . , n ) ∈ Y, as 1 , . . . , p ∈ X. By Lemma III.3, we can thus find (b1 , . . . , bn ) ∈ X n such that: A(b1 , . . . , bn ) = A∗∗ (1 , . . . , n ) and sup bj   (1 + δ) sup j  = 1 + δ . 1jn

1jn

The equality A(b1 , . . . , bn ) = A∗∗ (1 , . . . , n ) translates into: ⎧ b1 = 1 , . . . , bp = p ; ⎪ ⎪ ⎪ ⎪ ⎪ q ⎨ tj,l bj for 1  l  r ; bq+l = (3) ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎩ ϕk (bj ) = j (ϕk ) for 1  j  n, 1  k  m . Finally, we define the operator T : E → X by: T(j ) = bj ,

for 1  j  q .

The formulas (3) show that T is the identity on E ∩ X; moreover, for 1  l  r, we have: q q tj,l T(j ) = tj,l bj = bq+l , T(q+l ) = j=1

j=1

so T(j ) = bj for any j = 1, . . . , n. On the other hand, (3) shows that ϕk , Tj = j , ϕk for any j, k; as (j )1jn contains a basis of E and as (ϕk )1km contains a basis of F, then ϕ, T  = , ϕ for every  ∈ E and every ϕ ∈ F. It only remains to show that condition 1) of the theorem is satisfied if we choose δ = δ(ε) < 1 small enough. We note that T j  = bj   1 + δ, and that:

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5 Type and Cotype of Banach Spaces

T j  = bj   sup |ϕk (bj )| = sup |j (ϕk )|  |j (ϕj )|  kn

kn

1 , 1+δ

according to (1). Thus, T is (1 + δ)-isometric on the δ-net 1 , . . . , n . However, for every  ∈ SE , a standard argument allows us to write  = +∞ s−1 and λ = 1. It 1 s=1 λs s , with s ∈ {1 , . . . , n }, 0  λs  δ ensues that: T  

+∞

λs T

s 

 (1 + δ)

s=1

+∞ s=1

δ s−1 =

1+δ , 1−δ

whereas: T   T 1  −

+∞ s=2

δ s−1 T s  

+∞

1 − (1 + δ) δ s−1 1+δ s=2

1 δ(1 + δ) 5δ = := ρ . − =1− 1+δ 1−δ 1 − δ2 √ √ We now only need to adjust δ so that (1+δ)/(1−δ)  1 + ε and ρ  1 + ε to obtain T T −1   1 + ε, and thus the proof of the principle of local reflexivity is completed. Some applications of this theorem are given in Exercises VIII.8, VIII.9 and VIII.10.

III.2 Ultraproducts Ultraproducts provide a convenient framework to replace conditions of the type “for every ε > 0” by “ε = 0”: this is Non-Standard Analysis. Definition III.5 Let (Xi )i∈I be a family ofBanach spaces, and U a non-trivial 3 ultrafilter on I. Let X∞ = ∞ (Xi , i ∈ I) = i∈I Xi ∞ be the Banach space of bounded families x = (xi )i∈I , with xi ∈ Xi for every i ∈ I, equipped with the norm x = supi∈I xi Xi . Let N be the (closed) subspace of X∞ consisting of the families having limit zero along the ultrafilter U . The quotient Banach space X = X∞ /N is called the ultraproduct of the Xi ’s along the ultrafilter  U . We also denote X = i∈I Xi /U . If all the Xi ’s are equal to the same space X, X is called the ultrapower of X along U , and denoted X = X U . We can verify that if x = (xi )i∈I , then ˙xX = limU xi Xi . We choose a non-trivial ultrafilter, because if Ua is the ultrafilter of subsets containing a ∈ I, then the construction yields Xa . The interest of this new notion is shown in the following:

IV Type and Cotype of Banach Spaces

177

Proposition III.6 If a Banach space Y is finitely representable in a Banach space X, then Y is isometric to a subspace of an ultrapower of X. In particular, the principle of local reflexivity gives: Corollary III.7 The bidual X ∗∗ of any Banach space X is isometric to a subspace of an ultrapower of X. Proof of Proposition III.6 Let F be the set of finite-dimensional subspaces of Y. We equip I = F × R∗+ with the following order: (F, ε) (F , ε ) if F ⊆ F and ε  ε. This order is filtering and thus induces a filter on I. Let U be a finer ultrafilter on I. We will now show that Y is isometric to a subspace of X U : By hypothesis, for every i = (Ei , εi ) ∈ I, there exist a finite-dimensional subspace Fi of X and an operator Ti : Ei → Fi such that (1−εi ) y  Ti y  (1 + εi ) y for any y ∈ Ei . For every y ∈ Y, and any i = (Ei , εi ) ∈ I, set xi = Ti (y) if y ∈ Ei , and xi = 0 otherwise. As (1 − εi ) y  xi   (1 + εi ) y, we can define a mapping j : Y → X U by setting j(y) = limU (xi )i∈I and j is an isometry. For example, and in anticipation of Section V, note that: Proposition III.8 Any ultraproduct of Hilbert spaces is a Hilbert space. (It suffices to verify that the norm on X is associated with the scalar product defined by ˙x, y˙ = lim xi , yi Xi U

where x = (xi )i∈I and y = (yi )i∈I .) More generally, we could show that every ultraproduct of spaces Lp (μi ) is again an Lp (μ) space (1  p  ∞), but we skip the proof, not needing this result.

IV Type and Cotype of Banach Spaces IV.1 Introduction The Banach space structure, although useful in Analysis with the automatic consequences of completeness and of Baire’s theorem, is quite poor;   whenwe add a large number of vectors x1 , . . . , xn , the triangle inequality  ni=1 xi   n i=1 xi  quickly becomes imprecise. In contrast, the Hilbert space very rich, and,  is   structure  when the xi ’s are n 2 1/2 . x  orthogonal, we have exactly  ni=1 xi  = i i=1 More generally, we have the following result:

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5 Type and Cotype of Banach Spaces

Proposition IV.1 (Generalized Parallelogram Identity) Banach space; then:

Let X be a

1) X is isometric to a Hilbert space if and only if, for every x, y ∈ X: x + y2 + x − y2 = 2 (x2 + y2 ) . 2) We always have: 1 1 (x2 + y2 )  (x + y2 + x − y2 )  2 (x2 + y2 ) . 2 2 3) In any Hilbert space H, for every x1 , . . . , xn ∈ H, we have: n 1 2 θ x + · · · + θ x  = xi 2 . 1 1 n n 2n θi =±1

i=1

4) If X is isomorphic to a Hilbert space H, and if the Banach–Mazur distance dX = d(X, H)  C, then: n n 1 1 2 2 2 x   θ x + · · · + θ x   C xi 2 . i 1 1 n n 2n C2 θi =±1

i=1

i=1

Proof 1) The necessary condition requires no comment; the sufficient condition is a well-known easy exercise: if the space is real, for example, we set ϕ(x, y) = 14 (x + y2 − x − y2 ), and the parallelogram identity implies ϕ(x + x , 2y) = 2 [ϕ(x, y) + ϕ(x , y)]; we deduce that ϕ is separately additive and homogeneous, and clearly ϕ(x, x) = x2 . 2) The right-hand side follows from the triangle inequality, and the left-hand side from the change of variables x + y = u, x − y = v. 3) We proceed by induction on n: 1 θ1 x1 + · · · + θn+1 xn+1 2 n+1 2 θ1 ,...,θn+1 =±1

=

=

1 2n

1 2n



θ1 ,...,θn =±1



1 θ1 x1 + · · · + θn xn + xn+1 2 2 + θ1 x1 + · · · + θn xn − xn+1 2 (θ1 x1 + · · · + θn xn 2 + xn+1 2 )

θ1 ,...,θn =±1

= x1 2 + · · · + xn 2 + xn+1 2 . 4) This follows immediately from 3) .



IV Type and Cotype of Banach Spaces

179

Corollary IV.2

Let H be an infinite-dimensional Hilbert space. Then:  2 1) If (xn )n1 is summable in H, we have +∞ n=1 xn  < +∞. 2) This is optimal: for any sequence (λn )n1 of positive real numbers such  2 that +∞ n=1 λn < +∞, there exists a summable sequence (xn )n1 in H such that xn  = λn for any n  1. Proof

This is well known: for 1) , we have M=

sup

A⊆N∗ , A finite

   xn  < +∞ ; n∈A

hence θ1 x1 +· · ·+θn xn   2M, with θi = ±1, and x1 2 +· · ·+xn 2  4M 2 , by Proposition IV.1, 3) . For 2) , we take an orthonormal sequence (en )n1 of H and set xn = λn en ;     2 1/2 ; by the Cauchy criterion the family we have  n∈A xn  = n∈A λn (xn )n1 is hence summable. Proposition IV.1, while elementary, deserves some comments. 1) The converse of 4) is true: if we have the double inequality for any n  1, with a constant C, then X is isomorphic to a Hilbert space H, and more precisely d(X, H)  C. This is Kwapie´n’s theorem, which will be proved in the following section.  2) For the means θi =±1 , it is more convenient to write it using independent Bernoulli variables ε1 , . . . , εn (or Rademacher variables r1 , . . . , rn ); then: 1 θ1 x1 + · · · + θn xn 2 2n θi =±1 2     n  = E εi xi   = n=1



 2  n   εi (ω)xi    dP(ω) . n=1

The inequality in 4) can thus be written: 1 C

 n i=1

1/2 xi 

2

    n   εi xi   i=1

L2 ( ; X)

C

 n

1/2 xi 

2

.

i=1

In view of these inequalities, a couple of natural questions are raised: (a) What happens if we keep only one of these inequalities, with the xi ’s in a Banach space X? (b) What happens if we change the exponent 2?

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5 Type and Cotype of Banach Spaces

We are thereby led to the inequalities:  n  1/p  n   p   (1) ε x  C x  ; i i i  i=1

(2)

r

i=1

  1/q  n   n 1 q   εi xi   xi  .  C r i=1

i=1

Such inequalities provide tools to “measure” to what point a Banach space is “close to”, or “far from”, a Hilbert space. A first observation is that the exponent r plays a non-active role, because of (or thanks to) the Khintchine–Kahane inequalities. In general we take it equal to 1, 2 or p∗ (respectively q∗ ), where p∗ (respectively q∗ ) is the conjugate exponent of p (respectively q). A second observation is that necessarily p  2 and q  2 in (1) and (2); indeed, by taking r = 2, and the xi ’s all equal to the same vector x, of √ norm 1, we obtain, in (1): n  C n1/p , for any n  1, which is not possible unless p  2; similarly, (2) requires q  2. This restriction once made, the exponents p and q play an essential role. 3) A good idea could be to replace the Bernoulli variables by other random variables, for example, by Gaussian variables – which we will profitably do in the proof of Kwapie´n’s theorem – or by p-stable variables. The consequences are stated, without proof, in Subsection IV.4.

IV.2 Definitions and First Properties We proceed now to the notions of type and of cotype. Definition IV.3 Let X be a Banach space. 1) X is said to be of type p (1  p  2) if there exists a constant C  1 such that:   n 1/p  n   p   εi xi  C xi   2 i=1

L (X)

i=1

for every x1 , . . . , xn ∈ X, with (εn )n1 a Bernoulli sequence. The best constant C is called the type-p constant of X, and is denoted Tp (X). 2) X is said to be of cotype q (2  q  +∞) if there exists a constant C  1 such that:   n 1/q  n   1 q   εi xi   xi   C 2 i=1

L (X)

i=1

IV Type and Cotype of Banach Spaces

181

for every x1 , . . . , xn ∈ X. The best constant C is called the cotype-q constant of X, and is denoted Cq (X). Some examples are presented in the following subsection. Of course, every Hilbert space is simultaneously of type 2 and of cotype 2 (but this is the only case, according to Kwapie´n’s theorem). Remarks 1) The definitions of type and cotype involve only a finite, albeit arbitrary, number of vectors of X: these notions depend only on the finite-dimensional subspaces of X, and are hence “local” notions. More precisely, if X is of type p (respectively of cotype q) and if Y is crudely finitely representable in X, then Y is also of type p (respectively of cotype q). In particular, this is the case for the closed subspaces Y of X. Note right now that if a space is of type p, it is also of type p for any p  p, and that if it is of cotype q, it is also of cotype q for any q  q (see also Remark 2) below); the subspaces of X can thus be of a better type or cotype than X itself (it suffices to realize that X can be written as the direct sum of a “good” and a “bad” subspace). 2) From the Khintchine–Kahane inequalities, and from Remark 1 following Corollary V.3 of Chapter 4, it ensues that: (a) X is of type p if, and only if, for every sequence (xn )n1 of elements of X +∞  p such that +∞ n=1 xn  < +∞, the series n=1 εn xn converges almost surely. (b) X is of cotype q if, and only if, for every sequence (xn )n1 of +∞ elements of X such that the series n=1 εn xn converges almost surely (and in particular if it is unconditionally convergent), we have +∞ q n=1 xn  < +∞. Note that if X is of cotype q, then, in particular, for every weakly  unconditionally Cauchy series +∞ n=1 xn , i.e. satisfying  n     sup sup  θk xk   < +∞ ,

n1 θ=±1

+∞

k=1

we have n=1 xn q < +∞. Conversely, for 2 < q < +∞, if this property is satisfied, Talagrand [1992 c] showed that X is of cotype q. For p = 2, this is no longer the case, even though, according to the above, X is of cotype 2 + ε for any ε > 0; in fact, Talagrand [1992 a] (see also Talagrand [1994]) constructed a Banach space not of cotype 2, even though

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5 Type and Cotype of Banach Spaces

 2 each of its weakly unconditionally Cauchy series satisfies +∞ n=1 xn  < +∞, i.e. has the Orlicz property. 3) Every space is trivially of type 1 (this is simply the triangle inequality) and  of cotype +∞ (since  ni=1 εi xi 2  maxin xi , by the symmetry of the Bernoulli variables). X is said to have a non-trivial type if it is of type p > 1, i.e. when Tp (X) < +∞ and has a non-trivial cotype if it is of cotype q < +∞, i.e. Cq (X) < +∞. type: if (en )n1 For example, 1 does not have any non-trivial   is the canonical basis of 1 , then  nj=1 εj ej L2 ( ) = n, whereas 1  n  p 1/p = n1/p . Hence if  is crudely finitely representable e  j 1 j=1 in a Banach space X, this space in turn cannot have a non-trivial type. Similarly, c0 does not have any non-trivial cotype: if (en )n1 is its     n q 1/q = canonical basis, then  nj=1 εj ej L2 (c ) = 1, whereas j=1 ej  0

n1/q . Hence any space in which c0 is crudely finitely representable does not have any non-trivial cotype. Maurey and Pisier [1973] and [1976] showed that, in both cases (Pisier [1973 b] for the type), equivalences hold. For the type, the proof will be seen in Chapter 4 of Volume 2; for the cotype, the proof is fairly long, and we refer to Diestel–Jarchow–Tonge, Chapter 14, pages 283–298, or to Albiac–Kalton, Theorem 11.1.14. For further comments on the subject, we also refer to Subsection IV.4. 4) The type and cotype of a Banach space are inherited by its bidual. This follows from the local reflexivity principle. Indeed, assume that, for example, X is of type p, and let 1 , . . . , n ∈ X ∗∗ ; take E = span (1 , . . . , n ), and, with ε > 0, let T : E → X be the operator provided by the local reflexivity:   T   (1 + ε) for every  ∈ E. If xi = T i , we obtain:   n   n   n 1/p  n       p       εi i    εi T i  =  εi xi   Tp (X) xi   i=1

i=1

 (1 + ε) Tp (X)

i=1

 n

i p

i=1

1/p ;

i=1

thus Tp (X ∗∗ )  (1 + ε) Tp (X), and finally Tp (X ∗∗ ) = Tp (X), since, by Remark 1 and the inclusion X ⊆ X ∗∗ , clearly Tp (X)  Tp (X ∗∗ ). Similarly, Cq (X ∗∗ ) = Cq (X). Another possible argument is that X ∗∗ is isometric to a subspace of an ultrapower of X, and that such an ultrapower has the same type and cotype as X, as can easily be verified.

IV Type and Cotype of Banach Spaces

183

5) The type is inherited by quotients: if M is a closed subspace of X, and x˙ j = xj + M ∈ X/M, then, for every m1 , . . . , mn ∈ M:  n   n  1/p  n     p     εj x˙ j    εj (xj + mj )  Tp (X) xj + mj  ;  j=1

j=1

j=1

hence, by taking the lower bound on the mj ’s:  n   1/p n   p    T ε x ˙ (X) ˙ x  , j j p j  j=1

j=1

and finally Tp (X/M)  Tp (X). However the cotype is not inherited by quotients: in fact, we will see later that 1 is of cotype 2, whereas c0 is a quotient of it (as is any separable Banach space); however c0 does not have a non-trivial cotype. Before examining some examples, let us study to what extent the notions of type and cotype are dual of one another. Theorem IV.4 The dual X ∗ of a Banach space X of type p (with 1 < p  2) is of cotype q = p∗ , where p∗ is the conjugate exponent of p. Moreover: Cp∗ (X ∗ )  Tp (X) . This is also true for p = 1, but is of no interest (see Remark 3 above). The proof is based on the following classical lemma, stating that the dual of np (X) can be isometrically identified with np∗ (X ∗ ): Lemma IV.5 For 1 < p < +∞, and for ϕ1 , . . . , ϕn ∈ X ∗ , we have:  n  1/p∗   n  n  ∗ ϕi p = sup  ϕi (xi ) ; xi p  1 . i=1

Proof

If

n

i=1

p i=1 xi 

i=1

 1, we have:

 n  1/p∗  1/p  n n n   p∗ p    ϕ (x ) ϕ  x   ϕ  x  i i  i i i i  i=1

i=1

i=1



 n

ϕi p

∗

i=1

1/p∗ .

i=1

n p On the other hand, fix λ1 , . . . , λn  0, with i=1 λi = 1 such that ∗  n   n p∗ 1/p = i=1 ϕi  i=1 λi ϕi . Given ε > 0, let u1 , . . . , un ∈ X of norm 1 such that ϕi (ui )  0, and ϕi   ϕi (ui ) + ε. Set vi = λi ui . We obtain:

184

5 Type and Cotype of Banach Spaces  n

p∗

1/p∗

ϕi 

n

=

i=1

λi ϕi  

i=1  n

 = 

n

λi ϕi (ui ) + ε

i=1

λi

i=1

 n  ϕi (vi ) + ε λi

i=1

n

i=1

   n  n  n  sup  ϕi (xi ) ; xi p  1 + ε λi ; i=1

i=1

i=1

then, letting ε tend to 0, the result ensues. Proof of Theorem IV.4 Let ϕ1 , . . . , ϕn ∈ X ∗ , and let x1 , . . . , xn ∈ X such  that ni=1 xi p  1. First observe that, using orthogonality of independent Bernoulli variables, we have: 2      1  n n  n   n     n         E  E = ϕ (x ) ε ϕ , ε x ε ϕ ε x i i  i i i i  i i  i i    i=1

i=1

 n      εi ϕi   i=1

L2 (X ∗ )

    n   εi ϕi   i=1

i=1

L2 (X ∗ )

 n     εi xi    i=1

Tp (X)

  n     Tp (X)  ε ϕ i i  i=1

L2 (X ∗ )

i=1

L2 (X)

 n

1/p

xi 

p

i=1

;

then Lemma IV.5 implies:   n  1/p∗ n   p∗  ϕi   Tp (X)  εi ϕi   i=1

i=1

i=1

L2 (X ∗ )

,

which completes the proof. Remark Note that the cotype cannot be “dualized” so easily: we will see, in fact, that X = 1 is of cotype 2, but X ∗ = ∞ is not of type 2, and does not even have any type p > 1: indeed, like any separable Banach space, 1 is isometric to a subspace of ∞ ; however, as seen earlier, 1 does not have a non-trivial type. An “explicit” example of an isometric embedding of 1 into ∞ can be constructed as follows: set α1 , . . . , αn , . . . ∈ R such that 1, α1 , . . . , αn , . . . are rationally independent (for example αn = log pn , where pn is the n-th prime number); we define fj ∈ ∞ by fj (k) = cos 2π kαj , with k = 1, 2, . . .

IV Type and Cotype of Banach Spaces

185

If λ1 , . . . , λn ∈ C, Kronecker’s theorem (see Hardy–Wright, Chapter XXIII; see also Kahane 2, Chapter 9, § 2,3) shows that:     n   n   n     λj fj  = sup  λj cos 2π kαj  = |λj | ;  ∞

j=1

k∈N∗

j=1

j=1

thus ( fn )n1 is equivalent to the canonical basis of 1 . To “dualize” the cotype, we need an additional property for X, highlighted by Maurey and Pisier [1976]: Definition IV.6 A Banach space is said to be K-convex if there exists a constant C > 0 such that, for any n  1, and for every family (xA )A⊆{1,...,n} of vectors of X, we have:         n    εj xj   C wA xA  ,   j=1

L2 (X)

A⊆{1,...,n}

L2 (X)

 where (εj )j1 is a Bernoulli sequence, wA = j∈A εj the associated Walsh functions and xj = x{j} . The best constant C is called the K-convexity constant of X, and is denoted K(X). Pisier [1982] (see also Pisier [1986 b]; Milman–Schechtman, Chapter 14; Diestel–Jarchow–Tonge, Chapter 13; or Maurey [2003 a]) showed, by using the theory of holomorphic semi-groups, that a Banach space is K-convex if and only if it does not contain n1 ’s uniformly (see Chapter 4 of Volume 2); in other words, if and only if it has a non-trivial type p > 1. Theorem IV.7 Let X be a K-convex Banach space. Then, if X is of cotype q, its dual is of type p = q∗ , the conjugate exponent of q. Moreover: Tq∗ (X ∗ )  K(X) Cq (X) . Proof Fix n  1, and let ϕ1 , . . . , ϕn ∈ X ∗ . As n is fixed,  = n = {−1, +1}n can be chosen as the probability space, equipped with the normalized counting measure; thus  ∗ L2 (X ∗ ) = L2 (n , P; X ∗ ) = L2 (n , P; X) ; hence:  n     εk ϕk    k=1

2  1    n 2   = sup  εk ϕk , f  ; f ∈ L (X) and f L2 (X) = 1 . ∗

L2 (X )

k=1

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5 Type and Cotype of Banach Spaces

However, the Walsh functions wA , A ⊆ {1, . . . , n}, form an orthonormal basis of L2 = L2 (n , P) (see the Annex); hence every f ∈ L2 (X) can be written  f = A⊆{1,...,n} wA xA , with xA ∈ X, and consequently: n /

0

εk ϕk , f = Eω

k=1

/ n

εk (ω)ϕk ,



0 wA (ω)xA

=

n

ϕk (xk ) A⊆{1,...,n} k=1 1/p  1/q  n n p q ϕk  xk  k=1 k=1   1/p  n   n p  ϕk  Cq (X)  εk xk   2 L (X) k=1 k=1    1/p n    ϕk p Cq (X) K(X)  w x . A A  L2 (X) k=1 A⊆{1,...,n} k=1

  

  Taking the upper bound for  A⊆{1,...,n} wA xA L2 (X) = f L2 (X) = 1, we obtain:   1/p  n   n p   ε ϕ  ϕ  Cq (X) K(X) , k k k  2 ∗  k=1

L (X )

k=1

which completes the proof. To conclude this subsection, we show how we can replace the Bernoulli variables by Gaussian variables, without altering either the value, or the constant, of type or cotype. This will often prove useful in what follows, notably because of the invariance of the Gaussians under rotation. Theorem IV.8 Let X be a Banach space, and let (gn )n1 be a sequence of independent standard Gaussian variables. Then: 1) If X is of type p, 1 < p  2, we have:   n 1/p  n   p   Tp (X)  g x x  , j j j   j=1

2

2) If X is of cotype q, 2  q < +∞, we have:   1/q  n  n  1 q   gj xj   xj  ,  Cq (X) 2 j=1

for every x1 , . . . , xn ∈ X.

j=1

for every x1 , . . . , xn ∈ X.

j=1

More generally, these relations hold if (gn )n1 is replaced by any sequence (ϕn )n1 of symmetric independent random variables, with norm 1 in L2 (P).

IV Type and Cotype of Banach Spaces

187

Proof 1) For ω ∈  , the hypothesis is applied to the vectors ϕj (ω )xj : 2  2/p   n  n   2 p p  ε (ω)ϕ (ω )x dP(ω)  T (X) |ϕ (ω )| x  ; j j j p j j  

j=1

j=1

then, by using the symmetry of the ϕj ’s and Fubini’s theorem, we obtain:  

 n 2     ϕj (ω )xj  dP(ω )  j=1

 n

  Tp (X)2



|ϕj (ω )|p xj p

2/p

dP(ω ) .

j=1

Raising to the power p/2, we obtain:  p    n   n  p p p   ϕj xj   Tp (X)  |ϕj | xj    2

j=1

;

2/p

j=1

then, by Minkowski’s inequality in L2/p (note that 2/p  1), we get: p  n n   p   ϕ x  T (X)  |ϕj |p 2/p xj p j j p  2

j=1

= Tp (X)p

j=1 n

p

ϕj 2 xj p = Tp (X)p

j=1

n

xj p ,

j=1

as ϕj 2 = 1. This proves the first formula. 2) The second formula is shown similarly, with the reverse Minkowski’s inequality for the exponent 2/q  1:  q    n    n q q   ϕj xj    |ϕj | xj   Cq (X)  q

j=1

2



j=1 n

2/q

 |ϕj | 2/q xj  =

j=1

q

q

n

xj q .

j=1

In fact, for 2) , even if it means sacrificing a factor of 2, the use of the contraction principle (Chapter 4, Theorem IV.4) would suffice.

188

5 Type and Cotype of Banach Spaces

Remark Conversely, Maurey and Pisier [1976], Corollary 1.3, showed that there exists a constant C > 0 such that    n  n        gj xj   C εj xj    L2 (X)

j=1

L2 (X)

j=1

for every x1 , . . . , xn ∈ X if and only if c0 is not finitely representable in X, or, in other words, if and only if X has a non-trivial cotype (see also Pisier [1973 d] for a partial answer).

IV.3 Examples The fundamental example is provided by the Lebesgue spaces Lr = Lr (μ), where μ is a positive measure on a measurable space (T, T ). Theorem IV.9 Let 1  r < +∞; then: 1) Lr is of type inf(r, 2); 2) Lr is of cotype sup(r, 2); in particular L1 is of cotype 2. The space M(T, T ) of complex measures on (T, T ) is of cotype 2. This applies in particular to the spaces r . Moreover, r is isometric to a subspace of every infinite-dimensional space Lr (μ) (for example, the subspace of functions constant on a countable measurable partition of T): thus, by considering the canonical basis of r , we deduce that infinite-dimensional spaces Lr (μ) cannot have a better type or cotype. Proof Let f1 , . . . , fn ∈ Lr . Even though not absolutely essential, the Khintchine–Kahane inequalities are used here, and we work with  r 1/r  n     n      εj fj  = E εj fj    r

j=1

 instead of  n

r

j=1

  j=1 εj fj 2 . By the linearity of the integral, we have: r r       n   n    ε f = E ε (ω)f (t) dμ(t) E j j ω j j    j=1

T

r

j=1

r      n    ε (ω)f (t) dP(ω) dμ(t) = j j   T



  T

brr

j=1

 n j=1

|fj (t)|2

r/2 dμ(t) ,

IV Type and Cotype of Banach Spaces

189

where br is the constant of Khintchine’s inequalities (which we distinguish from Br , the constant of the Khintchine–Kahane inequalities). Hence:  n   n      2 1/2      . (∗) ε f  b |f | j j r j   r

j=1

r

j=1

It is now useful to bring out the following lemma: Lemma IV.10 1) If 1  r  2, the Lr -norm is r-convex: for every f1 , . . . , fn ∈ Lr :  n   1/r  n  2 1/2  r   |f |  f  . j j r   r

j=1

j=1

2) If 2  r < +∞, the is 2-convex: for every f1 , . . . , fn ∈ Lr :  n   1/2  n  2 1/2  2   |f |  f  . j j r   Lr -norm

r

j=1

 n

j=1

 2 1/2

Proof of the lemma Set S = ; S is the square function associj=1 |fj | ated with the fj ’s, to be extensively used in the study of the unconditionality of (scalar) martingale differences, in Chapter 7. 1) If r  2, we pointwise have: r/2  n n n  2 r/2 2 |fj | |fj | = Sr  = |fj |r ; j=1

j=1

hence:

 Sr dμ  T

j=1

n

fj rr ,

j=1

which is the desired result. 2) If r > 2, then: 1/2 r/2 1/r    n   n 2 |fj |2 dμ = |f | Sr = j   T



 n

j=1

 |fj |2 r/2

1/2 =

 n

j=1

according to Minkowski’s inequality in Lr/2 .

j=1

r/2

j=1

1/2

fj 2r

,

190

5 Type and Cotype of Banach Spaces

This lemma and (∗) prove the first assertion of the theorem, with the following precision: Tinf(r,2) (Lr )  br /Ar , where Ar is the constant of the Khintchine–Kahane inequalities, used at the beginning of the proof. For the second assertion, first take 1 < r < +∞, and let r∗ be the conjugate ∗ exponent; Lr is hence the dual of Lr (even if μ is not σ -finite). Moreover, ∗ according to the above, Lr is of type inf(r∗ , 2), thus its dual Lr is of cotype sup(r, 2), via Theorem IV.4. The case r = 1 (perhaps the most useful!) remains. We treat directly: 2    1/2    n   n     εj (ω)fj  dP(ω)  εj (ω)fj     dP(ω) 

j=1



1

 =

T×

 a1

   n   εj (ω)fj (t) dμ(t) dP(ω)  j=1

  n T

 a1

1

j=1

 n

1/2

|fj (t)|2

j=1

fj 21

dμ(t)

1/2 ,

j=1

where a1 is the constant of Khintchine’s inequalities. Here, an inequality has been used, opposite to that of Lemma IV.10, proved similarly and known as the 2-concavity of the L1 -norm. Finally, for the last assertion, it suffices to see that every finite-dimensional subspace E of M(T) is isometric to a subspace of a space L1 (μ): if (μ1 , . . . , μn ) is a basis of E, we set μ = |μ1 | + · · · + |μn |; as the μj ’s are absolutely continuous with respect to μ, by the Radon–Nikodým theorem,  n we can write μj = fj .μ, and the mapping nj=1 aj μj → j=1 aj fj is an isometry from E into L1 (μ). In fact, the space M(T) is itself an L1 (ν) space, by the Kakutani representation theorem (see Lacey, for example), but this has not been proved here. Remark Let H be a separable infinite-dimensional Hilbert space; for 1  r  +∞, Sr = Sr (H) is the Schatten class of index r, formed by the compact operators u whose sequence of singular numbers (sn )n1 is in r : sn is the distance between √ u and the operators of rank < n, or equivalently the n-th eigenvalue of u∗ u, written in non-increasing order; we set uSr =   +∞ r 1/r = (s ) n n1 r . n=1 (sn (u))

IV Type and Cotype of Banach Spaces

191

S∞ is the space K(H) of all compact operators and S1 is the space of nuclear operators. These spaces are the non-commutative analogues of the r spaces, with similar properties: (Sr )∗ = Sr∗ if 1 < r  +∞, (S1 )∗ = L(H). Nicole Tomczak-Jaegermann [1974] showed the non-commutative analogue of Theorem IV.9: if 1  r < +∞, Sr is of type inf(r, 2) and of cotype sup(r, 2). In particular, S1 is of cotype 2. In fact, as it is the dual of the C∗ -algebra K(H), this ensues from a more general theorem of Tomczak-Jaegermann [1974]: Theorem The dual of every C∗ -algebra is of cotype 2. A proof is seen in Exercise VIII.14, using imaginary Riesz products. Note that if T is a compact space, equipped with its Borel σ -algebra, M(T) is the dual of the commutative C∗ -algebra C(T). More generally, there also exist similar results for the non-commutative spaces Lr , of which Sr is a particular case (T. Fack [1987]).

IV.4 Complements The replacement of Bernoulli variables by Gaussian variables has already been studied. Their replacement by stable variables turns out slightly differently. First, we give the following definition: a Banach space X is said to be of stable type p if there exists a constant C > 0 such that:   n 1/p  n   p   θj xj  C xj   j=1

L1 (X)

j=1

for every x1 , . . . , xn ∈ X, where (θn )n1 is a sequence of independent p-stable variables. In contrast, the usual type p is sometimes called the Rademacher type p. Theorem (Pisier [1973 a]) For every Banach space X, and 1 < p  2, we have the following results: 1) If X is of stable type p, it is of type p. 2) If X is of type p, it is of stable type p for any p < p. In particular, for p = 2, if X is of stable type 2 (hence with Gaussians), it is of type 2. Moreover, in this case, the converse is also true, by Theorem IV.8 (which does not apply for p < 2 because then the p-stable variables are not in L2 ).

192

5 Type and Cotype of Banach Spaces

Note that for p < 2, X is of stable type p if and only if there exists a constant C > 0 such that:  n     εj xj     C (xj )jn p,∞ j=1

2

for every x1 , . . . , xn ∈ X (see Ledoux–Talagrand, pages 249–250). It is also a consequence of the theorem just below. Recall that (xj )j p,∞ = supj j1/p xj ∗ , where (xj ∗ )j is the sequence of norms rearranged in non-increasing order. See Marcus and Pisier [1984] for sharper results related to this phenomenon. Another difference from the Bernoulli case is that, if the spaces Lr have exactly the type inf(r, 2), there is no analogue in the stable case: Theorem (Pisier [1974 c], Maurey–Pisier [1976]) If a Banach space is of stable type p for some p < 2, it is also of stable type p for some p > p. The stable type p is sometimes more convenient than the Rademacher type p, in particular for the finite-representability of the r spaces. First, we note (see Chapter 6) that, for 1  r  +∞, every finite-dimensional subspace of Lr (μ), and thus in particular of r , is, for any ε > 0, (1 + ε)isomorphic to a subspace of nr for some n  1. Hence, saying that r (respectively c0 if r = ∞, since ∞ = c∗∗ 0 is finitely representable in c0 ) is crudely finitely representable in a Banach space X is equivalent to saying: There exists a constant λ > 0 such that, for every n  1, nr is (1 + λ)isomorphic to a subspace Xn of X. The space X is then said to contain nr ’s uniformly. Actually Krivine [1976] showed that if this occurs for some value λ0 > 0, then X contains nr ’s (1 + λ)-uniformly, for every λ > 0; the much easier cases r = 1 and r = ∞ trace back to James [1964 b]. The link with the notions of type and of cotype is due to Maurey and Pisier [1976] (see also Pisier [1986 b]; Milman–Schechtman, Chapter 13; and Maurey [2003 a]): Theorem (The Maurey–Pisier Theorem) Let X be an infinite-dimensional Banach space; set: p(X) = sup{p  1 ; X is of type p} q(X) = inf{q  +∞ ; X is of cotype q} . Then p(X) and q(X) are crudely finitely representable in X. For p < 2, X is of stable type p if and only if p is not crudely finitely representable in X.

V Factorization through a Hilbert Space; Kwapie´n’s Theorem

193

V Factorization through a Hilbert Space and ´ Theorem Kwapien’s V.1 Definition and Local Character of the Factorization Definition V.1 An operator u : X → Y is said to be factorizable through a Hilbert space if there exist a Hilbert space H and operators w : X → H and v : H → Y such that u = v w. X@ @@ @@ w @@ 

u

H

/Y ?   v 

For such a u, we set: γ2 (u) = inf{v w ; u = v w , with w : X −→ H , v : H −→ Y and H Hilbert} . We denote by 2 (X, Y) the space of operators factorizable through a Hilbert space; clearly γ2 is a norm on 2 (X, Y) that makes it a Banach space. In Chapter 6, the Pietsch factorization theorem will be seen to imply that every 2-summing operator is factorizable through a Hilbert space. The converse is false in general: the identity of an infinite-dimensional Hilbert space obviously admits such a factorization, but is not 2-summing (if (en )n1 w is an orthonormal sequence, en −−→ 0, whereas en  = 1). n→+∞ The local quality of factorization through a Hilbert space, and the link with the Banach–Mazur distance, are expressed by the following proposition: Proposition V.2 Let X, Y be two Banach spaces, and u ∈ L(X, Y) an operator from X into Y. Then:   1) If there exists a constant C  1 such that γ2 u|E  C for every finitedimensional subspace E of X, then u is factorizable through a Hilbert space, and γ2 (u)  C. 2) X is isomorphic to a Hilbert space if and only if IdX is factorizable through a Hilbert space; moreover, in that case, dX = γ2 (IdX ). Notation Recall that, if X and Y are two Banach spaces, we denote by d(X, Y) their Banach–Mazur distance. Clearly, if X is isomorphic to a Hilbert space H (i.e. d(X, H) < +∞), then, for every other Hilbert space H to which X is isomorphic, we have d(X, H ) = d(X, H); we denote this common value dX .

194

5 Type and Cotype of Banach Spaces

Proof 1) Let I be the set of finite-dimensional subspaces of X, ordered by increasing inclusion, and let F be the section filter on I, for which a base is composed of the sets IE of all finite-dimensional subspaces of X containing E. Let U be an ultrafilter finer than F. For each E ∈ I, we have a factorization u|E = vE wE : u|E

/Y EA > AA }} AA } } }} v wE AAA }} E HE  with wE   1 and vE   C. Let H = E∈I HE /U be the ultraproduct of the Hilbert spaces HE ; by Proposition III.8,  it is itself a Hilbert space. We define w : X → H by the formula w(x) = x˙ E E∈I , with:  xE =

0 wE (x)

if x ∈ /E if x ∈ E .

We have: w(x) = lim xE   sup xE  = sup wE (x)  x ; U

E∈I

E∈I

hence w  1. Moreover, with F(x) = {E ∈ I ; x ∈ E}, we have F(x) ∈ F ⊆ U , and E ∈ F(x) implies u(x) = vE wE (x); consequently u(x)  C wE (x) = C xE , and, by a passage to the limit, u(x)  C w(x). We can thus define v : w(H) → Y by v[wx] = u(x): v is continuous, and its norm is bounded by C; it can be continuously extended to v : w(H) = H0 → Y, hence a factorization: X@ @@ @@ w @@@

u

H0

/Y > ~~ ~ ~ ~~ v ~~

with w  1 and v  C. This ensues that u ∈ 2 (X, Y) and that γ2 (u)  v w  C. 2) For every isomorphism T : X → H, we have γ2 (IdX )  T T −1 , and hence γ2 (IdX )  dX . Conversely, if we have a factorization IdX = v w, with v : H → X and w : X → H, the inequality x  v w(x) shows that w is injective, with closed range. Replacing H by w(H) if necessary, we can assume w surjective; then v = w−1 , and dX  w w−1  = w v; hence dX  γ2 (IdX ).

V Factorization through a Hilbert Space; Kwapie´n’s Theorem

195

V.2 Subordination Criterion. Factorization Theorem Definition V.3 Let X be a Banach space, and (yi )ip , (xj )jq two finite families of elements of X. The family (yi )ip is said to be subordinate to (xj )jq or, equivalently, dominated by (xj )jq , when p

|ξ(yi )|  2

i=1

q

|ξ(xj )|2

j=1

for every continuous linear functional ξ ∈ X ∗ . This is denoted (yi )ip ≺ (xj )jq . Subordination is thus a property involving the “weak-2 norms” of (yi )ip p q and (xj )jq , rather than their “strong-2 ” norms i=1 yi 2 and j=1 xj 2 . If X is a Hilbert space, by Parseval’s identity, the two notions coincide. The following factorization theorem states a converse: an operator that somehow transforms weak domination into strong domination can be factorized through a Hilbert space. For every operator u : X → Y, the

Theorem V.4 (Factorization Theorem) following assertions are equivalent:

(a) u ∈ 2 (X, Y), and γ2 (u)  C; q p (b) If (yi )ip ≺ (xj )jq , we have i=1 uyi 2  C2 j=1 xj 2 . Proof (a) ⇒ (b) If u = v w, with w : X → H and v : H → Y, and if (ek )k1 is an orthonormal basis of H, we have: p

u(yi )2 =

i=1

p

v w(yi )2  v2

i=1

= v2

p

w(yi )2

i=1 p

| w(yi ), ek |2 = v2

k1 i=1

u(yi )2  v2

i=1

= v2

| w∗ (ek ), yi |2 ;

k1 i=1

hence, since (yi )ip ≺ (xj )jq and p

p

w∗ (ek )

q k1 j=1 q

∈

X∗,

we get:

| w∗ (ek ), xj |2 | w(xj ), ek |2 = v2

k1 j=1 q 2 2

 v w

q j=1

xj 2 ;

j=1

then a passage to the lower bound leads to (b) .

w(xj )2

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5 Type and Cotype of Banach Spaces

(b) ⇒ (a) This is the interesting implication. According to Proposition V.2 (passage from local to global), we can assume that X is finite-dimensional. The unit sphere K of its dual X ∗ is thus compact, and we transform the hypothesis into a property of disjunction of convex sets in the (real) space C(K); indeed, denote by C the set of functions  of the form: (ξ ) =

p

|ξ(yi )| − 2

i=1

q

|ξ(xj )|2 ,

∀ξ ∈ K ,

j=1

where (yi )ip and (xj )jq are finite families of vectors of X for which p

u(yi ) > C 2

i=1

2

q

xj 2 .

j=1

By concatenation, we see that C is a convex cone, and hypothesis (b) means that every  ∈ C takes on at least one non-negative value (otherwise there would exist a pair of sequences verifying (yi )ip ≺ (xj )jq , while q p 2 2 2 i=1 u(yi ) > C j=1 xj  , contrary to (b) ); thus the convex cone C is disjoint from the open convex cone C− of functions  ∈ C(K) such that supK  < 0. The Hahn–Banach separation theorem and the Riesz representation theorem hence provide a real non-zero measure μ ∈ M(K), such that:  (1)  dμ  0 for every  ∈ C, K (2)  dμ  0 for every  ∈ C− K

(indeed if μ|C  α and μ|C−  α, necessarily α = 0, as C and C− are cones). Note that (2) means that the measure μ is positive. We now construct a subspace of L2 (μ) through which u can be factorized. Denote by  x ∈ C(K) the function defined by  x(ξ ) = ξ(x) for every ξ ∈ K, and let w be the mapping w : x ∈ X →  x ∈ L2 (μ). Note that:   1/2  w = sup | x|2 dμ ; x  1 K

is > 0: indeed, if ξ0 ∈ supp μ, there exist first an x ∈ BX with |ξ0 (x)| = 1, then x(ξ )|  1/2 for$ξ ∈ V. Next, a neighborhood V of ξ0 in K such that |ξ(x)|$ = | μ(V) > 0, since ξ0 ∈ supp μ, and w2  K |ξ(x)|2 dμ(ξ )  V 41 dμ(ξ ) = 1 4 μ(V) > 0. If necessary multiplying μ by C/w, we can hence assume w = C.

V Factorization through a Hilbert Space; Kwapie´n’s Theorem

197

Now, if u(y) > C x, then the function  = | y|2 −| x|2 is in C, and hence, by (1):   | y|2 dμ  | x|2 dμ . u(y) > C x ⇒ K

K

As we have assumed w = C, this can also be written: 1/2  | y|2 dμ  C, u(y) > C ⇒ K

or, otherwise: (3)

1/2 | y|2 dμ .

 u(y)  K

We can now conclude: define v : w(X) → X by setting v( x) = u(x) for every x ∈ X; (3) shows that v  1, and with H = w(X) being the closure of w(X) in L2 (μ), v can be extended to H by continuity, while keeping a norm  1. Then u = v w, and hence γ2 (u)  v w  C, which completes the proof. An efficient application of Theorem V.4 requires a criterion of domination; here is one that fits well with the invariance of Gaussian variables under rotation: Criterion V.4 The following assertions are equivalent: (a) (yi )ip is dominated by (xj )jq . q p  1, represented by (b) There exists  operator A ∈ L(K , K ), of norm A  an q a matrix ai,j ip, jq ∈ Mp,q (K), and such that yi = j=1 ai,j xj for every i = 1, . . . , p.     Proof (b) ⇒ (a) is easy: if ξ ∈ X ∗ , we have ξ(yi ) ip = A ξ(xj ) jq , hence:        ξ(yi ) ip p  A  ξ(xj ) jq q   ξ(xj ) jq q . 2

2

2

Kq

that can be written of elements of (a) ⇒ (b) : Let S be the subspace hypothesis, an ξ(x1 ), . . . , ξ(xq ) , with ξ ∈ X ∗ . Using the    we can define  operator A : S → Kp , of norm  1, by A ξ(xj ) jq = ξ(yi ) ip . We can extend it to Kq without increasing its norm, by composing it with the orthogonal projection from Kq onto S. Then:   q q ξ(yi ) = ai,j ξ(xj ) = ξ ai,j xj j=1

for every ξ ∈ X ∗ , so that yi =

q

j=1 ai,j xj .

j=1

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5 Type and Cotype of Banach Spaces

By Lemma II.9, the unit ball of L(Rn ) (respectively L(Cn ) is the convex hull of the orthogonal (respectively unitary) matrices; this fact, together with the domination criterion, leads to a nicer expression of the factorization theorem. Then, Kwapie´n’s theorem easily ensues. Theorem V.5 (Final Form of the Factorization Theorem) the following assertions are equivalent:

For u ∈ L(X, Y),

(a) u can be factorized through a Hilbert space, and γ2 (u)  C. (b) For every integer n  1, every orthogonal (respectively unitary, if the   spaces are complex) matrix A = ai,j i,jn ∈ O(n) (resp. ∈ U(n)), and every x1 , . . . , xn ∈ X, we have 2 n n   n     C2 a u(x ) xj 2 . i,j j   i=1

j=1

j=1

Proof (a) ⇒ (b) is an immediate consequence of Theorem V.4 and of Criterion V.4 . (b) ⇒ (a) : Let (yi )ip be subordinate to (xj )jq . If necessary adding some zeros to the yi or to the xj , we can assume p = q, and, according to Criterion V.4 , we have yi =

p

ai,j xj

j=1

  for some A = ai,j i,jp in the unit ball B of L(Rp ). Then, if φ : B → R+ is the convex functional defined by 2 p   p   bi,j u(xj ) φ(B) =   , i=1

j=1

  for every B = bi,j i,jp , we have: φ(A) =

p

u(yi )2 .

i=1

p However, by hypothesis, φ(B) is bounded above by C2 j=1 xj 2 for every B ∈ O(p). As B is the closed convex hull of O(p), we thus also have φ(A)  p p p C2 j=1 xj 2 ; that is, i=1 uyi 2  C2 j=1 xj 2 . Theorem V.4 thus gives the result.

V Factorization through a Hilbert Space; Kwapie´n’s Theorem

199

´ Theorem V.3 Kwapien’s We have noted that, obviously, every space isomorphic to a Hilbert space is of type 2 and of cotype 2. Actually, there is no other case: Theorem V.6 (Kwapie´n’s Theorem) If a Banach space X is simultaneously of type 2 and cotype 2, it is isomorphic to a Hilbert space; more precisely; dX  T2 (X) C2 (X) . Proof By Proposition V.2, we know that dX = γ2 (IdX ). Thus we need According to to show that IdX can be factorized through a Hilbert space.  the factorization theorem (Theorem V.5), with A = ai,j i,jn ∈ O(n) and x1 , . . . , xn ∈ X, the following inequality must be proved: 2 n  n  n     T2 (X)2 C2 (X)2 a x xj 2 . i,j j   i=1

n

j=1

j=1

Let yi = j=1 ai,j xj , and consider a sequence g1 , . . . , gn of independent standard Gaussian variables. Thanks to 2) of Theorem IV.8, we have:   n 1/2  n   2  yi   C2 (X)  gi yi  .  i=1

i=1

We write: n

gi yi =

i=1

n i=1

gi

 n

 ai,j xj

=

j=1

L2 (X)

n  n j=1

 ai,j gi xj ,

i=1

  and set γj = i=1 ai,j gi . As A = ai,j i,jn ∈ O(n), the random vector (γ1 , . . . , γn ) has the same distribution as (g1 , . . . , gn ), by the rotation invariance of Gaussian vectors (Proposition II.8). In particular:  n    n        γj xj  = gj xj  ,   n

j=1

L2 (X)

j=1

L2 (X)

and hence, this time by 1) of Theorem IV.8:   n  1/2 1/2  n n   2 2   yi   C2 (X)  γj xj   C2 (X) T2 (X) xj  , i=1

j=1

L2 (X)

j=1

which completes the proof. Remark

The definition of type and cotype can be generalized to operators:

Definition V.7 Then:

Let u be an operator between two Banach spaces X and Y.

200

5 Type and Cotype of Banach Spaces

1) u is said to be of type p if, for every x1 , . . . , xn ∈ X, we have:  n  1/p  n   p   ε u(x )  C x  ; j j  j  L2 (X)

j=1

j=1

2) u is said to be of cotype q if, for every x1 , . . . , xn ∈ X, we have:   1/q  n   n q  u(xj )  C εj xj  .  j=1

j=1

L2 (X)

For u, the type-p constant and cotype-q constant are the smallest constants C > 0 satisfying the preceding inequalities. Hence, for a Banach space X, to be of type p (respectively of cotype q) is the same as, for IdX , to be of type p (respectively of cotype q). Kwapie´n’s theorem can then be generalized as follows (with the same proof): Theorem V.8 Let X, Y, Z be three Banach spaces, and let u : X → Y be an operator of type 2 and v : Y → Z be an operator of cotype 2. Then vu : X → Z can be factorized through a Hilbert space, and γ2 (vu)  T2 (u) C2 (v).

VI Some Applications of the Notions of Type and Cotype Two examples of the use of type and cotype are presented here. Theorem VI.1 Let K be a compact space and 1  p < 2. Then every operator u : C(K) → p is compact. Proof We use the fact that M(K) is of cotype 2. First consider the case p > 1. Let q = p∗ be the conjugate exponent of p. We have to prove that the adjoint u∗ : q → M(K) is compact. As q is reflexive, this amounts to showing that if w xn −−→ 0, then u∗ xn  −−→ 0. If this were not the case, first, modulo successive n→+∞

n→+∞

extractions, we could assume that u∗ xn   δ > 0. Then, by the Bessaga– Pełczy´nski selection theorem, (xn )n1 is equivalent to a block basis of the canonical basis of q and, consequently, is itself equivalent to this canonical basis (Chapter 2). In particular, we have:     N 1/q   θ x n n  C N  n=1

for every θn of modulus 1. Hence, if (εn )n1 is a Bernoulli sequence:    N  ∗ 1/q   ε (ω) u (x ) . n n   C u N  n=1

VI Some Applications of the Notions of Type and Cotype

201

By taking the square and then integrating over ω, we obtain:  N    ∗   ε u (x )  C u N 1/q . n n   L2 (M(K))

n=1

However, as M(K) is of cotype 2, we have:    1/2 N  N  ∗ ∗ 2   εn u (xn )  A1 u (xn )  A1 δ N 1/2 ,  n=1

L2 (M(K))

n=1

which is not compatible with the preceding inequality for N large enough, given that 12 > 1q . This contradiction shows the result. For p = 1, we consider again the adjoint u∗ : ∞ → M(K); we denote ∗ by (en )n1 the canonical basis of 1 , and by (e∗n )n1 that of c0 ⊆ ∞ = 1 . The   series n1 e∗n is weakly unconditionally Cauchy; hence so is n1 u∗ (e∗n ). But M(K) does not contain c0 (since it is of cotype 2; or, as we will see in Chapter 7, since it is weakly sequentially complete); the Bessaga–Pełczy´nski c0 theorem (Chapter 2) thus states that this series is in fact unconditionally  ∗ convergent. Therefore, with uN = N n=1 en (ux) en , we have, for x  1:  +∞   +∞ ∗ ∗ ∗ ∗  u(x) − uN (x)  | u (en ), x |  sup  λ u (e ) n n   |λn |1

n=N+1

n=N+1

    ∗ ∗    4 sup  u (en ) = εN ; I⊆{N+1,...} I finite

n∈I

hence u − uN   εN . By the unconditionality of the series, εN −−→ 0, so u is N→+∞

compact. Remark The theorem remains true if we replace C(K) by a C∗ -algebra A, because A∗ is of cotype 2, and only this point matters in the proof. However, it is obviously false for p = 2, because the Fourier transform F : C(T) √ → 2 (Z) ej − ek 2 = 2 for j = k; is not compact: en (t) = eint is of norm 1, whereas  hence ( en )n∈Z does not have any convergent subsequence. Theorem VI.2 Let (ϕn )n1 be an orthonormal sequence in L2 (0, 1) such that ϕn 1  δ, with δ > 0. $1 f (n) = 0 f ϕn dt. Then: For f ∈ L2 (0, 1), set:  1) (Paley’s Theorem) For every sequence (cn )n1 of complex numbers such +∞  2  that +∞ n=1 |cn f (n)| < +∞ for every f ∈ C([0, 1]), then n=1 |cn | < +∞. 2) (Carleman’s Theorem) There exists f ∈ C([0, 1]) such that  f ∈ / 2,1 . The space 2,1 is the Lorentz space consisting of all vanishing sequences a = (an )n1 such that:

202

5 Type and Cotype of Banach Spaces +∞

n−1/2 a∗n < +∞ ,

n=1

where (a∗n )n1

is the non-increasing rearrangement of (|an |)n1 . It is a Banach space for the norm: a2,1 =

+∞

n−1/2 a∗n

= sup σ

n=1

+∞ n=1

N∗ . We

where σ runs over the permutations of inclusions: p ⊂ 2,1 ⊂ 2

n−1/2 |aσ (n) | ,

clearly have the following strict

for p < 2 .

The dual of 2,1 is the Lorentz space 2,∞ (“weak-2 ”), consisting of all vanishing sequences b = (bn )n1 such that φ(b) = supn1 n1/2 b∗n < +∞. φ is not a norm on 2,∞ , but is equivalent to the norm: sup |A|−1/2 |bn | ; b2,∞ = A⊆N∗ , A finite

n∈A

indeed: φ(b)  b2,∞  2 φ(b) for any b ∈ 2,∞ . The duality between these two spaces is naturally provided  by a, b = +∞ n=1 an bn , for a ∈ 2,1 and b ∈ 2,∞ , because: +∞ n=1

|an bn | 

+∞

a∗n b∗n =

n=1

+∞

n−1/2 a∗n n1/2 b∗n  a2,1 b2,∞ .

n=1

Proof of the theorem 1) The closed graph theorem shows that the mapping u : C([0, 1]) → 1 f (n) n1 is continuous. defined by u( f ) = cn For its adjoint u∗ : ∞ → M([0, 1]), we have: u∗ (t1 , . . . , tN , 0, . . . , 0, . . .) =

N

cn tn ϕn ;

n=1

hence:

 N      c t ϕ n n n   u sup |tn | .  n=1

1

nN

We test this inequality with tn = εn (ω), where (εn )n1 is a Bernoulli sequence, and integrate with respect to ω, using the cotype 2 property of M([0, 1]); we obtain:

VII Comments

203

2 1/2     N  ε (ω) c ϕ u  E n n n  1

n=1

 A1

 N

1/2 |cn |2 ϕn 21

 A1 δ

n=1

 N

1/2 |cn |2

,

n=1

1/2  N 2  u/(A1 δ) for any N  1, which proves the and hence n=1 |cn | result. √ 2) Suppose that  f ∈ 2,1 for every f ∈ C([0, 1]), and take cn = 1/ n; the +∞ f (n)| < +∞. It follows sequence (cn )n1 is in 2,∞ , and hence n=1 |cn   2 < +∞, which is false; thus the result is proved by |c | from 1) that +∞ n n=1 contradiction. Remarks 1) The theorem obviously becomes false if we do not assume ϕn 1  δ. For example, if (ϕn )n1 is the sequence, extracted  from the Haar system, defined by ϕn (x) = 2n/2 1I(0,2−n−1 ) − 1I(2−n−1 ,2−n ) , we have ϕn 1 = 2−n/2 , f (n)|  2−n/2 f ∞ . and  f ∈ 1 for every f ∈ L∞ (0, 1), because | 2) The theorems of Paley and Carleman express that, for f ∈ C([0, 1]), we f 2 = f L2  f ∞ ). In can hardly say anything better than  f ∈ 2 (  f ∈ / p t) = e−t 1It>0 ), then, putting j = X1 + · · · + Xj ,  1/p we have: θ ∼ c +∞ j=1 σj / j , where the constant c depends only on p and on the distribution of the σj ’s (see also Pisier [1986 b]). This result is probably fairly old. It was first used in this context by Marcus and Pisier [1984], and they quoted R. Le Page for the scalar case. The books of Samorodnitzky–Taqqu and Zolotarev are also recent references concerning stable distributions. The article by Faraut and Harzallah [1974] contains a detailed presentation of kernels of positive and negative type, as well as numerous references and examples (see Exercise VIII.6). The interest in functions of negative type also comes from the following result of Schönberg: a metric space can be isometrically embedded in a Hilbert space if and only if the square of the distance is a kernel of negative type. Bretagnolle, Dacunha-Castelle and Krivine [1966] showed that, for 1  p  2, a Banach space is isometric to a subspace of Lp (μ) if and only if tp is a kernel of negative type, i.e. exp(−λtp ) is of positive type for any λ > 0. The technique of ultraproducts, used earlier by logicians, was introduced on this occasion in the study of Banach spaces (and further studied by Dacunha–Castelle and Krivine [1972]). See also Koldobsky [1992] and Koldobsky and Lonke [1999]. 2) The notion of Banach–Mazur distance already appears in the book by Banach, and it is studied in detail in the book by Tomczak–Jaegermann, with, notably, some examples of quasi-isometric spaces that are not isometric. We can show (Cambern [1968]) that d(c, c0 ) = 3, and thatthe compact sets K and L are homeomorphic if and only if d C(K), C(L) < 2 (Amir [1965] and Cambern [1967]). The important principle of local reflexivity is due to Lindenstrauss and Rosenthal [1969]. Many proofs have been given; we have followed that of Stegall (1980]. 3) The notions of type and cotype are due to Maurey [1972 b], [1972 c], [1973 a] and [1973 b] (see Maurey [2003 a]); a similar notion had been independently introduced by Hoffmann-Jørgensen [1973] and [1974], but it was the fundamental article by Maurey and Pisier [1976] (see also Maurey and Pisier [1973]) that truly marked their birth.

VIII Exercises

205

The notion of stable type p, underlying in Rosenthal [1973], was used by Maurey to establish his factorization theorems ([1973 b], [1973 c] and [1973 d], Maurey); see Chapter 4 of Volume 2 and Maurey [2003 a]. For a space, the equivalence between having a non-trivial type and not containing n1 ’s uniformly is due to Pisier [1973 b] and [1974 b]; these conditions are also equivalent (Giesy and James [1973]) to those for the space to be B-convex, a notion introduced by Beck [1962] for the validity of the strong law of large numbers in Banach spaces. For different variants of type and cotype (Fourier, Haar, Walsh), see Pietsch–Wenzel. The type and the cotype of Lorentz spaces were determined by Creekmore [1981]; those of the Schatten classes by Tomczak-Jaegermann [1974]; see also Cobos [1983]. For the extensions to the non-commutative spaces Lp (τ ), see Fack [1987]. The notion of K-convexity is due to Maurey and Pisier [1976]. Pisier later studied this in detail, providing an upper bound for the K-convexity constant of finite-dimensional spaces [1980], and showing the profound equivalence between being K-convex and not containing n1 ’s uniformly [1982]. 4) The theorem of factorization through a Hilbert space is due to Lindenstrauss and Pełczy´nski [1968]. Its use to show that spaces simultaneously of type 2 and cotype 2 are Hilbertian is due to Kwapie´n [1972]. The generalization of Theorem V.8 to operators was observed by Maurey (see Kwapie´n [1972]). 5) The theorem of Nazarov [1998] was extended by Lust-Piquard [1997] to the non-commutative case of operators on a Hilbert space:     If supi j |ai,j |2 and supj i |ai,j |2 are finite, we can find a matrix bi,j i,j , representing a continuous operator on 2 , such that |bi,j |  |ai,j | for every i, j.

VIII Exercises Exercise VIII.1 Let (X1 , X2 ) be a centered Gaussian vector ! with covariance   1 ρ 1−ρ ρ 1 , where |ρ| < 1. Show that E max(X1 , X2 ) = π . In particular, this expectation decreases with ρ; this is a special case of Slepian’s lemma in Chapter 1 (Volume 2).

206

5 Type and Cotype of Banach Spaces

Exercise VIII.2 Let H be a closed subspace of L2 (, P) composed of centered Gaussian variables (Gaussian space), and let A, B be two subspaces of H. Show that the following assertions are equivalent: (a) A and B are orthogonal; (b) The σ -algebras σ (A) and σ (B) generated respectively by the variables of A and of B are independent (in a Gaussian space, orthogonality is equivalent to independence). Exercise VIII.3 Let g be a standard Gaussian variable and ε a Bernoulli variable independent of g. Show that the vector (g, εg) is not Gaussian, even though each of its components is Gaussian. Exercise VIII.4

Let X be a real random variable, and  its characteristic 2(1 − Re (t) ) · Deduce that, if p > 2, function. Show that E(X 2 )  lim t2 t→0 p e−|t| is not a characteristic function. Exercise VIII.5 Let ψ : S × S → R such that ψ(x, x) = 0, ψ(x, y) = ψ(y, x)  and i,jn ci cj ψ(xi , xj )  0 for any real numbers c1 , . . . , cn with sum zero. Show that the inequality still holds if the ci are complex numbers with sum zero. Exercise VIII.6 Let ψ : S×S → R be a kernel of negative type, with positive values, and let α ∈ ]0, 1[. 1) Show that, for any u  0: uα =

α (1 − α)



+∞ 0

1 − e−tu dt . t1+α

ψα

2) Deduce that is again of negative type. 3) Let H be a Hilbert space, and p ∈ ]0, 2]. Set ψ(x, y) = x − yp . Show that ψ is of negative type. 4) Let X be a Banach space, and ψ(x, y) = x−y2 . Show that ψ is of negative type if and only if X is a Hilbert space. Exercise VIII.7 Let (pn )n1 , with p1 = 1, and (qn )n1 be two sequences of real numbers contained in[1, 3/2], dense in this interval, and disjoint. Let 3 3+∞ 5  +∞ 5 X= n=1 pn 2 and Y = n=1 qn 2 . Prove that d(X, Y) = 1, but that X and Y are not isometric, by showing that Y does not contain 51 isometrically. Exercise VIII.8 Use the local reflexivity theorem and the fact that the closed bounded subsets of X ∗∗ are weak∗ compact to show that there exists an

VIII Exercises

207

ultrapower X U of X containing X ∗∗ and a projection of norm 1 from X U onto X ∗∗ (follow the proof of Proposition III.6). Exercise VIII.9 Use the local reflexivity theorem to show that if the bidual X ∗∗ of X contains a non-zero element z such that, for some α > 0, we have x+z  α (x+z) for every x ∈ X, then X contains a subspace isomorphic to 1 . The converse is true (Godefroy [1989 a]; see also Deville–Godefroy– Zizler, Chapter III). Exercise VIII.10 We will use here the fact that any subspace of dimension n √ of a Banach space is n-complemented (see Chapter 6, Theorem IV.4). Let M be a closed subspace of codimension n of a Banach space X. 1) Show that there exist a basis (ϕ1 , . . . , ϕn ) of M ⊥ and x1∗∗ , . . . , xn∗∗ ∈ X ∗∗ n ∗∗ ∗ ⊥ such that Q(ϕ) = j=1 xj (ϕ) ϕj is a projection from X onto M with √ norm  n. 2) By using the local reflexivity theorem, show that, given α > 0, we can find x1 , . . . , xn ∈ X such that xj , ϕk = xj∗∗ , ϕk = δj,k , for 1  j, k  n, and such that, for every λ1 , . . . , λn ∈ R:       n  n   n   1  ∗∗  ∗∗        (1 + α) λ x λ x λ x j j  j j j j .    1+α j=1

j=1

j=1

n

3) Show that P0 (x) = j=1 x, ϕj xj defines a projection of X, with kernel M √ and with norm  (1 + α) n. 4) Show that, given ε > 0, we can choose α so that P = IdX −P0 is a √ projection from X onto M with norm  n + 1 + ε. Exercise VIII.11 Let X be a space of cotype q, and E a finite-dimensional subspace of X. Show that X/E is of cotype q. Exercise VIII.12 Let (rk )k1 be the sequence of Rademacher functions on n [0, 1]. For 1  k  n, define uk ∈ 2∞ (R) by:   2j − 1  uk = rk . 1j2n 2n+1 Show that, for every sequence (εk )1kn ∈ {−1, 1}n , there exists j0 (1   2j − 1  0 = εk for 1  k  n. Deduce that the j0  2n ) such that rk n+1 2 n 2 subspace of ∞ (R) generated by u1 , . . . , un is isometric to n1 (R), and then  n  1− 1 that Tp 2∞ (R)  n p . Show that if X ∗ contains n1 (R)’s uniformly, then X has no type p > 1.

208

5 Type and Cotype of Banach Spaces

 Exercise VIII.13 By considering the Riesz product nj=1 (1 + εj ), show that √ the K-convexity constant of n1 satisfies K(n1 )  δ log n, if n  2. Exercise VIII.14 Let A be a unitary C∗ -algebra, with unit I.  1) Let T1 , . . . , Tn ∈ A be self-adjoint elements of A with nj=1 Tj 2  1, and sequence. Show that the “imaginary” random let (ε1 , . . . , εn ) be a Bernoulli    √ Riesz product Rω = nj=1 I + i εj (ω) Tj satisfies Rω   e.  2) Let ϕ1 , . . . , ϕn ∈ A∗ ; set ω = nj=1 εj (ω) ϕj . Show that: E ω , Rω = i

n

ϕj , Tj .

j=1

3) Show that:  n

1/2 ϕj 

2

 n    n   = sup  ϕj (Tj ) ; Tj 2  1 .

j=1

j=1

j=1

4) Deduce that A∗ is of cotype 2 and that C2 (A∗ )  2

√

e.

Exercise VIII.15 Let (a1 , . . . , aN ) and (b1 , . . . , bN ) be two finite sequences of complex numbers. Let (a∗n )1nN and (b∗n )1nN be the non-increasing rearrangements of (|an |)1nN and (|bn |)1nN . Using an Abel summation by parts, show that:   N   N    a b a∗n b∗n n n  n=1

n=1

(we used this inequality for the duality of 2,1 and 2,∞ ). Exercise VIII.16 Let (T, μ) be a probability space and$(ϕn )n1 be an f (n) = T f ϕ n dμ. orthonormal sequence in L2 (T, μ). For f ∈ L2 (μ), set  1) Suppose that there exists C > 0 for which we have the following property: f (n)|  |an | for if a = (an )n1 ∈ 2 , there exists f ∈ L∞ (μ) such that | every n  1, and f ∞  C a2 . Then show ϕn 1  1/C := δ. In what follows, we study the converse; we thus assume ϕn 1  δ. First, the ϕn are also assumed to be real. n 2 2) Let a1 , . . . , an ∈ R be such that j=1 aj  1, and let φ : R → R be the C 2 function defined by φ(x) = x arctan x − 12 log(x2 + 1). For  ε = (ε1 , . . . , εn ) ∈ {−1, +1}n , we set fε = nj=1 εj aj ϕj . By maximizing

VIII Exercises

209

$ the functional ε → T φ( fε ) dμ, show that we can find α = (α1 , . . . , αn ) ∈ {−1, +1}n such that:   φ( fα ) dμ  φ( fα − 2 αj aj ϕj ) dμ T

T

for j = 1, . . . , n. In the following, we denote f = fα . 3) Let h = φ ( f ) ∈ L∞ (μ). Fix j ∈ {1, . . . , n}, and set g = f − 2 αj aj ϕj . Show that:  ϕj2 δ2  |aj | . dμ  |h( j)|  |aj | 2 2 3 T 1+f +g 4) Now let a = (an )n1 ∈ 2 (an ∈ R); show that there exists H ∈ L∞ (μ) ( j)|  |aj | for every j  1, with H∞  C a2 , where C such that |H depends only on δ. 5) Generalize 4) to complex an and ϕn . For the case T = [0, 1], show that, additionally, we can find H continuous.

6 p-Summing Operators. Applications

I Introduction The theory of operators on a Hilbert space H is well known to be rich and beautiful, and the class HS(H) = HS of Hilbert–Schmidt operators plays there a privileged role (kernel operators etc.). An equivalent definition of this class will be given, better suited for a generalization to non-Hilbertian spaces. First recall that, for an operator T : H → H, the expression T2HS =  2 i∈I Tei  does not depend on the choice of the orthonormal basis (ei )i∈I , and the Hilbert–Schmidt operators are those satisfying THS < +∞. Then: Proposition I.1 For every operator T ∈ L(H), the following assertions are equivalent: (1) T is Hilbert–Schmidt; (2) There exists a constant C > 0 such that, for any x1 , . . . , xN ∈ H: (∗)

 N n=1

1/2 Txn 2

 C sup

ξ 1

 N

1/2 | ξ , xn |2

.

n=1

Moreover, the best possible constant C in (∗) is THS . Proof (2) ⇒ (1) . Let (ei )i∈I be an orthonormal basis of H, and J a finite subset of I.  The relation (∗) and Bessel’s inequality imply i∈J Tei 2  C2 ; hence T2HS  C2 , when we take the upper bound over all finite subsets J of I. (1) ⇒ (2) . As every Hilbert–Schmidt operator is compact, T ∗ T is compact and self-adjoint; it thus possesses an orthonormal basis (ei )i∈I of eigenvectors: T ∗ Tei = si ei , with si  0. For x1 , . . . , xN ∈ H, we 210

II p-Summing Operators

211

 2 set σ 2 = supξ 1 N n=1 | ξ , xn | , and decompose each xn over  the basis (ei )i∈I : xn = i∈I an,i ei . It then ensues that: < N N N ; Txn 2 = T ∗ Txn , xn = an,i si ei , an,i ei n=1

n=1

=

N 

σ

2

|an,i |2 si



si = σ

i∈I 2

i∈I

 N  2 si |an,i | =

i∈I

n=1

i∈I

n=1



n=1

T2HS ,

i∈I

N  2 2 2 since N n=1 |an,i | = n=1 | ei , xn |  σ . The inequality (∗) is hence verified with C = THS , which completes the proof.

II p-Summing Operators II.1 Definition In view of Proposition I.1 in the Introduction, it is natural to extend Property (2) to operators T : X → Y between Banach spaces, and to replace the exponents 2 by p and q (1  p, q < +∞); we obtain the notion of a (q, p)-summing operator of X in Y: 1/q  1/p  N N q p Txn   C sup |ξ(xn )| , ξ ∈BX ∗

n=1

n=1

or, schematically: strong (Y) . T : weak p (X) −→ q

Here, we only treat the case p = q. Definition II.1 Let X and Y be two Banach spaces, and let T be an operator from X to Y. The operator T is said to be p-summing (1  p < +∞) if there exists a constant C > 0 such that, for any x1 , . . . , xN ∈ X:  1/p  1/p N N Txn p  C sup |ξ(xn )|p . n=1

ξ ∈BX ∗

n=1

The best possible constant C is denoted πp (T). The closed graph theorem shows that T : X → Y is p-summing if and only if T maps all weakly p -summable sequences into strongly p -summable sequences (i.e. their norms are p-summable):

212

6 p-Summing Operators. Applications  +∞

∀ ξ ∈ X∗

|ξ(xn )|p < +∞ ,

 ⇒

n=1

+∞

Txn p < +∞ .

n=1

It is easy to verify that πp is a norm, called the p-summing norm, on the space p (X, Y) of all p-summing operators from X into Y, making it a Banach space. Moreover, it is clear (take N = 1 in the definition) that the p-summing norm dominates the operator norm: πp (T)  T for every T ∈ p (X, Y). The following properties are immediate: Proposition II.2 Let T : X → Y be a p-summing operator. Then: 1) for all bounded operators U : W → X and V : Y → Z, the operator VTU : W → Z is p-summing, and πp (VTU)  V U πp (T) (ideal property); 2) if X1 ⊆ X and Y1 ⊆ Y are closed subspaces such that T(X1 ) ⊆ Y1 , then the T )  πp (T). restricted operator * T : X1 → Y1 is p-summing, with πp (* Of course, Proposition I.1 in the Introduction shows that the 2-summing operators on a Hilbert space are exactly the Hilbert–Schmidt operators. The fundamental example (“prototype”) of p-summing operator is given by: Proposition II.3 Let K be a compact space, and μ a regular probability measure on K. Then the natural injection jp : C(K) −→ Lp (K, μ) is p-summing, and πp ( jp ) = 1. Proof

Let f1 , . . . , fN ∈ C(K); we have: N

jp ( fn )pp

=

N 

|fn (t)| dμ(t) =

n=1 K N

n=1

 sup

p

K

|fn (t)|p = sup

t∈K n=1



sup

  N

N

 |fn (t)| dμ(t) p

n=1

| δt , fn |p

t∈K n=1 N

| ν, fn |p

ν∈BM(K) n=1

(this latter inequality is in fact an equality as BM(K) is the closed absolutely convex hull of the Dirac measures), which proves that jp is p-summing and that πp ( jp )  1. As jp  = 1, we conclude that πp ( jp ) = 1.

II p-Summing Operators

213

Remark It follows from 2) of Proposition II.2 that if Z is a closed subspace of C(K) and Zp denotes the closure of jp (Z) in Lp (K, μ), then the restriction * jp : Z → Zp is p-summing. In particular, if: A(D) = { f ∈ C(T) ;  f (n) = 0 , ∀ n < 0} and H 1 (D) = { f ∈ L1 (T) ;  f (n) = 0 , ∀ n < 0} , the natural injection * j1 : A(D) −→ H 1 (D) is 1-summing. This remark will be useful in the second proof we will give of Grothendieck’s theorem. Similarly: Corollary II.4 Let T : X → Y be an operator between the Banach spaces X and Y. If there exists a probability μ on the unit ball BX ∗ of X ∗ , equipped with the weak∗ topology, such that: Tx  C * xLp (μ) for every x ∈ X, where * x(x∗ ) = x∗ (x) for any x∗ ∈ BX ∗ , then T is p-summing, and πp (T)  C. Indeed, the mapping i : x ∈ X → i(x) = * x ∈ C(BX ∗ ) is an isometry; thus it suffices to use Proposition II.3 and 2) of Proposition II.2.

II.2 The Pietsch Factorization Theorem The following result is fundamental for p-summing operators: it means, via Corollary II.4, that the prototypical example given in Proposition II.3 is essentially the only one. Theorem II.5 (The Pietsch Factorization Theorem) Let T : X → Y be a p-summing operator (1  p < +∞). Then there exists a regular probability measure on the compact space K = (BX ∗ , w∗ ) obtained by equipping the unit ball of X ∗ with the weak∗ topology, such that:  1/p |ξ(x)|p dμ(ξ ) , ∀x ∈ X. Tx  πp (T) K

Remark This must be seen as a factorization theorem. Indeed, X is isometric to a subspace of C(K), by the map: i:

X x

−→ −→

C(K) * x

214

6 p-Summing Operators. Applications

where * x(ξ ) = ξ(x) for any ξ ∈ K = BX ∗ . If jp : C(K) −→ Lp (K, μ) is the natural injection, then the preceding inequality means that T defines a continuous operator from jp (iX) into Y, with norm  πp (T); it thus has a unique extension * T : Xp = jp (iX) → Y. Hence we have the following factorization diagram: X

T

/Y O * T

i

 i(X)

* jp

/ Xp

C(K)

Lp (K, μ)

Prior to the proof, let us first see the much nicer case p = 2. Recall (Chapter 5) that an operator T : X → Y can be factorized through a Hilbert space if there exists a Hilbert space H and operators A : X → H and B : H → Y such that T = BA. The norm γ2 (T) of the factorization through a Hilbert space is the lower bound of the products A B over all possible factorizations T = BA. Theorem II.6 Every 2-summing operator T : X → Y factorizes through a Hilbert space, and γ2 (T)  π2 (T). More precisely, with the notation of the preceding remark, we have: X

T

/Y O * T

i

 C(K)

j2

/ L2 (K, μ)

Proof Let P be the orthogonal projection of L2 (K, μ) onto X2 = j2 (iX); it suffices to take * T =* T ◦ P. Proof of the Pietsch factorization theorem This is a clever application of the Hahn–Banach separation theorem applied to the real space C(K) (the form given here is due to B. Maurey [1972 a]). Set:  C = φ : K −→ R ; ∃ x1 , . . . , xN ∈ X , φ(ξ ) = [πp (T)]p

N n=1

|ξ(xn )|p −

N n=1

 Txn p .

II p-Summing Operators

215

This is a convex cone of C(K) and supK φ  0 for every φ ∈ C, by definition of πp (T). It is thus disjoint from the open convex cone C− = {ψ ∈ C(K) ; supK ψ < 0} . The Hahn–Banach separation theorem guarantees the existence of a continuous linear functional on C(K), i.e. (by the Riesz representation theorem) of a nonzero measure ν ∈ M(K) that separates these two convex sets. However, as we are dealing with cones, ν is necessarily positive on one and negative on the other. By swapping ν and −ν if necessary, we thus have:  ν, ψ  0 , ∀ ψ ∈ C− ν, φ  0 ,

∀φ ∈ C.

The first inequality shows that ν is a positive measure, hence μ = ν/ν is a probability measure. Then, the second inequality, applied to φ(ξ ) = [πp (T)]p |ξ(x)|p − Txp , provides the conclusion. In the case where X is already a subspace of a space of continuous functions C(), we can reduce the enormous compact set (BX ∗ , w∗ ) to : Corollary II.7 If  is a compact space, and if X is a subspace of C(), then, for every p-summing operator T : X → Y, there exists a probability μ on  such that  1/p p |f (ω)| dμ(ω) Tf   πp (T) 

for any f ∈ X. Proof Indeed, in the preceding proof, in place of the unit ball (BX ∗ , w∗ ) as the compact space K, we can take any compact subset K of (BX ∗ , w∗ ) such that: x = sup |ξ(x)| ξ ∈K

for every x ∈ X. In particular, when X is a subspace of C(), we can take K = , by identifying the latter with the set of Dirac measures δω . Remark The dual X ∗ of X is sometimes difficult to identify; we can avoid referring to it in Definition II.1, thanks to the following identity, easy to establish (Exercise VII.1):   1/p   N  N  N p q   |ξ(xn )| = sup  an xn  ; |an |  1 , sup ξ ∈BX ∗

n=1

n=1

n=1

216

6 p-Summing Operators. Applications

where q is the conjugate exponent of p (the condition supnN |an |  1 for p = 1).

N

n=1 |an |

q 1

becomes

Corollary II.8 For 1  p < +∞, we have the following properties: 1) For 1  p  q < +∞, p (X, Y) ⊆ q (X, Y), and πq (T)  πp (T) for all T : X → Y. 2) Every p-summing operator is weakly compact. 3) Every p-summing operator is a Dunford–Pettis operator, i.e. it transforms weakly convergent sequences into sequences converging in norm. Hence, the most constraining condition is to be 1-summing. Proof 1) follows immediately from the factorization theorem and the xLq (μ) . inequality * xLp (μ)  * 2) follows, for p > 1, from the fact that T can be factorized through the reflexive space Xp ; for p = 1, it follows from 1) . 3) is a consequence of the factorization theorem and the Lebesgue dominated convergence theorem. Thanks to Khintchine’s inequalities, Proposition I.1 of the Introduction can be improved as follows: Theorem II.9 (The Pietsch–Pełczy´nski Theorem) operator T in a Hilbert space H is 1-summing, and: √ π1 (T)  2 THS .

Every Hilbert–Schmidt

Proof First, we assume that H is of finite dimension N. Let (en )N n=1 be an orthonormal basis of eigenvectors of T ∗ T: T ∗ Ten = sn en = λ2n en , with  2 2 λn  0. We have already seen that Tx2  N n=1 λn | x, en | . By Khintchine’s inequalities, this implies:  N  √  1   λn x, en rn (t) dt , Tx  2  0

where

(rn )N n=1

n=1

is the sequence of Rademacher functions. We set: N 1 ξt = λn rn (t)en , THS n=1

so that ξt  = 1 for any t ∈ [0, 1]; the inequality above can also be read as:  1  √ √ | ξt , x | dt = 2 THS | ξ , x | dμ(ξ ) , Tx  2 THS BH

0

where μ is the probability on BH defined, for f ∈ C(BH ), by:  1  f dμ = f (ξt ) dt . BH

0

III Grothendieck’s Theorem

217

Proposition II.3 then provides the result. In the general case, given x1 , . . . , xn ∈ H, let HN be the N-dimensional subspace generated by x1 , . . . , xn . We consider the restriction TN = T|HN of T to HN . As the range TN (HN ) is isometric to a subspace of HN , we can assume (by the ideal property 1) of Proposition II.2) that TN is a Hilbert–Schmidt operator of HN , with norm TN √HS  THS .√By the first part of the proof, it is 1-summing, and π1 (TN )  2 TN HS  2 THS . Consequently: n

Txk  =

k=1

n

TN xk  

√

2 THS sup

k=1

=

n

ξ ∈BHN k=1

| ξ , xk |

n √ 2 THS sup | ξ , xk | , ξ ∈BH k=1

which proves that T is 1-summing, and that π1 (T) 

√

2 THS .

III Grothendieck’s Theorem III.1 Grothendieck’s Inequality It had long been believed that everything was known about Hilbert spaces (but not about their operators. . . !). However, in the middle of the 1950s, Grothendieck [1956] found an apparently innocent property of the scalar product. It seemed innocent, but not for him, since he called it the “fundamental theorem of the metric theory of tensor products”. This property remained poorly understood for a long time, until Lindenstrauss and Pełczy´nski [1968] exposed its full power, in connection with the p-summing operators. Here is Grothendieck’s “innocent” property: Theorem III.1 There exists a smallest constant KG > 0, called the Grothendieck constant, so that, for every Hilbert space H of finite dimension1 and for every ε > 0, there exist functions fk , gk : S → R (S is the unit sphere of H), for k  1, such that: 1) x, y = 2)

+∞

+∞

fk (x) gk (y), for every x, y ∈ S;

k=1

fk ∞ gk ∞  KG (1 + ε).

k=1

1 We thank G. Pisier for pointing out this missing hypothesis in the French version.

218

6 p-Summing Operators. Applications

There are now several proofs of this result. The original proof of Grothendieck, clarified by Lindenstrauss and Pełczy´nski [1968], was based on an argument of mean over the n-dimensional Euclidean sphere equipped with its normalized rotation-invariant measure (see also Lindenstrauss– Tzafriri, Volume 1, page 68). The proof presented here is more probabilistic and is based on the following identity: Proposition III.2  Let (X, Y) be a centered Gaussian vector in R2 , with covariance matrix ρ1 ρ1 , where ρ = E(XY) = X, Y . Then:    π  E σ (X)σ (Y) , X, Y = sin 2 where σ is the sign function: σ (x) = 1 if x  0, and σ (x) = −1 if x < 0. Proof Let θ0 , −π/2  θ0  π/2 be such that ρ = sin θ0 . Note that, if g1 and g2 are two independent standard Gaussian variables, the centered Gaussian vectors (X, Y) and (g1 , g1 sin θ0 + g2 cos θ0 ) have the same distribution, as they have the same covariance. We thus obtain:     E σ (X) σ (Y) = E σ (g1 ) σ (g1 sin θ0 + g2 cos θ0 )  x2 +y2 1 = σ (x) σ (x sin θ0 + y cos θ0 ) e− 2 dxdy 2π R2  2π −θ0 +∞ 1 2 σ (cos θ ) σ (sin θ0 cos θ + cos θ0 sin θ ) e−r /2 rdr dθ = 2π −θ0 0  2π −θ0   1 = σ (cos θ ) σ sin(θ + θ0 ) dθ . 2π −θ0   We set ϕ(θ ) = σ (cos θ )σ sin(θ + θ0 ) , and examine the values of ϕ: θ

−θ0

π − θ0

π/2

3π/2

2π − θ0

σ (cos θ )

+1

−1

−1

+1

σ sin(θ + θ0 )

+1

+1

−1

−1

ϕ(θ )

+1

−1

+1

−1

Hence we have:  π  π  π    2 1  π +θ0 − −θ0 + +θ0 − −θ0 = θ0 , E σ (X)σ (Y) = 2π 2 2 2 2 π which is the claimed result. We now proceed to the proof of Theorem III.1:

III Grothendieck’s Theorem

219

Proof of Theorem III.1 Let A be the algebra of functions f : S×S → R that can be written: +∞ uk (x)vk (y) , f (x, y) = k=1

where uk and vk are bounded on S, and satisfy: +∞

uk ∞ vk ∞ < +∞ .

k=1

A is a unitary Banach algebra for the norm   +∞ +∞ f ∧ = inf uk ∞ vk ∞ ; f (x, y) = uk (x)vk (y) , k=1

k=1

with 1I∧ = 1 and fg∧  f ∧ g∧ , as can be immediately verified. A is in fact the projective tensor product of ∞ (S) by itself, but this remark is not used here. The Hilbert space H can now be represented as a Gaussian space, i.e. as a space consisting of centered Gaussian random variables (for this, it suffices to associate a family of standard independent Gaussian variables to an orthonormal basis of H). For X, Y ∈ S, and ω ∈  fixed, we set:     ϕω (X, Y) = σ X(ω) σ Y(ω) .     Clearly ϕω ∈ A and ϕω ∧  σ X(ω) ∞ σ Y(ω) ∞ = 1; hence, if we set:  ϕ(X, Y) = Eω ϕω (X, Y) =

ϕω (X, Y) dP(ω) , 

we obtain ϕ∧  Eω ϕω ∧  1. Now note that f ∞  f ∧ for every f ∈ A, so convergence in norm in A implies pointwise convergence; thus the series +∞  π 2k+1 ϕ 2k+1 , (−1)k 2 (2k + 1)! k=0

which converges in A, has sin ( π2 ϕ) for its sum. As sin ( π2 ϕ) = X, Y , by Proposition III.2, we deduce: X, Y ∈ A, and  X, Y ∧ 

+∞ (π/2)2k+1 k=0

(2k + 1)!

ϕ2k+1  ∧

We thus obtain Theorem III.1, with KG  sh

+∞ (π/2)2k+1 k=0

π · 2

(2k + 1)!

= sh

π · 2

220

6 p-Summing Operators. Applications

Remark The proof of Theorem III.1 can clearly be adapted to complex Hilbert spaces (with complex Gaussian variables, or more brutally by separating the real and imaginary parts), but the constant KG changes. Hence there is a real Grothendieck constant KGR and a complex Grothendieck constant KGC . Note that these are effectively different; indeed, Grothendieck showed that KGR  π/2, while Pisier [1978 c] showed that KGC < e1−γ = 1.527 · · · < π/2. From now on, we ignore the distinction, and simply write KG .   Theorem III.3 (Grothendieck’s Inequality) Let ai,j ip,jq be a rectangular matrix with scalar coefficients. Assume that:       a x y i,j i j   sup |xi | sup |yj |  ip

ip,jq

jq

for any scalars x1 , . . . , xp , y1 , . . . , yq . Then, for every Hilbert space H:       a x , y

i,j i j   KG sup xi  sup yj   ip

ip,jq

jq

for any x1 , . . . , xp , y1 , . . . , yq ∈ H. Proof Let x1 , . . . , xp , y1 , . . . , yq ∈ H, all assumed non-zero (w.l.o.g). By considering the subspace generated by these vectors, we can assume that H is of finite dimension. By Theorem III.1, we can write: xi , yj = xi  yj 

   +∞  yj xi gk , fk xi  yj  k=1

with

+∞ k=1

fk ∞ gk ∞  KG (1 + ε); hence:

       +∞    yj  xi    gk ai,j xi , yj  =  ai,j xi  yj  fk  x  y   i,j

i

i,j k=1



+∞  k=1

 sup xi  yj  fk ∞ gk ∞ i,j

 sup xi  sup yj  KG (1 + ε) , i

j

which gives the result, as ε > 0 is arbitrary. We now come to the fundamental result of this section.

j

III Grothendieck’s Theorem

221

III.2 Grothendieck’s Theorem This theorem is actually the reformulation of Grothendieck’s inequality in terms of p-summing operators, due to Lindenstrauss and Pełczy´nski. Theorem III.4 (Grothendieck’s Theorem) 1-summing, and:

Every operator u : 1 → 2 is

π1 (u)  KG u . p Proof Let x1 , . . . , xp ∈ 1 be such that i=1 |ξ(xi )|  ξ ∞ for every ξ ∈ ∞ . After a perturbation if necessary, the xi ’s can be assumed to have finite supports: q ai,j ej , xi = j=1

  where (ej )j1 is the canonical basis of 1 . The matrix ai,j ip,jq satisfies the hypothesis of Grothendieck’s inequality; indeed, with u1 , . . . , up and v1 , . . . , vq ∈ R, denoting u = (u1 , . . . , up ) and v = (v1 , . . . , vq ), we have:    q  p p           a u v |u | a v u∞ |v(xi )|  u∞ v∞ , i,j i j  i  i,j j   i,j

i=1

j=1

i=1

according to the conditions on the xi ’s. Then, by Grothendieck’s inequality:      ai,j yi , u(ej )   KG sup yi 2 sup u(ej )2  KG u sup yi 2  i

i,j

j

i

for any y1 , . . . , yp ∈ 2 . Selecting these last vectors in the unit ball of 2 , and norming u(x1 ), . . . , u(xp ), we thus obtain: p i=1

u(xi )2 =

p yi , u(xi ) = ai,j yi , u(ej )  KG u , i=1

i,j

which proves that u is 1-summing and that π1 (u)  KG u. Remark In Section V of Chapter 7, a second proof of this theorem will be presented, due to Pełczy´nski and Wojtaszczyk (see Pełczynski, ´ pages 19–20, or Wojtaszczyk, pages 201–203 and 230). Instead of Grothendieck’s inequality, it uses the existence of a particular 1-summing map, following from a theorem in Harmonic Analysis due to Paley. We can give a more general formulation: Theorem III.5 (Grothendieck’s Theorem) For any space L1 (μ) = L1 (S, S, μ), every operator u : L1 → H into a Hilbert space H is 1-summing, and π1 (u)  KG u.

222

6 p-Summing Operators. Applications

This follows from a local property of L1 (μ), and more generally of the spaces Lp (μ), expressed by saying that Lp (μ)-spaces are Lp -spaces. Proposition III.6 Let (S, T , μ) be a measure space, and 1  p  ∞. For every finite-dimensional subspace E of Lp (μ) and for any ε > 0, there exists a subspace F of Lp (μ), of finite dimension M, containing E, such that d(F, M p ) 1 + ε. Proof Let {x1 , . . . , xn } be a basis of E such that xk p = 1 for 1  k  n. We can find simple functions ϕ1 , . . . , ϕn ∈ Lp (μ) such that xk − ϕk p  α for any k, with α > 0, depending on ε and n, to be determined later. Next, there exists a measurable partition {S1 , . . . , SM } of S such that each function ϕ1 , . . . , ϕn is constant on each Sj , for 1  j  M. Let F1 be the subspace of Lp (μ) consisting of all functions ϕ ∈ Lp (μ), constant on each Sj . This space F1 is isometric to M p .  Now we move F1 in Lp (μ) so that it contains E. If x = nk=1 ak xk ∈ E, we have:  n   n  n         a ϕ − x = a (ϕ − x )  |ak |. α  Cn xp α k k k k k     k=1

p

p

k=1

k=1

(since all spaces of dimension n are isomorphic); hence:     n  a ϕ (1 − Cn α)xp   k k   (1 + Cn α)xp .  k=1

p

Thus, with α < 1/Cn , the spaces E and E1 = [ϕ1 , . . . , ϕn ] ⊆ F1 are isomorphic. In particular, {ϕ1 , . . . , ϕn } is a basis of E1 . Moreover, as:  n  n  Cn   |al |  |ak |  Cn xp  ak ϕk    , 1 − Cn α p k=1

k=1

F1∗

there exist linear functionals φ1 , . . . , φn ∈ such that φl (ϕk ) = δl,k and Cn p φl   · The mapping T : F1 → L (μ) defined by: 1 − Cn α Ty = y +

n

φk (y) (xk − ϕk )

k=1

then satisfies: 1) Tϕk = xk for 1  k  n, hence F = T(F1 ) ⊇ E; Cn α yp , hence: 2) Ty − yp  n 1−C nα     n Cn α n Cn α yp  Typ  1 + yp , 1− 1 − Cn α 1 − Cn α

III Grothendieck’s Theorem

223

and n Cn α 1 + (n − 1) Cn α 1 − Cn α d(F, F1 )   1 + ε, = n Cn α 1 − (n + 1) Cn α 1− 1 − Cn α provided that α was chosen small enough. 1+

Remark In particular, this holds for the spaces p , 1  p  ∞. Similarly, c0 is a L∞ -space. Using a partition of unity, we could also show that the spaces C(K) are L∞ -spaces. Corollary III.7 Let Y be a Banach space and 1  p  ∞. Assume that there exists a constant K > 0 such that, for any M  1, every operator u : M p →Y satisfies πq (u)  K u. Then any operator U : Lp (μ) → Y is q-summing, and πq (U)  K U. Proof Let x1 , . . . , xn ∈ Lp (μ), and consider the subspace E generated by these vectors. Proposition III.6 tells us that there exists a subspace F of Lp (μ), M containing E, such that d(F, M p )  1 + ε. Let j : p → F be an isomorphism such that j = 1 and j−1   1+ε. Then U ◦j : M p → Y satisfies πq (U ◦j)  K U ◦ j  K U; hence: n k=1

Uxk q =

n

  (U ◦ j) j−1 (xk ) q

k=1

 K U sup q

q

= K U sup q

q

 n k=1  n

∗

 (xk ) |q ; y∗ ∈ B(Mp )∗

j

−1

∗−1

∗

|y |j



k=1

 K q (1 + ε)q Uq sup

 

∗

(y )(xk )| ; y ∈ B(Mp )∗

 n

q

|z∗ (xk )|q ; z∗ ∈ BF∗



k=1

= K (1 + ε) U sup q

q

q

 n

∗

∗



|z (xk )| ; z ∈ B(Lp (μ))∗ q

k=1

by the Hahn–Banach theorem, which completes the proof, since ε > 0 was arbitrary. Remark In this corollary, the spaces Lp (μ) can of course be replaced by any Lp -space. In particular, for p = ∞, by C(K) or by c0 . Proof of Theorem III.5 It suffices to note that Theorem III.4 holds if we replace 2 by an arbitrary Hilbert space H, as any operator u : 1 → H takes

224

6 p-Summing Operators. Applications

its values in a separable Hilbert subspace, hence isometric to 2 . Next, it still M holds if we replace 1 by M 1 because 1 is 1-complemented in 1 and hence u : 1 → 2 with any operator u : M 1 → 2 can be extended to an operator * the same norm. Finally, it remains to use 2) of Proposition II.2, and then Corollary III.7.

III.3 The Dual Form of Grothendieck’s Theorem Another version of Grothendieck’s theorem, known as the dual form, is seen here. In truth, while duality appears naturally within the proof, it is not apparent in the statement of the theorem. Theorem III.8 (Dual Form of Grothendieck’s Theorem) Let (S, S, μ) and (T, T , ν) be two measure spaces. Then every operator u : L∞ (ν) → L1 (μ) is 2-summing, with π2 (u)  KG u. Remark According to the Remark following the proof of Corollary III.7, this result holds more generally for any operator u : C(K) → L1 (μ), and also for any operator u : c0 → L1 (μ). A Banach space Y is said to satisfy Grothendieck’s theorem, or to be a GT space, if every operator u : Y → 2 is 1-summing. We have just seen that every space L1 (μ), or more generally every L1 -space, is a GT space. Then Theorem III.8 takes on the following more general form: a Banach space X is a GT space if and only if every operator v : X ∗ → L1 is 2-summing (see Pisier 1, Proposition 6.2). A lemma is required. N Lemma III.9 Let u : M ∞ → 1 . Then: Q

π2 (u) = sup{π2 (uv) ; v : 2 −→ M ∞ , v  1 , Q  1} . Proof Denote the right-hand side by s. Obviously, by the ideal property, we have π2 (u)  s. Now let x1 , . . . , xQ ∈ M ∞ , (e1 , . . . , eQ ) be the canonical basis Q Q M of 2 , and let v : 2 → ∞ be defined by v(eq ) = xq , 1  q  Q. Of course: π2 (uv)  s v. Now, by definition of π2 (uv), we have:  Q q=1

1/2 uv(eq )21

 π2 (uv) ,

III Grothendieck’s Theorem

since



1/2 Q ∗ (e )|2 |a q q=1

225

 Q ∗ = a∗ 2 for any a∗ ∈ 2 . Hence:

 Q

1/2 u(xq )21

 s v ,

q=1

which shows that π2 (u)  s, because:    Q   Q 2  v = sup  αq xq  ; |α |  1 q  q=1

= sup

 Q

∞

q=1

1/2 |ξ(xq )|

2

 ; ξ ∈ B(M∞ )∗ = BM . 1

q=1

Proof of Theorem III.8 To avoid problems of duality with non-reflexive spaces, we reduce to the finite-dimensional case, by noting that it suffices N to show π2 (u)  KG u for any operator u : M ∞ → 1 . Indeed, first, by Corollary III.7, any operator U : L∞ (ν) → N 1 will then be 2-summing and π2 (U)  KG U. Next, if V : L∞ (ν) → L1 (μ) and if x1 , . . . , xn ∈ L∞ (μ), by Proposition III.6, for any ε > 0 there exists a subspace F of L1 (μ) with finite dimension N, containing Vx1 , . . . , Vxn and such that d(F, N 1 )  1 + ε. is an isomorphism such Then, as in the proof of Corollary III.7, if j : F → N 1 N −1 ∞ that j = 1 and j   1 + ε, the use of U = j ◦ V : L (ν) → 1 leads to π2 (V)  KG (1 + ε). We hence obtain π2 (V)  KG , since ε > 0 is arbitrary. N Hence let u : M ∞ → 1 . Lemma III.9 will be brought into play. Consider Q an arbitrary operator v : 2 → M ∞ . By Grothendieck’s theorem, and 2) of Q Proposition II.2, the adjoint operator v∗ : M 1 → 2 is 1-summing; hence the composition: M (uv)∗ = v∗ u∗ : N ∞ −→ 1 −→ 2

Q

is also 1-summing. The Pietsch factorization theorem thus provides a probability measure σ on {1, . . . , N} (indeed N ∞ can be seen as the space of continuous functions on the compact set {1, . . . , N}) with the factorization: N ∞

(uv)∗

/ Q = 2 CC | CC || | CC || w j CC || ! N 1 (σ )

226

6 p-Summing Operators. Applications

where:   w  π1 (uv)∗  u∗  π1 (v∗ )  KG u∗  v∗  = KG u v , j being the canonical injection (recall that π1 ( j) = 1). By duality, we obtain the diagram: / N 2 C = 1 CC { { CC { { C {{ ∗ w∗ CC {{ j ! N ∞ (σ ) uv

Q

We are going to verify that j∗ is 1-summing and π1 ( j∗ )  1, which will complete the proof of the theorem, since then: π2 (uv)  w∗  π2 ( j∗ )  w π1 ( j∗ )  KG u v , and hence π2 (u)  KG u, by Lemma III.9. N First, for every y ∈ N ∞ (σ ) and every x ∈ ∞ , we have: j∗ y, x (N ,N∞ ) = y, jx (N∞ (σ ),N (σ )) = 1

N

1

σn xn yn ;

n=1

hence j∗ y = (σn yn )1nN . Now let y(1) , . . . , y(K) ∈ N ∞ (σ ); we have: K

j∗ (y(k) )N =

K N

1

k=1

|σn y(k) n |.

k=1 n=1

N When σn = 0, we ∗ define ξn (y) = yn for y = (yj )jN ∈ ∞ (σ ); then ξn ∈ N N 1 (σ ) = ∞ (σ ) and ξn N (σ ) = 1; hence: 1

K

∗

j (y )N = (k)

N

1

k=1



n=1 N n=1

σn

 K 

σn

 |ξn (y )| (k)

k=1

sup

K

ξ ∈BN (σ ) k=1 1

 |ξ(y )| = sup (k)

K

ξ ∈BN (σ ) k=1 1

|ξ(y(k) )|,

 ∗ as N n=1 σn = 1, σ being a probability. This means that j is 1-summing and ∗ π1 ( j )  1, as announced.

IV Some Applications of p-Summing Operators

227

IV Some Applications of p-Summing Operators IV.1 The Dvoretzky–Rogers Theorem A well-known theorem of Riemann states that in a finite-dimensional space, unconditional convergence coincides with absolute convergence. On the contrary, in 1950, Dvoretzky and Rogers showed the existence in every infinitedimensional space of summable, but not absolutely summable, sequences. In Chapter 1 of Volume 2, a more precise version of this result will be seen; however, the theory of p-summing operators provides a very simple proof for the following qualitative form: Theorem IV.1 (The Dvoretzky–Rogers Theorem) Every infinite-dimensional Banach space contains unconditionally convergent series that are not absolutely convergent. Proof Let X be a Banach space in which every unconditionally convergent series is absolutely convergent. Then X does not contain c0 (see the Example following Proposition II.2 in Chapter 3). Then, by the Bessaga–Pełczy´nski theorem (more precisely, Theorem IV.3 of Chapter 3), every w.u.C. series is unconditionally convergent, and hence absolutely convergent, thanks to our hypothesis. This means that the identity operator IdX of X is 1-summing. By Corollary II.8, IdX is thus weakly compact, and hence X is reflexive. But Corollary II.8, again, also tells that IdX is a Dunford–Pettis operator, i.e. it transforms the weakly convergent sequences into strongly convergent sequences. Since, clearly, any Dunford–Pettis operator defined on a reflexive space is compact, IdX is thus compact, which means that X is finite-dimensional. We now examine a local form of the preceding result.

IV.2 2-Summing Norm of the Identity in Finite-Dimensional Spaces Theorem IV.2 For every Banach space E of finite dimension n, the identity operator IdE satisfies: √ π2 (IdE ) = n . The upper bound of π2 (IdE ) is based on an inequality of independent interest. Lemma IV.3 Let E be a subspace of dimension n of a space L2 (μ), and let ( fl )l1 be a bounded sequence of elements of E. Then:     √  sup |fl |   n sup fl 2 .   l1

2

l1

228

6 p-Summing Operators. Applications

Proof Let (u1 , . . . , un ) be an orthonormal basis of E, and consider the  n 2 1/2 . Then, for each l  1, |u | associated square function S = j=1 j  fl = nj=1 fl , uj uj , so: 1/2  1/2  n n |fl |  | fl , uj |2 |uj |2 = fl 2 S ; j=1

j=1

hence, by taking the upper bounds and then integrating:      sup |fl |   S2 sup fl 2 ,   l1 l1 2 √ which provides the result, since S2 = n. Proof of Theorem IV.2 Let x1 , . . . , xN ∈ E. By adding null-vectors if necessary, we may assume N  n. Modulo the choice of a basis, E is then a subspace of L2 (μ) = N 2 , where μ is the counting measure on {1, . . . , N}. For ξ running through a countable dense subset of the unit ball of the dual E∗ of E, let fξ (k) = ξ(xk ), 1  k  N. Since supξ ∈BE∗ |fξ (k)| = xk  and   N 2 1/2 , an application of the lemma gives: fξ 2 = k=1 |ξ(xk )|  1/2  1/2 N N √ 2 2 xk   n sup |ξ(xk )| , k=1

√

ξ 1

k=1

and consequently π2 (IdE )  n. The reverse inequality would be easy if we used the notion of trace and the nuclear norm ν, as well as the inequality ν(uv)  π2 (u) π2 (v); we would then have:  2 n = tr(IdE )  ν(IdE ) = ν(IdE IdE )  π2 (IdE ) , √ thus π2 (IdE )  n. However, we will give a proof which does not use these notions, based on the Pietsch factorization theorem. There exists a factorization through a Hilbert space H: E? ?? ?? U ?? 

IdE

H

/E ?     V

with π2 (U)  1 and V  π2 (IdE ). As VU = IdE , U is injective and we can assume dim H = dim E = n. Then V = U −1 , and IdH = UV. We take the 2-summing norms, and Proposition I.1 of the Introduction gives: √ n =  IdH HS = π2 (IdH )  π2 (U) V  V  π2 (IdE ) , which completes the proof of Theorem IV.2.

IV Some Applications of p-Summing Operators

229

Here is an important application of this theorem: Theorem IV.4 results:

Every Banach space E of dimension n satisfies the following

√ 1) (John’s Theorem) d(E, n2 )  n; 2) (The Kadeˇc–Snobar Theorem) For any Banach space X containing E, there √ exists a projection P : X → E with norm  n. Proof 1) Since IdE is 2-summing, it can be factorized as: T T −1 E −−→ n2 −−−−→ E , √ √ with T T −1   π2 (IdE ) = n; thus d(E, n2 )  T T −1   n. 2) Let K be the unit ball BE∗ , and μ a Pietsch measure for IdE . Hence IdE can be factorized as: j j∞ v i E −→ C(K) −−−→ L∞ (μ) −→ L2 (μ) −−→ E, where i, j∞ and j are the natural injections and v  π2 (IdE ) = particular:

√

n. In

T S IdE : E −−→ L∞ (μ) −−→ E , √ with T  1, and S  n. Now we have the following lemma: Lemma IV.5 For every measure μ, the space L∞ (μ) has the extension property: for any Banach space X, every operator T : Y → L∞ (μ) from a subspace Y of X can be extended to an operator * T : X → L∞ (μ) such that * T  = T. Using this lemma, we can form P = S* T : X → E, which is an extension T of IdE , in other words a projection from X onto E, such that P  S * √  n. We will see in the exercises that this bound is almost optimal. Proof of Lemma IV.5 It suffices to repeat the proof of the Hahn–Banach theorem word for word. In the real case, we replace the order relation in R by that of L∞ (μ) (note that L∞ (μ) has the 2-ball property, namely: every family of closed balls of this space that intersect pairwise has a non-empty intersection). In the complex case we take * T x = T1 (x) − iT1 (ix), where T1 is

230

6 p-Summing Operators. Applications

the extension to the underlying real spaces; indeed, for every x ∈ X of norm 1, and (almost) all s ∈ S, we have:     T x)(s) = * T eiθx,s x (s) = T1 eiθx,s x (s) |(* T x)(s)| = eiθx,s (*  T1  = T .

IV.3 Unconditional Bases of 1 and c0 We use Grothendieck’s theorem to prove the following result: Theorem IV.6 Every normalized unconditional basis (un )n1 of 1 (respectively c0 ) is equivalent to the canonical basis of 1 (respectively c0 ). In fact, as may be easily seen using the parallelogram identity, this is also true for 2 . Actually, Lindenstrauss and Zippin [1969] showed that this property of uniqueness (up to equivalence) of unconditional bases characterizes these three spaces isomorphically. Proof We first consider the case of 1 , and call C the unconditional constant  of (un )n1 ; with x = +∞ n=1 an un ∈ 1 , then:  N     sup  θ a u n n n   C x1 , 

θn =±1

n=1

for any N  1. If ε1 , . . . , εN are independent Bernoulli variables, an integration and the cotype-2 property of 1 provide:     N  εn an un  C x1  E   1 √ 2 It follows that:

n=1  N

1

1/2 an un 21

n=1

 +∞

1/2 |an |2

C

1/2  N 1 2 =√ |an | . 2 n=1 √ 2 x1 ,

n=1

and hence an operator T : 1 → 2 can be defined by:   T(x) = (an )n1 = u∗n (x) n1 . By Grothendieck’s theorem, this operator is 1-summing, and thus, for any scalars a1 , . . . , aN , we have:

V Sidon Sets N n=1

|an | =

N

T(an un )1  π1 (T)

n=1

 C KG

√

 C KG 2

231

sup

N

ξ ∞ =1 n=1

|ξ(an un )|

   N   2 sup  θ a u n n n 

√

|θn |1

n=1

2 x1 .

+∞ Obviously x1  n=1 |an |, as (un )n1 is normalized, so (un )n1 is √ 2 (C KG 2)-equivalent to the canonical basis of 1 . Before considering the case of c0 , note that the condition “(un )n1 is normalized” can obviously be replaced by: 0 < inf un 1  sup un 1 < +∞. n1

n1

Now let (vn )n1 be a normalized unconditional basis of c0 . As trivially c0 does not contain 1 , James’ theorem (Chapter 3) ensures that this basis is shrinking; hence the coordinate linear functionals form a basis (v∗n )n1 of 1 ; this basis is itself unconditional, and it is quasi-normalized: 1  infn1 v∗n 1  supn1 v∗n 1 < +∞ (this upper bound depends on the basis constant of the basis (vn )n1 ). The first part of the proof tells us that (v∗n )n1 is equivalent to the canonical basis of 1 and consequently (vn )n1 is equivalent to that of c0 .

V Sidon Sets The notion of Sidon sets provides a perfect illustration of the results of this text, not only in this chapter, but also in later chapters. For an overview of Sidon sets up to 1983, we refer to M. Déchamps [1984]. Older references are Lopez– Ross, and Lindahl–Poulsen. Edwards, Graham–McGehee or Rudin 1 are more general references. See also Kashin–Saakyan.

V.1 Definitions Let G be a metrizable compact$ Abelian group, and  =  G its countable dual group. We denote by  μ(γ ) = G γ (−x) dμ(x) the Fourier transform at γ ∈  of μ ∈ M(G). Recall that via the identification of f ∈ L1 (G) with the measure f .m, where m is the Haar measure of G, L1 (G) is isometrically embedded in M(G). For these notions, refer to the Annex.

232

6 p-Summing Operators. Applications

When μ ∈ M(G), the spectrum of μ is defined as the set: sp (μ) = {γ ∈  ;  μ(γ ) = 0} . If sp (μ) ⊆ ⊆ ,  μ is also said to be carried on . For ⊆ , if X(G) is one of the spaces C(G), Lp (G) for 1  p  ∞, or M(G), we set: X (G) = { f ∈ X(G) ; sp (f ) ⊆ } = { f ∈ X(G) ;  f (γ ) = 0 , ∀ γ ∈ / } . Similarly, we denote by P the set of trigonometric polynomials with spectrum in . Definition V.1 A subset ⊆  is said to be a Sidon set if every function  f (γ )| = f ∈ C (G) has an absolutely convergent Fourier series, i.e. γ ∈ |   γ ∈ |f (γ )| < +∞. This is indeed a series because  is countable, and then there is no ambiguity  f (γ ) γ since the convergence is unconditional about the convergence of γ ∈  in C(G). If is a Sidon set, the Banach–Steinhaus theorem guarantees the existence of a constant C > 0 such that: | f (γ )|  C f ∞ γ ∈

for every f ∈ C (G). The smallest constant satisfying this inequality is called the Sidon constant of and is denoted S( ). Conversely, if the preceding inequality holds for every f ∈ C (G), clearly is a Sidon set; in fact, it suffices for the inequality to hold for every trigonometric polynomial p ∈ P : indeed, if (Kn )n1 is an approximate identity, then, for every f ∈ C (G), the trigonometric polynomial f ∗ Kn has its spectrum in , and converges to f in C(G). Remark  In other  words, is a Sidon set whenever the Fourier transform F : f →  f (γ ) γ ∈ realizes an isomorphism between C (G) and 1 ( ). It is remarkable that any isomorphism between C (G) and 1 ( ) ensures the “Sidon-ness” of (Theorem V.15), as was shown by Varopoulos [1976], answering a question of Pisier. His proof is based on Grothendieck’s theorem. More generally, is a Sidon set as soon as C (G) is of cotype 2 (see Theorem V.24 later in this chapter), as was shown independently by Pisier [1978 b] and Kwapie´n and Pełczy´nski [1980]. In fact, it is even sufficient that C (G) has a finite cotype (Bourgain and Milman [1985]). We will see this in Chapter 6 of Volume 2.

V Sidon Sets

233

V.2 Examples The following example turns out to be fundamental (see Chapter 4 of Volume 2, Subsection III.2). Proposition V.2 The set R = {εk ; k  1} of Rademacher variables is a Sidon set, with constant S(R) = π/2, in the dual of the Cantor group  = ∗ {−1, +1}N . Proof A separation of real and imaginary parts leads easily to S(R)  2. To obtain π/2, take a1 , . . . , aN ∈ C, and write an = ρn eiθn ; we will show that:   N  N  an εn (ω) = sup ρn | cos(θ + θn )| . (∗) sup  ω∈ n=1

θ∈R n=1

With this latter sum denoted f (θ ), it ensues that:  N   2π  2π N N   2 dθ dt   = = an εn (ω)  f (θ ) ρn | cos t| ρn , sup  2π 2π π ω∈ 0 0 n=1

n=1

n=1

which proves that S(R)  π/2. As for showing (∗), on the one hand, for every ω ∈ , there exists θ0 ∈ R such that:  N   N  N   iθ0   = a ε (ω) a ε (ω) e = ρn ei(θ0 +θn ) εn (ω) n n n n   n=1

=

n=1 N

n=1

ρn cos(θ0 + θn )εn (ω)  f (θ0 )  sup f (θ ) . θ∈R

n=1

On the other hand, for every θ ∈ R, there exists ω0 ∈  such that εn (ω0 ) = sgn(cos(θ + θn )) for any n = 1, . . . , N; hence:   N N iθ f (θ ) = ρn cos(θ + θn ) εn (ω0 ) = Re an e εn (ω0 ) n=1

n=1

     N    N iθ    an e εn (ω0 )  sup  an εn (ω) ,  ω∈ n=1

n=1

which proves (∗). To prove that S(R) = π/2, select δ > 0 such that:   N  N   sup  an εn (ω)  δ |an | ω∈ n=1

n=1

for any a1 , . . . , aN ∈ C. We test this inequality with the normalized N-th roots of unity:

234

6 p-Summing Operators. Applications

zn =

2iπ n 1 , 1nN; exp N N

we obtain, via (∗):  N   2π n  1  sup  cos θ + N   δ . θ∈R N 

n=1

However, the Riemann sum on the left-hand side of this inequality tends to  2π 1 | cos t| dt 2π 0 as N tends to infinity, and this uniformly with respect to θ ; we thus obtain:  2π 2 1 | cos t| dt = · δ 2π 0 π It is often more convenient to use the following dual characterization of Sidon sets: Proposition V.3 The set is a Sidon set, of constant S( )  C, if and only if, for every bounded sequence a = (aγ )γ ∈ , there exists a measure μ ∈ M(G) such that μ  C a∞ and  μ(γ ) = aγ ,

∀γ ∈

(μ is said to interpolate a). Proof

First, if such a measure μ exists, then, for f ∈ P :           f (γ )aγ   sup  f (γ ) μ(γ ) | f (γ )| = sup  γ ∈

a∞ 1 γ ∈

μC γ ∈

       C sup  f (γ ) μ(γ )γ (0) μ1 γ ∈

= C sup |( f ∗ μ)(0)|  C f ∞ . μ1

Conversely, as the Fourier transform F : C → 1 ( ) is an isomorphism, its adjoint F ∗ : ∞ ( ) → C ∗ = M/M(− ) is also an isomorphism; hence, for every a ∈ ∞ ( ), there exists ξ ∈ C ∗ such that a = ( ξ , γ )γ ∈ and ξ   S( ) a∞ . By the Hahn–Banach theorem, there exists ν ∈ C ∗ = M(G) extending ξ and such that ν = ξ   S( ) a∞ . Then, if μ ∈ M(G) is defined by μ(A) = ν(−A) for every Borel set A, we indeed obtain, for any γ ∈ :  μ(γ ) = μ, γ = ν, γ = ξ , γ = aγ .

V Sidon Sets

235

The following arithmetical example, due to Sidon [1927], can then be deduced: Proposition V.4 The set = {3k ; k  1} is a Sidon set in Z. More generally, every Hadamard set is a Sidon set in Z. A subset = {λ1 < λ2 < · · · } of N is called a Hadamard set if there exists a q > 1 such that λk+1 /λk  q for every k  1. Note an important consequence: Corollary V.5

Every infinite subset of Z contains an infinite Sidon set.

This is also the case in the dual of every compact Abelian group G. Hence, if is infinite, C always contains 1 . In Chapter 5 of Volume 2, a more general example of Sidon sets, namely the quasi-independent sets, will be studied. Proof of Proposition V.4

We proceed in two steps.

1) Assume first that q  3. Let a = (ak )k1 ∈ ∞ . We are going to to construct a measure μ ∈ M(T) such that  μ(λk ) = ak for every k  1, with μ  2 a∞ . Note that we can assume a∞  1/2; the measure μ will appear as a Riesz product. Such products were introduced (obviously!) by F. Riesz in 1918, and provide a very important tool for the study of lacunary sets. Consider the trigonometric polynomial (with ak = |ak | eiϕk ): Pn (t) =

n n       1 + ak eiλk t + ak e−iλk t = 1 + 2 |ak | cos (λk t + ϕk ) . k=1

k=1

When we expand this product, we see that the spectrum of Pn is the set of words of length  n:   n θk λk ; θk = −1, 0 or 1 . n = k=1

Here is the crucial point: since q  3, whenever a number N ∈ Z can be written: n N= θk λk , k=1

with n  1 and θk ∈ {−1, 0, 1}, then this representation is unique. Therefore  Pn (0) = 1 and, for 1  k  n:  Pn (λk ) = ak .

236

6 p-Summing Operators. Applications

As the condition a∞  1/2 implies Pn (t)  0, then:  Pn 1 = Pn (t) dm(t) =  Pn (0) = 1 . T

The sequence (Pn )n1 hence has a weak∗ cluster point μ in M(T). Then μ  1, and  μ(λk ) = ak for every k  1, since Pn (λk ) = ak if n  k. The case q  3 is thus completed. Even though this is not used here, note that, for j  n:  0 if N ∈ / n  Pj (N) =   n θk =1 ak . θk =−1 ak if N = k=1 θk λk ∈ n , and hence:



0  μ(N) = 

θk =1 ak

.



 if N ∈ /  = n1 n  if N = nk=1 θk λk ∈  ;

θk =−1 ak

therefore the sequence (Pn )n1 has only a single cluster point, and thus μ(0) = 1, then μ = 1. converges weak∗ to μ. Moreover, as  2) For q > 1 arbitrary, we choose an R  1 sufficiently large so that: a) Q = qR  3; b) (Q − 2)/(Q − 1) > 1/q; c) Q/(Q − 1) < q, and we decompose the sequence (λk )k1 into R subsequences: (λjR+r )j0 ,

1  r  R.

Each of these subsequences is a Hadamard sequence of constant Q  3. Hence, given a = (ak )k1 ∈ ∞ such that a∞  1/2, there exists, for μr (λjR+r ) = ajR+r for j  0. For each r  R, a Riesz product μr such that  fixed r, we denote lj = λjR+r , and note that if N = lj + θ1 lj−1 + · · · + θj−1 l1 + θj l0 , then:



1 1 N  lj 1 − − 2 − · · · Q Q and

  lj

Q−2 1 > lj Q−1 q

  1 1 Q < lj q . N  lj 1 + + 2 + · · ·  l j Q Q Q−1

It follows that N ∈ / sp μr for r = r. Consequently, if we set: μ = μ1 + · · · + μR , then μ  R and  μ(λk ) = ak for any k  1.

V Sidon Sets

237

Note that we have proved that the union of the Sidon sets r = {λjR+r ; j  0} for 1  r  R is a Sidon set. Later on (Theorem V.20), we will see that any finite union of Sidon sets is again a Sidon set (Drury’s theorem). However, that proof is notably more difficult.

V.3 Smallness Properties of Sidon Sets Sidon sets must be considered as small sets; we will describe three aspects: functional, set theoretic and arithmetical. First: ∞ admits a Proposition V.6 If is a Sidon set, then every element of L ∞ continuous representative; in short, L = C .

Note that Rosenthal [1967] gave examples of sets ⊆ N satisfying this property but which were not Sidon sets; since then, such sets have been called Rosenthal sets. ∞ , we have Proof Let (Kn )n1 be an approximate identity. For every f ∈ L f ∗ Kn ∈ P ⊆ C , and hence: | f (γ )|  Kn (γ )  S( ) f ∗ Kn ∞  S( ) f ∞ ; γ ∈

as  Kn (γ ) −−→ 1 for any γ ∈ , Fatou’s lemma leads to: n→+∞



| f (γ )|  S( ) f ∞ .

γ ∈

The function tative of f .





γ ∈ f (γ ) γ

is thus a continuous function, which is a represen-

For the other two properties, we restrict ourselves to the Sidon sets in Z. Proposition V.7 If is a Sidon set in Z, it is not possible to have A + B ⊆ with A and B infinite. We choose to use (although this is not indispensable) the following result (Lefèvre [1999 a]), interesting in itself: Lemma V.8 If is a Sidon set in Z and if A + B ⊆ with B infinite, then the Fourier transform f ∈ 1 ( ) FA : f ∈ CA (T) −→  is 1-summing and π1 (FA )  S( ).

238

6 p-Summing Operators. Applications

Proof Let g1 , . . . , gn ∈ CA ; by density, we can assume that they are trigonometric polynomials in PA . As B is infinite, there exist b1 , . . . , bn ∈ B such that the translated spectra sp (g1 ) + b1 , . . . , sp (gn ) + bn are pairwise disjoint. Then, with ebj (t) = eibj t : n

 gj 1 ( ) =

j=1

n

| gj (l)| =

j=1 l∈sp (gj )

=

n

n



| gj (λ − bj )|

j=1 λ∈sp (gj )+bj

    n   |e= e= bj gj (λ)| =  bj gj 



j=1 λ∈sp (gj )+bj

1 ( )

j=1

(since the translated spectra are pairwise disjoint)    n    S( )  e g (since sp (gj ) + bj ⊆ ) bj j   ∞

j=1

 S( ) sup

n

|eibj t gj (t)| = S( ) sup

t∈T j=1

n

|gj (t)| .

t∈T j=1

Hence FA is 1-summing, as claimed. Proof of Proposition V.7 Given N distinct elements a1 , . . . , aN in A, we consider the trigonometric polynomials:   N 1 2π ijk iak t e , exp fj (t) = √ N N k=1 √ 1 whose spectra are in A. For any j,  fj 1 = √ N = N; hence: N N

 fj 1 = N 3/2 .

j=1

Moreover, the Walsh matrix   2π ijk   1 √ exp N N 1j,kN is unitary; hence, if ζ1 , . . . , ζN are complex numbers of modulus 1, and with:   N 1 2π ijk ζj , zk = √ exp N N j=1

V Sidon Sets

then

N

k=1 |zk |

sup

N

t∈T j=1

2

=

N

j=1 |ζj |

2,

239

and consequently:

 N  N       iak t     |fj (t)| = sup sup  ζj fj (t) = sup sup  zk e  t∈T ζj ∈U

 sup sup

 N

t∈T ζj ∈U

= sup

 N

ζj ∈U

t∈T ζj ∈U k=1

j=1

|zk |2

k=1

1/2  N k=1

1/2

|ζj |2

1/2 |eiak t |2

×

√

N=

√ √ N× N =N.

j=1

By applying Lemma V.8 for g1 = f1 , . . . , gN = fN , we obtain N 3/2  S( ) N; that is, N  S( )2 , and |A|  S( )2 ensues. Proposition V.9 If is a Sidon set in Z, then cannot contain arithmetic progressions of arbitrarily large length. For this, we use a sequence of special polynomials, called the Rudin–Shapiro polynomials; examples were first constructed by Shapiro in his thesis in 1951, and later again by Rudin [1959]. Their special property is that their uniform norm  . ∞ and their quadratic norm  . 2 are of the same order on the unit circle U. See Kahane [1980] for a sharper result. Lemma V.10 There exist polynomials Pj , j ∈ N, such that: 2j −1 a) Pj (z) = m=0 δm zm , with δ0 = 1, δ2n = δn and δ2n+1 = (−1)n δn ; b) |Pj (z)|2 + |Pj (−z)|2 = 2j+1 for |z| = 1. Proof

Set P0 (z) = 1, and define: Pj+1 (z) = Pj (z2 ) + z Pj (−z2 ) .

Then P1 (z) = 1 + z and P2 (z) = 1 + z + z2 − z3 . Condition a) is easily shown by induction; b) follows from the parallelogram identity: |u + v|2 + |u − v|2 = 2(|u|2 + |v|2 ).

Proof of Proposition V.9

Set: Rj (t) = Pj (eit ) ,

so that Rj ∞  2

j+1 2

(in other words Rj ∞ 

√ 2 Rj 2 ). Now suppose:

a, a + b, a + 2b, . . . , a + Nb ∈ .

240

6 p-Summing Operators. Applications

By replacing with − a, and changing to (− ) if necessary, we can assume a = 0 and b  1. For N  2j , we then have Qj ∈ P , if: Qj (t) = Rj (bt) . But then: 2j =



| Qj (n)|  S( ) Qj ∞ = S( ) Rj ∞  S( )

√ √ 2 2j ;

n∈

hence 2j  2 S( )2 , and thus N  4 S( )2 . In fact, a more precise result can be achieved. For this, we introduce the following notation: Notation For ⊆ Z, we denote by α (N) the maximum number of elements of that can belong to an arithmetic progression of length N. In other words: " # α (N) = max | ∩ {a + b, a + 2b, . . . , a + Nb}| ; a, b ∈ Z . The following result is known as the mesh condition: Theorem V.11 that:

If ⊆ Z is a Sidon set, there exists a constant C > 0 such α (N)  C log N .

In particular: | ∩ [1, N]|  C log N. Note that this is optimal, since, with log(2n ) = {2, 22 , . . . , 2n , . . .}, we have | ∩ [1, 2n ]| = n = . The proof log 2 could rely on the Rudin–Shapiro polynomials, as in Kahane 1, pages 34–35; but here we prefer a probabilistic method. Recall the theorem of majorization of Rademacher processes: If ε1 , . . . , εn are independent Bernoulli variables, and if aj,k ∈ C for 1  j  N  and 1  k  n, then, with Xj = nk=1 aj,k εk , we have:    E sup |Xj |  C sup Xj 2 log N jN

jN

(Chapter 1, Theorem IV.5). To use this result, we need the following classical inequality: Lemma V.12 (Bernstein’s Inequality) For every trigonometric polynomial  ikt P(t) = N k=−N ck e of degree less than or equal to N, we have: P ∞  N P∞ .

V Sidon Sets

241

In fact, in Chapter 6 of Volume 2, a slightly more general version of this inequality will be required, valid for functions that are only almost periodic. Two alternative proofs of Bernstein’s inequality are presented as exercises. Proposition V.13 Let λ1 < · · · < λn be real numbers. For every function  f : R → C of the form f (t) = nk=1 ck eiλk t for t ∈ R, we have: f ∞  δ f ∞ , where δ = sup1kn |λk |. Proof

The idea is to express f (t) = f (t) =

n

n

k=1 iλk ck

eiλk t in the form:

iψ(λk ) ck eiλk t ,

k=1

where ψ : R → R is a continuous periodic function interpolating the λk ’s. For this, we consider the 4δ-periodic function ψ0 on R such that ψ0 (x) = 2δ − |x|, for |x|  2δ. Then ψ(x) = ψ0 (x − δ) − δ is equal to δ − |x − δ| when −δ  x  3δ, and in particular, ψ(x) = x when |x|  δ. Here is the essential point: when seen as defined on R/4δZ, the function ψ0 is – up to a positive constant – the convolution of 1I[−δ,δ] by itself, and thus has non-negative Fourier coefficients:  2δ  iπ  1 0 (l) = dx  0 ψ ψ0 (x) exp − 4δ −2δ 2δ for any l ∈ Z (it is easy, although unnecessary, to calculate these coefficients  0 (l) = ψ0 (0) = 2δ, we obtain: explicitly). As l∈Z ψ ilπ 0 (l) e 2δ (x−δ) , ψ ψ(x) = ψ0 (x − δ) − δ = l=0

0 (0) = δ. Hence, we can write: since ψ ilπ al e 2δ x , iψ(x) = l∈Z

0 (l) if l = 0 and a0 = 0, and, consequently: with al = i e−ilπ/2 ψ     N ilπ lπ λk iλk t 2δ e . ck al e = al f t + f (t) = 2δ k=1

Then: f ∞ 



l∈Z

|al | f ∞ =

l∈Z

and the proof is complete.

l=0

l∈Z

0 (l) f ∞ = (2δ − δ) f ∞ = δ f ∞ , ψ

242

6 p-Summing Operators. Applications

Remark Bernstein’s inequality is optimal, as shown by P(t) = eiNt . In fact, in the probabilistic estimations, the quantity P∞ often appears via its logarithm, and hence the following easy estimation suffices, up to a loss of a constant factor, say  3: P ∞  N(N + 1) P∞ (it suffices to write: P ∞ 

N

|k| |ck |  2P∞

k=−N

N

N = N(N + 1)P∞

k=1

by using |ck |  P∞ ). Corollary V.14

If P is a trigonometric polynomial of degree  N, then: P∞  5 sup |P(t)| , t∈FN

where FN = {jπ/2N ; 0  j  4N − 1} is the set of (4N)-th roots of unity. Proof Let t0 ∈ T be such that |P(t0 )| = P∞ . There exists t1 ∈ FN such that |t1 − t0 |  π/4N; hence: P∞  |P(t0 ) − P(t1 )| + |P(t1 )|  |t0 − t1 | P ∞ + |P(t1 )| π N P∞ + sup |P(t)| ,  4N t∈FN which leads to: P∞ 

1 1−

π sup |P(t)|  5 sup |P(t)| . t∈FN t∈FN 4

Proof of Theorem V.11 Let a + b, a + 2b, . . . , a + Nb be an arithmetic progression of length N, and consider ∩ {a + b, a + 2b, . . . , a + Nb}. We can assume a = 0, and write: ∩ {b, 2b, . . . , Nb} = {λ1 b, . . . , λn b} . Let: P(t) =

n

eiλk t .

k=1

We are going to inject into P independent Bernoulli variables ε1 , . . . , εn in order to form a random polynomial Pω (t) =

n k=1

εk (ω) eiλk t .

V Sidon Sets

243

If now we set: Qω (t) = Pω (bt) , we obtain a trigonometric polynomial Qω ∈ P with spectrum in . Hence: n=

n

| Qω (λk b)|  S( ) Qω ∞ = S( ) Pω ∞ .

k=1

Thus, by taking the expectation, we obtain: n  S( ) Eω Pω ∞ . However, as Pω is a trigonometric polynomial of degree  N, the corollary of Bernstein’s inequality gives: Pω ∞  5 sup |Pω (t)| . t∈FN

If now, for t ∈ FN , we set: Xt (ω) = Pω (t) =

n

eiλk t εk (ω) ,

k=1

the conditions of application of the majorizing theorem (recalled following the statement of Theorem V.11) are satisfied; hence:    √  E sup |Pω (t)|  C sup Pω (t)L2 () log |FN | = C n log(4N) . t∈FN

t∈FN

Finally, we obtain: n  5 C S( )

√  n log(4N) ,

that is: n  25 C2 S( )2 log(4N) , which completes the proof of Theorem V.11.

V.4 Sidon Sets and Spaces of Cotype 2 Before proving the main result of this subsection, we present a special case, as a direct application of p-summing operators: Theorem V.15 Proof

If the space C (G) is isomorphic to 1 , is a Sidon set.

Let f ∈ C , and consider the convolution operator: Cf : μ ∈ M(G) −→ f ∗ μ ∈ C .

244

6 p-Summing Operators. Applications

In fact, as the spectrum of f is in , Cf defines an operator: U : M(G)/M c −→ C . Thus we obtain the factorization: q U Cf : M(G) −−→ M(G)/M c −−→ C , where q is the canonical surjection. By hypothesis, C is isomorphic to 1 , and: ∗  M(G)/M c = C(− ) ≈ (1 )∗ = ∞ ; therefore the dual form of Grothendieck’s theorem implies that U is 2-summing, and hence U can be factorized through a Hilbert space. That ends the proof, thanks to the following proposition. Proposition V.16 Let f : G → C be a continuous function. The convolution operator Cf : μ ∈ M(G) → f ∗ μ ∈ C(G) can be factorized through a Hilbert  f (γ )| < +∞. space if and only if γ ∈ |  Then γ2 (Cf ) = γ ∈ | f (γ )|.  f (γ )| < +∞, let aγ be complex numbers such that a2γ = Proof If γ ∈ |   2  f (γ ); we have γ ∈ |aγ |2 < +∞. Then h = γ ∈ aγ γ is in L (G) and f = h ∗ h. The following factorization ensues: C

f /C ME = EE zz z EE z E zz Ch EE zz Ch " L2 (G)

and γ2 (Cf )  Ch M→L2 Ch L2 →C  h2 h2 =

γ ∈

|aγ | =



| f (γ )|.

γ ∈

Conversely, assume that Cf can be factorized through a Hilbert space H: MB BB BB B B BB

Cf

H

/C ?     A

By setting hγ = B(γ ) ∈ H (recall that γ and the measure γ .m are identified),  P(γ ) γ , we have: then, for every trigonometric polynomial P = γ             P(γ ) γ   A  P(γ ) hγ  Cf (P)∞ = Cf γ

∞

γ

 A B PM .

H

V Sidon Sets

245

Expressing these inequalities with the translated polynomial P−t instead of P, we obtain:     1   Cf (P)∞   P(γ ) γ (t) hγ    B PM ; A H

γ

hence: 1 Cf (P)∞  A

  2 1/2    P(γ ) γ (t) hγ  dt  B PM .  G

H

γ

Now: 2       ) (t) dt h , h      P (γ ) γ (t) h P (γ ) P (γ dt = γ (t)γ γ γ γ  G

H

γ

G

γ ,γ

=



| P(γ )|2 hγ 2H .

γ

Therefore, we have: 

| P(γ )|2 hγ 2H

1/2  B PM

γ

for any trigonometric polynomial P. Now, for P, we select Kn , an approximate unity, and then: 1/2  hγ 2H  B . (∗) γ ∈

On the other hand: 1 Cf (P)∞  A



| P(γ )|2 hγ 2H

1/2

γ

for any trigonometric polynomial P. Choosing P = PF = being any finite subset of sp ( f ), we obtain: 1/2  1  (∗∗) |f (γ )|  hγ 2H . A γ ∈

Finally, (∗) and (∗∗) give:

and consequently



γ ∈

| f (γ )|  A B ,

γ ∈



γ ∈ |f (γ )|

 γ2 (Cf ).

 f (γ ) γ, F  |f (γ )| γ ∈F

246

Remark

6 p-Summing Operators. Applications

The proof provides the following quantitative estimate: S( )  KG [d(C , 1 )]2 ;

indeed:  f 1 = γ2 (Cf )  γ2 (U)  π2 (U)  KG U [d(C , 1 )]2  KG [d(C , 1 )]2 f ∞ . With the same proof we could similarly show that is a Sidon set when C is of cotype 2 if we had the following more general version of Grothendieck’s theorem, or more precisely its dual form: For every space Y of cotype 2, and for every compact space K, every operator T : C(K) → Y is 2-summing. But this proof is rather long (see Pisier 1, Theorem 5.14), and we prefer an alternative path, with Harmonic Analysis and Probability (essentially making use of Rider’s theorem) instead of Banach space methods. Surprisingly enough for its time, Rider’s theorem provides a characterization of Sidon sets thanks to random Fourier series; its purpose was a clearer presentation of Drury’s theorem stating that the union of two Sidon sets is again a Sidon set. These two results triggered the renewal of the study of Sidon sets in the period 1975– 1985 (see M. Déchamps [1984]). Rider’s theorem also provides a link with the last chapters of Volume 2 of this book. But first, some notation is required: ∗

Notation Let  = {−1, 1}N be the Cantor group, and P its Haar measure, seen as a probability measure on , and let (εn )n1 be the sequence of Rademacher functions on . Denote  = {γn ; n  1}. For any trigonometric N  ω polynomial P = N n=1 an γn , set P = n=1 εn (ω) an γn , and: [[P]]R = EPω ∞ . This clearly defines a norm on the set of trigonometric polynomials, such that [[P]]R  P2 ; indeed, for any ω ∈ : 1/2  Pω ∞  Pω 2 = |εn (ω) P(γn )|2 n1

=



| P(γn )|2

1/2 = P2 .

n1

However, in general, this is not comparable with the uniform norm  . ∞ . We refer to Chapter 6 of Volume 2 for further developments. First, consider the following simple result: Proposition V.17

If is a Sidon set, then: P∞  S( ) [[P]]R

for every P ∈ P .

V Sidon Sets

Proof

247

Pω ∈ P for any P ∈ P ; since is a Sidon set, then: P∞  | P(γn )| = |εn (ω)  P(γn )| n1

=



n1 ω (γ )|  S( ) Pω  ; 6 |P n ∞

n1

hence, by taking the expectation: P∞  S( ) [[P]]R .   Note that we in fact get n1 |P(γn )|  S( ) [[P]]R . Rider’s theorem provides the converse. Theorem V.18 (Rider’s Theorem) If there exists a constant C > 0 such that: | P(γn )|  C [[P]]R n1

for every P ∈ P , then is a Sidon set, and S( )  α C3 . Prior to the proof of this theorem, we present two immediate consequences. A first consequence, the converse of Proposition V.17, shows that on P(G), the norm  . ∞ is not dominated by [[ . ]]R . In fact, [[ . ]]R is not dominated  inx (where by  . ∞ either, as shown by a consideration of Q(x) = N n=0 δn e and δ2n+1 = (δn )n0 is the Rudin–Shapiro sequence:√δ0 = 1, δ2n = δn  (−1)n δn ), for which we have Q∞  4 N + 1 and [[Q]]R ≈ N log N (see Lemma V.10; or Chapter 6 (Volume 2), Proposition III.13 and the Marcus– Pisier theorem, Theorem IV.5). If P∞  C [[P]]R for every P ∈ P , then is a Sidon set. N We write = {λn ; n  1}, and let P = n=1 an λn . Since

Corollary V.19 Proof N

= P(t)+P(−t) , by considering the real and imaginary parts 2 of an , we can assume that an ∈ R. Hence, if θn = sign(an ) (with θn = 1 if an = 0): n=1 (Re an )λn (t)

N

N     |an | =  θn an λn 

n=1

n=1

=C

N

∞

an λn

n=1

R

C

N n=1

θn an λn

R

= C [[P]]R ,

since the sequence (θn εn )nN has the same distribution as (εn )nN . Theorem V.20 (Drury’s Theorem) Sidon set.

The union of two Sidon sets is again a

248

6 p-Summing Operators. Applications

Proof Let 1 and 2 be two Sidon sets, which can be assumed disjoint. As seen in the preceding corollary, for every choice of signs θn = ±1 we have:  N

 =

θn an γn

 N

R

n=1

 an γn

; R

n=1

thus, by this 1-unconditionality, we obtain, for P ∈ P 1 ∪ 2 : | P(γ )| = | P(γ )| + | P(γ )| γ ∈ 1 ∪ 2

γ ∈ 1

 S( 1 )



γ ∈ 2

 P(γ )γ

 + S( 2 )



R

γ ∈ 1

 P(γ )γ

γ ∈ 2

 R

   S( 1 ) [[P]]R + S( 2 ) [[P]]R = S( 1 ) + S( 2 ) [[P]]R .  3 Hence 1 ∪ 2 is a Sidon set, and S( )  α S( 1 ) + S( 2 ) . Proof of Rider’s Theorem We follow the proof given by Pisier [1978 a]. For this, we show the following finite version, which of course suffices to obtain Rider’s theorem. Theorem V.21

G such that: Let = {λ1 , . . . , λN } be a finite set in  =  N

|an |  C

n=1

 N

 an λn

n=1

; R

then S( )  42 C3 . We begin with two lemmas: Lemma V.22 Let = {λn ; n  1} be a subset of , and (εn )n1 be a Bernoulli sequence. Suppose that there exist two constants A > 0 and δ < 1 such that: for every ω ∈ , there exists a measure σω ∈ M(G) such that σω   A and such that: | σω (λn ) − εn (ω)|  δ ,

∀n  1.

Then is a Sidon set and S( )  2A/(1 − δ).  Proof Let P = N n=1 an λn . The an ’s can be assumed real. Then let ω ∈  such that εn (ω) = sign(an ), 1  n  N (with sign(0) = 1). We have: N n=1

|an | =

N n=1

εn (ω)an =

N n=1

 σω (λn )an +

N   εn (ω) −  σω (λn ) an , n=1

V Sidon Sets

249

and as: N

 σω (λn )an =

n=1

N

 σω (λn ) P(λn ) = σω , P = (Pˇ ∗ σω )(0) ,

n=1

ˇ = P(−t), we obtain: with P(t) N

|an |  A P∞ + δ

n=1

N

|an | ,

n=1

which is the stated result. For every f1 , . . . , fN ∈ C(G) and γ1 , . . . , γN ∈ , we have:  N   N          ε (ω) f (γ )γ  4 E ε (ω)f Eω  n n n n ω n n . 

Lemma V.23

∞

n=1

Proof

∞

n=1

For any n  1 and for any x ∈ G, with (τt f )(x) = f (x − t), we have:   (τt fn )(x)γn (t) dm(t) = fn (x − t)γn (t) dm(t) G

G

= ( fn ∗ γn )(x) =  fn (γn )γn (x) ;

hence, for every ω ∈ :  N      εn (ω)fn (γn )γn    n=1

∞

    N    εn (ω)γn (t)(τt fn )   dm(t) G

∞

n=1

    N   εn (ω)γn (t)fn  =   dm(t) . G

∞

n=1

Hence, an integration with respect to ω and an interchange of the integrals provide:  N   N           Eω  ε (ω) f (γ )γ  E ε (ω)γ (t)f n n n n ω n n n  dm(t)  n=1

∞

G

  G

∞

n=1

   N   4 Eω  εn (ω)fn   dm(t) ∞

n=1

(contraction principle)   N    = 4 Eω  εn (ω)fn   . n=1

L1

∞

First note that  the norm [[ . ]]R defined on P(G) is the norm of the space , P; C(G) . To avoid problems of measurability, given that we only use the

250

6 p-Summing Operators. Applications

first N Rademacher functions, we replace the Cantor group  by its “subgroup” N = {−1, 1}N , which is finite. We divide the proof into three steps:

Step 1. Pisier’s modification   We consider the subspace E of L1 N ; C(G) formed of functions  : N → C(G) defined by: (ω) = ε1 (ω)f1 + · · · + εN (ω)fN , for f1 , . . . , fN running over C(G). We can define a linear functional on E by setting: L() =

N

 fn (λn ) .

n=1

However, by hypothesis: N

|an |  C

 N

n=1

 an λn

, R

n=1

so we obtain: |L()| 

N

| fn (λn )|  C

 N

n=1

 fn (λn )λn

 R

n=1

 N      = C Eω  εn (ω)fn (λn )λn   n=1

 N      4 C Eω  εn (ω)fn   , n=1

∞

∞

by Lemma V.23. This means: |L()|  4 C L1 (N ; C (G)) , i.e. L is continuous on E. By the Hahn–Banach   theorem, this continuous linear C(G) functional can be extended to L1 N ; C(G) . However, even  though  ∗ = does not possessthe Radon–Nikodým property, we have L1 N ; C(G) ∞ L N ; M(G) , because N is finite. Thus there exist measures μω ∈ M(G) such that μω   4 C and such that, for any f1 , . . . , fN ∈ C(G):  μˇ ω , (ω) dP(ω) = N

N n=1

 fn (λn ) .

V Sidon Sets

251

In particular, by taking (ω) = εn (ω)λj , we obtain:   1 if n = j (1)  μω (λj )εn (ω) dP(ω) = 0 if n =  j. N

Step 2. Drury’s convolution device the measures Drury’s idea was to construct new measures νω ∈ M(G) from  νA (λj ) A⊆{1,...,N} , μω ∈ M(G), while keeping under control the 1 -norm of  for each fixed λj . For this, a convolution in the auxiliary group N must be performed, and this, not only  for scalar  functions but for functions with values in the convolution algebra M(G), ∗ . We thus define  μωω ∗ μω dP(ω ) . (∗) νω = N

Note that, as N is finite, there is no problem with the definition of this integral, which is simply the average over the 2N possible choices of signs. Since μω   4 C, it follows that νω   16 C2 . We have   μω = A⊆{1,...,N} μA wA (ω) and νω = A⊆{1,...,N} νA wA (ω), where the “Fourier coefficients” μA and νA of μω and νω are given by:   μω wA (ω) dP(ω) and νA = νω wA (ω) dP(ω) , μA = N

N

with wA the Walsh function associated with A. Since:   μA (λj ) =  μω (λj )wA (ω) dP(ω) , N

Parseval’s identity in L2 (N , P) gives:  2 | μA (λj )| = | μω (λj )|2 dP(ω)  16 C2 . N

A⊆{1,...,N}

However the definition (∗) and the equality wA (ω) wA (ωω ) = wA (ω)2 wA (ω ) = wA (ω ) imply νω =



A (μA

∗ μA ) wA (ω); hence: νA = μA ∗ μA .

Consequently: (2)

A⊆{1,...,N}

| νA (λj )| =

A⊆{1,...,N}

| μA (λj )|2  16 C2 ,

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6 p-Summing Operators. Applications

and, with the choice A = {n}:   2  νω (λj )εn (ω) dP(ω) =  ν{n} (λj ) =  μ{n} (λj ) , N

so: (3)



 1 if n = j  νω (λj )εn (ω) dP(ω) = 0 if n = j , N

by (1).

Step 3. We now obtain the measures σω intervening in Lemma V.22 by multiplying, and not by convolving, νω with a Riesz product, in the auxiliary group N . For 0 < a  1, define:  N  N     1  1 + aεn (ω) − 1 − aεn (ω) . R(ω) = 2a n=1

n=1

Thanks to the independence of the εn ’s, and the fact that 1 ± a εn (ω)  0, we have:  1 |R(ω)| dP(ω)  · RL1 (N ) = a N Set:



R(ωω )νω dP(ω ) .

σω = N

Then:

 σω   16 C

|R(ωω )| dP(ω ) 

2 N

16 C2 · a

Next, by expanding the products within R(ω), we obtain: R(ω) =

N n=1

and hence:





εn (ω) +

a|A|−1 wA (ω) ,

A⊆{1,...,N} |A|3, |A| odd

R(ωω ) νω (λj ) dP(ω )

 σω (λj ) = N

=

N 

εn (ωω ) νω (λj ) dP(ω )

n=1 N

+



a|A|−1

|A|3 |A| odd



wA (ωω ) νω (λj ) dP(ω ) . N

V Sidon Sets

253

However, by (3):  N  N εn (ωω ) νω (λj ) dP(ω ) = εn (ω) n=1 N

εn (ω ) νω (λj ) dP(ω ) N

n=1

= εj (ω) , and:

    |A|−1 a wA (ωω ) νω (λj ) dP(ω )    N |A|3 |A| odd

  =  a|A|−1 νA (λj )wA (ω)   |A|3 |A| odd

 a2



| νA (λj )| as 0 < a  1

A⊆{1,...,N}

 16 C2 a2 ,

by (2).

Consequently: | σω (λj ) − εj (ω)|  16 C2 a2 . As σω   16 C2 /a, Lemma V.22 provides the following upper bound for the Sidon constant of : 2 (16 C2 /a) S( )  · 1 − 16 C2 a2 √ An optimization with a = 1/(4 3 C) gives: √ S( )  192 3 C3  333 C3 , which completes the proof of Rider’s theorem. The following result can now be proved: Theorem V.24 Sidon set.

Any set ⊆  =  G for which C (G) is of cotype 2 is a

We need to use Gaussian sequences instead of Bernoulli sequences, so as to exploit their rotation invariance. We begin with the following lemma, of independent interest. Complex Gaussian sequences were defined in Chapter 5, Definition II.7. Lemma V.25 Let E be a complex Banach space of cotype 2. Let (Z1 , . . . , ZN ) be a standard complex Gaussian sequence, and let (ϕ1 , . . . , ϕN ) be an orthonormal system in a space L2 (X, μ). Then, for x1 , . . . , xN ∈ E:   1/2   1/2 N N 2 2     ϕn (t)xn  dμ(t)  C2 (E) Zn (ω)xn  dP(ω) .   X

n=1



n=1

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6 p-Summing Operators. Applications

Proof First, assume that the ϕn ’s are simple functions on a common partition A1 , . . . , AJ of X: ϕn =

J

αj,n 1IAj .

j=1

Then, by Theorem IV.8 of Chapter 5, which allows us to replace the Bernoulli variables in the inequality of the definition of cotype by symmetric independent variables of norm 1 in L2 :   N J N 2 1/2  2 1/2      ϕn xn  dμ = μ(Aj )  αj,n xn   X

n=1

j=1

n=1

 J  N ! 2 1/2   = μ(Aj ) αj,n xn   j=1

n=1

 N !    J   Zj  C2 (E)  μ(Aj ) αj,n xn   j=1

n=1

. L2 (;E)

Hence:     N 2 1/2   N     * Z ϕ x dμ  C (E) x  n n 2 n n  X

n=1

n=1

,

L2 (;E)

with: * Zn =

J !

μ(Aj ) αj,n Zj .

j=1

However, as the system (ϕ1 , . . . , ϕN ) is orthonormal, the matrix C of coeffi cients cj,n = μ(Aj ) αj,n , is “unitary,” meaning C∗ C = IN . By the (complex) ZN ) rotation invariance of complex Gaussians, the Gaussian vector (* Z1 , . . . , * has the same distribution as (Z1 , . . . , ZN ); hence the preceding inequality becomes:     N 2 1/2   N    ϕn xn  dμ  C2 (E)  Z x ;  n n   X

n=1

n=1

L2 (;E)

thus the lemma is proved when the ϕn ’s are simple functions. In the general case, we approach ϕ1 , . . . , ϕN by normalized simple functions ψ1 , . . . , ψN , and orthonormalize them with the Schmidt procedure, obtaining (χ1 , . . . , χN ):

V Sidon Sets

255

ψ2 − ψ2 , ψ1 χ1 , ··· ψ2 − ψ2 , ψ1 χ1  n As the Schmidt procedure is “triangular”: χn = k=1 κk,n ψk , and as the ψn ’s are almost orthonormal, the χn ’s are still close to the ψn ’s; hence every orthonormal system can be approached by an orthonormal system of simple functions, which completes the proof of the lemma. χ1 = ψ1 ,

χ2 =

We will later need a Gaussian version of the Khintchine–Kahane inequalities. Theorem V.26 (Fernique’s Theorem on the Integrability of Gaussian Vectors) Let (xn )n1 be a sequence of elements of a Banach space and (Zn )n1 a standard sequence of Gaussian variables. n Set Sn = j=1 Zj xj and M = supn1 Sn . Then, as soon as the series  n1 Zn xn is almost surely bounded, i.e. P(M < +∞) = 1, M possesses moments of every order, and more precisely M ∈ L2 , the Orlicz space 2 associated with the function 2 (x) = ex − 1. Note that the Gaussians are taken real or complex, according to the real or complex nature of the space itself. Proof (1)

Set q(t) = P(M > t). The key lies in the following inequality:    t − s 2 if 0 < s < t . P(M  s) q(t)  q √ 2

For this, let (Zn(1) )n1 and (Zn(2) )n1 be two independent copies of the sequence (Zn )n1 . (1) (2) √ (1) (2) √ Denoting Zn = (Zn − Zn )/ 2 and Zn = (Zn + Zn )/ 2, by rotation invariance of Gaussians (see Chapter 5), the sequences (Zn )n1 and (Zn )n1 are independent copies of (Zn )n1 . Let M and M be the maximal functions   (i.e. the analogues of M) associated with the series n1 Zn xn and n1 Zn xn . They are independent and have the same distribution as M. Hence, by definition of q(t): P(M  s) q(t) = P(M  s) P(M > t) = P(M  s, M > t) .  (α) (α) However, denoting Sn = nj=1 Zj xj (α = 1, 2), we have the inclusion:   t−s t−s (2) (1) , sup Sn  > √ · (2) (M  s, M > t) ⊆ sup Sn  > √ 2 n1 2 n1 Indeed, if M  s and M > t, we can find n  1 such that: √ √ √ Sn(1) + Sn(2)  > t 2 , while Sn(1) − Sn(2)   M 2  s 2 ,

256

6 p-Summing Operators. Applications

and (2) ensues by the triangle inequality. But now, (1) follows from (2), by the (1) (2) independence of the two sequences (Sn )n1 and (Sn )n1 . Then fix s0 > 0 such √that P(M  s0 ) > 0. The change of t to t 2 + s0 > s0 in (1) yields: √ P(M  s0 ) q(t 2 + s0 )  [q(t)]2 ; or, by setting r(t) =

q(t) : P(M  s0 ) √ r(t 2 + s0 )  [r(t)]2 .

Finally, fix t0 > 0 such that√r(t0 ) = e−δ < 1, and define (tn )n0 by the 2 induction relation tn+1 = tn 2 + s0 . We have: r(t √n+1n)  [r(tn )] ; hence nδ −2 . As the order of magnitude of tn is ( 2) and as the√function r(tn )  e r is non-increasing, an interpolation of t between two powers of 2 gives: 2 q(t)  ae−bt , where a and b are two positive constants. Then, if 0 < λ < b, an integration by parts leads to:  +∞  +∞  2 2 2 2λteλt q(t) dt  1 + 2aλte(λ−b)t dt < +∞ , E eλM = 1 + 0

0

which completes the proof. As  . 2 ≈ supp1  . p /p1/2 , we can thus deduce: Corollary V.27 Assume that the Gaussian series S = almost surely. Then, for 0 < p < +∞:

+∞ n=1

Zn xn converges

Ap S2  Sp  Bp S2 , √ with Bp = O( p). We can now prove Theorem V.24: Proof of Theorem V.24 Denote = {λn ; n  1}, and apply the lemma with E = C , L2 (X, μ) = L2 (G), ϕn = λn and xn = cn λn (cn ∈ C). Since the λn ’s are characters, then:        N  N     λ (t) c λ = c λ n n n n n   ∞

n=1

n=1

∞

for any t ∈ G; Lemma V.25 thus provides:     1/2 N 2  N        cn λn   C2 C Zn (ω) cn λn  dP(ω) ,   n=1

∞



n=1

∞

V Sidon Sets

257

or, thanks to Khintchine’s inequalities for Gaussian vectors (Corollary V.27)):  N    N          C c λ  a C Z (ω) c λ  n n 2 n n n  dP(ω) ,  ∞

n=1



∞

n=1

where a > 0 is a numerical constant. By rotation invariance of Gaussian vectors, the right-hand side of this expression is unchanged when we replace each cn by its modulus |cn |; thus, as:   N N   N  |cn | = |cn | λn (0)   |cn | λn   , n=1

n=1

∞

n=1

we obtain: N

  |cn |  a C2 C

  N    Zn (ω) cn λn  dP(ω) .  

n=1

∞

n=1

We are almost ready to apply Rider’s theorem, but this requires a return to Bernoulli variables. This last stage of the proof can be done with the next result (a more general version, the Marcus–Pisier theorem, will be presented in Chapter 6 of Volume 2). Proposition V.28 Let (Zn )n1 be a standard complex Gaussian sequence, and let λn , n  1 be elements of the dual group . Assume that there exists a constant C > 0 such that, for any n  1 and every a1 , . . . , an ∈ C:   n   n  |aj |  C E  a Z λ j j j .  j=1

j=1

∞

√ Then, there exists a constant C = O(C log C) > 0 such that, for any n  1 and every a1 , . . . , an ∈ C:   n   n  |aj |  C E  aj εj λj   . j=1

j=1

∞

Hence = {λn , ; n  1} is a Sidon set, with constant S( )  c C4 , where c > 0 is a numerical constant.   Proof Let M  1 be such that E |Z1 |1I{|Z1 |>M}  1/(2C). Since  ∞  ∞   r 2 2 2 E |Z1 |1I{|Z1 |>M} = √ √ e−r /2 r dr = e−x dx  e−M , 2 M 2 M  we can choose M  c logC, with c a positive numerical constant. By equidistribution, we have E |Zn |1I{|Zn |>M}  1/(2C) for any n  1.

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6 p-Summing Operators. Applications

Set Zn = Zn 1I{|Zn |M} and Zn = Zn 1I{|Zn |>M} . As the Zn ’s are symmetric, the contraction principle gives:    n  n n         |aj |  C E  aj Zj λj  + C E  aj Zj λj  j=1

∞

j=1

  n     2 MC E  aj εj λj   j=1

∞

   n    2 MC E  aj εj λj   j=1

thus: n j=1

∞

+C +

1 2

j=1 n

∞

|aj |E(|Zj |)

j=1 n

|aj | ;

j=1

    n  |aj |  4 MC E  a ε λ j j j .  j=1

∞

The last assertion comes now from Rider’s theorem:  S( )  α (4 MC)3  c ( log C C)3  c C4 .

VI Comments The notion of the p-summing operator and the factorization theorem are due to Pietsch [1967]. An exhaustive study of these operators and of their applications, notably to Harmonic Analysis, can be found in Lindenstrauss and Pełczy´nski [1968]. For the (q, p)-summing operators, one could consult Diestel–Jarchow–Tonge, which also contains a proof of Grothendieck’s theorem based only on Khintchine’s inequalities (for this, see also TomczakJaegermann). The proof presented in the text essentially follows Pisier 1, the best reference on the subject. As previously mentioned in the text, Grothendieck’s theorem was first published in Grothendieck [1956], but it took 12 years before Lindenstrauss and Pełczy´nski [1968] explained its depth and potential, thanks to the notion of the p-summing operator introduced by Pietsch [1967]. The best estimations currently known for the Grothendieck constants KGR and KGC seem to be: π π 1.338 < KGC  1.41 < < KGR  √ = 1.782 . . . ; 2 2 2 log(1 + 2) 2 Krivine conjectured that this last upper bound is the best one, but this was disproved by

Braverman, K. Makarychev, Y. Makarychev and Naor [2011]: The Grothendieck Constant is Strictly Smaller than Krivine’s Bound, 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 453–462, arXiv:1103.6161. See Appendix C.

VI Comments

259

for more information on these bounds, due to Krivine [1977] and [1979], Davie (unpublished work), Haagerup [1987] and König [1990], the reader could consult Pisier 1, Section 5.e, and Diestel–Jarchow–Tonge, page 29. In Pisier 1 an exhaustive study of GT spaces can also be found, with the general version of the dual form of Grothendieck’s theorem, as well as the following theorems: quotients of L1 by reflexive subspaces are GT and of cotype 2 (Kisliakov [1978], Pisier [1978 d]) and so is L1 /H 1 (Bourgain [1984 a]). A generalization of this last result was given by Kisliakov [1991]. Theorem IV.2 is due to Garling and Gordon [1971]. The proof given here, using the Pietsch factorization theorem, is due to Kwapie´n (see Pisier 2, page 40). Item 1) of Theorem IV.4 in fact follows from a more general result, concerning the existence of an ellipsoid of maximal volume (John [1948]); its construction was generalized by Lewis [1979] (see Chapter 1 of Volume 2); item 2) is due to Kadeˇc and Snobar [1971]; essentially, it cannot be improved (Szarek [1983]). The terminology of “Sidon sets” was chosen by Kahane and Rudin (see M. Déchamps [1984], page 10). The first systematic study of their properties was made by Rudin [1960]; therein can be found generalizations of Proposition V.9 and of Theorem V.11, the latter being due to Steˇckin [1956] (Rudin [1960], page 214, indicates that Helson had also informed him of this result in 1956). This will be seen in Chapter 4 of Volume 2, along with the study of the (p)-sets. In locally compact groups, the notion of a Sidon set is replaced by that of a topological Sidon set; see Déchamps–Gondim [1972] for a deep study. The construction of the Rudin–Shapiro polynomials given here in Lemma V.10 is a variant due to Brillhart and Carlitz [1970] (see Kahane 2, Chap. 6, § 5). Theorem V.15 is an answer to a question by Pisier, due to Varopoulos [1976]; he used the techniques of Q-algebras, but his proof was also based on Grothendieck’s theorem. The proof presented in the text follows Pisier 1, Section 4.b. Theorem V.18 was proved by Rider [1975] in order to give a clearer presentation of Theorem V.20 due to Drury [1970]; the proof given here follows that of Pisier [1978 a], which also includes Theorem V.24, shown independently by Kwapie´n and Pełczy´nski [1980] (Theorem 3.1). A proof of Drury’s theorem can be found in the new edition of Kahane–Salem. Theorem V.26 (integrability of Gaussian vectors) is due to Fernique [1970] and [1971]. Bourgain and Milman [1985], solving a conjecture of Pisier [1981 a], showed that is a Sidon set as soon as C possesses a finite cotype. This result will be proved in Chapter 6 (Volume 2).

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6 p-Summing Operators. Applications

We already mentioned in Chapter 5 the result of Maurey and Pisier [1976] stating that any space having no finite cotype contains n∞ ’s uniformly; this leads to the following dichotomy for the subsets ⊆  G: either is a Sidon set, and hence C is isomorphic to 1 , or else C contains n∞ ’s uniformly, i.e. (n) (n) for every integer n  1, there exist bump functions f1 , . . . , fn ∈ C such that, for any a1 , . . . , an ∈ C, we have:   n  (n)  −1  C sup |ak |   ak fk    C sup |ak | . 1kn

k=1

∞

1kn

Thus the “uncertainty principle” (see Havin–Jöricke) can here be circum(n) vented; in other words, the spectrum of the fk ’s, contained in , can be controlled at the same time as the supports of the functions, which are almost disjoint. Li, Queffélec and Rodríguez-Piazza [2002] showed that, for random subsets of Z (in the sense of Erdös–Bourgain selectors), the dichotomy is even stronger: either almost surely is a Sidon set, and hence C is isomorphic to 1 , or else C almost surely contains c0 : there exists an infinite sequence of bump functions fn so that:     n −1  C sup |ak |   ak fk    C sup |ak | , 1kn

k=1

∞

1kn

for any a1 , . . . , an ∈ C and every n  1. In Lefèvre, Li, Queffélec and Rodríguez-Piazza [2002], recent results can be found comparing the arithmetical properties of with the Banach properties of B for various function spaces B. See also Déchamps–Gondim [1987]. Other types of properties of Sidon sets, or rather of their complements, are given in Kalton and Pełczy´nski [1997]: If S is a Sidon set, then LS1c (G) has the Dunford–Pettis property; however, if S is infinite, LS1c (G) is not complemented in its bidual, is not isomorphic to a complemented subspace of a Banach lattice and is not a L1 -space. We mention that it is not known, for an infinite Sidon set S, whether L1 (G)/LS1 can be isomorphic to a subspace of L1 .

VII Exercises Exercise VII.1 1) Let X be a Banach space, and let 1  p < + ∞. Show that, for x1 , . . . , xN ∈ X:  N  1/p   N   N p p∗   sup |ξ(xn )| = sup  an xn  ; |an |  1 , ξ ∈BX ∗

where

p∗

n=1

is the conjugate exponent of p.

n=1

n=1

VII Exercises

261

2) Show that, for every operator T : X → Y: Q

πp (T) = sup{πp (TU) ; U : p∗ −→ X , U  1 , q  1} (follow the proof of Lemma III.9). Exercise VII.2 1) Show that every operator T : X → Y whose biadjoint T ∗∗ : X ∗∗ → Y ∗∗ is p-summing is itself p-summing, and that πp (T)  πp (T ∗∗ ). 2) Show that if T is p-summing, then so is T ∗∗ , and that πp (T ∗∗ ) = πp (T) (use the Pietsch factorization theorem; if μ is a Pietsch measure associated with T, embed C(K) into L∞ (μ); then, after two dualizations in the diagram, remark, on one hand, that T ∗∗ has values in Y, and on the other, that there exists a projection P : L∞ (μ)∗∗ → L∞ (μ) of norm 1, as L∞ (μ) is a dual; in the case p = 1, also use the existence of a projection Q : L1 (μ)∗∗ → L1 (μ)). Exercise VII.3 1) Let 1  q < +∞ and u : X → N whose adjoint is 2 be an operator k q-summing. Let x1 , . . . , xk ∈ X be such that supξ ∈BX∗ j=1 |ξ(xj )|  1. a) Show that: k j=1

1 uxj   A1

 0

N 1 

 

  rn (t) u∗ (en ) dt ,

n=1

where A1 is the constant in Khintchine’s inequalities and where (en )nN is the canonical basis of N 2. b) By using the Pietsch factorization theorem, and then Fubini’s theorem and Khintchine’s inequalities, deduce that: k j=1

uxj  

Bq πq (u∗ ) . A1

2) Show that every operator u : X → H with values in a Hilbert space and with a q-summing adjoint is itself 1-summing (Kwapie´n [1970 a]). Exercise VII.4 1) Show that the adjoint of every 2-summing operator u : H → H from a Hilbert space into itself is also 2-summing. 2) Let X be a Banach space. Assume that the adjoint u∗ of every 2-summing operator u : X → H from X into a Hilbert space H is also 2-summing.

262

6 p-Summing Operators. Applications

Let K be the unit ball of X ∗ equipped with the weak∗ topology, and let iX : X → C(K) be the canonical isometry.  2 a) Let μn ∈ M(K) be measures such that supξ ∈BM(G)∗ +∞ n=1 |ξ(μn )|  1.   For f ∈ C(K), set w( f ) = μn , f n1 . Show that the operator wiX is 2-summing. b) Deduce that i∗X w∗ is 2-summing, and then so is i∗X . c) Deduce that i∗X can be factorized through a Hilbert space, then that X ∗ is isomorphic to this Hilbert space. d) Conclude that X is isomorphic to a Hilbert space (Kwapie´n [1970 b]). Exercise VII.5 1) Let 1 < p  2, and let Z1 , . . . , ZN , . . . be independent (symmetric) p-stable variables, such that Zk 1 = 1. Show that the space they generate in L1 () is isometric to p (see Chapter 5, Theorem II.10). 2) Deduce that, for 1  p  2, every operator u : C(K) → Lp (μ) is 2summing, and that π2 (u)  KG u. Exercise VII.6 This exercise presents an alternative proof of Bernstein’s inequality, based on an idea of de la Vallée-Poussin (see Bojanov and Naidenov [1999], page 273). 1) Let Q be a real trigonometric polynomial, of period 2π and of degree  N, such that Q∞ < 1. For a fixed t ∈ R, set Q(t) = cos(Nα). Suppose that |Q (t)| > N | sin(Nα)|. a) Show that we can assume 0 < α < π/N and Q (t) < 0. b) Set f (x) = cos(Nx) − Q(x + t − α). Show that f vanishes at least three times on ]0, π/N[ . c) For 1  j  2N −1, specify the sign of f ( jπ/N); deduce that f possesses at least 3 + (2N − 1) zeros in [0, 2π [ . d) Deduce that f is identically null (note that f stems from an (ordinary) polynomial of degree  2N), in contradiction to f (0) > 0. 2) Let P be a trigonometric polynomial of degree  N. a) Deduce from 1) that if P∞ < 1, then P ∞  N (we can assume P ∞ = P (t0 ) ∈ R+ ; then take Q = Re P). b) Deduce P ∞  N P∞ . Exercise VII.7 (A complex method for Bernstein’s inequality) Let D be the open unit disk, and λ a complex number of modulus > 1. For any polynomial

VII Exercises

263

N n P(z) = degree N, we denote by Q its reciprocal polynomial: n=0 an z of  N n Q(z) = z P(1/z) = N n=0 aN−n z . We have |Q(z)| = |P(z)| for |z| = 1. 1) Let f , F be two polynomials of degree N such that all roots of F belong to D (or D) and such that |f (z)|  |F(z)| if |z| = 1. a) Show that all roots of λF−f belong to D (or D). (Hint: use the maximum modulus principle to show that |f (z)|  |F(z)| for |z|  1, or use Rouché’s theorem.) b) Show that the derivative of a polynomial whose roots lie in a convex set again has its roots in this convex set. Deduce that |f (z)|  |F (z)| for |z| = 1. 2) Let P be a polynomial of degree N satisfying sup|z|=1 |P(z)| = 1.

a) Use 1) to show that |P (z)|  N if |z| = 1. N−1 | for |z| = 1, and b) Use 1) again to show that  |Q (z)|  |P (z) − λNz conclude that sup|z|=1 |P (z)| + |Q (z)| = N (Malik’s inequality). c) We moreover assume that P is zero-free in D. Show that |P (z)|  N/2 for |z| = 1 (Lax’s inequality). d) Let ρ  1. We now assume P to be zero-free in ρ D. Show that, for |z| = 1, we have ρ |P (z)|  |Q (z)|; deduce from this: |P (z)|  N/(1 + ρ).  inθ be a trigonometric polynomial of degree  N, 3) Let T(θ ) = N n=−N cn e and let P be the polynomial defined by P(eiθ ) = eiNθ T(θ ) (so d◦ P  2N). Assume T∞ = 1. a) Assume that T is real; then show that P is self-reciprocal. Deduce the Schaake–Van der Corput inequality: T (θ )2 + N 2 T(θ )2  N 2 . b) In the general case where T is complex, show that T ∞  N (Bernstein’s inequality). Exercise VII.8 Let PN be the space of trigonometric polynomials of degree  N, and let  .  be a translation-invariant norm on PN (in particular  .  =  . ϕ , for any Orlicz function ϕ). From the proof of Bernstein’s theorem given in the text, deduce: P   N P for every P ∈ PN . Exercise VII.9 1) Let k  1 be a fixed integer. Show that, for n  2, the Sidon constant of n . n = {1, 2k , . . . , nk } is  δk log n

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6 p-Summing Operators. Applications

2) Give an example of a compact group G and of an infinite Sidon set in its dual, with a constant equal to 1. 3) Show that, in Z, S( ) > 1 as soon as | |  3; show that, for any ε > 0, we can find infinite sets in Z whose Sidon constant is  1 + ε. Exercise VII.10 Show that = {γn ; n  1} is a Sidon set if and only if (γn )n1 is an unconditional basis of C . Exercise VII.11 1) Let G be a compact group. Show that if ⊆ , and if$ C is complemented in C by a projection P, then the formula Q( f ) = G (τ−θ Pτθ )( f ) dm(θ ) defines a translation-invariant projection. 2) Deduce that there exists a measure μ ∈ M(G) such that  μ(λ) = 1 when μ(λ) = 0 when λ ∈  (use Q∗ ). λ ∈ and  3) Deduce that if C is complemented in C for every ⊆ , then is a Sidon set (use Lemma V.22, noting that there is no need to control the norm of the measure: it suffices to know that the Fourier series of every f ∈ C is absolutely convergent). Exercise VII.12 Consider, for μ ∈ M(G), the convolution operator T = Cμ : C(G) −→ C(G), and assume that it is 1-summing. Using translation invariance, show that if μ is a Pietsch$measure for T, then the measure ν defined, for every Borel set A, by ν(A) = G μ(A + t) dm(t) is again a Pietsch measure; deduce:  Tf ∞  π1 (T) |f (t)| dm(t) , G

and then: f ∗ μ∞  π1 (T) f 1 for any f ∈ L1 (G). Then show that δx ∗ μ∞  π1 (T) for any x ∈ G, and finally that μ admits a density h ∈ L∞ (G) with h∞  π1 (T). Show that this last inequality is in fact an equality. Exercise VII.13 Let G be a compact (metrizable) Abelian group, and a finite subset of its dual. Show that the Banach–Mazur distance between √ | | | | (draw inspiration from the proof of the equality C and 2 is  √ d(n∞ , n2 ) = n). Exercise VII.14 1. An operator T : C(T) → L2 (T) is said to be a multiplier if, for any a ∈ T, [T( fa )] = [T( f )]a , where fa (t) = f (t − a). Set en (t) = eint for n ∈ Z. a) Let T be a multiplier of C(T) in L2 (T). Show that there exists a sequence (λn )n∈Z such that T(en ) = λn en for any n ∈ Z.

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b) Assume now that T is 1-summing. i) Show that there exists a probability μ on T such that:  T( f )2  π1 (T) |f (t)| dμ(t) T

for every f ∈ C(T). ii) By using the property of multipliers and Fubini’s theorem, deduce: T( f )2  π1 (T) f 1 . iii) By testing this inequality on the Fejér kernel, deduce that λ = (λn )n∈Z ∈ 2 (Z) and λ2  π1 (T). c) Conversely, show that any multiplier T : C(T) → L2 (T), with T(en ) =  λn en , such that n∈Z |λn |2 < +∞, is 1-summing. 2. Let Lip1/2 be the Banach space of functions f : T → C for which there exists a constant C > 0 such that: |f (s) − f (t)|  C |eis − eit |1/2 for every s, t ∈ T. We norm it by: f  = sup

eis =eit

|f (s) − f (t)| + f ∞ . |eis − eit |1/2

We admit that there exists a continuous operator K : C(T) → Lip1/2 such that K(en ) = |n|−1/2 en for every n ∈ Z∗ , and that K( fa ) = (Kf )a for any  2 cos(nt) a ∈ T (the convolution operator with the function ϕ(t) = +∞ √ n=1 n  works). Assume that n∈Z | f (n)| < +∞ for any f ∈ Lip1/2 , so that the Fourier transform F maps Lip1/2 into 1 (Z).  a) With j(x) =  n∈Z xn en ∈ L2 (T) for every x = (xn )n∈Z ∈ 1 (Z), and φ( f ) =  f (n) n∈Z for f ∈ Lip1/2 , show that the operator T = jφK : C(T) → L2 (T) is 1-summing. b) Show that T is a multiplier, and that T(e0 ) = 0 and T(en ) = |n|−1/2 en for every n ∈ Z∗ .  f (n)| = +∞ c) Deduce that there exists f ∈ Lip1/2 such that n∈Z | (Bernstein’s theorem).

7 Some Properties of Lp -Spaces

I Introduction This chapter presents a selection of properties of the Lp -spaces. Section II focuses on L1 . After defining the notion of uniform integrability, we give a condition for a sequence of functions to be uniformly integrable (the Vitali–Hahn–Saks theorem), which allows us to show that the spaces L1 (m) are weakly sequentially complete. Then a characterization is seen of the weakly compact subsets of L1 as the weakly closed and equi-integrable subsets (the Dunford–Pettis theorem). We end this section by showing that L1 cannot be a subspace of a space with an unconditional basis. The study of L1 will be continued in Chapter 4 of Volume 2; more specifically we will emphasize the structure of its reflexive subspaces. In Section III we show that the trigonometric system forms a basis of Lp (0, 1) for p > 1. In fact, this is an immediate consequence of the Marcel Riesz theorem, stating that the Riesz projection, as well as the Hilbert transform, is continuous on Lp for p > 1; the main part of Section III is devoted to the proof of this result. Rather than a direct proof of this result, a reasoning by interpolation has been chosen: indeed, this gives us the opportunity to show the Marcinkiewicz theorem, which is at the origin of real interpolation, as well as Kolmogorov’s theorem, stating that the Riesz projection is of weak type (1, 1) (Theorem III.6). A result of Orlicz (Corollary III.9) concludes this section: it states that the unconditional convergence of a series in Lp , for 1  p  2, implies the convergence of the sum of squares of the norms; as a consequence the trigonometric system is unconditional only for L2 . In Section IV, we show that, in contrast, the Haar basis is unconditional in Lp (0, 1) for 1 < p < +∞. This unconditionality is linked to the fact that the Haar basis is a martingale difference; thus a few complements on martingales are first presented, notably on the behavior in Lp of the square function of a 266

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martingale (Theorem IV.6). The proof consists in starting with the easy case, p = 2, and then p is successively doubled to treat the cases p = 4, 8, 16, . . .; it finishes with an interpolation based on the Riesz–Thorin theorem, shown beforehand in Subsection IV.1. The end of this section presents a particular property of the Haar basis, which in a way makes it “extremal”; in the course of this study, we prove Lyapunov’s theorem, stating that the range of vector measures with values in Rn is convex. Finally, the goal of Section V is to give an alternative proof of Grothendieck’s theorem, as a simple consequence of a theorem of Paley stating that +∞ 1  k 2 k=1 | f (2 )| < +∞ for every function f ∈ H (T). For this, we succinctly develop the theory of H p -spaces, and then prove the factorization theorem H 1 = H 2 H 2 and the theorem of Frédéric and Marcel Riesz. Grothendieck’s theorem  k  then follows from the fact that the application f ∈ A(D) −→  f (2 ) k1 ∈ 2 is 1-summing and surjective (Theorem V.6). In the Comments, we show that there is essentially only one space L1 (m), if the space is assumed separable and the measure m atomless.

II The Space L1 II.1 The Dunford–Pettis Theorem An important property of L1 -spaces is that they are weakly sequentially complete. This follows from the Vitali–Hahn–Saks theorem, which also leads to the Dunford–Pettis theorem characterizing the weakly compact subsets of L1 . Throughout this section, we consider a measure space (S, T , m), and assume that m(S) < +∞. However, note that if m is only assumed σ -finite, there exists a function h0 ∈ L1 (m) which is strictly positive m-almost everywhere. Then the measure h0 .m is finite, and the application f ∈ L1 (m) → f /h0 ∈ L1 (h0 .m) is an isometry. Moreover, in the general case, if ( fn )n1 is a sequence of functions  of L1 (m), the set S0 = n1 supp( fn ) has a σ -finite measure. The theorems to come, concerning weak compactness or weak sequential completeness, always involve the consideration of a sequence of functions; thus nothing is lost if the measure m is assumed to be finite. Definition II.1 A subset H of L1 (m) is said to be uniformly integrable (or equi-integrable) if:    sup | f | dm = 0 . lim a→+∞

f ∈H | f |a

A sequence of integrable functions ( fn )n1 is said to be uniformly integrable if the set { fn ; n  1} is uniformly integrable.

7 Some Properties of Lp -Spaces

268

The following proposition provides examples of uniformly integrable subsets; its proof is easy and is left as an exercise. Proposition II.2 If there exists g ∈ L1 (m) such that | f |  g for every f ∈ H, then H is uniformly integrable. In particular: 1) any finite subset of L1 (m) is uniformly integrable; 2) any H ⊆ L1 (m) that is contained in L∞ (m) and bounded in L∞ (m) is uniformly integrable. For the second assertion, recall that m(S) < +∞. Proposition II.3 H is uniformly integrable if and only if H is bounded in L1 (m) and is uniformly absolutely continuous (also known as equicontinuous): for every ε > 0, there exists δ > 0 such that:  m(A)  δ ⇒ sup | f | dm  ε . f ∈H A

Proof For every positive measurable function g:     g dm = g dm + g dm  a m(A) + A

A∩{ga}

A∩{g>a}

{g>a}

g dm

for any measurable subset A and any a > 0; consequently:   | f | dm . sup | f | dm  a m(A) + sup f ∈H A

f ∈H {| f |>a}

Hence, if H is uniformly integrable, we see that, on one hand, by the choice A = S, H is bounded, and on the other hand, H is $uniformly absolutely continuous: for any ε > 0, we can select a > 0 such that {| f |>a} | f | dm  ε/2; $ then, if m(A)  δ = ε/(2a), we have supf ∈H A | f | dm  ε. Conversely, the Markov–Tchebychev inequality shows that: sup m({| f | > a}) 

f ∈H

1 sup  f 1 −−→ 0 , a→+∞ a f ∈H

since H is bounded. The uniform absolute continuity of H hence allows $the choice of a > 0 such that supf ∈H m({| f | > a})  δ; therefore {| f |>a} | f | dm  ε for every f ∈ H, and thus H is uniformly integrable. Proposition II.4 A sequence ( fn )n1 in L1 (m) converges in norm (to f ) if and only if it is uniformly integrable and converges in measure (to f ). Proof According to the Markov–Tchebychev inequality, convergence in norm implies convergence in measure. Moreover, { fn ; n  1} is bounded

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and uniformly absolutely continuous; indeed, for any ε > 0, there exists N $  1 such that $ fn − f 1  ε/2 for $n  N; then select δ > 0 such that A | f | dm  $ ε/2, A | f1 $| dm  ε , . . . , A | fN−1 | dm $ ε for m(A)  δ; the inequality A | fn | dm  A | f | dm +  fn − f 1 implies A | fn | dm  ε for any n  1 when m(A)  δ. The set { fn ; n  1} is hence uniformly integrable, by Proposition II.3. Conversely, assume that ( fn )n1 converges in measure to f and that the set { fn ; n  1} is uniformly integrable. First we show that the function f is integrable. Indeed, by Proposition II.3, the set { fn ; n  1} is bounded, and, moreover, there exists a subsequence ( fnk )k1 converging almost everywhere to f ; Fatou’s lemma hence implies:   | f | dm  lim inf | fnk | dm  sup  fn 1 < +∞ . S

k→+∞ S

n1

Therefore, for every ε > 0:   | fn − f | dm = S



{| fn −f |ε}

| fn − f | dm +



 ε m(S) +

{| fn −f |>ε}

{| fn −f |>ε}

| fn − f | dm



| fn | dm +

{| fn −f |>ε}

| f | dm

−−→ ε m(S),

n→+∞

since m({| fn − f | > ε}) −−→ 0 and the set { fn ; n  1} is uniformly absolutely n→+∞

continuous. Thus  fn − f 1 −−→ 0. n→+∞

We can now state the following theorem: Theorem II.5 (The Vitali–Hahn–Saks Theorem) integrable functions such that:  fn dm lim

Any sequence ( fn )n1 of

n→+∞ A

exists (and is finite) for every measurable set A is uniformly integrable. Moreover, ( fn )n1 converges weakly to a function f ∈ L1 (m). A consequence follows: Theorem II.6 For every measure space (S, T , m), the space L1 (m) is weakly sequentially complete. For every measurable space (S, T ), the space M(S) of measures on (S, T ) is weakly sequentially complete. Proof The first assertion follows immediately from the Vitali–Hahn–Saks theorem. For the second, if (μn )n1 is a weak Cauchy sequence in M(S), we can assume that μn = 0 for n large enough, and hence for any n  1

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7 Some Properties of Lp -Spaces

without loss of generality. We introduce the probability P =

+∞

1 2n μn 

|μn |,

n=1 μn = fn .P.

and, thanks to the Radon–Nikodým theorem, we can write Since 1 the map f ∈ L (P) → f .P ∈ M(S) is an isometry, the sequence ( fn )n1 is weakly Cauchy in L1 (P), hence weakly convergent; thus it is the same for the sequence (μn )n1 . In fact, if we choose in M(S) a maximal  of pairwise  3family1 (μα )α∈A singular measures, then M(S) is isometric to α∈A L (|μα |) 1 . Two auxiliary results are now required: Proposition II.7 On the σ -algebra T of measurable subsets of S, the relation defined by m(AB) = 0 (where  denotes the symmetric difference), is an equivalence relation, and the formula dm (A, B) = m(AB) = 1IA − 1IB L1 (m) defines a distance on the quotient T /m which makes it$ a complete metric space. Moreover, for every f ∈ L1 (m), the map If : A → A f dm is uniformly continuous; more precisely:        f dm − f dm  | f | dm .   A

B

AB

Proof It suffices to note that if 1IAn − f L1 (m) −−→ 0, then f can only take on n→+∞ the values 0 and 1 (as a subsequence of (1IAn )n1 converges almost everywhere to f ); f is hence an indicator function 1IA . The uniform continuity of If follows from the inequality and from the absolute continuity of the integral of f . Note (see the remark following Corollary III.7 in the Annex, Subsection III.2) that an atom for the measure m is any measurable subset A such that m(A) = 0, and for which m(B) = 0 or m(B) = m(A) for every measurable B ⊆ A. Lemma II.8 For any r > 0, there exists a finite measurable partition (Si )ij of S such that m(Si )  r for i  j0 , and such that Sj0 +1 , . . . , Sj are atoms with m(Si ) > r. Remark It follows from Lemma II.8 that, when m is atomless, every uniformly absolutely continuous set is automatically bounded, thus uniformly integrable. Sub-Lemma II.9 Let m be a finite positive measure on (S, T ); if m is atomless, then, for every a < m(S), there exists A ∈ T such that m(A) = a.

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This sub-lemma is in fact a special case (the 1-dimensional case) of Lyapunov’s theorem, to be seen later (Theorem IV.10): by the way, this theorem could be proved by induction with a start from this case (see Lindenstrauss–Tzafriri II, Theorem 2.c.9). Proof of the sub-lemma We can assume a > 0. We work in the quotient space T /m, considered as a subspace of L1 (m), equipped with the following induced order: A  B means m(A ∩ Bc ) = 0. First, we select a maximal measurable subset A such that m(A)  a. This is possible as the set E of measurable subsets B such that m(B)  a is inductive. Indeed, let B be a totally ordered class of measurable subsets such that m(B)  a for every B ∈ B, and set a = supB∈B m(B). Let (Bn )n1 be an increasing sequence of elements of B  such that m(Bn )  a − 1/n for every n  1, and B∞ = n1 Bn . Then m(B∞ ) = a and B∞ ∈ E. If B∞ is not a majorant of B, there exists B0 ∈ B such that B0  Bn for an infinite number of n (since otherwise, for every B ∈ B, we would have B  Bn for n large enough, so that B  B∞ ). It follows that B0  B∞ and that m(B0 ) = a . But then B  B0 for every B ∈ B (otherwise we would have a subset B ∈ B such that B0 < B and m(B) > m(B0 ) = a ); hence B0 is a majorant of B. Now, if we had m(A) < a, the set C of measurable subsets C such that m(C) > 0 and m(A ∩ C) = 0 would have a minimal element. Indeed, we would have infC∈C m(C) > 0, as otherwise there would exist a C0 ∈ C such that 0 < m(C0 ) < a−m(A), and the set A∪C0 would contradict the maximality of A, since the condition m(A∩C0 ) = 0 would lead to m(A) < m(A)+m(C0 ) = m(A∪C0 ) = m(A)+m(C0 )  a. Then the same reasoning as above shows that C is inductive for the “decreasing order of inclusion”. Such a minimal element can only be an atom, and thus the proof of the sub-lemma is complete. Proof of Lemma II.8 First, note that two distinct atoms are disjoint, in the quotient T /m; those of measure > 1/n are thus finite in number, and consequently there is only a countable quantity of atoms Ak , k  1. The set  S = S  k1 Ak does not contain any atoms; so, thanks to the sub-lemma, S can be decomposed into a finite (finite because m(S ) < +∞) number of subsets S1 , . . . , Sj1 with measure  r. Denote by Sj0 +1 , . . . , Sj ( j0  j1 ) the atoms of measure > r. These can be assumed to be A1 , . . . , Ak0 . As  k>k0 m(Ak ) < +∞, and m(Ak )  r for k > k0 , the Ak ’s, for k > k0 , can be pulled together in a finite number of blocks such that each union   Sj1 +1 = k∈I1 Ak , . . . , Sj0 = k∈I Ak has measure  r, which completes j the proof. Remark Whenever the (finite) measure m is atomless, the space L1 (m) is infinite-dimensional. Indeed, we have a decreasing sequence of measurable

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272

subsets S1 = S ⊇ S2 ⊇ S3 ⊇ . . . such that m(Sn  Sn+1 ) > 0 for any n  0; the functions 1ISn Sn+1 are linearly independent. Proof of the theorem

The functions can be assumed real.

a) For any ε > 0, the sets

         A ∈ T ;  ( fn − fp ) dm  ε FN = A

n,pN

are closed in T /m, thanks to the continuity of the applications Ifn −fp . As the  hypothesis in the statement of the theorem means that we have N1 FN = T /m, Baire’s theorem implies the existence of N0  1 such that FN0 has a non-empty interior. Hence there exist a measurable subset A0 ∈ T and an r > 0 such that:     m(AA0 )  r ⇒  ( fn − fp ) dm  ε , ∀ n, p  N0 . A

Now (A0 ∪ B)A0 ⊆ B and (A0 ∩ Bc )A0 ⊆ B, and    f dm = f dm − f dm ; A0 ∪B

B

thus:

A0 ∩Bc

     m(B)  r ⇒  ( fn − fp ) dm  2 ε ,

∀ n, p  N0 .

B

By applying this inequality to B ∩ { fn  fp } and to B ∩ { fn < fp }, we obtain:  m(B)  r ⇒ | fn − fp | dm  4 ε , ∀ n, p  N0 . B

b) As every finite set is uniformly absolutely continuous, it is possible to choose r > 0 such that we moreover have:  | fp | dm  ε B

$ for every p  N0 when m(B)  r. In particular B | fN0 | dm  ε if m(B)  r, which gives:  | fn | dm  5 ε , ∀ n  N0 . m(B)  r ⇒ B

Finally:

 | fn | dm  5 ε ,

m(B)  r ⇒ B

∀n  1.

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c) To show the uniform integrability, it remains to see that supn1  fn 1 < +∞. On one hand, thanks to Lemma II.8, the above implies:  | fn | dm < +∞ sup n1 Si

for i  j0 . On the other hand, for j0 < i  j, the fn ’s are constant on the Si ’s, as the latter are atoms; we thus have:       fn dm = | fn | dm  Si Si $ for these values of i. Hence, since lim Si fn dm exists, we have n→+∞ $ supn1 Si | fn | dm < +∞ also for j0 < i  j. $ j Because S = i=1 Si , it follows that supn1 S | fn | dm < +∞. d) Finally, set:  fn dm ; ν(A) = lim n→+∞ A

we obtain a σ -additive measure ν that is absolutely continuous with respect to m, thanks to the uniform absolute continuity of the sequence ( fn )n1 . The Radon–Nikodým theorem hence implies the existence of a function f ∈ L1 (m) such that ν = f .m. Then, easily, the weak convergence of ( fn )n1 to f in L1 (m) ensues. The characterization of weakly compact subsets now follows: Theorem II.10 (The Dunford–Pettis Theorem) A subset H of L1 (m) is weakly relatively compact if and only if it is uniformly integrable. Proof 1) If H is not uniformly integrable, there exists an ε > 0 and functions fn ∈ H, n  1, such that  | fn | dm  ε {| fn |n}

for any n  1. No subsequence of ( fn )n1 can then be uniformly integrable, and hence, by the Vitali–Hahn–Saks theorem, no subsequence of ( fn )n1 can be weakly convergent. Indeed, if such a subsequence ( fnk )k1 was weakly convergent, for every measurable subset A, there should be a $ limit limk→+∞ A fnk dm = limk→+∞ fnk , 1IA . From the Eberlein–Šmulian theorem, it thus follows that H is not w-relatively compact. w∗ 2) Conversely, if H is uniformly integrable, we consider the closure H of H in the bidual (L1 )∗∗ = (L∞ )∗ for the topology w∗ = σ (L1∗∗ , L∞ ). If F ∈ w∗ H , there exists a filter ( fα )α of elements of H such that

274

7 Some Properties of Lp -Spaces  F(g) = lim α

fα g dm S

for every g ∈ L∞ . However, as H is uniformly absolutely continuous, for every ε > 0 we can find a δ > 0 such that  sup | f |1IA dm  ε f ∈H S

whenever m(A)  δ. Therefore, if we set: ν(A) = F(1IA ) , then ν is a σ -additive measure, and is absolutely continuous with respect to m. The Radon–Nikodým theorem hence implies that F ∈ L1 (m). Thus w∗ H ⊆ L1 , which means that H is weakly relatively compact.

II.2 Unconditionality It is easy to see that L1 (0, 1) is not isometric to a dual space: otherwise, the Krein–Milman theorem (see the Preliminary Chapter, Theorem II.7), applied to the w∗ -topology, would provide extreme points in the unit ball of L1 (0, 1), but there are none. It is a bit more difficult to show that in fact L1 (0, 1) is not isomorphic to a dual space (Gelfand [1938]): one argument is that it does not possess the Radon–Nikodým property, unlike every space isomorphic to a separable dual space: see Chapter 4, Exercise VII.10. By the second James theorem (Chapter 3, Theorem V.4), it thus ensues that L1 (0, 1) does not have an unconditional basis. Actually, more is true (Pełczy´nski [1961]): Theorem II.11 (Pełczy´nski) The space L1 (0, 1) is not isomorphic to any subspace of a space with an unconditional basis. As every separable Banach space is isometric to a subspace of C([0, 1]), we can thus deduce the following: Corollary II.12 The space C([0, 1]) does not have an unconditional basis. Note that this corollary also follows from Theorem V.7 of Chapter 3 and from the weakly sequentially complete character of M([0, 1]) (Theorem II.6 above). The existence of a basis for C([0, 1]) was used to show that every Banach space contains a subspace with a basis. This corollary ruins the hope of using a similar argument to obtain unconditional basic sequences in every Banach space. The question of whether every Banach space contains an unconditional basic sequence long remained open, and was only resolved in 1991, in the

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275

negative, by Gowers and Maurey (independently). Refer to Section VI of Chapter 3 for more information on the subject. The following proof is due to Milman [1971 a]. It uses two preliminary remarks. Let (rn )n1 be the sequence of Rademacher functions: rn (t) = sign[sin(2n π t)] . Then: Lemma II.13 The sequence (rn )n1 of Rademacher functions is a sequence of independent random variables taking on the values −1 and 1 with probability 1/2. Proof The function rn takes alternately the values +1 and −1 on the dyadic intervals of order n; the second assertion is thus clear. For the first assertion, the following remark is useful: let t ∈ [0, 1[, and write its (proper) dyadic expansion: t=

+∞ εn (t)

2n

n=1

,

where εn (t) = 0 or 1. One moment of thought allows us to see that εn (t) = 1−rn (t) . Thus it boils down to showing that the sequence (εn )n1 is indepen2 dent, and this can be done by induction: assume that P(ε1 = a1 , . . . , εn = an ) = 1/2n for every a1 , . . . , an = 0 or 1 (P is the Lebesgue measure), and then take a1 , . . . , an , an+1 ∈ {0, 1}; the essential remark is that I = {t ; ε1 (t) = a1 , . . . , εn (t) = an } is an interval, and that I ∩ {εn+1 = 0} is its first half while I ∩ {εn+1 = 1} is its second half, so that each is of measure (probability) 1/2n+1 : this completes the induction and the proof of the lemma. Now, for every x ∈ L1 (0, 1), two facts must be noted: Fact 1

w

rn x −−→ 0. n→+∞

Proof By the lemma, the sequence (rn )n1 is orthonormal in L2 (0, 1), w because it is independent and centered. Hence rn x −−→ 0 for every x ∈ n→+∞

L2 (0, 1), and thus, by approximation, for every x ∈ L1 (0, 1). Fact 2 Proof

x + rn x1 −−→ x1 . n→+∞

Indeed, 1 + rn  0; hence:  1   x + rn x1 = |x(t)| 1 + rn (t) dt 0

 =

0

by Fact 1.

1

 |x(t)| dt + 0

1

rn (t) |x(t)| dt −−→ x1 , n→+∞

7 Some Properties of Lp -Spaces

276

Proof of the theorem Suppose that L1 (0, 1) ⊆ Y and that Y has an unconditional basis (ek )k1 . We can assume that ek  = 1 for every k  1. Denote by (Pk )k1 the sequence of projections associated with this basis, and by K the basis constant. Take x1 ∈ L1 such that x1 1 = 1. As Pk x1 −−→ x1 (in Y), there k→+∞

exists k1 > 1 such that: x1 − Pk1 x1  

1

·

26 K

w

We have Pk1 (rn x1 ) −−→ 0 because Pk1 has finite rank and rn x1 −−→ 0, by n→+∞ n→+∞ Fact 1. From this and Fact 2, there exists an n1 such that:   1 1 and  x1 + rn1 x1 1 − 1  · Pk1 (rn1 x1 )  7 2 K 4 Set x2 = rn1 x1 ∈ L1 . Then x2 1 = x1 1 = 1, and there exists k2 > k1 such that: 1 x2 − Pk2 x2   7 · 2 K By Facts 1 and 2, there exists n2 such that:   1 Pk2 rn2 (x1 + x2 )   8 2 K and

   x1 + x2 + rn (x1 + x2 )1 − x1 + x2 1   1 · 2 8

Set x3 = rn2 (x1 + x2 ) ∈ L1 . Continuing like this, we construct by induction a sequence (xj )j1 in L1 such that: a) xj − Pkj xj  

1 2j+5 K

for any j  1;

1 b) Pkj−1 (xj )  j+5 for any j  2; 2 K   c)  x1 + · · · + xj+1 1 − x1 + · · · + xj 1   xj+1 1 = x1 + · · · + xj 1

for any j  1.

1 2j+1

and

By c) , for any j  1: 1  xj+1 1 = x1 + · · · + xj 1  2 . 2 If we now set u1 = Pk1 x1 , and, for j  2: uj = Pkj xj − Pkj−1 xj , the sequence (uj )j1 is a sequence of blocks of (ek )k1 and, thanks to conditions a) and b) , we have u1 −x1   1/(26 K)  1/(25 K), and, for j  2:

III The Trigonometric System

uj − xj Y  Pkj xj − xj Y + Pkj−1 xj Y 

1 2j+5 K

277

+

1 2j+5 K

=

1 2j+4 K

·

As K  1, this implies: 1 1 1 −  · 2 2j+4 4 Next, as the constant of the basic sequence (uj )j1 is also less than or equal to to K, the constant of (ek )k1 , we have u∗j  uj   2K, and hence u∗j   8K. We obtain: uj Y  xj 1 −

+∞

1

2j+4

u∗j  uj − xj  

j=1

+∞



8K ·

j=1

1 1 = < 1. 2 2j+4 K

The basic sequences (uj )j1 and (xj )j1 are thus equivalent. In particular, (xj )j1 is unconditional, and with C its unconditional constant, we have:       n   n     a x  C sup(|a |, . . . , |a |) x j j 1 n  j   2 C sup(|a1 |, . . . , |an |) .  j=1

1

j=1

1

Moreover, as xk∗   2 C/xk 1  4 C, we also have:    n  |a |   n 1  ∗  k    , x a x  a x j j j j =  k  4 C 4 C 1 j=1

and thus

j=1

    n 1  aj xj    4C sup(|a1 |, . . . , |an |) ·  1 j=1

The sequence (xj )j1 is hence equivalent to the canonical basis of c0 . However, this cannot be possible, as L1 does not contain c0 (because, for example, L1 is w.s.c., or because L1 has cotype 2 while c0 does not).

III The Trigonometric System In this section, the trigonometric system is shown to be a basis of Lp = Lp (0, 2π ) = Lp (T) for 1 < p < +∞. In fact this is an immediate consequence of the Marcel Riesz theorem. Hence we first study the Riesz projection and the Hilbert transform.

III.1 The Riesz Projection Definition III.1 The Riesz projection is the transformation associating to every trigonometric polynomial f the trigonometric polynomial Rf defined by:

7 Some Properties of Lp -Spaces

278

Rf (t) =



 f (n) eint .

n0

The Hilbert transform H f of f is defined by: (−i) sign(n) f (n) eint . H f (t) = n∈Z

In the preceding definition, by convention sign(0) = 0. These two transformations are linked by the following relations: Rf =

 1  f (0)1I + ( f + iH f ) 2

  H f = i f + f (0)1I − 2Rf .

and

The Hilbert transform is related to the notion of conjugate function: to every real harmonic function u on the unit disk D = {z ∈ C ; |z| < 1} corresponds a unique real harmonic function v with v(0) = 0 such that h = u + iv is holomorphic in D; v is called the conjugate function of u. Now let f be a continuous real-valued function on T, and let Pr (t) =

+∞

r|n| eint

n=−∞

be the Poisson kernel (0  r < 1). Then, for z = reiθ , the function u defined by:  dt Pr (θ − t)f (t) u(z) = (Pr ∗ f )(θ ) = 2π T is harmonic in the unit disk D, and lim u(reiθ ) = f (θ )

r→1

for every θ ∈ T. Moreover, the function h defined by  2π it 1 e + z it h(z) = f (e ) dt 2π 0 eit − z is holomorphic in D and u = Re h. As +∞

eit + z e−int zn , =1+2 it e −z n=1

we have: h(z) =  f (0) + 2

+∞ n=1

 f (n)zn .

III The Trigonometric System

279

Hence, for every real-valued trigonometric polynomial f : h∗ (θ ) ≡ lim h(reiθ ) = f (θ ) + i(H f )(θ ) , r→1

and H f thus appears as the radial limit of v. For all this, we refer, for example, to Rudin 2, Chapter 11 (see also the beginning of Section V). Note, however, that when f is a trigonometric polyno int mial, all these properties are immediate; for example, if f (t) = N n=−N an e , N then h(z) = a0 + 2 n=1 an zn . Now we have the following result (M. Riesz [1927]): Theorem III.2 (The Marcel Riesz Theorem) For 1 < p < +∞, there exists a constant Cp > 0 such that, for every trigonometric polynomial f : Rf p  Cp  f p . The Riesz projection and the Hilbert transform can thus be extended, for 1 < p < +∞, as continuous operators on Lp (T). Note that for p = 2, R is clearly the orthogonal projection of L2 (T) onto H 2 (T) = { f ∈ L2 (T) ;  f (n) = 0 ∀ n < 0} . An immediate consequence is the following: Corollary III.3 of Lp (0, 2π ).

For 1 < p < +∞, the trigonometric system forms a basis

III.2 The Marcinkiewicz Theorem and the Marcel Riesz Theorem A direct proof of Theorem III.2 could be given (see Rudin 2, Chapter 17; Rudin 1, § 8.7.4; Lasser, Theorem 8.6; or Zygmund, Chapter VII, for example); however, a reasoning by interpolation is more instructive. The next section treats the Riesz–Thorin theorem, the foundation of complex interpolation; here the Marcinkiewicz theorem is used, the basis of the theory of real interpolation. Definition III.4 Let (S, T , m) be a measure space, and T a linear map sending Lp (m) into L0 (m), with 1  p < +∞. The operator T is said to be of weak type (p, p) if there exists a constant M > 0 such that, for any a > 0: M  f pp . ap Clearly, by the Markov–Tchebychev inequality, every continuous operator of Lp (m) into itself is of weak type (p, p). m({x ∈ S ; |(Tf )(x)| > a}) 

7 Some Properties of Lp -Spaces

280

Theorem III.5 (The Marcinkiewicz Theorem) If T is an operator simultaneously of weak type (p1 , p1 ) and of weak type (p2 , p2 ), with 1  p1 < p2 < +∞, then T defines a continuous operator from Lp (m) into itself, for every p with p1 < p < p2 . To prove the Marcel Riesz theorem, we use its triviality for p = 2, as well as the following theorem: Theorem III.6 (Kolmogorov’s Theorem) type (1, 1).

The Riesz projection is of weak

Recall that the Riesz projection does not define a continuous operator from L1 (T) into itself, as the norm of the Dirichlet kernel tends to infinity (see Chapter 4 of this volume). First we prove this theorem of Kolmogorov. We follow Lasser, Chapter XIII. Proof First of all, we have to show that Rf makes sense for every f ∈ L1 (T). Admit that we have found a constant M > 0 such that, for any a > 0: M  f 1 a for every trigonometric polynomial f . Let Fn be the Fejér kernel of order n.    1 As Fn ∗ f −−→ f , the sequence R( fn ∗ f ) n1 is Cauchy for the convergence m({|Rf | > a}) 

(∗)

n→+∞

in measure, and hence converges in measure in L0 (T) (see Chapter 1 of this volume). It is this limit that is called Rf . Clearly R thus extended is linear, and coincides on L2 (T) with the orthogonal projection onto H 2 (T). Moreover, the weak type (1, 1) inequality is clearly preserved for this extension. We turn now to the proof of (∗). 1) First consider the case where f is a non-negative trigonometric polynomial, not identically null. Note that, as f is non-negative,  f (0) =  f 1 > 0. By homogeneity, we may assume  f (0) = 1. We use the polynomial defined in the preceding section:   f (n) zn = 1 + 2 f (n) zn . h(z) =  f (0) + 2 n1

n1

As f  0, f ≡ 0 and Pr (t) > 0 for every t ∈ T, then:  iθ Re h(re ) = Pr (t)f (θ − t) dm(t) > 0 T

for any r < 1. Next, for any λ > 0, we can define a function ϕλ , holomorphic in a neighborhood of the closed unit disk D, by: ϕλ (z) =

h(z) − λ + 1. h(z) + λ

III The Trigonometric System

281

w−λ + 1 is a conformal map from the half-plane w+λ {w ∈ C ; Re w > 0} onto the disk {ζ ∈ C ; |ζ − 1| < 1}, and it sends the set {w ∈ C ; Re w > 0 , |w| > λ} onto {ζ ∈ C ; |ζ − 1| < 1 , Re ζ > 1}. Hence, on one hand, Re ϕλ (eiθ )  0 for any θ ∈ [0, 2π ] and, on the other hand, Re ϕλ (eiθ )  1 whenever |h∗ (θ )| = |h(eiθ )| > λ. Thus, by Cauchy’s formula:   ∗ iθ Re ϕλ (e ) dm(θ )  Re ϕλ (eiθ ) dm(θ ) m({|h | > λ})  {|h∗ |>λ} T   = Re ϕλ (eiθ ) dm(θ ) = Re ϕλ (0) = ϕλ (0) The function w →

T

2 1−λ +1= · = 1+λ 1+λ However h∗ = 2 Rf − 1I, and hence {|Rf | > a} ⊆ {|h∗ | > 2a − 1} (note that Re h∗ > 0 implies Re Rf > 1/2, so that only the case a > 1/2 needs to be considered). So: m({|Rf | > a})  m({|h∗ | > 2a − 1}) 

1 2 = , 1 + (2a − 1) a

which is (∗) since  f 1 = 1. 2) Let f be a non-negative continuous function on T. As the Fejér kernel Fn is positive, f ∗ Fn is a positive-valued trigonometric polynomial; then, by 1) : m(|R( f ∗ Fn )| > a) 

1  f ∗ Fn 1 . a

However, as f ∈ L2 (T), the projection Rf exists, and R( f ∗ Fn ) =  2 Fn ∗ Rf −−→ Rf ; in particular we have convergence in measure; hence: n→+∞

m({|Rf | > a}) = lim m({|R(Fn ∗ f )| > a}) n→+∞

1 1 lim Fn ∗ f 1 =  f 1 . a n→+∞ a 3) Finally, when f is a real-valued trigonometric polynomial, we decompose it into f = f + − f − , and apply 2) to f + and to f − . When f is arbitrary, we apply the above to the trigonometric polynomials Re f and Im f . 

Remark

For 0 < α < 1, set  f α =  . αα



1/α | f | dm . α

T

is a distance on Lα . Moreover, denote:    f 1,∞ = supt>0 t m {| f | > t} .

This is not a norm, but

7 Some Properties of Lp -Spaces

282

Hence: Proposition III.7 For 0 < α < 1, there exists a constant Cα > 0 such that  f α  Cα  f 1,∞ ; consequently, the Riesz projection R is continuous from L1 (T) into Lα (T). Proof With the formula of integration by parts seen in Chapter 1 (Proposition IV.2), we write, for a > 0:  +∞  α | f (x)| dm(x) = α tα−1 m({| f | > t}) dt T 0  a =α tα−1 m({| f | > t}) dt 0  +∞ +α tα−1 m({| f | > t}) dt a  a  +∞ α−1 α t dt + α tα−2  f 1,∞ dt 0

a

α aα−1  f 1,∞ . =a + 1−α α

Now, selecting a =  f 1,∞ , we obtain  f α  1/(1 − α)1/α  f 1,∞ , as stated. We can now present the proof of the Marcinkiewicz theorem: Proof of the Marcinkiewicz theorem to estimate the function:

Let f ∈ Lp (m), and let t > 0; our goal is

σ (t) = m({x ∈ S ; |(Tf )(x)| > t}). Set:

 h(x) =

f (x) if | f (x)| > t 0

if | f (x)|  t

and g(x) = f (x) − h(x). We have:   p1 |h| dm = | f |p1 dm S {| f |>t}   1 1 p  p−p | f | dm  p−p | f |p dm , t 1 {| f |>t} t 1 S hence h ∈ Lp1 (m). Likewise:    |g|p2 dm = |g|p |g|p2 −p dm  tp2 −p | f |p dm , S

S

S

hence g ∈ Lp2 (m). Then, using the weak types (p1 , p1 ) and (p2 , p2 ) of T, we obtain:

III The Trigonometric System

283

σ (t)  m({|(Th)(x)| > t/2}) + m({|(Tg)(x)| > t/2}) 2p1 M1 2p2 M2  p hpp11 + p gpp22 . t1 t2 The formula of integration by parts (Chapter 1, Proposition IV.2) provides:  +∞ p tp−1 σ (t) dt Tf p = p 0   +∞ p−1   t p1 p1  p 2 M1 | f (x)| dm(x) dt t p1 {| f (x)|>t} 0   +∞ p−1   t p2 p2 | f (x)| dm(x) dt + p 2 M2 t p2 {| f (x)|t} 0   | f (x)|   tp−p1 −1 dt dm(x) = p 2p1 M1 | f (x)|p1 S 0   +∞   tp−p2 −1 dt dm(x) + p 2p2 M2 | f (x)|p2 S | f (x)|  p p1 p = 2 M1 | f (x)| dm(x) p − p1 S  p + 2p2 M2 | f (x)|p dm(x) p2 − p S = Mp  f pp . Proof of the Marcel Riesz theorem For 1 < p  2, the theorem follows from the above. For 2  p < +∞, we argue by duality. Indeed, clearly, if f and g are two trigonometric polynomials: f ∗ (Rg) = (Rf ) ∗ g (it suffices to examine their Fourier coefficients); thus:    f (x) Rg (−x) dm(x) = [f ∗ (Rg)](0) T   = [(Rf ) ∗ g](0) = Rf )(x) g(−x) dm(x) . T

Consequently, with p∗ the conjugate exponent of p:       Rgp = sup  f (x) Rg (−x) dm(x) = 

 f p∗ =1

T

 f p∗ =1

T

      sup  Rf )(x) g(−x) dm(x)

sup Rf p∗ gp  Cp∗ gp .

 f p∗ =1

7 Some Properties of Lp -Spaces

284

III.3 Conditionality of the Trigonometric System To end this section, we see that, except for p = 2, the trigonometric system is not unconditional. We begin with the following: Proposition III.8 If X is a space of cotype q < +∞, then the unconditional +∞  q convergence of +∞ n=1 xn implies n=1 xn  < +∞. As Lp (m) is of cotype 2 when 1  p  2, we obtain: Corollary III.9 (Orlicz’s Theorem) If 1  p  2 and if  2 unconditionally in Lp (m), then +∞ n=1 xn  < +∞. Proof

+∞

n=1 xn

converges

Let (rn )n1 be the sequence of Rademacher functions. Then:  N

1/q xn 

q

n=1

  Cq (X)

 N 1 

0

 

n=1

 N      Cq (X) sup  rn (t)xn   0t1

since supN1 supθn =±1

2 1/2  rn (t)xn   dt

n=1

 Cq (X) C ,  N   θn xn   C. n=1

Proposition III.10 The trigonometric system is not an unconditional basis of Lp (0, 2π ), except when p = 2. Proof When p = 1, we already know that it is not even a basis. When 1 < p < +∞, we can even assume 1 < p  2, thanks to Proposition III.8 of Chapter 3. Indeed, with 1/p + 1/q = 1, the system of coordinate linear functionals of the trigonometric system in Lq (0, 2π ) is the trigonometric system in Lp (0, 2π ). But then, by the Orlicz theorem, the unconditional   int in Lp (0, 2π ) implies that 2 convergence of f (t) = n∈Z an e n∈Z |an | < 2 +∞, i.e. that f ∈ L (0, 2π ).

IV The Haar Basis in Lp This section presents a closer study of the Haar basis in Lp (0, 1) when 1 < p < +∞, and in particular a proof of its unconditionality. To achieve this, we will see that this basis appears as a sequence of martingale differences. So we first prove results on martingales in Lp , focusing particularly on their maximal functions and their square functions. For this, it is convenient to first handle the

IV The Haar Basis in Lp

285

cases of special values of p, and then to conclude by an interpolation between those values. This explains why we start with a proof of the Riesz–Thorin interpolation theorem, the first and most elementary result on interpolation.

IV.1 The Riesz–Thorin Interpolation Theorem The following result, a simple version of what is known as the Phragmén– Lindelöf principle, is needed here. Theorem IV.1 (The Three-Lines Theorem) Let B be the open vertical strip {z ∈ C ; 0 < Re z < 1}, and f : B → C a continuous function, holomorphic and bounded in B, not identically null. When 0  θ  1, set: Mθ = sup | f (θ + iy)| . y∈R

Then, for any θ ∈ [0, 1]: 0 < Mθ  M01−θ M1θ . Proof

We first show that: sup | f (z)| = sup | f (z)| .

(1)

z∈∂B

z∈B

For this, fix z0 ∈ B, as well as an ε > 0. We can obviously assume that f (z0 ) = 0. Consider the function: fε (z) =

f (z) · 1 + εz

On one hand, for z ∈ B, |1 + εz|  Re(1 + εz) = 1 + ε Re z  1, hence | fε (z)|  | f (z)|. On the other hand, since f is bounded, fε tends to 0 as |z| tends to infinity; thus there exists T > | Im z0 | such that | fε (z)| < | fε (z0 )| as soon as | Im z|  T. With BT = {z ∈ B ; | Im(z)| < T}, we thus obtain: | fε (z0 )|  sup | fε (z)| = max | fε (z)|, z∈∂BT

z∈BT

thanks to the maximum principle. But this maximum cannot be attained on the horizontal sides of ∂BT , because of the choice of T. It is hence attained on one of the vertical sides, and so: max | fε (z)|  sup | fε (z)|  sup | f (z)| = max{M0 , M1 } .

z∈∂BT

z∈∂B

z∈∂B

In particular | fε (z0 )|  max{M0 , M1 }. Then, letting ε tend to 0, we obtain | f (z0 )|  max{M0 , M1 }, thus (1).

286

7 Some Properties of Lp -Spaces

Now let M0 > M0 and M1 > M1 , so that M0 > 0 and M1 > 0 (in fact, we will see later that M0 and M1 are themselves > 0), and consider the real numbers a = log M1 − log M0 and b = log M0 , whereby aθ + b = (1 − θ ) log M0 + θ log M1 . We apply the inequality (1) to g(z) = f (z) e−(az+b) . As M0 e−b = 1 and M1 e−(a+b) = 1, we obtain Mθ e−(aθ+b) = supy∈R |g(θ + iy)|  1, or Mθ  eaθ+b = M0 1−θ M1 θ . Letting M0 tend to M0 and M1 to M1 , we obtain Mθ  M01−θ M1θ , at least when 0 < θ < 1. To conclude, the principle of isolated zeros shows that Mθ > 0 for 0 < θ < 1: indeed, if we had Mθ = 0, f would be null on the whole line Re z = θ , and hence everywhere null. Then the previous inequality shows that M0 > 0 and M1 > 0 (and then Mθ  M01−θ M1θ also for θ = 0 or 1; without this remark, this would be true by setting 00 = 1). We now proceed to the Riesz–Thorin interpolation theorem. The framework is the following: Consider a measure space (S, , μ) with μ a σ -finite positive measure, and n a linear map T, defined on the space of simple functions f = k=1 ak 1IAk (μ(Ak ) < +∞) of (S, , μ), and with values in L0 (S, , μ). As the space of simple functions is dense in each Lp for p < +∞, if T has a continuous extension (again denoted T) to such an Lp , this extension is unique. We have (the spaces being complex): Theorem IV.2 (The Riesz–Thorin Theorem) Let 1  p0 < p1  +∞ and 1  q0 < q1  +∞, and suppose that the operator T continuously maps Lp0 into Lq0 , with norm M0 , and Lp1 into Lq1 , with norm M1 . Then, for every θ ∈ ]0, 1[, T continuously maps Lp into Lq , with norm  M01−θ M1θ , where p and q are defined by 1−θ θ 1 = + p p0 p1

1−θ 1 θ = + · q q0 q1

and

Proof Denote by r∗ the conjugate exponent of r ∈ [1, +∞]. Let f ∈ Lp ∗ and g ∈ Lq be simple functions such that  f p = gq∗ = 1. We can write ∗ f = u F 1/p and g = v G1/q , where F and G are strictly positive and with  integral 1, and where |u|  1 and |v|  1. Indeed, if f = nk=1 ak 1IAk , with  ak = 0 for any k, it suffices to take F = |ak |p on Ak and 1 on ( k Ak )c , and  u = eiαk on Ak and 0 on ( k Ak )c . Proceed similarly for g. Then the brackets of duality can be written:     ∗ Tf , g = (Tf )g dμ = T uF 1/p v G1/q dμ . S

S

IV The Haar Basis in Lp

As

287

1 1−θ θ = + ∗ , we obtain: ∗ ∗ q q0 q1 Tf , g = (θ ) ,

where:

 (z) =

1−z z  1−z + z  ∗ + ∗ T u F p0 p1 v G q0 q1 dμ .

S

This function  clearly satisfies the hypothesis of the three-lines theorem. Now:  1−iy iy  1−iy + iy  ∗ + ∗ (iy) = T u F p0 p1 v G q0 q1 dμ = T( fy ), gy , S

where the simple functions fy and gy are such that | fy |  F 1/p0 and |gy |  ∗ G1/q0 , and hence such that  fy p0  1 and gy q∗0  1 (since F and G have integral 1). Therefore: |(iy)|  T( fy )q0 gy q∗0  TLp0 →Lq0  fy p0 gy q∗0  M0 . Similarly: |(1 + iy)|  M1 . The three-lines theorem thus implies: | Tf , g | = |(θ )|  M01−θ M1θ . Taking the upper bound over all possible g, we get Tf q  M01−θ M1θ , which is the stated result. Remark This result remains true for real spaces provided that we increase the ˜ f + ig) = upper bound M01−θ M1θ by a factor 2: it suffices to complexify T by T( ˜ Tf + iTg and to note that T  T  2 T. Actually, if p0  q0 and p1  q1 , this extra factor 2 can be omitted, by using another method (see Bennett–Sharpley, Chapter 4, Theorem 1.7). Corollary IV.3 (The Hausdorff–Young Theorem) For 1  p  2, the Fourier transform F, defined by:  e−2π i x,y f (y) dy, Ff (x) = Rn

p∗

continuously maps Lp (Rn ) into L (Rn ), and has norm  1. Proof It is well known (and immediate) that F maps L1 (Rn ) into L∞ (Rn ) continuously, with norm 1. Moreover, by Plancherel’s theorem, F is an isometry from L2 (Rn ) into itself. An application of the Riesz–Thorin theorem with p0 = 1, q0 = ∞ and p1 = q1 = 2 thus suffices to conclude because if (1 − θ ) θ 1 θ 1 1−θ = + , then ∗ = + . p 1 2 p ∞ 2

7 Some Properties of Lp -Spaces

288

Remark Beckner [1975] showed that FLp →Lp∗ is < 1 for 1 < p < 2, and gave an explicit formula for this norm. See also Brascamp and Lieb [1976 b] and Barthe [1998].

IV.2 The Martingale Square Function Let (Mn )1nN be a martingale, finite for now, adapted to a filtration A1 ⊆ . . . ⊆ AN . Set M0 = 0 and denote by dn = Mn − Mn−1 (1  n  N) the differences of this martingale. The associated square function is: S=

 N

1/2 dn2

.

n=1

We will now see that the Lp -norms of S and of MN have the same behavior. A few preliminaries are necessary. Theorem IV.4 (Doob) For every finite martingale M1 , . . . , MN ∈ Lp , with 1 < p < +∞, the maximal function M ∗ = max(|M1 |, . . . , |MN |) satisfies: M ∗ p  p∗ MN p . Proof This is a consequence of Doob’s maximal inequality (Theorem V.1 of Chapter 1) and of the following lemma: indeed, by Doob’s inequality, X = M ∗ and Y = |MN | satisfy the hypothesis of this lemma. Let X, Y ∈ Lp be non-negative random variables such that:   P(X > t)  (1/t)E Y1I(X>t)

Lemma IV.5

for any t > 0. Then Xp  p∗ Yp . Proof

We have: 

+∞

E(X ) = p



p tp−1 P(X > t) dt

0



+∞

pt

p−2

    E Y1I(X>t) dt = E Y

0



X

pt

p−2

dt

0

 1/p∗ = p∗ E(YX p−1 )  p∗ Yp E(X p ) . 1/p∗  . The result ensues after a division by E(X p )

The following theorem plays a major role in the study of the behavior of the square function.

IV The Haar Basis in Lp

289

Theorem IV.6 For every p with 1 < p < +∞, there exists a constant Cp  1 such that: Cp−1 MN p  Sp  Cp MN p

(∗)

for any p-integrable martingale (M1 , . . . , MN ). Moreover, we can find Cp = O(p2 ). Proof The method, due to Cotlar [1955], follows this plan: first, we prove the inequality for p = 2, 4, 8, . . ., then for any p  2 by linearization and Riesz–Thorin interpolation, and finally by duality for 1 < p < 2. From now on, we denote M = MN . a) The case p = 2 is easy: the differences dn are orthogonal; indeed, for k < n we have:     E(dk dn ) = E EAk (dk dn ) = E dk EAk (dn ) = E(dk . 0) = 0. So then: E(M ) = 2

N

E(dn2 )

=E

 N

n=1

=S +2 2

n=1



dk dn

1k 0 such that, for any N  1, every sequence (a1 , . . . , aN ) of scalars and every choice of signs θ1 , . . . θN ∈ {−1, 1}:      N   N     θn an dn   Kp  an dn    . n=1

p

n=1

p

But this follows immediately from Theorem IV.6, as the martingales defined   by Mn = nk=1 ak dk and Mn = nk=1 θk ak dk for 1  n  N have the same  N  2 2 1/2 . We thus have: square function S = n=1 an dn MN p  Cp Sp  Cp2 MN p . In particular, we obtain the following result: Theorem IV.8 (Paley) For 1 < p < +∞, the Haar basis (hn )n1 of Lp (0, 1) is unconditional. Proof Let An = σ (h1 , . . . , hn ) be the σ -algebra generated by h1 , . . . , hn . In particular, since h1 = 1I, A1 is the trivial σ -algebra, and (hn )n1 is adapted to the filtration (An )n1 , i.e. hn is An -measurable for any n  1. It thus suffices to show that the hn ’s $are martingale differences, i.e. that EAn (hn+1 ) = 0. This means showing that A hn+1 (t) dt = 0 for every A ∈ An . The case when A is an atom of An suffices. For this, we write n + 1 = 2k + l, with 1  l  2k , and  2l − 2 2l  set I = . We distinguish two cases: , 2k+1 2k+1 m m + 1 a) either A is an interval j , j with j  k; then either A ∩ I = ∅ or 2 2 $ A ⊇ I; as the$support of hn+1 is contained in I and as I hn+1 (t) dt = 0, we indeed have A hn+1 (t) dt = 0 in both cases;  m m + 1 b) or A = k+1 , k+1 , with m + 1 < 2l − 2; in this case A ∩ I = ∅, hence 2 2 $ hn+1 = 0 on A, and consequently A hn+1 (t) dt = 0. Remark Burkholder [1984] showed that the unconditional constant for the Haar system is better than the constant provided by Theorem IV.6: it is exactly equal to max{p − 1, p∗ − 1}, instead of Kp = O(p4 ) in the general case. The proof of Burkholder [1988] is elementary and interesting, but is based on the use of a biconcave function whose introduction is hard to grasp. The first proof of Theorem IV.8 was given by Paley [1932].

7 Some Properties of Lp -Spaces

294

IV.4 The Extremal Character of the Haar Basis The Haar basis possesses a remarkable property of “reproducibility”, which makes it extremal among all unconditional bases of Lp . Theorem IV.9 Let (un )n1 be an unconditional basis of Lp (0, 1), with 1 < p < +∞, and K its unconditional constant. Then, for any ε > 0, (un )n1 has a block basis (vn )n1 which is (1 + ε)-equivalent to the Haar basis (hn )n1 . In particular, Kp  K, where Kp is the unconditional constant of (hn )n1 in Lp (0, 1). The last assertion follows from the fact that the unconditional constant of (vn )n1 is  K; hence, if θk ∈ {−1, 1} and if a1 , . . . , an are scalars:  n     n  n             θ a h  (1 + ε) θ a v  K(1 + ε) a v k k k k k k k k    k=1

p

k=1

p

   n  2  K(1 + ε)  ak hk   , k=1

k=1

p

p

which gives Kp  K(1 + ε)2 ; thus Kp  K, by letting ε tend to 0. For the essential part of the proof, we need a classical theorem of convexity due to Lyapunov. Theorem IV.10 (Lyapunov’s Theorem) Let μ1 , . . . , μn be atomless real measures on a measurable  space (S, T ). Then  the range of the map  : T → Rn defined by (A) = μ1 (A), . . . , μn (A) is convex and compact in Rn . In particular, for every A ∈ T , there exists B ∈ T such that μk (B) = μk (A)/2 for any k = 1, . . . , n. Proof Set ν = |μ1 |+· · ·+|μn |. This is a finite positive measure. The Radon– Nikodým theorem guarantees the existence of functions ϕk ∈ L1 (ν) such that μk = ϕk .ν. We define a linear mapping T : L∞ (ν) → Rn by:   T( f ) = f ϕk dν . 

1kn

This map is w∗ -continuous. Let P = { f ∈ L∞ (ν) ; 0  f  1I} be the positive part of the unit ball of L∞ (ν). It is w∗ -closed (since f ∈ P if and only if  f ∞  1 and 1I−f ∞  1), hence w∗ -compact. Thus T(P) is a compact and convex subset of Rn . We now show that (T ) = T(P) to complete the proof. Clearly, (T ) ⊆ T(P): indeed, if A ∈ T , then 1IA ∈ P and (A) = T(1IA ). To prove the reverse inclusion, we select x ∈ T(P), and define: Px = { f ∈ P ; T( f ) = x}.

IV The Haar Basis in Lp

295

This entails showing that Px contains an indicator function 1IA . Now Px is clearly convex, and as T is w∗ -continuous, Px is also w∗ -compact. By the Krein–Milman theorem, Px possesses extreme points. We will show that every extreme point of Px is an indicator function. Suppose f ∈ Px is not an indicator function. This means that f = f 2 , or f (1I − f ) = 0. Hence there exist A ∈ T and r > 0 such that ν(A) > 0 and r  f  1 − r on A. We consider the subspace X of L∞ (ν) of functions that are zero (almost everywhere) outside A. As ν has no atoms, X is infinitedimensional (since every measurable subset of A with measure > 0 can be decomposed into two other subsets of measure > 0). The restriction of T to X cannot be injective, and thus there exists g ∈ X, non-zero, such that T(g) = 0. We can multiply g by a constant in order to have g∞  r. But then f + g and f − g are in Px , and hence f is not extremal. Proof of Theorem IV.9 Set w1 = h1 = 1I. As (un )n1 is a basis of Lp , we can 1 ak uk such that: find a v1 = kk=1 εw1 p · v1 − w1   23 K Let m be the Lebesgue measure on (0, 1), and let u∗k ∈ Lq = (Lp )∗ , for k  1, be the coordinate linear functionals of the basis (un )n1 . We apply Lyapunov’s theorem to the measures m and u∗k .m of density u∗k on the interval $ [0, 1], for 1  k  k1 . We can find A ⊆ [0, 1] such that m(A) = 1/2 and A u∗k dm = $ + − 1 1 ∗ 2 0 uk dm for 1  k  k1 . Set I0 = A and I0 = [0, 1]  A; then:  1 1 and m(I0+ ) = m(I0− ) = u∗k (1II + − 1II − ) dm = 0 , 1  k  k1 . 0 0 2 0 If we set w2 = 1II + − 1II − , the function w2 does not have any components on 0 0 k 2 the basis (un )n1 before the index k1 ; thus there exists v2 = k=k1 +1 ak uk such that: εw2 p · v2 − w2 p  24 K We continue by induction in the same way: suppose we have constructed k j blocks v1 , . . . , vn−1 , with vj = k=kj−1 +1 ak uk and functions w1 , . . . , wn−1 taking only the three values ±1, 0 such that: m(wj = −1) = m(wj = 1) = 2−rj when j = 2rj + sj  2, 0  sj  2rj − 1, and such that: vj − wj p 

εwj p · 2j+2 K

7 Some Properties of Lp -Spaces

296

To perform step n, we write n = 2r + s with 0  s  2r − 1 and:   s s+1 ; In = r , r 2 2 we apply Lyapunov’s theorem to In and to the measures m and u∗k .m, with 1  k  kn−1 , to find two subsets In+ and In− ⊆ In such that: m(In+ ) = m(In− ) = and

 0

1

1 m(In ) 2

u∗k (1IIn+ − 1IIn− ) dm = 0 ,

1  k  kn−1 .

Let wn = 1IIn+ −1IIn− ; the function wn does not have any component on the basis k n (un )n1 before the index kn−1 , and thus there exists vn = k=kn−1 +1 ak uk such that: εwn p (1) vn − wn p  n+2 · 2 K This completes the proof: indeed, (vn )n1 is a sequence of blocks of (un )n1 , and, by (1), (wn )n1 is 1+ε 1−ε -equivalent to (vn )n1 . Finally, the construction of (wn )n1 shows that this sequence is isometrically equivalent to the Haar basis (hn )n1 , since its construction has been mimicked by that of (hn )n1 : they have the same distribution.

V Another Proof of Grothendieck’s Theorem An alternative proof of Grothendieck’s theorem is given here: it is due to Pełczy´nski and Wojtaszczyk, and it makes this theorem appear as a relatively simple consequence of classical results in Harmonic Analysis. We begin by showing them.

V.1 The Spaces Hp Let H(D) be the space of holomorphic functions in the open unit disk D of C. For 1  p  +∞, denote:   2π    dθ 1/p H p (D) = F ∈ H(D) ;  f p = sup | F(reiθ )|p < +∞ 2π 0 r 0, the uniform continuity of Bn on D gives an r < 1 such that |Bn (z)|  1 − ε for r  |z|  1. Consequently, as F/Bn is holomorphic in D, we obtain:      2π    F(0)   2π F(reiθ dθ   F(reiθ )  dθ 1  =     B (0)    B (reiθ )  2π  1 − ε F1 . iθ 2π  B (re n n n 0 0 Hence: n 

|zj | = |Bn (0)|  (1 − ε)

j=1

|F(0)| ; F1

thus we obtain the convergence of the infinite product Blaschke condition.



n1 |zn |,

and the

Proof of Theorem V.1 By the lemma, if k is the order of the zero of F at 0, and (zn )n1 the sequence of its non-null zeros, counted with their multiplicities, then the infinite product: B(z) = zk

+∞  n=1

zn zn − z |zn | 1 − zn z

converges uniformly on every compact subset of D; the function K = F/B is then holomorphic and does not vanish in D. Hence there exists G ∈ H(D) such that F/B = G2 . Setting H = BG, we have F = GH. It remains to see that G 1/2 and H ∈ H 2 (D) with G2 = H2 =  f 1 . Since F1  G2 H2 (by the Cauchy–Schwarz inequality), and since |B(z)|  1 for any z ∈ D implies H2  G2 , it suffices to show that G22  F1 , i.e. K1  F1 . For this, set Kn (z) = F(z)/zk Bn (z). Fix r < 1. As in the lemma, for any ε > 0, we can find an s with r  s < 1 such that |Bn (seiθ )|  1 − ε for any θ . Then, because M1 (Kn ) is increasing:  2π  2π 1 dθ dθ   F1 . |Kn (reiθ )| |Kn (seiθ )| 2π 2π 1 − ε 0 0

V Another Proof of Grothendieck’s Theorem

299

With ε tending to 0, this gives:  2π dθ  F1 , |Kn (reiθ )| 2π 0 and then:

 M1 (K, r) =

2π

|K(reiθ )|

0

dθ  F1 , 2π

by letting n tend to infinity, and using the uniform convergence of Kn to K on every compact subset of D. Finally K1  F1 , as claimed. Theorem V.3 Let Pr be the Poisson kernel on D. For 1  p  +∞, the map J : H p (T) → H p (D), sending f ∈ H p (T) to the function F defined by F(reiθ ) = (Pr ∗ f )(θ ), is an isometric isomorphism. Moreover, for 1  p < +∞, we have Pr ∗ f − f p − → 0. r→1

+∞

f (n)rn einθ . We first show that the map Proof Recall that (Pr ∗ f )(θ ) = n=0  p J is an isometry. Fix f ∈ H (T). For 1  p < +∞, the translations τf : t → ft (where ft (θ ) = f (θ − t)) are continuous. Since Pr is a probability density, then:  2π dθ |( f ∗ Pr )(θ ) − f (θ )|p 2π 0   2π  2π   2π  dt dt p dθ  − f (θ ) f (θ − t)Pr (t) Pr (t) =  2π 2π  2π 0 0 0   2π   2π dt dθ  | f (θ − t) − f (θ )|p Pr (t) 2π 2π 0 0 (by Hölder’s inequality for the measure with density Pr )  2π   2π dt p dθ Pr (t) | f (θ − t) − f (θ )| = 2π 2π 0 0  2π dt =  ft − f pp Pr (t) − → 0, r→1 2π 0 

p

as t →  ft − f p is continuous. p p In particular, Mp (F, r) =  f ∗ Pr p − →  f p . r→1

When p = ∞, there is no longer convergence in norm, but instead weak∗ convergence: f = w∗ - limr→1 Pr ∗ f in H ∞ (T) (thanks to the convergence in norm in H 1 ); this is sufficient to imply:  f ∞  lim inf Pr ∗ f ∞ = lim M∞ (Jf , r) = Jf ∞ . r→1

r→1

As the inequality Pr ∗ f ∞   f ∞ is evident, we obtain the equality Jf ∞ =  f ∞ .

7 Some Properties of Lp -Spaces

300

It remains to see that J is surjective. We begin with the reflexive case 1 < p < +∞.  n With F ∈ H p (D), we write F(z) = +∞ n=0 an z . Let (rk )k1 be a sequence of p positive numbers tending to 1 such that Mp (F, rk ) −−→ Fp . For each k  1, k→+∞  n inθ we have F(rk eiθ ) = +∞ n=0 an rk e , and the convergence is uniform in θ . We thus have: p  2π   2π  N    n inθ   F(rk eiθ )p dθ = lim an rk e  dθ .  N→+∞ 0

0

n=0

Hence there exists a strictly increasing sequence of integers (nk )k1 such nk an rkn einθ , the trigonometric polynomials fk satisfy that, setting fk (θ ) = n=0  fk p −−→ Fp . In particular, the sequence ( fk )k1 is bounded in Lp (T), and k→+∞

there exists a subsequence converging weakly to a function f ∈ H p (T). As f (n) = an for n  0 and  f (n) = 0 rk −−→ 1, the Fourier coefficients of f are  k→+∞ for n < 0. Hence Jf = F. When p = ∞, we can use the same argument, taking a weak∗ limit of the sequence ( fk )k1 in L∞ (T). The case p = 1, where there is no compactness, is still pending. However, we can use the factorization theorem: every F ∈ H 1 (D) can be written F = GH, with G, H ∈ H 2 (D). By the above, there exist g, h ∈ H 2 (T) such that G =   = nk=0  g(k) h(n − k) Jg and H = Jh. The function gh is in L1 (T), and gh(n) 1  for n  0, gh(n) = 0 for n < 0; in particular, gh ∈ H (T). Consequently:  +∞ +∞  n n   J(gh)(z) = gh(n)z =  g(k)h(n − k) zn = F(z) ; n=0

n=0

k=0

hence J is indeed surjective. Remark The end of the proof shows that every function f ∈ H 1 (T) can be 1/2 written f = gh, with g, h ∈ H 2 (T), and g2 = h2 =  f 1 . Since every function F ∈ H 1 (D) originates from a function f ∈ H 1 (T) defined on the boundary T of D, the following theorem easily ensues: Theorem V.4 (The Frédéric and Marcel Riesz Theorem) Every measure μ on T with spectrum in N ( μ(n) = 0 for n < 0) is absolutely continuous with 1 (T). respect to the Haar measure. In short: MN (T) = LN Proof

For such a measure μ, set, for |z| < 1:  +∞ dμ(t) =  μ(n) zn . F(z) = −it T 1 − ze n=0

V Another Proof of Grothendieck’s Theorem

301

This function F is analytic in D and, as  μ(n) = 0 for n < 0, we obtain:  F(reiθ ) = Pr (θ − t) dμ(t) , T

thus:

 M1 (F, r) 

 T

2π

d|μ|(t)

Pr (θ − t)

0

dθ = μ . 2π

Hence F ∈ and, by Theorem V.3, there exists f ∈ H 1 (T) such that  iθ n inθ  μ(n) =  f (n) for any F(re ) = (Pr ∗ f )(θ ) = +∞ n=0 f (n) r e . We thus obtain  n ∈ Z, and consequently μ is the measure with density f . H 1 (D),

V.2 An Alternative Proof of Grothendieck’s Theorem Recall that this theorem involves showing that every operator T : 1 → 2 is 1-summing. This alternative proof is based on the following result (Paley [1933]): Theorem V.5 (Paley’s Theorem) There exists a constant C > 0 such that: 1/2  +∞ k 2  |f (2 )|  C  f 1 k=1

for every f ∈ H 1 (T). In other words, the operator: P:

−→ −→

H 1 (T) f

 2 k   f (2 ) k1

is continuous. Proof We use the factorization theorem, with the Remark following the proof of Theorem V.3:    +∞ +∞ int int , bn e cn e f (t) = n=0

n=0

with: +∞ n=0

|bn |2 =

+∞

|cn |2 =  f 1 .

n=0

Then:  f (2k ) =

r+s=2k

br cs ,

7 Some Properties of Lp -Spaces

302

hence: +∞

| f (2k )|2 =

k=1

k 2 +∞  2k 2 +∞   2    br c2k −r   |br c2k −r | 

k=1

2

r=0

k=1

r=0

k−1 +∞  2

2



k=1

2

|br c2k −r |

+

|br c2k −r |

r=2k−1 +1

r=0

k−1 +∞  2

k=1

2 

k

2

|br |

2

 2k

r=0

 |cs |

2

s=2k−1



k

2

+

|br |

2

 2k−1 −1

r=2k−1 +1

2

 +∞

|br |

2

 +∞  2k

r=0

k=1

+2 2

 +∞

|br |2

6

 +∞

 +∞

|br |2

|cs |

2

 +∞ 

s=0

k=1

|cs |2 +

+∞

r=2k−1 +1

|c2k |2

|cs |2

s=0

|cs |2



 +∞ 

2

k=1

r=2k−1 +1

= 6  f 21 .

s=0

A = { f ∈ C(T) ;  f (n) = 0 , ∀ n < 0} = CN (T) . The operator:

is 1-summing and surjective.

A f

−→  2  −→  f (2k ) k1



k

Let:

Q:

|br |2



The crucial point is now: Theorem V.6



k

2

k=1

 +∞

 +∞

r=0



|cs |2

s=0

+2

|cs |

s=0

s=2k−1

 +∞

r=0

 2

|br |2

V Another Proof of Grothendieck’s Theorem

303

Proof 1) The canonical injection j1 : C(T) → L1 (T) is the prototype of a 1summing application; as it sends A into H 1 (T), it induces a 1-summing map * j1 : A → H 1 (T). It only remains to note that Q is the composition of the Paley projection P of Theorem V.5 with* j1 to obtain that Q is 1-summing. 2) Proving that Q is surjective reduces to showing the existence of a constant δ > 0 such that: Q∗ xA∗  δ x2 for every x ∈ 2 . For this, we first note that, for every measure ν ∈ M(T) = C(T)∗ such that ν|A = Q∗ x, we have: ν, f = Q∗ x, f = x, Qf =

+∞

 f (2k )xk

k=1

for any f ∈ A. Consequently, with f (t) = ∈ N, we obtain:  ν(−n) = 0 ν(−2k ) = xk , for any n  0 and = 2k (i.e. ν ∈ MN∗ ∪{−2k ; k1} ), and  k  1. Consider then: +∞ k xk ei2 t . gx (t) = eint , n

k=1

This function is in

L2 (T),

Q∗ xA∗ =

and ν − gx ∈ A⊥ ; hence: inf

h∈A⊥ ν|A =Q∗ x

ν + h1 = inf gx + h1 . h∈A⊥

However: A⊥ = {μ ∈ M(T) ; μ, f = 0 , ∀ f ∈ A} = {μ ∈ M(T) ;  μ(−n) = 0 ∀ n ∈ N} = {μ ∈ M(T) ;  μ(n) = 0 ∀ n  0} = MN∗ ; so the Frédéric and Marcel Riesz theorem implies: A⊥ ⊆ L1 (T). Thus gx + h ∈ L1 (T) for every h ∈ A⊥ . Now the Riesz projection R maps gx + h to gx , and, by Kolmogorov’s theorem, this projection is of weak type (1, 1), hence continuous from L1 (T) into Lα (T) for 0 < α < 1 (Proposition III.7): there exists a constant Cα > 0 such that: gx α  Cα inf gx + h1 = Cα Q∗ xA∗ . h∈A⊥

To conclude, it now suffices to use the following:

7 Some Properties of Lp -Spaces

304

For any α < 1, there exists a constant Cα > 0 such that:

Lemma V.7

g2  Cα gα for every function g of L2 (T) with spectrum in {2k ; k  1}. This indeed implies: x2 = gx 2  Cα Cα Q∗ xA∗ . Proof of Lemma V.7 We first show that the norm  . 2 is equivalent to the norm  . 4 . Writing: g(t) =

+∞

k

ak ei2 t ,

k=1

we have: |g(t)|2 =



ak al ei(2

k −2l )t

k,l1

=

+∞

|ak |2 +



ak al ei(2

k −2l )t

= g22 + u(t) .

k=l

k=1

As the numbers 2k − 2l , for k = l, are all distinct, then, by Parseval’s formula: 2  +∞ 2 2 2 u2 = |ak al |  |ak | = g42 . k=l

k=1

By the triangle inequality, it follows that: g24 =  |g|2 2  g22 + u2  2 g22 , √ and hence g4  2 g2 . Now, 2 can be seen as the barycenter of α and 4: 2 = θ α + (1 − θ ) 4, with 0 < θ < 1; by Hölder’s inequality for the conjugate exponents 1/θ and 1/(1 − θ ), then:   |g|2 dm = |g|θα |g|(1−θ)4 dm g22 = T

 

T

therefore gθα 2 = g2  4

1−θ θα

T

θ   1−θ √ 4(1−θ) 4(1−θ) α 4 |g| dm |g| dm  gθα 2 g2 ; α T

2−4(1−θ) g2

gα = 2

2−α α

gα .

 41−θ gθα α , and we obtain, as wanted:

VI Comments

305

Remark In Chapter 4 of Volume 2, we will see that this lemma is a special case of the fact that every Sidon set is a (2)-set. Theorem V.6 easily leads to Grothendieck’s theorem. Indeed, let T : 1 → 2 be an arbitrary operator. With (εn )n1 the canonical basis of 1 , using the surjectivity of Q : A → 2 , we can find, for every n  1, an element fn ∈ A such that Q( fn ) = T(εn ), with  fn ∞  (1/δ) Tεn 2  (1/δ) T (where δ is the constant introduced in the proof of Lemma V.7). We can then define an operator: * T : 1 −→ A by setting * T (εn ) = fn for any n  1, so that we have a factorization T = Q ◦ * T: 1 ? ?? ?? ?? * T 

T

A

/ 2 ?     Q

and consequently T is 1-summing, since Q is 1-summing.

VI Comments 1) For a survey of the Lp -spaces, see Alspach and Odell [2001]. We will return to L1 (and in particular to its reflexive subspaces) in more detail in Chapter 4 (Volume 2). 2) There is essentially only one space L1 ; indeed: Theorem VI.1 If m is an atomless measure on a measure space (, A), and if the space L1 (m) is separable, then L1 (m) is isometric to L1 (0, 1). When m possesses atoms, its atomic part generates a subspace 1 1 complemented and isometric to N 1 or to 1 . When it is separable, L (m) is N 1 1 thus isometric, either to L (0, 1), or to L (0, 1) ⊕1 1 with 1  N < +∞, or to L1 (0, 1) ⊕1 1 . We will in fact prove a bit more: if (, A, P) is an atomless probability space, complete (i.e. every negligible set is measurable), and separable (i.e. the space L1 (P) is separable, or, equivalently, the metric space A/P, used in Section II in the proof of the Vitali–Hahn–Saks theorem, is separable), then, modulo the negligible sets, it is isomorphic to the probability space [0, 1] equipped with the σ -algebra of Lebesgue measurable sets and with the Lebesgue measure. Hence, for 1  p  +∞, the space Lp (, A, P) is isometric to Lp (0, 1). The proof is taken from Malliavin, § 6.4.

7 Some Properties of Lp -Spaces

306

Proof If the measure is not σ -finite, the space L1 (m) is not separable; as in the beginning of Section II, we can thus reduce to the case where m is finite, and even assume that m is a probability P. As L1 (P) is separable, it contains a dense sequence of functions gn that we can take as simple functions. Let: An = σ (g1 , . . . , gn ) be the σ -algebra generated by the first n functions. As the sequence (gn )n1 is dense, then EA∞ (L1 ) = L1 ; thus we can assume A∞ = A (here A is the P-completion of the σ -algebra A∞ ). Since each σ -algebra An is finite because the gk ’s are simple functions, it is generated by a finite measurable partition An,1 , . . . , An,kn of . Some subsets may be of measure zero: in this case we group them with subsets of non-zero measure. Shrinking the σ -algebras if necessary, we may thus assume that P(An,j ) > 0 for any n  1 and 1  j  kn . Moreover, as An+1 ⊇ An , the subsets An,k can be assumed to be inductively numbered so that: jk+1  An+1,j , An,k = j=jk +1

with 0 = j1 < j2 < · · · < jkn +1 = kn+1 (note that the sequence ( j1 , . . . , jkn +1 ) depends on n: j2 = j2 (n), . . .). Set an,k = P(An,k ), and define: kn   1 an,k + an,l 1IAn,k . fn = 2 l 0 and P(C  D) > 0; but we would then have: P(An ∩ D) a.s. P(D) 1IAn −−→ 1IC , EAn 1ID = n→+∞ P(C) P(An ) which is not possible, since EAn 1ID −−→ 1ID . n→+∞ Now, for every continuous function u on [0, 1],  kn  1 E[u( fn )] = an,k u an,k + an,l 2 a.s.

k=1

l 0, there exists a Borel set U0 with measure m(U0 )  ε such that ϕr = ν ∗ kr tends to 0 in mean on U0c = G  U0 , i.e.:  ϕr (t) dm(t) − →0 . U0c

r→0

For this, let N be a Borel set such that m(N) = 0 and ν(A) = ν(A ∩ N) for every Borel set A. We can find an open U0 such that N ⊆ U0 and m(U0 )  ε. This U0 is exactly where we test the convergence stated in the preceding Fact. To do so, we introduce: Ur = {x ∈ G ; B(x, r) ⊆ U0 } . This set is contained in U0 . The two essential points are: (1) ν(Ur ) − → ν(U0 ), r→0 (2) t ∈ / U0 ⇒ Ur ∩ B(t, r) = ∅. 

Indeed, we then have:       ν B(t, r) ν B(t, r) ∩ U0   dm(t) =   dm(t) ϕr (t) dm(t) = m B(0, r) U0c U0c m B(0, r) U0c       ν B(t, r) ∩ U0 ν B(t, r) ∩ U0   dm(t) −   dm(t) . = m B(0, r) m B(0, r) G U0

Next, as soon as m(B) > 0, the use of convolution and of Fubini’s theorem leads to the equality:    ν A ∩ (B + t) dm(t) ; (∗) ν(A) = m(B) G

7 Some Properties of Lp -Spaces

312

  ν B(t, r) ∩ U0   hence dm(t) = ν(U0 ), and: m B(0, r) G     ν B(t, r) ∩ Ur   dm(t) ϕr (t) dm(t)  ν(U0 ) − m B(0, r) U0c U0    ν B(t, r) ∩ Ur   dm(t) = ν(U0 ) − m B(0, r) G 

= ν(U0 ) − ν(Ur ) −−→ 0 r→0

since Ur ⊆ U0 by (2)

by (∗) again

thanks to (1) :

the preceding Fact, and hence Lemma VI.4, are proved. It remains to show (1) and (2) . For (1) : set δ > 0, and let K0 be a compact set contained in U0 such that ν(U0  K0 )  δ; when r is small enough (r < dist(K0 , U0c )), then K0 + B(0, r) ⊆ U0 , hence K0 ⊆ Ur , and ν(U0  Ur )  ν(U0  K0 )  δ. For (2) : if Ur ∩ B(t, r) = ∅, we can write u = t + b, with u ∈ Ur and b ∈ B(0, r); but then t = u + (−b), with u ∈ Ur and −b ∈ B(0, r), and therefore, given the definition of Ur , t ∈ Ur + B(0, r) ⊆ U0 . Remark In the case where the Haar measure of the group G is doubling, i.e. there exists a constant C > 0 such that     m B(x, 2r)  C m B(x, r) for every x ∈ G and any r > 0 (in particular in the case of the circle T), a somewhat shorter proof can be given, even implying convergence almost everywhere. In fact, if we set:     |μs | B(x, r) 1  > , Aj = x ∈ G ; lim sup  j m B(x, r) r→0 it suffices to see that m(Aj ) = 0 for any j  1. However, as μs is singular, there exists a Borel set A of G such that m(A) = 0 and |μs |(A) = μs . The regularity of the measure μs provides, for any ε > 0, an open set  ⊇ Ac such that |μs |() < ε. Let K be a compact set contained in Ac ∩ Aj . For every x ∈ K, there exists rx < dist(K, c ) such that    1  |μs | B(x, rx ) > m B(x, rx ) . By compactness, there exist x1 , . . . , xp ∈ j K such that K ⊆ B1 ∪. . .∪Bp ⊆ , with Bk = B(xk , rxk ). Moreover we can assume that rx1  rx2  · · ·  rxp . Then, starting with k1 = 1, we denote by k2 the first index such that Bk2 ∩Bk1 = ∅, and continue like this until we q / l=1 Bkl , there exists, by construction, obtain Bkq . Then, if x ∈ Bk but x ∈ a kl < k such that Bk ∩ Bkl = ∅; therefore d(x, xkl )  2 rxk + rxkl  3 rxkl .

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p q Consequently k=1 Bk ⊆ l=1 B(xkl , 3 rxkl ) (this is the Vitali covering lemma). Thanks  made on the group G, we have    to the hypothesis 2 m B(x, 3r)  m B(x, 4r)  C m B(x, r) , and we obtain: q q q   2 2 m B(xkl , 3 rxkl  C m(Bkl )  C j |μs |(Bkl ) m(K)  l=1

= C2 j |μs |

 q

l=1

 Bkl

l=1

 C2 j |μs |()  C2 j ε ,

l=1

which completes the proof of Lemma VI.4. This is another path leading to the F. and M. Riesz theorem. Proposition VI.5 (Godefroy) Proof

N is a Shapiro set in Z.

We want to show that every subset ⊆ N is nicely placed. L1/2

Thus let fn ∈ BL1 (T) be such that fn −−→ f . In order to prove that the n→+∞



spectrum of f is contained in , we use the space H 1 (D). To simplify the notation, for every F ∈ H 1 (D), denote by F ∗ the “boundary value” function J −1 ( f ) ∈ H 1 (T). By Cauchy’s formula: sup |F(z)| 

|z|r

1 F ∗ 1 ; 1−r

the functions in the unit ball BH 1 (D) of H 1 (D) are hence uniformly bounded on every compact subset of D, and consequently form a normal family of holomorphic functions. As BH 1 (D) is closed for the topology τ of uniform convergence on every compact of D, it is τ -compact. By extractτ ing a subsequence if necessary, we can assume that Fn = J( fn ) −−→ G ∈ n→+∞

fn (k) for any k ∈ N, the H 1 (D). Since, by Cauchy’s formula, Fn (0) = k!  ∗ spectrum of G is contained in . It only remains to see that f = G∗ . (k)

L1/2

Now, with Hn = Fn − G, then Hn∗ = fn − G∗ −−→ f − G∗ ; hence, for n→+∞ any ε > 0, there exists an integer N  1 such that, for n, k  N, we have:  |Hn∗ (t) − Hk∗ (t)|1/2 dm(t)  ε . T

As |Hn − Hk |1/2 is subharmonic, we obtain, for any r < 1:  2π dt  ε. |Hn (reit ) − Hk (reit )|1/2 2π 0 $ 2π τ dt Therefore if Hk −−→ 0 we have 0 |Hn (reit )|1/2 2π  ε. k→+∞

314

7 Some Properties of Lp -Spaces

Now, using Theorem V.3, we have a sequence rh −−→ 1 such that h→+∞ $ a.s. Hn (rh eit ) −−→ Hn∗ (t); we hence get T |Hn∗ (t)|1/2 dm(t)  ε. Then, by h→+∞ letting n tend to infinity:  | f (t) − G∗ (t)|1/2 dm(t)  ε . T

As ε > 0 is arbitrary, this indeed gives f = G∗ . To this day, only one method is known to show that a set is not a Riesz set: it consists of showing that the set contains a set of words, i.e. a set of the type:  n   εk γk ; εk = −1, 0 or 1 , = n1

k=1

G, sufficiently lacunary so that where (γk )k1 is a sequence in  =   the representation nk=1 εk γk of elements of  is unique. Such a set {γk ; k  1} is said to be dissociate (to simplify the notation, we assume that  does not contain any element of order 2). In this case, for every sequence (an )n1 such that |an |  1/2, we can construct the Riesz product μ=

+∞ 

  1 + an (γn + γ n ) ,

n=1

which is a positive measure whose spectrum is contained in , and is singular when (an )n1 ∈ / 2 . So it might be possible that every non-Riesz set contains, up to a translation, such a set of words. For this to be – or not to be – true, seems a very difficult question. If it were true, it would be, in the words of Brassens, “extraordinaire, et pour tout dire, inespéré” (extraordinary, and frankly, unexpected)! Nonetheless, Host and Parreau [1979] showed that if ⊆  does not contain any translation of a set of words, then every measure μ ∈ c0 ()}. A with spectrum in is necessarily in M0 = {μ ∈ M(G) ;  bit later, Graham, Host and Parreau [1981] showed that such a measure is even in Rad(L1 ), the set of measures whose image in the quotient algebra M(G)/L1 (G) has a spectral radius equal to zero. 9) Maurey [1980 a] showed that H 1 (D) possesses an unconditional basis (however, it does not have a completely unconditional basis, in the sense of operator spaces (Ricard [2000])). His proof was very difficult and indirect: by using the method of decomposition of Pełczy´nski, he showed that H 1 (D) is isomorphic to the space H 1 (δ) of dyadic martingales (i.e. adapted to the filtration induced by the dyadic intervals) whose

VII Exercises

315

maximal functions are integrable; for this space, the Haar system is an unconditional basis. Soon afterwards, Carleson [1980] provided an explicit unconditional basis of H 1 (D); then Wojtaszczyk [1982] showed that the Franklin system is an unconditional basis of H 1 (D). There are experts who consider Maurey’s article a posteriori as one of the founding articles of the Theory of Wavelets. 10) The Banach–Saks theorem (Exercise VII.4) is no longer valid in L1 ; in fact every Banach space with the Banach–Saks property (where every bounded sequence contains a subsequence whose Cesaró means converge) is reflexive. This can easily be seen with the use of James’ theorem: a Banach space is reflexive if and only if every continuous linear functional on this space attains its norm (James [1964 a], then James [1972] for a simplified proof; see Megginson, § 1.13; the separable case is easier, and was proved by James [1957]; another proof of the separable case can be found in Simons [1972 b]; see also Simons [1972 a] and Godefroy [1987]). Nonetheless, in L1 , Szlenk’s theorem ensures that, from every weakly convergent sequence, we can extract a subsequence whose Cesaró means converge in norm (Szlenk [1965]; see Diestel, pages 111–113). We also point out Komlós’ theorem : from every bounded sequence of elements of L1 (0, 1), we can extract a subsequence whose Cesaró means converge almost everywhere, hence in measure (Komlós [1967]).

VII Exercises Exercise VII.1 1) Show that if a sequence of elements of L1 converges weakly and in measure, then the two limits are equal (and there is also convergence in norm; see Proposition II.4). 2) $Let F = {g1 , . . . , gn } be a finite subset of L∞ (0, 1). Set g0 = 1I and αk = 1 0 gk (x) dx. Show that, $ for any ε > 0, there exists a measurable subset A = Aε,F such that A gk (x) dx = ε αk for 0  k  n (for example, use Lyapunov’s theorem for the measure (gk dxk )0kn ). With fε,F = λ(A1ε,F ) 1IAε,F , deduce that the filtering family ( fε,F )ε,F (which is in the unit ball of L1 ) converges weakly to 1I, but nonetheless converges in measure to 0. It ensues that every weak neighborhood of 1I intersects every neighborhood of 0 for the topology of convergence in measure. 3) Let An be the dyadic σ -algebra of order n on [0, 1], and let: " Fn = f ∈ L1 ; f is An -measurable, f  0,  f 1 = 1, # and mes({ f = 0})  1/n .

7 Some Properties of Lp -Spaces

316

 Show that n1 Fn is relatively compact in measure, but that 1I is adherent to it for the weak topology. Exercise VII.2 1) (Dunford and Pettis [1940]) For any Banach space X, show that every weakly compact operator T : L1 ([0, 1]) → X is representable (see Chapter 4, Exercise VII.7; the proof presented in Exercise VII.10 of Chapter 4 can be adapted here). 2) Deduce that all reflexive Banach spaces share the Radon–Nikodým property. 3) Let g ∈ L∞ ([0, 1]; X) and Tg : L1 ([0, 1]) → X the operator defined by:  1 Tg (ϕ) = ϕ(t) g(t) dt , 0

and let K ⊆

L1 ([0, 1])

be weakly compact.

a) Let (gn )n1 be a sequence of simple functions converging almost everywhere to g, and let ε > 0. Using the uniform integrability of K, show the existence of a measurable subset A ⊆ [0, 1] such that: (i) gn −−→ g uniformly on A;  n→+∞ ε (ii) |ϕ(t)| dt  g ∞+1 [0,1]A (note that Egorov’s theorem, with the same proof, still holds for vector-valued functions). b) Show, by using (i) , that the set of values taken on by g1 = 1IA g is relatively compact in norm in X, and deduce that the operator Tg1 $1 defined by Tg1 (ϕ) = 0 ϕ(t) g1 (t) dt is compact. c) By writing:   Tg (ϕ) =

ϕ(t) g(t) dt + A

ϕ(t) g(t) dt , [0,1]A

deduce that Tg (K) can be covered by a finite number of balls of radius 2 ε. d) Deduce that every representable operator T : L1 ([0, 1]) → X maps the weakly compact subsets into norm-compact subsets (such an operator, formerly known as a completely continuous operator, is called a Dunford–Pettis operator). 4) Show that, for every weakly compact operator T : L1 ([0, 1]) → L1 ([0, 1]), the operator T 2 is compact. Exercise VII.3 (Bochner and Taylor [1938]) 1) Show that, for every  p Banach space ∗ X, there exists an isometry from ∗ p ∗ L ([0, 1]; X ) into L ([0, 1]; X) , when 1  p < +∞.

VII Exercises

317

2) Show that if X ∗ possesses the Radon–Nikodým ∗property, then this  isometry is surjective: to every  ∈ Lp ([0, 1]; X) , we can associate T : L1 ([0, 1]) → X ∗ defined by [T(ϕ)](x) = (ϕ x). 3) Conversely, if there exists a value of p, with 1  p < +∞, for which the isometry is surjective, show that X ∗ has the Radon–Nikodým property. Exercise VII.4 With 1 < p < +∞, let (xn )n1 be a bounded sequence in Lp (0, 1). The purpose is to show that a subsequence (xnk )k1 can be 1 K xn converge in norm extracted from (xn )n1 , whose Cesaró means K k=1 k (the Banach–Saks theorem). 1) Show that we may assume that (xn )n1 converges weakly, to an element x ∈ Lp (0, 1), but not in norm. 2) Under these conditions, show that, from (xn )n1 , a subsequence can be extracted such that (xnk − x)k1 is an unconditional basic sequence. 3) Using the fact that Lp (0, 1) is of type s = min(2, p), deduce that this subsequence satisfies:    1/s K   K s   (xnk − x)  C xnk − x .  k=1

k=1

4) Conclude. Exercise VII.5 1) Show that, for 1  p < +∞, the space En generated by the n first Rademacher functions r1 , . . . , rn in Lp (0, 1) is isomorphic to the space n2 = (Rn ,  . 2 ). 2) Let P be the orthogonal projection of L2 (0, 1) onto the space generated by all of the Rademacher functions r1 , r2 , . . . We assume p  2. a) Show that P is continuous from Lp (0, 1) into itself. n b) Show that there exists a subspace Fn of Lp (0, 1), isometric to 2p = n (R2 ,  . p ), such that Fn ⊇ En and PFn = En . c) Deduce that the space (12 ⊕ 22 ⊕ · · · ⊕ n2 ⊕ · · · )p is isomorphic to a complemented subspace of p (note that p is isometric to n (2p ⊕ 4p ⊕ · · · ⊕ 2p ⊕ · · · )p ). 3) Show that this holds for 1 < p < 2. 3 n 4) Deduce that n1 2 p is isomorphic to p for 1 < p < +∞.  3 n 5) Show that the canonical basis of n1 2 p provides an unconditional canonical basis of p for p = 2. basis  is not equivalent to the 3  3of p that n n 6) Is n1 2  isomorphic to 1 ? Is n1 2 c isomorphic to c0 ? 1

0

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7 Some Properties of Lp -Spaces

Exercise VII.6 √ 1 = 2 and that 1−fn 1  2+1 1) Set fn (t) = 2 eint +e2int √ . Show that 1+fn√ (use the inequality 1 + a2 − 1  a2 ( 2 − 1), valid for any real a with |a|  1). 2) Let X be a space with an unconditional basis containing H 1 (T). Using the Bessaga–Pełczy´ nski selection theorem, show that the unconditional √ constant of X is  ( 2 + 1)/2 > 1, 2 (Wojtaszczyk [1982]). Note that H 1 is universal for the subspaces of L1 with an unconditional basis (Maurey [1980 a], page 119). Exercise VII.7 1) For 1  p < 2, show that (1 + x)p + (1 − x)p  2 (1 + xp ) for 0  x  1, with equality only for x = 0. Deduce that |a + b|p + |a − b|p  2 (|a|p + |b|p ) for any a, b ∈ R, with equality only when a = 0 or b = 0. Show similarly that, for 2 < p < +∞, we have |a + b|p + |a − b|p  2 (|a|p + |b|p ), with equality only when a or b is null. 2) Let 1  p < +∞ and p = 2. p

p

a) Show that, for f , g ∈ Lp (S, T , m), we have  f + gp +  f − gp = p p 2 ( f p + gp ) if and only if fg = 0 m-almost everywhere (see also Chapter 2, Exercise V.5). b) Deduce that if X is a subspace of Lp (m) isometric to p , then there exists a sequence (Sn )n1 of measurable subsets of S, essentially disjoint, such that X is the space of functions in Lp (m) which are constant on each Sn . c) Show that there exists a projection of norm 1 from Lp (m) onto X (reduce to the case where m is a probability P, and use the conditional expectation with respect to the σ -algebra generated by the subsets Sn , n  1). Exercise VII.8 ϕ (n) for 1) Set ϕ(t) = i (π − t) e−it for 0 < t < 2π ; compute ϕ∞ and  n ∈ Z.   f (n)   f 1 ϕ∞ . f  0, show that +∞ 2) If f ∈ H 1 (T) and  n=0 n+1 3) Show that, for every f ∈ H 1 (T), there exists F ∈ H 1 (T) such that | f|   F and  F1   f 1 (use the factorization theorem, Theorem V.1). +∞  |f (n)|  π  f 1 for every f ∈ H 1 (T). 4) Prove Hardy’s inequality: n+1 n=0 √ 5) Show that, for every f ∈ H 1 (T), ( f (n)/ n + 1)n0 ∈ 2 (use Hardy’s inequality).

VII Exercises

319

6) Show that the operator T : H 1 (T) → 2 thus defined is not 1-summing. 7) Conclude that H 1 is not a complemented subspace of L1 (T) (see also Exercise VII.11).

Exercise VII.9 1) Show that the operator T : L1 (0, 1) → C([0, 1]) defined by:  x f (t) dt Tf (x) = 0

is not 1-summing. (Hint: show that T is not weakly compact by considering en (x) = nxn−1 and conclude with Corollary II.8 in Chapter 6.) The rest of this exercise is devoted to a quantitative finite-dimensional version of this result. n sin nt + rn (t) (|t|  π ) be the Dirichlet kernel of eikt = 2 2) Let Dn (t) = t k=−n

order n. Recall that supn1 rn ∞ < +∞. Let D∗N (t) = sup1kN |Dk (t)| be the associated maximal function. By adjusting δ in: 

π 0

sup | sin kt| kN

dt  t



δ

Nt 0

dt + t

 δ

π

dt , t

show that D∗N 1 = O(log N) (the norm is that of the space L1 (−π , π ), with the normalized Lebesgue measure). sk = x1 + · · · + xk . Show that if X(t) = 3) Let x = (x1 , . . . , xN ) and  π N 1 ikt N x e , then s = X(t) Dk (t) dt. If σ : N k k k=1 1 → ∞ is defined 2π −π by σ (x) = (s1 , . . . , sN ). Then deduce:  π 1 |X(t)| D∗N (t) dt . σ (x)∞  2π −π 4) Show that π1 (σ )  D∗N 1  C log N.   5) Let A = aj,k j,k1 be the Hilbert matrix, defined by aj,k = and aj,j = 0.

1 if j = k, j−k

a) Show that there exists ϕ ∈ L∞ (T) of norm ϕ∞  π such that aj,k =  ϕ ( j − k), and deduce that A defines a bounded operator 2 → 2 , with norm  π . Let (e1 , . . . , eN ) be the canonical basis of N 1 ; consider the N vectors  N. a e (1  j  N) of  fj = N k=1 j,k k 1

320

7 Some Properties of Lp -Spaces

b) Show that, if ξ = (ξ1 , . . . , ξN ) ∈ N ∞ , we have: N

√ |ξ( fj )|  N π

j=1

 N

1/2 |ξk |

2

.

k=1

c) Show that there exists a constant δ > 0 such that σ ( fj )∞  δ log j for 1  j  N. d) Conclude that π1 (σ )  c log N, where c is a numerical constant. N.B. – More precisely, we have, by the above: 1 π1 (σ )  lim inf N→+∞ log N π

and

lim sup N→+∞

π1 (σ ) 2  ; log N π

we are going to see that π1 (σ )/ log N −−→ 1/π . For this, we admit (see N→+∞

Duren, Chapter 8, Exercise 8, page 144), that there exists a trigonometric polynomial fN such that:  2N int int (1) fN (t) = N n=0 e + n=N+1 cn,N e ;  2π 1 1 log N. (2)  fN 1 = | fN (t)| dt ∼ N→+∞ π 2π 0 N First note that the operator σ : N 1 → ∞ can be defined by the matrix (bj,k )j,kN , where  1 if k  j bj,k = 0 if k > j,

 so that σ (x) = (s1 , . . . , sN ), with sj = N k=1 bj,k xk . N Consider, for B(t) : N 1 → ∞ given by the  0  t  2π , the operator  −i( j−k)t . We have the following three matrix bj,k (t) j,kN , with bj,k (t) = fN (t) e properties (see Wojtaszczyk § III.H. 24):   Point 1. For any t, B(t) is of rank 1, and hence π1 B(t) = B(t) =  ikt supj,k |bj,k (t)|. Indeed, if y = B(t)x, then yj = e−ijt N k=1  e fN (t), and hence  −it −2it −Nit . As for the equality y is proportional to the vector e , e , . . . , e B(t) = supj,k |bj,k (t)|, it is evident via a test on the extreme points of the unit ball of N 1.  2π 1 Point 2. We have σ = B(t) dt. 2π 0 $ 2π 1 Let B = 2π 0 B(t) dt and (βj,k )j,kN its matrix. We have, for any j, k  N, $ 2π 1  βj,k = 2π 0 bj,k (t) dt = fN ( j − k). If j < k, this means βj,k = 0, as fN is of analytic type; if j  k, we have 0  j − k  j  N, and thus βj,k = 1 by (1) above. Hence we indeed have βj,k = bj,k for any j, k  N.

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321

Point 3. We have π1 (σ )   fN 1 . In fact, Points 1 and 2 imply:  2π  2π   1 1 π1 B(t) dt = B(t) dt π1 (σ )  2π 0 2π 0  2π  2π 1 1 = sup |bj,k (t)| dt = | fN (t)| dt =  fN 1 , 2π 0 2π 0 j,k as |bj,k (t)| = | fN (t)| for all the indices j, k. It is now easy to conclude: Point 3 and relation (2) imply the inequality 1 1 1 (σ ) limN→+∞ πlog N  1/π ; moreover, as σ ( fk )∞  1+ 2 +· · ·+ k − 1 ∼ log k, 1 (σ ) Question 5) of Exercise VII.9 shows that limN→+∞ πlog N  1/π . Thus the stated result. Exercise VII.10 When 0 < p < q  +∞, a multiplier from H p into +∞ n ∈ H q as soon as H q is any sequence (λn )n0 such that n=0 λn an z +∞ n p p q n=0 an z ∈ H . Denote by M(H , H ) the space of those multipliers. A theorem of Hardy and Littlewood [1932] states that (λn )n0 ∈ M(H p , H q ) 1 (n + 1) 1 = n−α + O(n−α−1 ), with α = − · We will when λn = (n + 1 + α) p q see that the simple condition λn = O(n−α ) is in general not sufficient (Duren [1969 b]). 1) (Caveny [1966]) Let 1 < p < +∞ and let p∗ be the conjugate exponent of p.  n p∗ p ∞ a) Show that if +∞ n=0 λn z = g(z) ∈ H , then (λn )n0 ∈ M(H , H ). 1 < γ < 1 (consider the Deduce that (n−γ )n1 ∈ M(H p , H ∞ ) if p  n −γ +O(n−γ −1 )). function g(z) = (1 − z)γ −1 = +∞ n=0 bn z , with bn = n ∗ b) Conversely, if (λn )n0 ∈ M(H p , H ∞ ), show that there exists g ∈ H p such that λn =  g(n) for any n  0 (use the Marcel Riesz theorem to obtain an operator from Lp into H ∞ ).  2/p |λ |2 = O(N 2 ) (test 2) Show that if (λn )n0 ∈ M(H p , H 2 ), then N n n=1 n +∞ − p1 −1 n = n=0 an r zn , with r < 1 and the hypothesis on f (z) = (1 − rz) 1/p an ∼ n ). 1 1 3) Assume 0 < p < q < 2. Let α = − · p q  +∞ n p −α 2 a) Show that there exists +∞ a n=0 n z ∈ H such that n=1 |an n | = +∞.  −α n b) Show that there exist signs εn = ±1 such that +∞ n=1 an (εn n )z has a −α / M(H p , H q ) radial limit almost nowhere, and deduce that (εn n )n1 ∈ (see Theorem V.1 of Chapter 6 of Volume 2).

322

7 Some Properties of Lp -Spaces 1 1 − · Suppose p q ∈ M(H p , H q ) for every choice of signs εn = ±1. Set

4) Assume 2 < p < q < +∞, and again denote α =

that (εn n−α )n1 1 1 γ = −α > · 2 q Show that (εn n−1/2 )n1 ∈ M(H p , H ∞ ) for every choice of signs  −1/2 zn ∈ H p∗ (see Question 1) , and show εn = ±1. Deduce that +∞ n=1 εn n that this leads to a contradiction). / M(H p , H q ). It Hence, there exists a choice of signs such that (εn n−α )n1 ∈ can be shown (Duren) that if 0 < p  2  q < +∞ and λn = O(n−α ), then (λn )n0 ∈ M(H p , H q ). Exercise VII.11 Let G be an infinite metrizable compact Abelian group, and m its Haar measure. Recall (see the Annex) that m is atomless and L1 (G) is thus isometric to L1 ([0, 1]), by Theorem VI.1. 1) Show that, for every f ∈ L1 (G), the convolution operator Cf : L1 (G) −→ L1 (G) , defined by Cf (ϕ) = ϕ ∗ f is representable (see Chapter 4, Exercise VII.7). 2) Let μ ∈ M(G) and assume that the convolution operator Cμ : L1 (G) −→ L1 (G) is representable: there exists$ a function g : G → L1 (G), essentially bounded, such that ϕ ∗ μ = G ϕ(t) g(t) dm(t) for every ϕ ∈ L1 (G). Set h(t, s) = [g(t)](s). (γ )(γ ) for γ , γ ∈  =  G, show that  h(γ , γ ) = a) By calculating C μ

 μ(γ ) if γ = −γ , and = 0 if γ = −γ . b) Deduce that there exists f ∈ L1 (G) such that h(t, s) = f (t − s) for any t, s ∈ G, and that μ has density f with respect to m (Costé, quoted in Diestel–Uhl, page 90). 3) (Lust-Piquard Lust-Piquard [1976]; see also [1978]) Deduce that ⊆  1 (G) has the Radon–Nikodým property, and is a Riesz set if and only if L 1 if and only if L (G) is isomorphic to a separable dual space (use Exercise VII.10 of Chapter 4). 4) Similarly, show that C (G) has the Radon–Nikodým property if and only if it is a Rosenthal set (see the remark following Proposition V.6 in Chapter 6). 1 is not complemented 5) Let ⊆  be an infinite Riesz set; show that L 1 , it would be 1 1 in L (G) (if there were a projection P : L (G) → L $ 1 ) such that Pf = representable: there would exist g ∈ L∞ (G; L G gf dm;  we would have |g(x, .)(−γ )| = 1 for almost all x ∈ G, which contradicts the Riemann–Lebesgue lemma).

VII Exercises

323

Exercise VII.12 (Dor’s Theorem2 ) The purpose is to prove the following: Theorem (Dor)

Let μ and ν be two positive measures and

T : L1 (μ) −→ L1 (ν)   be an isomorphism between L1 (μ) and T L1 (μ) such that √ T T −1  = λ < 2.   Then there exists a projection P : L1 (ν) → T L1 (μ) , and moreover P  λ2 /(2 − λ2 ). Part I 1) Let 1  p < 2, and let f1 , . . . , fn ∈ Lp (0, 1) be such that  fk p  1. Assume that 4 n 51/p   n   p  ak fk  |ak | ,  θ  p

k=1

k=1

for any a1 , . . . , an ∈ C, with 0 < θ  1. Let (rk )k1 be the sequence of Rademacher functions. Show that: 2 p/2  1  1  n  n  p p  θ |ak |  ak rk (s)fk (t) ds dt  0

k=1

 

0

1

k=1

max |ak fk (t)|(2−p)p/2

0 1kn

 

1

max |ak fk (t)| dt p

 n

p/2 |ak fk (t)|p

k=1 (2−p)/2  n

0 1kn

dt p/2

|ak |

p

.

k=1

2) Similarly, show that, for 2 < p < +∞, if   1/p  n   n 1 p   ak fk   |ak |  θ p k=1

k=1

for any a1 , . . . , an ∈ C, then:

 1/p n     |ak |p .  max |ak fk |   θ 2/(p−2) 1kn

p

k=1

Part II Let 0 < c  1, and let g1 , g2 , . . . be non-negative functions in L1 (0, 1) such that

2 L. Dor [1975 a]; see also Maurey [1975 b]; another approach can be found in Godefroy, Kalton

and Li [2000]. An analogous result for p > 1 can be found in Schechtman [1979].

7 Some Properties of Lp -Spaces

324



1 0

max ak gk (t) dt  c kn

n

ak

k=1

for all a1 , . . . , an  0, and any n  1. Consider the space L∞ equipped with its weak∗ topology, and, in ∞ (L × L∞ × · · · ) equipped with the corresponding product topology, the compact convex set:   +∞ ϕk  1 p.p. . D = (ϕk )k1 ; ϕk  0, ∀ k  1, and 1) Show that the set D of





1

k=1

for (ϕk )k1 ∈ D is a convex

gk (t)ϕk (t) dt 0

k1

and weak∗ compact subset of ∞ .  1 2) Suppose that inf gk (t) ϕk (t) dt < c for all sequences (ϕk )k1 ∈ D. k1 0

a) By separating D and A = {(ck )k1 ; ck  c, ∀ k  1} ⊆ ∞ , show that there exists a convergent series of positive numbers ak such that +∞ $ 1 +∞ a  1 and such that a k=1 k k=1 k 0 gk (t) ϕk (t) dt  c < c for all (ϕk )k1 ∈ D.  b) Let n be such that nk=1 ak > c /c, and let E1 , . . . , En be disjoint measurable subsets such that maxln al gl (t) = ak gk (t) for t ∈ Ek . Show that there exists (ϕk )k1 ∈ D such that:  1  1 n n max ak gk (t) dt = ak gk (t) ϕk (t) dt < c ak . 0

kn

0 k=1

k=1

c) Conclude that the initial hypothesis was false. 3) Determine the set of extreme points of D. 4) Deduce $ that there exist disjoint measurable subsets A1 , A2 , . . . of [0, 1] such that Ak gk (t) dt  c for any k  1. Part III Show that, for 1  p < +∞ and p = 2, if f1 , . . . , fn ∈ Lp (0, 1) satisfy the condition of I, 1) or of I, 2) , then there exist disjoint measurable subsets A1 , . . . , An such that  fk 1IAk p  θ 2/|p−2| . Part IV between n1 and T(n1 ) such that 1) Let T : n1 → L1 (0, 1) be an isomorphism √ −1 T = 1 and T  = λ < 2. Set δ = λ2 /(2 − λ2 ). Show that there exists a projection P : L1 (0, 1) → T(n1 ) of norm P  δ. 2) Show that the preceding result holds if we replace the measure space [0, 1] by an arbitrary probability space (, A, P) (reduce to the separable case with a conditional expectation, then use Theorem VI.1). 1 1 3) Let (S, T , μ) be an arbitrary measure √ space, and T : L (μ) → L (P) such −1 that T = 1 and T  = λ < 2

VII Exercises

325

a) Use 2) and the L1 character of L1 (μ), seen in Chapter 6, to construct ∼ operators Sα : L1 (P) → Eα = n1α such that Sα T = IdEα and Sα   δ. b) Deduce the existence of an operator S : L1 (P) → L1 (μ)∗∗ such that ST = IdL1 (μ) and S  δ. c) Show that there exists a projection P0 : L1 (μ)∗∗ → L1 (μ) with norm 1 (L∞ (μ) is a commutative and unitary C∗ -algebra: use this fact to represent it as C(K) for some compact space K; introduce the positive measure * μ ∈ $M(K) associated with the positive linear functional 1 ϕ ∈ L∞ (μ) → 0 ϕ dμ, then use the Radon–Nikodým theorem). 1 d) Deduce  there exists a projection P of norm from  δ of L (P) onto  1 that T L (μ) . Exercise VII.13 Let 0 < c $1, and let g1 , . . . , gn be non-negative functions 1 of norm 1 in L1 (0, 1) such that 0 maxkn gk (x) dx  c n. 1) Let Aj be the set of x ∈ [0, 1] such that j is the smallest integer k  n for  which gk (x) = max1ln gl (x), and let J be the set of j  n such that gj (x) dx  c/2. Show that |J|  c n/2. Aj

2) Let (ξj )j∈J be selectors, i.e. (0, 1)-valued independent random variables, defined on a probability space , of mean c/16, and let: Iω = {j ∈ J ; ξj (ω) = 1} .  For any x ∈ [0, 1], set ϕj (ω, x) = k∈J{j} ξk (ω)1IAk (x), for j ∈ J, then:  ψ(ω, x) = j∈J ξj (ω) ϕj (ω, x) gj (x). Show that:  1  c 2 ψ(ω, x) dP(ω) dx  |J| 16 0  (note that the Aj , j ∈ J, are disjoint), and deduce that there exists an ω0 ∈  $1 such that |Iω0 |  c |J|/32 and such that 0 ψ(ω0 , x) dx  c |Iω0 |/8.  3) For j ∈ Iω0 , set Bj = k∈Iω {j} Ak , and let: 0    gj (x) dx  c/4 . D = j ∈ Iω0 ; Bj

Show that$ |D|  c |J|/64  c2 n/128 and that, for any j ∈ D,  k∈D{j} Ak gj (x) dx  c/4. 4) Show that:   1      dx  c a g (x) |aj | j j   4 0

j∈D

j∈D

for every sequence (aj )j∈D of scalars (Bourgain [1982]; see also Tomczak– Jaegermann, § 31).

8 The Space 1

I Introduction The space 1 plays a crucial role in the study of Banach spaces. This is because its absence from a Banach space X implies nice properties of weak convergence for bounded sequences. This aspect is clarified with precision in Rosenthal’s 1 theorem. To prove it, in the real case, we first establish the Rosenthal– Bourgain–Fremlin–Talagrand theorem: this studies the simple convergence of sequences of continuous functions on a Polish space. For this we use a method due to G. Debs. The complex case does not follow immediately, and several auxiliary results due to A. Pajor are needed. For separable spaces, it boils down to saying that the elements of the bidual X ∗∗ , seen as functions on the unit ball BX ∗ of the dual equipped with the weak∗ topology, are all of Baire-1 class, and hence “quite regular.” The Odell–Rosenthal theorem targets this. To conclude, we show that the spaces containing 1 are those whose duals contain the space of measures on [0, 1]. This result is due to A. Pełczy´nski.

II Rosenthal’s 1 Theorem This remarkable and fundamental theorem was discovered by Rosenthal in 1974. Its statement is remarkably simple: it says that every Banach space satisfies the following dichotomy: Theorem II.1 (Rosenthal’s 1 Theorem) Let X be a Banach space, and (xn )n1 a bounded sequence in X. Then, from (xn )n1 , a subsequence can be extracted, which is: 1) either weakly Cauchy 2) or equivalent to the canonical basis of 1 . 326

II Rosenthal’s 1 Theorem

327

These two conditions are mutually exclusive since they are hereditary (if a sequence possesses either of these properties, then so do all its subsequences), and because the canonical basis of 1 is not w-Cauchy, as we have already seen. Here is an immediate consequence: Corollary II.2 contains 1 .

Every non-reflexive weakly sequentially complete space

In particular, this is the case for any space with the Schur property. Proof If the space X does not contain 1 , from each bounded sequence we can extract a w-Cauchy subsequence. If X is w.s.c., this subsequence is w-convergent. By the Eberlein–Šmulian theorem, BX is thus w-compact, and hence X is reflexive.

II.1 Rosenthal–Bourgain–Fremlin–Talagrand Recall that a Polish space P is a separable topological space for which there exists a distance defining the topology of P and turning P into a complete metric space. Theorem II.3 (The Rosenthal–Bourgain–Fremlin–Talagrand Theorem) Let P be a Polish space, and ( fn )n1 a uniformly bounded sequence of continuous real-valued functions on P. Then, for the product topology τ (topology of pointwise convergence) in RP : 1) either ( fn )n1 has a subsequence with no Borel cluster value; 2) or every function f in the closure of { fn ; n  1} is the limit of a subsequence of ( fn )n1 . Remark This no longer holds for complex-valued functions: indeed, between the real and imaginary parts, the first property could apply to one and the second to the other. Nonetheless, we have the following weaker version: if ( fn )n1 is a uniformly bounded complex-valued sequence of continuous functions on P, then one of the two following conditions holds true: 1) ( fn )n1 has a subsequence with no Borel cluster value; 2) ( fn )n1 has a convergent subsequence. Indeed, if ( fn )n1 does not have a convergent subsequence, there exists an infinite subset M0 of N∗ such that either (Re fn )n∈M0 or (Im fn )n∈M0 does not have a convergent subsequence (otherwise, for every infinite subset M of N∗ , (Re fn )n∈M and (Im fn )n∈M would have a convergent subsequence, so, from a convergent subsequence (Re fn )n∈M1 , another convergent subsequence

328

8 The Space 1

(Im fn )n∈M2 could be extracted with M2 ⊆ M1 , and hence ( fn )n∈M2 would be convergent). By replacing fn by ifn , we can assume that it is (Re fn )n∈M0 that has no convergent subsequence. By Theorem II.3, no cluster value of (Re fn )n∈M0 can be Borel. But then, it is the same for ( fn )n∈M0 . Proof (Debs [1987]) First, we can assume that fn ∞ 1. Consider a complete distance on P. Denote by F(P) the set of closed subsets of P, and let U0 be a non-trivial ultrafilter on N. For u, v ∈ Q, with u < v, we define the maps: Du,v : F(P) −→ F(P) as follows: for every closed set F of P, we denote by Du,v (F) the set of x ∈ F such that, for any neighborhood V of x: # " n ∈ N ; infV∩F fn < u < v < supV∩F fn ∈ U0 (we have quantified the fact that the corresponding functions fn are highly oscillatory on V ∩ F). Then:  Du,v (F) ⊆ F Du,v (F) is closed. Indeed, x ∈ Du,v (F) if and only if either x ∈ F or, since U0 is a ultrafilter, there exists a neighborhood V of x such that: {n ∈ N ; infV∩F fn  u} ∈ U0

or

{n ∈ N ; supV∩F fn  v} ∈ U0 .

We can then define a derivation on P by iteration: for any ordinal α, set: ⎧ (0) P =P ⎪ ⎪ ⎪ ⎪ ⎨P(1) = Du,v (P)   ⎪ P(α+1) = Du,v P(α) ⎪ ⎪ ⎪ ⎩ (α)

P = β 0. We can find u < v ∈ Q such that v − u < ε and such that: {n ∈ N ; u < fn (x) < v} ∈ U0 . Indeed, by the previous assumption fn (P) ⊆ [−1, 1], we can find ur < vr ,  1  r  R, with vr − ur < ε, such that [−1, 1] ⊆ Rr=1 [ur , vr ], and N = R r=1 {n ∈ N ; ur < fn (x) < vr }. However, as U0 is an ultrafilter, one of the sets {n ∈ N ; ur < fn (x) < vr } is in U0 .

II Rosenthal’s 1 Theorem

333

Now select u , v ∈ Q such that u < u < v < v and v − u < ε, and apply to the pair (u , u): there exists p  1 such that:

( -- )

fn (x)  u , ∀ n ∈ Hp

or

fn (x)  u , ∀ n ∈ Hp .

The second possibility can never hold, because, in this case, we would have: Hp ∩ {n ∈ N ; u < fn (x) < v} = ∅ , which is excluded, since Hp ∈ U0 and {n ∈ N ; u < fn (x) < v} ∈ U0 . Consequently, we have: fn (x)  u ,

∀ n ∈ Hp .

Similarly, there exists an index q  1 such that: fn (x)  v ,

∀ n ∈ Hq .

Hence, if n ∈ Hp ∩ Hq , we have: u  fn (x)  v . As Hp ∩ Hq ∈ F, and as v − u < ε, the filter ( fn (x))F is thus Cauchy, and hence g(x) = limF fn (x) exists. It only remains to show that g is the limit of a subsequence. This ensues

from the existence of a countable base for F. Indeed, define Hk = kl=1 Hl ; these sets again form a base of the filter F, and, removing some of them ⊂ H . For each k  1, denote if necessary, we can assume that Hk+1 = k ). The sequence (n ) nk = min (Hk  Hk+1 k k1 is strictly increasing, and:

{n1 , n2 , . . .} ∩ Hk = {nk , nk+1 , . . .} ; therefore limF fn = limk→+∞ fnk , and the proof of the theorem is hence complete. In fact, this proof of Theorem II.3 implies the following: Corollary II.5 Let P be a Polish space, and ( fn )n1 a uniformly bounded sequence of continuous real-valued functions on P. Then, if every function in the closure (for the product topology) of {fn ; n  1} is Borel measurable, this closure consists in fact of Baire-1 functions, and each of its elements is the limit of a subsequence ( fnk )k1 . Moreover, every subsequence of ( fn )n1 has a convergent subsequence. Proof By hypothesis, the functions in the closure are all Borel, so the first case of the theorem does not hold. Now, for every g in the closure of {fn ; n  1}, there exists an ultrafilter U0 on N such that g = limU0 fn . The proof of the second case provides a filter F with a countable base, coarser than U0 , for which limF fn exists, and is equal to the limit of a subsequence of ( fn )n1 . As this limit is necessarily g, since U0 is finer than F, the stated result is proved.

334

8 The Space 1

Remark The proof of the theorem highlights the following fact. In the first case, we found rationals u < v ∈ Q, a subsequence ( fnk )k0 and a continuous injective map ϕ : {0, 1}N → P such that, if ε ∈ {0, 1}N , then:    fnk ϕ(ε) > v ⇐⇒ εk = 1   fnk ϕ(ε) < u ⇐⇒ εk = 0 . If to each ε ∈ {0, 1}N we associate the set I = {k ∈ N ; εk = 1}, then the above condition can be re-written:  fnk (x) > v for k ∈ I ∀ I ⊆ N , ∃ x = ϕ(ε) ∈ P : fnk (x) < u for k ∈ N  I . Thus, if we define: Ak = {x ∈ P ; fnk (x) < u}

and

Bk = {x ∈ P ; fnk (x) > v} ,

these sets satisfy the property in the following definition: Definition II.6 Let P be a set, and (Ak )k1 and (Bk )k1 two sequences of subsets of P. They are said to be Boolean independent if: 1) Ak ∩ Bk = ∅ for any k  1;



2) k∈I Ak ∩ k∈J Bk = ∅ for all finite disjoint subsets I, J ⊆ N∗ . A more concrete example of Boolean independent sequences is obtained with P = [0, 1[ and       j j+1 j j+1 , , and Bk = . Ak = 2k 2k 2k 2k k k 0j2 −1 j even

0j2 −1 j odd

The proof of the theorem hence provides the following: Proposition II.7 Let P be a Polish space, and ( fn )n1 a uniformly bounded sequence of continuous real-valued functions on P without any convergent subsequence. Then there exist a subsequence ( fnk )k0 and rationals u < v ∈ Q such that, if: Ak = {x ∈ P ; fnk (x) < u}

and

Bk = {x ∈ P ; fnk (x) > v} ,

the two sequences (Ak )k1 and (Bk )k1 are Boolean independent.

II Rosenthal’s 1 Theorem

335

II.2 Proof of Rosenthal’s 1 Theorem in the Real Case The proof will be deduced from that of the Rosenthal–Bourgain–Fremlin– Talagrand theorem. By replacing the space X by the (closed) subspace generated by the xn ’s, n  1, we can assume that X is separable. The unit ball BX ∗ of its dual is then a metrizable compact set for the weak∗ topology, and hence a Polish  space. The elements xn of X are continuous functions on this space K = BX ∗ , w∗ . By the Rosenthal–Bourgain–Fremlin–Talagrand theorem: – either every point g ∈ RK adherent to {xn ; n  1} is the pointwise limit of a subsequence (xnk )k1 , and then, in particular, this subsequence is weakly Cauchy; – or there exist a subsequence  (xnk )k1 and numbers u < v ∈ Q, such  that, with  Ak = {y ∈ K = BX ∗ , w∗ ;xnk (y)< u}andBk = {y ∈ K = BX ∗ , w∗ ; xnk (y)>v}, the two sequences (Ak )k1 and (Bk )k1 are Boolean independent. In the latter case, the sequence (xnk )k1 is equivalent to the canonical basis of 1 : a proof of this follows from Proposition II.8 below. Proposition II.8 (Rosenthal) Let K be a set, and let ( fk )k1 be a sequence of real functions, bounded in   ∞ (K) = f : K −→ R ; sup |f (t)| < +∞ . t∈K

Suppose that there exist two Boolean independent sequences (Ak )k1 and (Bk )k1 of subsets of K, and real numbers δ > 0 and r ∈ R such that:  fk (t)  r − δ for t ∈ Ak fk (t)  r + δ

for

t ∈ Bk .

Then, the sequence ( fk )k1 is equivalent in ∞ (K) to the canonical basis of the real space 1 . Proof

Obviously:    n   ak fk    k=1

∞

 sup fk ∞ . k1

n

|ak | .

k=1

For the converse, we set:  P = {k ∈ {1, . . . , n} ; ak > 0} N = {k ∈ {1, . . . , n} ; ak < 0} .

336

8 The Space 1

By Boolean independence:     Ak ∩ Bk = ∅ k∈N

Take: t1 ∈

k∈P



 Ak ∩

We have:    n    a f k k 

∞



   Ak ∩ Bk = ∅ .

k∈P



k∈N

k=1

and

 and t2 ∈

Bk

k∈P

k∈N



 Ak ∩

k∈P



 Bk .

k∈N

   1  sup  ak fk (s) − fk (t)  2 (s,t)∈K×K n



k=1

 1   ak fk (t1 ) − fk (t2 ) 2 k=1 5 4    1  = ak fk (t1 ) − fk (t2 ) + ak fk (t1 ) − fk (t2 ) 2 n

k∈P

δ

n

k∈N

|ak | .

k=1

II.3 Proof of Rosenthal’s 1 Theorem in the Complex Case The first proof of the complex case is due to L. Dor [1975 b]. Here we give that of A. Pajor (see Pajor), which derives the complex case from the real case. Theorem II.9 Let X be a complex Banach space, and let (zn )n1 be a sequence equivalent, in the underlying real space, to the canonical basis of the real space 1 . Then (zn )n1 contains a subsequence equivalent to the canonical basis of the complex space 1 . The proof necessitates several intermediate results. The first one is in fact a variant of Proposition II.8 in more combinatorial terms. Proposition II.10 Let S ⊆ {−1, 0, +1}N with the following property: For every infinite subset M ⊆ N, there exists an element s = (sn )n∈N ∈ S such that the two subsets {n ∈ M ; sn = −1} and (∗) {n ∈ M ; sn = +1} are infinite. Then, there exists an infinite subset M0 ⊆ N such that PI (S) ⊇ {−1, +1}I, for every finite subset I ⊆ M0 , where PI is the natural projection of {−1, 0, +1}N onto {−1, +1}I . Proof First, S can be assumed closed; in fact, if S satisfies the condition (∗), obviously so does its closure S. Apply the proposition to S: for every finite

II Rosenthal’s 1 Theorem

337

subset I of M0 , and for any ε = (εk )k∈I , there exists s ∈ S such that sk = εk for any k ∈ I. However, since I is finite, there exists s ∈ S such that s k = sk for any k ∈ I; consequently, we have PI (S) ⊇ {−1, +1}I . Hence S is a Polish space. Consider the projections fn : S → {−1, 0, +1} ⊆ R, s → fn (s) = sn : they are continuous. The hypothesis on S implies that the sequence ( fn )n∈N has no convergent subsequence. Consequently, there exist a subsequence ( fnk )k∈N and rational numbers u < v ∈ Q such that the sequences (Ak )k∈N and (Bk )k∈N , with Ak = {s ∈ S ; snk < u}

and

Bk = {s ∈ S ; snk > v} ,

are Boolean independent. Thanks to the conditions of the proposition, the rational numbers u and v can be chosen so that Ak and Bk are: Ak = {s ∈ S ; snk = −1}

and

Bk = {s ∈ S ; snk = +1} ;

then the Boolean independence of the two sequences is exactly the stated result, with M0 = {nk ; k ∈ N}. For this, we return to the proof of Theorem II.3. We had a non-trivial ultrafilter U0 on N. For each s ∈ S, a base of neighborhoods of s is provided by: VN = {σ ∈ S ; σj = sj , ∀ j  N} ,

N  1.

We will show that, for every s ∈ S and for any N  1: (-)

{n ∈ N ; infσ ∈VN σn = −1 and supσ ∈VN σn = +1} ∈ U0 .

Then, since the elements of S take on the values −1, 0 and 1 only, if we take −1 < u < 0 < v < +1: {n ∈ N ; infσ ∈VN fn (σ ) < u < v < supσ ∈VN fn (σ )} = {n ∈ N ; infσ ∈VN σn = −1 and supσ ∈VN σn = +1} ∈ U0 , and thereby Du,v (S) = S. With the rest of the construction remaining unchanged, we obtain a subsequence (nk )k0 such that the sequences (Ak )k0 and (Bk )k0 defined above are Boolean independent. Suppose now that (-) is not true. Then there exist an s ∈ S and an integer N  1 such that, for the neighborhood VN of s, we would have: {n ∈ N ; infσ ∈VN σn = −1 and supσ ∈VN σn = +1} ∈ / U0 . As U0 is an ultrafilter, it follows that the complement of this set is in U0 : {n ∈ N ; infσ ∈VN σn = 0 or 1

or

supσ ∈VN σn = −1 or 0} ∈ U0 .

Again because U0 is a ultrafilter, this implies: {n ∈ N ; infσ ∈VN σn = 0 or 1} ∈ U0 or {n ∈ N ; supσ ∈VN σn = −1 or 0} ∈ U0 .

338

8 The Space 1

Consider, for example, the first case, and denote by M the set appearing there. It is infinite, since it is in U0 , and, according to the condition (∗), there exists a σ ∈ S such that {n ∈ M ; σn = −1} is infinite. Set σj = sj if 1  j  N and σj = σj for j  N + 1; then σ ∈ VN , and this contradicts the definition of M. Proposition II.11 (Pajor) Let E be a finite set, and (E− , E+ ) a partition of E. Define the map Q : EN → {−1, +1}N , by:  (Qσ )n = −1 if σn ∈ E− (Qσ )n = +1

if σn ∈ E+ ,

where σ = (σn )n∈N . Let  be a subset of EN such that Q() = {−1, +1}N . Then, there exist an infinite subset M0 of N and a couple (e− , e+ ) ∈ E− × E+ such that, with n− = {σ ∈  ; σn = e− }

and

n+ = {σ ∈  ; σn = e+ } ,

the sequences (n+ )n∈M0 and (n− )n∈M0 are Boolean independent. Proof First, we prove the existence of a pair (e− , e+ ) ∈ E− × E+ and of an infinite subset M of N such that, for every infinite subset L ⊆ M, there exists σ ∈  with {n ∈ L ; σn = e− } and {n ∈ L ; σn = e+ } infinite. Indeed, if this were not the case, then, since E is finite, we could by iteration construct an infinite subset L such that, for every couple (e− , e+ ) ∈ E− × E+ and for every σ ∈ , one of the sets {n ∈ L ; σn = e− } or {n ∈ L ; σn = e+ } would be finite. However, for every infinite set L, since Q() = {−1, +1}N , there exists a σ ∈  such that {n ∈ L ; σn ∈ E+ } and {n ∈ L ; σn ∈ E− } are infinite. + − + But, as the set E is finite, there exists a pair (e− 0 , e0 ) ∈ E × E such that − + {n ∈ L ; σn = e0 } and {n ∈ L ; σn = e0 } are infinite, which is a contradiction. Now define the application  :  → {−1, 0, +1}M by: ⎧ − ⎪ ⎪ ⎨−1 if σn = e   (σ ) n = +1 if σn = e+ ⎪ ⎪ ⎩ 0 otherwise , and set S = () ⊆ {−1, 0, +1}M . Then Proposition II.10 implies the result. Proposition II.12 Let (zn )n0 be a sequence of complex-valued functions, uniformly bounded by 1 on a set T. If the real parts xn = Re zn satisfy, for some δ > 0:      an xn (t)  δ |an | sup  t∈T

n∈I

n∈I

II Rosenthal’s 1 Theorem

339

for every finite subset I of N and every sequence (an )n∈I of real numbers, then there exists an infinite subset M of N such that:     δ  cn zn (t)  |cn | sup  2 t∈T n∈I

n∈I

for every finite subset I of M and every sequence (cn )n∈I of complex numbers. First let us see how Theorem II.9 ensues. Proof of Theorem II.9 Let T be the unit ball of X ∗ and isometrically embed X in ∞ (T, C). If (zn )n1 is a sequence equivalent to the canonical basis of the real space 1 , it does not have any weakly Cauchy subsequence. Consequently, there exists an infinite subset M0 of N∗ such that one of the two sequences (xn )n∈M0 = (Re zn )n∈M0 and (yn )n∈M0 = (Im zn )n∈M0 does not have any weakly Cauchy subsequence (as otherwise, for every infinite set M, both sequences (xn )n∈M and (yn )n∈M would have one; starting with M = N∗ , we would obtain an infinite subset M1 of N∗ such that (xn )n∈M1 is weakly Cauchy; next, with M = M1 , we could find an infinite subset M2 of M1 such that (yn )n∈M2 is weakly Cauchy; the sequence (zn )n∈M2 would then be weakly Cauchy). By replacing zn by izn if necessary, we can assume that it is (xn )n∈M0 . By the real case of Rosenthal’s theorem, (xn )n∈M0 has a subsequence equivalent to the canonical basis of the real space 1 . Then Proposition II.12 states that the corresponding subsequence (znk )k1 itself possesses a subsequence equivalent to the canonical basis of the complex space 1 . Proof of Proposition Let K be the w∗ -closed balanced (in the real sense)   II.12 convex hull of { xn (t) n0 ; t ∈ T} ⊆ ∞ (T, R). By convexity, for every sequence (an )n0 ∈ 1 (N, R), we have:        sup  an κn  ; κ = (κn )n0 ∈ K  δ |an | . n∈N

n∈N

So the map j : 1 (N, R) → C(K) sending a = (an )n0 ∈ 1 (N, R) to  the function κ = (κn )n0 → n0 an κn is a δ-isomorphism. Its adjoint ∗ j : M(K) → ∞ is thus surjective, and j∗ (δκ ) = κ for every κ ∈ K, δκ being the Dirac mass at κ. Identifying δκ and κ, we obtain: {−δ, δ}N ⊆ K . Now consider the w∗ -closed balanced (in the real sense) convex hull H of   { zn (t) n0 ; t ∈ T} ⊆ ∞ (T, C). We have Re H = K; hence there exists a

subset H of H such that Re H = {−δ, δ}N .

340

8 The Space 1

  Then let 0 < α < π4 δ, and divide the interval [−1, 1] into N = α2 + 1 disjoint intervals I1 , . . . , IN of length  α. For any u ∈ [−1, 1], we denote by k(t) the unique integer k  N such that u ∈ Ik . Next, the application  : H → {−N, . . . , −1, +1, . . . , N}N is defined as follows: if κ = (ζn + iξn )n∈N ∈ H , set, for any n  0:   (κ) n = sign(ζn ) k(ξn ) . Then set  = (H ), take E =  {−N, . . . ,−1, +1, . . . , N} and define  Q : EN → {−1, 1}N by Q (θn )n = (sign θn )n . Since Re H = {−δ, δ}N , we have Q() = {−1, 1}N , and we can thus apply Proposition II.11: there exist an infinite subset M0 of N and two distinct integers k, l ∈ {1, 2, . . . , N} such that, for every finite subset I of M0 , and every subset of J of I, κ ∈ H can be found so that:  (κ )n ∈ (−Ik ) for n ∈ J (κ )n ∈ Il

for n ∈ I  J .

Replacing J by I  J, we can also find a κ ∈ H so that:  (κ )n ∈ (−Ik ) for n ∈ I  J (κ )n ∈ Il

for n ∈ J .



Setting κ = κ +κ 2 , we thus have, by convexity of H, an element κ = (ζn +iξn )n0 ∈ H such that, a being the center of the interval 12 (−Ik +Il ), then:  ζn = δ and |ξn − a|  α/2 for n ∈ J ζn = −δ

and

|ξn + a|  α/2

for n ∈ I  J .

Note that, in particular, |ξn − a sign(ζn )|  α/2 for any n ∈ I. Now, for every family (cn )n∈I of complex numbers, we have:              cn zn   sup  cn τn  ; (τn )n0 ∈ H  ∞ (T) n∈I n∈I      cn θn κn   sup  θn =±1

n∈I

θn =±1

n∈I

       sup  cn θn ζn + ia sign(ζn )  −



|ξn − a sign(ζn )| |cn |

n∈I

    α  cn (δ + ia) εn  − |cn |  sup  2 εn =±1 n∈I n∈I   α 2 |δ + ia| − |cn | ,  π 2 n∈I

III Further Results on Spaces Containing 1

since:

341

    2  max  wn εn   |wn | εn =±1 π n∈I

n∈I

for every sequence (wn )n∈I of complex numbers, since the Rademacher functions form a Sidon set of constant π/2 in the Cantor group (see Chapter 6 in this volume). A fortiori we thus have:       2 α   c z  |cn | ; δ − n n  π 2 ∞ (T)

n∈I

4

n∈I

 hence, by taking α = π − 1 δ, we obtain:     δ   c z  |cn | , n n  2 ∞ (T) n∈I

n∈I

which completes the proof. Remark In Chapter 3 of Volume 2, we will see a finite-dimensional version of Theorem II.9.

III Further Results on Spaces Containing 1 III.1 The Odell–Rosenthal Theorem Definition III.1 An element of the unit ball of the bidual X ∗∗ of a Banach space X is said to be of first class if it is the weak∗ limit of a sequence of elements of the unit ball BX of X. Then the following result holds (Odell and Rosenthal [1975]): Theorem III.2 (The Odell–Rosenthal Theorem) If X is a separable Banach space, then X does not contain 1 if and only if every element of the unit ball BX ∗∗ of its bidual is of first class. Proof Suppose that X does not contain 1 . We first consider the case where X is a real Banach space. As X is separable, we can find a sequence (xn )n1 which is dense (in norm) in the unit ball of X. By the discussion preceding Proposition II.8, and because X does not contain 1 , every element of the weak∗ closure of {xn ; n  1} is the weak∗ limit of a subsequence of (xn )n1 (the weak∗ topology on BX ∗∗ coincides with that of pointwise convergence on the metrizable compact set (BX ∗ , w∗ )). However this closure is equal to the unit ball BX ∗∗ of X ∗∗ , hence the result. In the case where X is a complex Banach space, first, by Theorem II.9, the underlying real space XR also does not contain the (real) space 1 .

342

8 The Space 1

Every element of the unit ball of (XR )∗∗ is hence the limit of a sequence of elements uk ∈ BX . Now we have an R-linear surjective isometry J : ψ ∈ X ∗ → ϕ = Re ψ ∈ (XR )∗ , since ψ(x) = (Re ψ)(x) − i(Re ψ)(ix) for every x ∈ X. To each  ∈ X ∗∗ , we can hence associate an element  ∈ (XR )∗∗ by setting, for any ϕ ∈ (XR )∗ : (ϕ) = (Re )(ψ) = (Re )(J −1 ϕ), so , Re ψ = Re , ψ . If  ∈ BX ∗∗ , then  ∈ B(XR )∗∗ , and the real case of the proof gives a sequence of elements uk ∈ BX , weak∗ convergent to : uk , ϕ −−→ , ϕ k→+∞ for every ϕ ∈ (XR )∗ . In other words: uk , Re ψ = uk , ϕ −−→ , ϕ = Re , ψ k→+∞

for every ψ ∈ X ∗ . Replacing ψ by iψ and using the equality Re(iψ) = − Im ψ, we obtain: uk , Im ψ −−→ − Re , iψ ; k→+∞

therefore, by summing: uk , ψ −−→ Re , ψ − i Re , iψ = , ψ , k→+∞

and hence (uk )k1 is weak∗ convergent to . For the converse, note first that the elements of the unit ball of ∗∗ 1 which w∗

are not in 1 are not of first class. Indeed, if xn ∈ 1 and xn −−→ x∗∗ , n→+∞ the sequence (xn )n1 is weakly Cauchy. As 1 has the Schur property (see Chapter 2), it is in particular weakly sequentially complete (also a consequence of the Dunford–Pettis theorem, Chapter 7), and hence x∗∗ ∈ 1 . Further “explicit” examples of elements that are not of first class can be given: if (en )n1 is the canonical basis of 1 , for every non-trivial ultrafilter U on ∗ N∗ , we can define an element eU ∈ ∗∗ 1 by eU = w -limU en ; eU maps x = (xn )n1 ∈ ∞ to limU xn . However such an element cannot be of first class since it is not measurable on the Polish space (B∞ , w∗ ): indeed, for every subset I of N∗ , we see that eU (1II ) = 1 if and only if I ∈ U ; as the map that sends I ∈ P(N∗ ) to 1II ∈ B∞ is a homeomorphism, the non-measurability of eU ensues from Sierpinski’s lemma. Suppose now that X contains a subspace Y isomorphic to 1 . If all the elements of X ∗∗ were of first class, so would be those of Y ∗∗ , according to the following lemma; but by the previous study, this is not possible. Lemma III.3 Let X be a Banach space, and Y a subspace of X. Each firstclass element of X ∗∗ , which is in Y ⊥⊥ , is the limit of a sequence of elements in the unit ball of Y. In other words, if we identify Y ⊥⊥ and Y ∗∗ , this element is of first class in Y ∗∗ . When X = C(K), where K is the unit ball of Y ∗ equipped with the

III Further Results on Spaces Containing 1

343

weak∗ topology, Y can be identified with the space A0 (K) of affine continuous functions on K and which are null at 0. The lemma can be reformulated as ∼ follows: each first-class function which is in Y ∗∗ = A0 (K)⊥⊥ is of affine first class, i.e. it is the limit of a sequence of continuous affine functions. Remarkably, each affine function that is of first class is of affine first class (see Comments, Section IV). Proof Assume that x∗∗ = limn→+∞ xn , with xn ∈ BX , and denote A = w∗ {xn ; n  1}; we have x∗∗ ∈ A . On the other hand, by hypothesis, w∗ x∗∗ ∈ BY ⊥⊥ = BY . However: Sub-Lemma III.4 Let X be a Banach space, and A, B ⊆ X two subsets of X. w∗ w w∗ If A ∩ B = ∅, then 0 ∈ A − B . w

Consequently, 0 ∈ A − BY , and a fortiori, 0 ∈ convw (A − BY ). However convex sets closed in norm are weakly closed; therefore 0 ∈ conv  (A − BY ). Thus there exist a subsequence (xnk )k1 , some yN,k ∈ BY kN+1 −1 and some coefficients λN,k  0 such that k=kN λN,k = 1 with   kN+1 −1   kN+1 −1 w∗  k=kN λN,k (xnk − yN,k )  1/N. Now k=kN λN,k xnk −−→ x∗∗ , and N→+∞ kN+1 −1 ∗ converge to x∗∗ . λ y ∈ B also weak hence the elements k=k N,k N,k Y N Proof of the sub-lemma w∗

w∗

Let ϕ1 , . . . , ϕq ∈ X ∗ and ε > 0. For every x∗∗ ∈

A ∩ B , there exist a ∈ A and b ∈ B such that | ϕj , x∗∗ − a |  ε/2 and | ϕj , x∗∗ − b |  ε/2, for 1  j  q; then | ϕj , a − b |  ε. We thus indeed w obtain 0 ∈ A − B .

III.2 Duals of Spaces That Do Not Contain 1 The following result will be proved (Pełczy´nski [1968]): Theorem III.5 (Pełczy´nski) properties are equivalent:

For every Banach space X, the following

(1) X contains 1 ; (2) X ∗ contains L1 = L1 (0, 1); (3) X ∗ contains the space M([0, 1]) of measures on [0, 1]. Moreover, when X is separable, these conditions are also equivalent to: (4) C([0, 1]) is a quotient of X. We need the following elementary result:

344

8 The Space 1

Proposition III.6 Every separable Banach space is a quotient of 1 . Proof Denote such a space by Z. Let (zn )n1 be a dense sequence in the unit ball of Z. With (en )n1 the canonical basis of 1 , define S : 1 → Z by setting Sen = zn for any n  1; by the density of (zn )n1 , S is surjective. Proof of theorem (1) ⇒ (3) . Let (xn )n1 be a sequence in X, equivalent to the canonical basis of 1 , and let Y = [xn ; n  1] be the closed subspace generated by this sequence. Proposition III.6 provides a continuous and surjective operator T : Y → C = C([0, 1]). If we could extend T to a surjective operator * T : X → C, everything would be over; however C does not have the extension property: operators with values in C cannot in general be extended to super-spaces of the original space. Nonetheless, this is “almost” possible as the bidual C ∗∗ = M∗ of C does possess this property. Lemma III.7 For every measure space (S, T , μ), the space L∞ (μ) possesses the extension property: for every Banach space X and for every continuous operator T : Y → L∞ (μ) from a T: X → subspace Y of X into L∞ (μ), there exists an extension * T  = T. L∞ (μ) such that * In particular, C ∗∗ has this property. Proof The first assertion was already seen in Chapter 6, Lemma IV.5. The second assertion comes from M being isometric to a space L1 (ν); indeed, if (μα )α is a maximal family of dis∼ joint measures, then, by the Radon–Nikodým theorem, M = 3 1 ∼ 1 1 L (|μα |) = L (μ), where μ is the direct sum of the meaα

sures μα . Then let us continue the proof of the theorem: if j : Y → X and i1 : C → C ∗∗ are the canonical injections, there exists an extension * T  = T, which makes the following T : X → C ∗∗ such that * diagram commutative: Y

T

/ C

* T

 / C ∗∗

i1

j

 X

III Further Results on Spaces Containing 1

345

By dualizing, we obtain: M = C∗

i2

/ C ∗∗∗

∗ * T

i∗1

/ X∗ j∗

 C∗

T∗

 / Y∗

where i2 : C ∗ = M → C ∗∗∗ is the canonical injection. The ∗ operator J = * T i2 : M → X ∗ is then the desired embedding. In fact, i∗1 i2 = IdC ∗ ; hence: ∗

T ∗ = T ∗ i∗1 i2 = j∗* T i2 = j∗ J . Now T is surjective, and hence, by the open mapping theorem, there exists a constant C > 0 such that: C μM  T ∗ μY ∗  j∗ JμY ∗  JμX ∗  T μM . (2) ⇒ (1) . Assume we have an embedding T : L1 → X ∗ . Then T ∗ : X ∗∗ → hence there exists a constant c > 0 such L∞ is surjective;   that T ∗ BX ∗∗  ⊇ c BL∞ . By Goldstine’s theorem, the weak∗ closure of T ∗ BX contains the ball c BL∞ . Let ε > 0: for any n  1, we consider an (ε/2c)-net {z1 , . . . , zM } of the unit circle U = {z ∈ C ; |z| = 1} (for reasons of notation, the spaces are assumed complex). Set: Tn = {(1, m1 , . . . , mn−1 ) ∈ Nn ; 1  mj  M , 1  j  n − 1} , and define the tree: T =



Tn .

n1

Note that if α = (1, m1 , . . . , mn−1 ) ∈ Tn , then: Tn+1 = {(α, mn ) ; α ∈ Tn and 1  mn  M} , where we have set: (α, m) = (1, m1 , . . . , mn−1 , m) . Let n  1 be fixed for now. We are going to define, by induction, a family (Aα )α∈Tn of disjoint measurable subsets of [0, 1] and a function fn ∈ T ∗ BX such that, for α ∈ Tn : (∗)

M  m=1

A(α,m) ⊆ Aα ⊆ [0, 1] ,

346

8 The Space 1 and, for 1  m  M: (∗∗)

|fn − c zm | 

ε on A(α,m) . 2

To start the induction, we take: A(1) = [0, 1] , and assume that we have constructed disjoint sets: {Aα ; α ∈ Tn } of positive measure. For every α ∈ Tn , we divide Aα into disjoint sets D(α,1) , . . . , D(α,M) of positive measure. We consider the function gn ∈ BL∞ defined by:  zm if t ∈ D(α,m) and α ∈ Tn gn (t) = 0 otherwise .  w∗   As T ∗ BX ⊇ c BL∞ , there exists fn ∈ T ∗ BX sufficiently approximating gn to ensure that the sets: A(α,m) = {|fn − c zm |  ε/2} ∩ D(α,m) all have positive measure. This completes the induction. For every n  1, we now choose xn ∈ BX such that T ∗ (xn ) = fn . Let us prove that this sequence (xn )n1 is equivalent to the canonical basis of 1 . For this, we take scalars a1 , . . . , aN , and aN ∈ U by: define the numbers * a1 , . . . ,* ⎧ ⎨ |an | if a = 0 n * an = an ⎩ 1 if an = 0 , so that |* an | = 1 for every n. For 1  n  N, there hence exists mn ∈ {1, . . . , M} such that: |* an − zmn |  ε/2c · Set α = (1, m1 , . . . , mN ). As α ∈ TN+1 , the assertions (∗) and (∗∗) give: |fn − c zmn |  ε/2 on Aα

III Further Results on Spaces Containing 1

347

for 1  n  N. Consequently:       N    ∗ N    T  an xn   T an xn   n=1

n=1

  N     c an* an 1IAα   c

∞

∞

n=1 N

n

n=1

n=1

|an | − ε

    N  = an fn   n=1

∞

  N    − an (c* an − fn ) 1IAα  

∞

n=1

|an | = (c − ε)

N

|an | ,

n=1

and this completes the proof of the first part of the theorem, since the implication (3) ⇒ (2) is trivial. For the second part, first note that (4) ⇒ (3) , whether X is separable or not. It remains to see that (1) ⇒ (4) when X is separable. We keep the notation and the diagram: X∗

j∗

/ Y∗ O T∗

M = C∗ of the proof of (1) ⇒ (3) . With δt as the Dirac mass at t ∈ [0, 1], the set: F = {T ∗ (δt ) ; t ∈ [0, 1]} ⊆ Y ∗ , equipped with the weak∗ topology, is homeomorphic to [0, 1] and equivalent to the canonical basis of 1 ([0, 1]), since T ∗ is an embedding and is weak∗ continuous. Moreover, j∗ (BX ∗ ) = BY ∗ , by the Hahn–Banach theorem. It follows that the weak∗ compact set: E = j∗ −1 (F) ∩ T BX ∗ satisfies j∗ (E) = F. We are thus ready to apply the following lemma: Lemma III.8 Let E and F be two compact spaces. Suppose that F is perfect (without isolated points), and that there exists a continuous surjective map φ : E → F. Then, there exists a subset  of E, dense-in-itself, such that the restriction φ| :  → F is bijective. Recall that a topological space is dense-in-itself if all of its points are limit points.

348

8 The Space 1

We thus have a subset:  = {γt ; t ∈ [0, 1]} of E, dense-in-itself for the weak∗ topology, such that j∗ (γt ) = T ∗ (δt ) for any t ∈ [0, 1]. Moreover, this set is equivalent to the canonical basis of 1 (); indeed, for any finite family (at )t∈I of scalars, we have:         ∗ ∗    |at |  c  at T (δt ) = c  at j (γt )  t∈I

t∈I

      c at γt   t∈I

X∗

Y∗

 c T

t∈I



Y∗

|at | .

t∈I

Now, as X is separable, the unit ball BX ∗ of X ∗ is metrizable for the weak∗ topology, and hence separable, since compact. As  is a bounded subset of X, it follows that we can extract from it a countable subset {γ1 , γ2 , . . .} dense-initself. The following proposition then completes the proof of Theorem III.5, since with  the Cantor set, C([0, 1]) and C() are isometric, as seen in Chapter 2. Proposition III.9 (Rosenthal) Let X be a separable Banach space, and (ϕn )n1 a bounded sequence in X ∗ , equivalent to the canonical basis of 1 and dense-in-itself for the weak∗ topology. Then there exists a subset  of X ∗ , weak∗ homeomorphic to the Cantor set , such that the natural map T : X → C(), defined by (Tx)(ω) = ω(x), is surjective. Proof of Lemma III.8 Using Zorn’s lemma and the compactness of E, we can find a minimal closed subset E0 of E such that φ(E0 ) = F. As F is perfect, so is E0 , thanks to its minimality. Now for each y ∈ F, we select an element xy ∈ E0 such that φ(xy ) = y. Let  = {xy ; y ∈ F}. Obviously φ() = F, and hence, as E0 is minimal,  = E0 . Finally, the conjunction of “E0 is perfect” and “ is dense in E0 ” implies “ is dense-in-itself”. Proof of Proposition III.9 Let D = {ϕ1 , ϕ2 , . . .}, and let K be the weak∗ closure of D. We construct a subset  of K, homeomorphic to the Cantor set, such that the application T defined in the statement of the theorem is surjective. As X is separable, K is compact and metrizable for the weak∗ topology. We set K10 = K. For n  0, assume that we have constructed subsets K1n , . . . , K2nn of K, disjoint, compact and with non-empty interiors. Let Fn+1 be a (1/2n+3 )n+1 net of the unit ball of 2∞ containing the vectors of the canonical basis of n+1 2∞ . Since D is dense-in-itself, for any j, we can choose distinct elements d2j−1 (n) = d2j−1 and d2j (n) = d2j ∈ D ∩ Kjn . Next, as the space generated

III Further Results on Spaces Containing 1

349

n+1

by d1 , d2 , . . . , d2n+1 is isomorphic to 21 , for every f ∈ Fn+1 , there exists an element uf ∈ X such that uf   C (where C is a constant depending on the constant of equivalence between (ϕn )n1 and the canonical basis of 1 ), and such that dj (uf ) = f ( j) for 1  j  2n+1 . Note that, because Fn+1 is finite, the sets:  (∗) Vj = {k ∈ Kjn ; |k(uf ) − dj (uf )| < 1/2n+3 } f ∈Fn+1

are open and non-empty, for 1  j  2n . Moreover, as Fn+1 contains the n+1 canonical basis of 2∞ , the sets V1 , . . . , V2n are disjoint. n+1 Then, for 1  j  2n , we can find compact disjoint neighborhoods K2j−1 n+1 and K2j of d2j−1 and d2j respectively, whose diameters are at most 1/2n+3 , and are contained in Vj . This construction provides, by induction, the sets Kjn and Fn for any n  0 and 1  j  2n . Now we define:

=

+∞ 2n 

Kjn .

n=0 j=1

By construction, this set  is homeomorphic to the Cantor set . It remains to show that the map T : X → C() is onto. Begin by fixing n  0 and scalars c1 , . . . , c2n , such that: n

φn =

2

cj 1IKjn ∩

j=1

is of norm 1 in C(). We can choose f ∈ Fn such that |f ( j) − cj |  1/2n+2 for any j. For each j, (∗) shows that if k ∈ Kjn ∩ , then |k(uf ) − f ( j)|  1/2n+2 ; consequently, T(uf ) − φn ∞  1/2n+1 . Now let ψ ∈ C(), with norm 1. We can find φ1 of the above form, with norm  1, and such that ψ − φ1 ∞  1/22 . The preceding reasoning shows that there exists u1 = uf1 ∈ X, with norm  C, such that T(u1 ) − φ1 ∞  1/22 . Start again with ψ1 = ψ − T(u1 ) instead of ψ; as ψ1 ∞  T(u1 ) − φ1 ∞ + ψ − φ1 ∞  1/21 , there exists φ2 , with norm  1/21 , such that ψ1 − φ2 ∞  1/23 , and thus, by homogeneity, there is u2 ∈ X, of norm  C/21 , such that T(u2 ) − φ2 ∞  1/23 , hence T(u1 ) + T(u2 ) − ψ∞  1/22 . By continuing likewise, for every n  1, we obtain elements un ∈ X, with norm  C/2n−1 such that: T(u1 ) + · · · + T(un ) − ψ∞  Then, setting u =

+∞

n=1 un ,

1 . 2n

we conclude that T(u) = ψ.

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IV Comments 1) H.P. Rosenthal [1974 b] only proved his theorem for real Banach spaces. The complex case does not follow immediately from the real case; L. Dor [1975 b] modified Rosenthal’s proof to adapt it to the complex case. The method used here, allowing the deduction of the complex case from the real case is due to A. Pajor [1983] (see Pajor). The original proof of H.P. Rosenthal in spite of the use of a transfinite construction, remains “elementary”, in the sense that it does not depend on any external result. Even though we did not follow it for our presentation, we highly recommend a reading of it (see also Morrison). J. Farahat [1974] later replaced this transfinite construction by an argument of infinite combinatorics, the Nash–Williams theorem, and this proof is the one usually presented (see Diestel, pages 192–218, Guerre-Delabrière, Lindenstrauss–Tzafriri). The proof given here is due to G. Debs [1987], and our presentation follows that of G. Godefroy from a graduate course in Paris in 1989. An interesting variant, using a technique of re-norming, can be found in Deville–Godefroy–Zizler, Chapter III § 3. 2) The Rosenthal–Bourgain–Fremlin–Talagrand theorem, as formulated here, is due to Rosenthal [1974 a]. It was generalized by Bourgain, Fremlin and Talagrand [1978] (see also Talagrand, Theorems 7-3-1, 14-1-8 and 14-2-1). This enables us to give the complete version of the Odell–Rosenthal theorem: Theorem (The Odell–Rosenthal Theorem) Let X be a separable Banach space. Then, the following assertions are equivalent: (a) X does not contain 1 ; (b) every element of the unit ball of X ∗∗ is of first class; (c) every bounded sequence of elements of the bidual X ∗∗ possesses a weak∗ convergent subsequence. The result fails if X is not separable. It suffices to take X = c0 (), where  is an uncountable set; then X does not contain 1 , but the element 1I ∈ X ∗∗ = ∞ () is not the weak∗ limit of a sequence (xn )n1 of elements of X, since such a limit would necessarily have countable support. 3) In the proof of Theorem III.2, we used the fact that the elements of ∗∗ 1  1 , as functions on (B∞ , w∗ ), are not of first class; actually they are not measurable (Christensen). The fact that every affine function of first class is of affine first class was shown by Mokobodzki, based on results by Rogalski (see Rogalski [1968],

IV Comments

351

Theorem 80); the proof relies essentially on a theorem of Choquet [1962], stating that every affine function f of first class on a compact convex set K (contained in a locally convex space) satisfies the barycenter calculus (a proof of this theorem is given in Alfsen, Theorem I.2.6, or Phelps, Section 12). If Y is a (real) Banach space, K the unit ball of Y ∗ equipped with the weak∗ topology and i : Y → C(K) the canonical injection, we call the barycenter of the probability measure μ the element bμ = i∗ (μ) of the function g : K → R is said to satisfy the unit ball K = BY ∗ ; a measurable $ barycenter calculus if K g dμ = g(bμ ) for every probability measure μ on K. Every function g ∈ A0 (K) = i(Y) satisfies it. Thanks to the barycenter calculus, we can show that if f : K → R is affine and of first class, null at 0, then it is in A0 (K)⊥⊥ ⊆ C(K)∗∗ (and hence of affine first class, by Lemma III.3). Let us see this (this proof was communicated to us by G. Godefroy). First, as f is of first class, it has at least one point of continuity, and is thus bounded on a neighborhood of this$ point. As it is affine, it is bounded on K. Hence, by the formula * f , μ = K f dμ for μ ∈ M(K) = C(K)∗ , f ∗∗ * defines an element f ∈ C(K) (of first class, by the dominated convergence theorem, since f is a function of first class). We are going to construct a filter of elements yF of Y such that i(yF ) ∈ A0 (K) converges pointwise to f . This will prove the stated result; indeed, even though we cannot use the dominated convergence theorem for a filter, the fact that f satisfies the barycenter calculus allows us to write, for every probability measure μ:   * f dμ = f (bμ ) = lim yF (bμ ) = lim i(yF ) dμ f , μ = K

F

F

K

= lim i(yF ), μ , F

so i(yF ) is convergent to * f in C(K)∗∗ . Of course, Y is assumed infinite-dimensional; otherwise this is without interest. In this case, the set F of finite-dimensional subspaces F of Y ∗ is a filtering ordered set (see the Preliminary Chapter). For every F ∈ F, the restriction * f|BF is the image by i of a linear functional zF on F, continuous because F is finitedimensional. As the dual F ∗ is isometric to Y/F⊥ , there exists yF ∈ Y such that zF , ϕ = yF , ϕ for every ϕ ∈ F, and yF  = zF   f ∞ . Clearly, i(yF ) converges pointwise to f on K. weak∗

However, an element of the bidual of a Banach space X can satisfy the barycenter calculus and be of second class on the unit ball of the dual X ∗ , equipped with the weak∗ topology, without being the iterated limit of a double sequence of elements of X (Talagrand [1984 b]; his space X thus constructed has the Schur property, and hence is w.s.c.).

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8 The Space 1

4) There exist many other characterizations of spaces containing 1 (see van Dulst for a systematic overview). K. Musial [1979] and L. Janicka [1979] showed that a Banach space X does not contain 1 if and only if its dual has the weak Radon–Nikodým property (see also Talagrand, Corollary 7-3-8); it boils down to saying that the dual has the weak∗ Radon–Nikodým property (Talagrand, Proposition 7-4-3), or also that it is strongly regular (Ghoussoub–Godefroy– Maurey–Schachermayer, by their Remark VI.17 and Corollary VI.18). For separable spaces, another characterization uses the ball topology, i.e. the coarsest topology for which norm-closed balls remain closed (Godefroy and Kalton [1989]). With techniques not developed here (the Choquet integral representation, or the boundary method of Godefroy [1987]), we can deduce another result from Rosenthal’s theorem and the Krein–Milman theorem: Theorem If X is a separable Banach space not containing 1 , then every convex w∗ -compact subset of X ∗ is the closed convex hull in norm of the set of its extreme points. This property is actually equivalent to not containing 1 . This holds even if X is not separable (R. Haydon [1976]; see Ghoussoub–Godefroy– Maurey–Schachermayer, Theorem VII.1). A Banach space X contains 1 if and only if there exists a symmetric type on this space, i.e. a non-null element g of the bidual X ∗∗ such that g + x = g − x for every x ∈ X (Maurey [1983]; see also Rosenthal [1984]). Another characterization (G. Godefroy [1989 a]; see also Deville–Godefroy–Zizler, Chapter III) is the existence of an equivalent norm, called octahedral, i.e. for every finite-dimensional subspace F of X, and every ε > 0, there exists an element x0 ∈ X, of norm 1, such that x + x0   (1 − ε)(x + 1) for every x ∈ F. 5) Theorem III.5 as stated here is actually due to Hagler [1973]. An isometric version of Pełczy´nski’s theorem (Theorem III.5) was recently given by Dilworth, Girardi and Hagler [2000]: a dual X ∗ contains L1 isometrically if and only if the space X contains subspaces asymptotically isometric to 1 . It is their method that we follow for the proof. Lemma III.8 is taken from Hagler and Stegall [1973], and Proposition III.9 from Rosenthal [1972]. We mention the following result of Stegall [1975]: if X is separable, but X ∗ is not, then X contains a bi-orthogonal system of cardinality c. A question which dates back to Banach is: in a separable Banach space whose dual is not separable, can there exist a bounded sequence without a weakly Cauchy subsequence (Banach, page 243, § 9)? In other words, if

V Exercises

353

E is separable, but E∗ is not, does E contain 1 ? We have seen in Chapter 3 that the answer is positive if the space E possesses an unconditional basis. But it is negative in general. The first example, called the James tree space, denoted JT, was constructed by R. James [1974]; a continuous analogue, the James function space JF is due to Lindenstrauss and Stegall [1975]. The reader may see Fetter–Gamboa. J. Hagler [1977] gave another example: a separable Banach space JH whose dual is not separable, for which every infinite-dimensional subspace contains c0 , and every infinite-dimensional subspace of its dual contains 1 . T. Gowers [1995] also constructed a space X not containing 1 , even though every infinite-dimensional subspace of X has a non-separable dual.

V Exercises Exercise V.1 Let X be a Banach space and z ∈ X ∗∗ , z = 0, such that z + x = z + x for every x ∈ X. Show that X contains 1 (Hint: use the local reflexivity principle to construct a sequence equivalent to the canonical basis of 1 ). Exercise V.2 1) (Lohman [1976]) Let X be a Banach space, and Y a subspace of X not containing 1 . Show that every weak Cauchy sequence in the quotient X/Y possesses a subsequence that can be raised to a weakly Cauchy sequence in X. (Hint: if (ξn )n1 is a weakly Cauchy sequence in w X/Y, then ξ2n − ξ2n−1 −−→ 0; then take convex combinations tending to n→+∞ 0 in norm; hence deduce that if (ξn )n1 does not have a subsequence that can be lifted to a weakly Cauchy sequence, then Y contains a sequence equivalent to the canonical basis of 1 .) 2) A Banach space X is said to have the Dunford–Pettis property if, for every sequence (xn )n1 converging weakly to 0 in X and every sequence (xn∗ )n1 converging weakly to 0 in X ∗ , xn∗ (xn ) −−→ 0. Show that if X has the n→+∞ Dunford–Pettis property and if Y is a subspace of X not containing 1 , then X/Y also has the Dunford–Pettis property (Diestel [1980]). 3) Show that if X has the Dunford–Pettis property and does not contain 1 , then X ∗ has the Schur property (Fakhoury [1977]). Exercise V.3 Let X and Y be two Banach spaces. Denote by Lw∗ (X ∗ , Y) the space of continuous linear maps u : (X ∗ , w∗ ) → (Y, w). Assume that X has the Schur property.

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1) Show that every u ∈ Lw∗ (X ∗ , Y) is a compact operator. 2) Show that Lw∗ (X ∗ , Y) is isometric to a subspace of the space of continuous functions on the compact (BX ∗ , w∗ ) × (BY ∗ , w∗ ). 3) Show that, if we now assume Y weakly sequentially complete, so is Lw∗ (X ∗ , Y) (Lust-Piquard [1975]). Exercise V.4 When X is separable, give a direct proof of the implication (4) ⇒ (1) of Theorem III.5 by using the Odell–Rosenthal theorem. Exercise V.5 (Lust-Piquard [1979]) Let G be a (metrizable) compact Abelian ∞ (G) has the Schur property. group and ⊆  =  G. Assume that the space L 1 , where =   (− ), does not 1) Show that the predual space L1 (G)/L contain 1 . 2) Deduce that there exists an approximate identity (Kn )n1 in L1 (G) whose 1 is weakly Cauchy. image in L1 (G)/L ∞ and any x ∈ G. 3) Deduce that Kn ∗ f (x) converges for every f ∈ L 4) Deduce that is a Rosenthal set (see the remark following Proposition V.6 in Chapter 6, and Exercise VII.11 in Chapter 7).

Lust-Piquard [1989] showed that the sets P of prime numbers and S of squares (or more generally of k-th powers, k  2) are not Rosenthal sets. Exercise V.6 (see Godefroy [1978], [1979], [1981], [1983], [1984 a] and [1989 b]) 1) a) Let P be a Polish space and let fn : P → R be continuous functions converging pointwise to f : P → R. Show that, for every open subset  of R, the set f −1 () is an Fσ

 subset of P (write f −1 (]a, b[) = n1 kn fk−1 ([a − 1n , b + 1n ])). b) Let Y be a Banach space and Z ⊆ Y ∗ a closed subspace. Show that the three following conditions are equivalent: (i) y|Z  = y for every y ∈ Y (recall that y|Z  = dist(y, Z ⊥ )); (ii) BZ is w∗ -dense in BY ∗ ; (iii) y   − y for every y ∈ Y and  ∈ Z ⊥ ⊆ Y ∗∗ . c) Let E be a separable Banach space that does not contain 1 , and let M be a closed subspace of E∗ . Using the Odell–Rosenthal theorem, show that the ball BM is w∗ -closed if it is w∗ -sequentially closed: if fn ∈ BM , w∗ n  1, and fn −−→ f , then f ∈ BM (use the argument from the proof of n→+∞ Lemma III.3). A Banach space X is said to have the PIf ,∞ property if, for every family

(Bα )α∈I of closed balls of X such that α∈I Bα = ∅, there exists a finite

subset F ⊆ I such that α∈F Bα = ∅.

V Exercises

355

2) Show that, for every Banach space Y, there exists a norm-1 projection π : Y ∗∗∗ → Y ∗ (use the restriction to Y). 3) Let X be a Banach space for which there exists a norm-1 projection π : X ∗∗ → X. Show that X has the PIf ,∞ property. For every  ∈ X ∗∗ , set: P() = {x ∈ X ; x − x  x −  , ∀ x ∈ X} . 4) Show that P() is convex, and that the conditions of 1) b) are satisfied if and only if 0 ∈ P() for every  ∈ Z ⊥ . Denote M = { ∈ X ∗∗ ; 0 ∈ P()}. 5) Show that  ∈ M if and only if ker  ∩ BX ∗ is w∗ -dense in BX ∗ (use 1) b) ). 6) Let Y be a Banach space and let  ∈ Y ∗∗ . Show that, for every y1 , . . . , yn ∈ Y and every r1 , . . . , rn > 0 such that ri > yi − , 1  i  n, we

have ni=1 BY (yi , ri ) = ∅ (use the local reflexivity principle, Chapter 5, Subsection III.1). 7) Let X be a Banach space. For  ∈ X ∗∗ , denote by B the set of all balls B(x, r) = x + r BX , for x ∈ X and r > x − . Show that if X has the PIf ,∞

property, then B∈B B = ∅. Deduce that P() = ∅. 8) Now assume X separable, not containing 1 and having the PIf ,∞ property. a) Show that M is a linear subspace of X ∗∗ . (Hint: use the Odell–Rosenthal theorem and 1) a) to see that the set ker  ∩ BX ∗ is a Gδ subset of (BX ∗ , w∗ ) for every  ∈ X ∗∗ , and 5) to see that it is dense if  ∈ M; then use Baire’s theorem to show that if 1 , 2 ∈ M and λ1 , λ2 ∈ R, then ker(λ1 1 + λ2 2 ) ∩ BX ∗ is w∗ -dense, and conclude by again using 5) . b) Show that M is w∗ -closed in X ∗∗ . (Hint: by 1) c) , and the Banach– Dieudonné theorem (see Exercise, Preliminary Chapter), it suffices to show that every weak∗ limit of a sequence of elements of BM is in BM ; then use 5) and Baire’s theorem.) c) Show that X ∗∗ = X ⊕ M, that the associated projection π : X ∗∗ → X with kernel ker π = M has norm 1, and that, if Y = M⊥ ⊆ X ∗ , then X is isometric to Y ∗ . d) Show that Y is the unique isometric predual of X: if Y ∗ is isometric to X, then Y is isometric to Y (for every projection π : X ∗∗ → X with norm 1 and with kernel ker π w∗ -closed, verify that ker π ⊆ M, and conclude that π = π ). This result is still true when X is not separable; but the techniques are then different ((BX ∗ , w∗ ) is no longer metrizable), and the proof is more difficult (Godefroy and Kalton [1989]). 9) By using the preceding techniques, show that, for every Banach space X, if  ∈ X ∗∗ is an element of first class, then P() has at most one element (if

356

8 The Space 1

x1 , x2 ∈ P(), then x1 and x2 coincide on a dense Gδ subset of (BX ∗ , w∗ ) and are hence equal). Deduce that if X is L-summand in its bidual, i.e. X ∗∗ = X ⊕1 Xs (every  ∈ X ∗∗ can be written  = x + s , with x ∈ X and s ∈ Xs and  = x + s ), then X is weakly sequentially complete.

Annex Banach Algebras. Compact Abelian Groups

I Introduction In this book, we essentially use Fourier analysis only on the circle T and the Cantor group . Nonetheless, a general framework is useful: it is presented in this annex. Prior to this, it is necessary to discuss Banach algebras (used for themselves in other parts of this book).

II Banach Algebras In this section, certain results about Banach algebras are recalled, those that will be useful later. We suppose that A is a complex Banach algebra, commutative, with a unit element e. The commutativity is in fact seldom needed (if a ∈ A, we can often replace A by the closed sub-algebra generated by a), but is convenient. We suppose that the norm of A satisfies: ab  a b

and

e = 1 .

An equivalent norm satisfying this can always be found.

II.1 Invertible Elements and Maximal Ideals The following lemma, albeit very simple, is fundamental. Lemma II.1 If a < 1, then e − a is invertible.  n n Proof It suffices to note that the series +∞ n=0 a converges, since a   n n n+1 a , and then that (e − a)(e + a + · · · + a ) = e − a − −→ e; hence n→+∞ +∞ n −1 n=0 a = (e − a) . 357

358

Banach Algebras. Compact Abelian Groups

Therefore when a0 ∈ A is invertible, then a is also invertible whenever a − a0  < 1/a−1 0 ; hence: Theorem II.2 The set of invertible elements of A is open. Corollary II.3 Every maximal ideal of A is closed. Proof If I does not intersect the open set of invertible elements, then neither does I; in particular I = A. As the closure of an ideal I is itself an ideal, we obtain I = I if I is a maximal ideal.

II.2 Spectrum of an Element The spectrum of an element a ∈ A is the set sp(a) = {λ ∈ C ; (a − λe) non-invertible}. The fundamental result is: Theorem II.4 sp(a) is a non-empty compact subset of C. Proof

We introduce the resolvent Ra : C  sp(a) → A of a, defined by: Ra (λ) = (a − λe)−1 .

Then, if λ0 ∈ C  sp(a) and |λ − λ0 | < 1/Ra (λ0 ), Lemma II.1 implies that λ ∈ sp(a). Hence sp(a) is closed. Moreover, this lemma also implies that sp(a) is contained in the closed disk of center 0 and of radius a. To show that the spectrum is not empty, we use the following lemma. Lemma II.5 Ra is holomorphic in C  sp(a), and tends to 0 at infinity. Then, if sp(a) were empty, for every continuous linear functional  ∈ A∗ , the function  ◦ Ra : C → C would be entire and would tend to 0 at infinity; it would thus be null by Liouville’s theorem. Hence we would have Ra = 0, clearly false. Proof of the lemma a) If λ ∈ sp(a), and |λ − λ0 | < 1/Ra (λ0 ), we have: (a − λe) − (a − λ0 e) = |λ − λ0 | < hence e − (a − λe)Ra (λ0 ) < 1, which implies:

1 ; Ra (λ0 )

II Banach Algebras 

(a − λe)Ra (λ0 )

−1

= = =

+∞ 

e − (a − λe)Ra (λ0 )

n=0 +∞ 

359 n

n   (a − λ0 e) − (a − λe) Ra (λ0 )

n=0 +∞

(λ − λ0 )n Ra (λ0 )n ;

n=0

thus: Ra (λ) =

+∞

(λ − λ0 )n Ra (λ0 )n+1 .

n=0

Therefore Ra is analytic in C  sp(a). b) It remains to note that, for |λ| > a, we have: +∞  1 n 1 −1 =− a −−−→ − e . λRa (λ) = − e − a λ λn |λ|→+∞ n=0

Corollary II.6 (The Gelfand–Mazur Theorem) Every commutative complex Banach algebra that is a field has dimension 1. Proof Let a ∈ A. We know that its spectrum sp(a) is not empty. Let λ ∈ sp(a). Then a − λe is not invertible, and hence, since A is a field, a − λe = 0, i.e. a = λe. Corollary II.7 Every maximal ideal of A is a closed hyperplane of A. Proof First note that every ideal I is a vector subspace: if a ∈ A, and λ ∈ C, then λa = (λe) a ∈ I. Now, if I is a maximal ideal, A/I is a field. Moreover, we have seen that I is necessarily closed. Hence A/I is a Banach algebra for the quotient norm. By the preceding corollary, A/I is thus of dimension 1, i.e. I is a hyperplane.

II.3 Spectral Radius We have seen that sp(a) ⊆ D(0, a); now we improve this result. The spectral radius of a is defined as the number: r(a) = sup{|λ| ; λ ∈ sp(a)} . We have already seen that r(a)  a. In fact: Theorem II.8

r(a) = limn→+∞ an 1/n = infn→+∞ an 1/n .

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Remarks 1) The existence of the limit figures among the conclusions of the theorem. 2) If B is a Banach sub-algebra of A and a ∈ B, in general the spectrum of a with respect to B differs from the spectrum of a with respect to A. But the theorem shows that the spectral radius is nonetheless the same in A and in B. Proof a) Set l = infn→+∞ an 1/n . Then evidently l  lim infn→+∞ an 1/n . Let ε > 0. There exists N  1 such that aN 1/N  l + ε. For n  N, the Euclidean division of n by N provides: n = pn N + qn , with 0  qn  N − 1. Then: an   aN pn aqn ; hence: an 1/n  aN pn /n aqn /n . However, qn /n −−→ 0, since 0  qn  N − 1; consequently pn /n −−→ N n→+∞ n→+∞ and hence: lim sup an 1/n  aN 1/N  l + ε . n→+∞

This proves the existence of the limit and the second equality. b) For λ ∈ sp(a), the equality: an − λn e = (a − λe)(an−1 + λan−2 + · · · + λn−2 a + λn−1 e) shows that an − λn e is not invertible (otherwise a − λe would be invertible). Thus λn ∈ sp(an ), and consequently |λn |  an . It ensues that r(a)  infn1 an 1/n = l. c) Finally, note that, for |λ| > a:   +∞ 1 −1 1 n 1 =− a ; Ra (λ) = − e − a n+1 λ λ λ n=0

thus we can integrate term by term on the circle Cr of center 0 and radius r > a:  1 λk Ra (λ) dλ = −ak , k  0. 2π i Cr   As Ra is holomorphic for |λ| > r(a), since sp(a) ⊆ D a, r(a) , by definition of r(a), Cauchy’s theorem implies that this formula still holds for any r > r(a). Consequently: ak   rk+1 sup Ra (λ) ; |λ|=r

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361

thus: l = lim ak 1/k  r . k→+∞

As this is true for any r > r(a), we indeed obtain l  r(a).

II.4 Characters of a Commutative Algebra If B is a commutative algebra on a (commutative) field K, a character of B is a non-identically-null algebra homomorphism χ : B → K. If A is a commutative Banach algebra with unit (and if the norm satisfies the conditions given at the beginning of this section), we have: Theorem II.9

Every character χ of A is continuous and χ  = 1.

Proof We show that |χ (a)|  1 if a = 1; then we have χ  = 1 since e = 1 and χ (e) = 1. In fact this ensues from Lemma II.1: if there existed a0 ∈ A of norm 1 such that |χ (a0 )| > 1, then, with λ = χ (a0 ), the invertibility of e − (1/λ)a0 would ensue, since (1/λ)a0  < 1. But this is not possible, as χ e − (1/λ)a0 = χ (e) − (1/λ)χ (a0 ) = 1 − 1 = 0. Theorem II.10 The mapping χ → ker χ is a bijection between the set X(A) of characters of A and the set of its maximal ideals. Proof As χ is a character, ker χ is an ideal. It is maximal since ker χ is a hyperplane, as χ is in particular a linear functional. Conversely, if I is a maximal ideal, then, by the Gelfand–Mazur theorem, there exists a field isomorphism α : A/I → C; then α ◦ σ : A → C is a character of A with kernel I, where σ : A → A/I is the canonical surjection. Theorem II.11 For every a ∈ A: sp(a) = {χ (a) ; χ ∈ X(A)} . Proof a) For every χ ∈ X(A), we have:   χ a − χ (a) e = χ (a) − χ (a)χ (e) = 0 ; hence a − χ (a) e cannot be invertible, and thus χ (a) ∈ sp(a). b) Conversely, if λ ∈ sp(a), then a − λe is not invertible. It is hence contained in a maximal ideal I. Thanks to Theorem II.10, can write I = ker χ with χ ∈ X(A). Hence χ (a − λe) = 0, so that λ = χ (a).

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II.5 Involutive Banach Algebras Let A be a Banach algebra (not necessarily commutative) possessing a unit element e. The algebra A is said to be involutive if there exists an algebra anti* = λ* a, and homomorphism a ∈ A → * a ∈ A i.e. (a + b) = * a +* b, (λa)  > > * (a b) = b* a that is involutive: (* a) = a, and such that: * a = a. Usually an involution is denoted a → a∗ , but we prefer to use the above notation to avoid confusion with the elements of the dual. a)  0 for every A continuous linear functional L ∈ A∗ is positive if L(a* a ∈ A. In fact, it can be shown that the hypothesis of continuity is redundant (see Loomis, pages 96–97). Note that, since: > ? > * ea =* e (* a) = (* a e) = (* a) = a , we have * e = e, and hence L(e) = L(e* e)  0. Thus the following theorem shows, in particular, that if L = 0, then L(e) > 0. Theorem II.12 (The Bochner–Weil–Raikov Theorem) Let A be a commutative involutive Banach algebra with unit e. If L ∈ A∗ is a positive linear functional, then: 1) |L(a)|  L(e) r(a) for every a ∈ A. In particular: L  L(e). 2) If L(a) =  0, there exists χ ∈ X(A) such that χ (a) = 0. Proof 1) The map (a, b) → L(a * b) is a positive Hermitian form. The Cauchy– Schwarz inequality thus holds: a)L(b * b) . |L(a * b)|2  L(a* In particular, for b = e, and by iterating, we obtain:  1/4 a)2 a)1/2  L(e)1/2+1/4 L(a*  ··· |L(a)|  L(e)1/2 L(a* n   n n−1 1/2 a)2 ;  L(e)1/2+1/4+ · · · +1/2 L(a* hence: a)2 |L(a)|  L(e)1/2+1/4+ · · · +1/2 L1/2 (a* n

n

n−1

n

1/2 ,

and then, letting n tend to +∞ leads to: |L(a)|  L(e) r(a* a)1/2 . However, thanks to Theorem II.11: r(a* a) = sup{|χ (a* a)| ; χ ∈ X(A)}

 2 = sup{|χ (a)χ (* a)| ; χ ∈ X(A)}  sup{|χ (a)| ; χ ∈ X(A)} ,

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363

since the mapping χ * : x → χ (* x) defines a character of A; hence r(a* a)  r(a)2 : the first assertion is proved. 2) Now, if L(a) = 0, necessarily r(a) = 0, i.e. sp(a) = {0}, and hence, by Theorem II.11, there exists χ ∈ X(A) such that χ (a) = 0. Definition II.13 A C ∗ -algebra is defined as a complex involutive Banach algebra (with unit element) such that a* a = a2 . Theorem II.14 Every commutative C ∗ -algebra A is isometric to the algebra of continuous functions on the compact space X(A). Example Every space L∞ (S, T , μ) is hence isometric to a space of continuous functions on a compact space; in particular, this is true for ∞ . But in this case, we can see directly that ∞ is isometric to C(βN), where βN is the set of ˇ ultrafilters on N, i.e. the Stone–Cech compactification of N. Proof By Theorem II.9, the spectrum X(A) of A is a subset of the unit ball of the dual A∗ ; hence we can equip it with the weak∗ topology. Again thanks to Theorem II.9, we see that it is a w∗ -closed subset; hence it is compact for this topology. We define the Gelfand transform: G : A −→ C[X(A)]

  by Ga (χ ) = χ (a). Clearly, by Theorem II.9, Ga∞  a. In fact, Theorem II.11 even states that Ga∞ = r(a). We now wish to show that G is a surjective isometry. For this, we use the following notion: an element h ∈ A is said to be Hermitian if * h = h. It suffices to show: Lemma II.15 Let A be a C∗ -algebra. For every Hermitian element, the following properties hold: 1) r(h) = h; 2) sp(h) ⊆ R. Indeed, since a* a is Hermitian for every a ∈ A, r(a* a) = a* a = a2 . However, as seen at the end of the proof of the preceding theorem, r(a* a)  r(a)2 ; hence a  r(a), and thus r(a) = a, since the reverse inequality is always true; consequently Ga∞ = a. Next, as a* a and a + * a are Hermitian, the numbers χ (a)χ (* a) ∈ sp(a* a) and χ (a) + χ (* a) ∈ sp(a + * a) are real, by 2) of the lemma, so that χ (* a) = χ (a), a, and the Stone–Weierstrass for every character χ ; this means that Ga = G* theorem then implies the density of GA in C[X(A)]; hence GA = C[X(A)].

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Proof of the lemma As the closed sub-algebra generated by h is a commutative C∗ -algebra, A itself can be assumed commutative. n n 1) We have h2 = h * h, hence h2  = h * h = h2 ; this implies h2 1/2 = h, and thus r(h) = h. 2) Set: exp(ih) =

+∞ n n i h n=0

n!

·

It is clear that [exp(ih)]* = exp(−ih) = [exp(ih)]−1 , i.e. u = exp(ih) is unitary . However, for any unitary element u, we have u = u* u1/2 = the closed unit disk u u−1 1/2 = e1/2 = 1 thus sp(u) is contained  in −1 in place of u, sp(u) of C. As this is also the case for sp(u−1 ) = sp(u) is thus contained in the unit circle U. It ensues from Theorem II.11 that exp(iλ) ∈ U for every λ ∈ sp(h), in other words λ ∈ R.

III Compact Abelian Groups A topological Abelian group is an Abelian group equipped with a topology for which the group operations: (x, y) → x + y and x → (−x) are continuous. The study is limited to compact Abelian groups, additionally assumed to be metrizable. Then, on this group, there exists a translation-invariant distance (see, for example, N. Bourbaki, Topologie Générale, Chapter IX, § 3, Proposition 2).

III.1 Haar Measure The fundamental result concerning locally compact groups is the existence of the Haar measure; a proof of its existence in the case of compact metrizable groups is given in Theorem III.1 below; it is simpler than in the general case (see Weil). The commutativity is assumed, as we only work with Abelian groups, but it is not indispensable. Theorem III.1 For every (metrizable) compact Abelian group G, there exists a unique positive measure m, of total mass 1, that is translation-invariant. Examples

We mostly use two groups:

1) The unit circle or torus T = R/Z. Its Haar measure is the Lebesgue measure, or rather the image of the Lebesgue measure on R by the canonical surjection R → R/Z. At times, we consider T as R/2π Z; its Haar measure is then (the image of) the normalized Lebesgue measure: dm = dx/2π .

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∗

2) The Cantor group  = {−1, 1}N . This is a compact Abelian group for term-by-term multiplication. Its Haar measure is the product-measure of the measure on {−1, +1} with the mass 1/2 attributed to −1 and to +1. At times, rather than the infinite Cantor group, we consider the “subgroup” N = {−1, 1}N (N  1). Proof of the theorem 1) Unicity. If m and m are two translation-invariant probabilities, then, since m is translation-invariant, for every continuous function f on G, we have:   f (t) dm(t) = f (t + x) dm(t) . G

Hence: 

G

 

 f (t + x) dm(t) dm (x) G G    f (t + x) dm (x) dm(t) = G G    f (u) dm (u) dm(t) =

f (t) dm(t) = G

G

G

(translation invariance of m )



f (u) dm (u) ,

= G

and consequently m = m . 2) Existence. Since G is compact and metrizable, the space C(G) of continuous functions on G is separable. Thus there exists a total sequence ( fn )n1 such that fn ∞  1/2n . We consider the set P(G) of probabilities on G, i.e. : P(G) = {μ ∈ M(G) = C(G)∗ ; μ = μ(1I) = 1} . It is convex and w∗ -compact. Define on P(G): +∞  2 ϕ(μ) = μ( fn ) . n=1

As the series is absolutely convergent on P(G), and as each function 2 μ → μ( fn ) is w∗ -continuous, ϕ is itself w∗ -continuous. Moreover, it is convex, and even strictly convex. Indeed, if we have:   ϕ(μ) + ϕ(ν) μ+ν , = ϕ 2 2

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 +∞  μ( fn ) + ν( fn ) 2 n=1

2

=

+∞ [μ( fn )]2 + [ν( fn )]2

2

n=1

,

so that: +∞  2 μ( fn ) − ν( fn ) = 0 . n=1

We thus obtain μ( fn ) = ν( fn ) for any n  1, and hence μ = ν since the sequence ( fn )n1 is total. We then set: (μ) = sup ϕ(τx μ) , x∈G

where (τx μ)(A) = μ(A + x) for any Borel set A of G. However, the w∗ w∗ mappings μ → τx μ are w∗ -continuous: if μj −−→ μ, then τx μj −−→ τx μ j→+∞

since, for every f ∈ C(G), we have τ−x f ∈ C(G), and thus:

j→+∞

τx μj , f = μj , τ−x f −−→ μ, τ−x f = τx μ, f . j→+∞

The mapping  is hence w∗ -lower semi-continuous (w∗ -lsc). Moreover, it is clearly convex; in fact it is even strictly convex. Indeed, suppose:   (μ) + (ν) μ+ν = ·  2 2 Let σ = (μ + ν)/2; note that the mapping x ∈ G → τx σ ∈ P(G) is w∗ -continuous, since the mapping x ∈ G → τ−x f ∈ C(G) is continuous in norm (thanks to the uniform continuity of f on G). Therefore the mapping x ∈ G → ϕ(τx σ ) is continuous. By the compactness of G, there exists x0 ∈ G such that: (σ ) = ϕ(τx0 σ ) . However, by hypothesis: (σ ) =

 1  1 (μ) + (ν)  ϕ(τx0 μ) + ϕ(τx0 ν) . 2 2

As ϕ is strictly convex, this is only possible if: τx0 σ = τx0 μ = τx0 ν , i.e. σ = μ = ν. The map  is thus indeed strictly convex. It now remains to note that a strictly convex and w∗ -lsc function on a set itself convex and w∗ -compact (here P(G)) attains its minimum at a unique point m ∈ P(G). Clearly:

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(τx μ) = (μ) for every μ ∈ P(G) and every x ∈ G, so the uniqueness of the minimum implies τx m = m for every x ∈ G, i.e. the translation invariance. Remark In this proof, the metrizability is essential. It can be shown that, when G is a compact group, the existence on P(G) of a w∗ -lsc strictly convex function forces the metrizability of G (Godefroy and Li [1998]). In contrast, the commutativity is not essential; in fact it suffices to modify  by setting (μ) = supx,y∈G ϕ(Lx μ Ry ), where Ry f (t) = f (ty) and Lx f (t) = f (xt), to obtain the bi-invariance of the Haar measure. The following application of this remark was indicated to us by Jean SaintRaymond: Let E be a finite-dimensional real vector space such that the group G of its (linear) isometries operates transitively on the unit sphere SE of E. Then E is E. isometric to dim 2 For this, $ we consider a positive definite quadratic form Q0 on E, and set Q(x) = G Q0 (Ux) dm(U), where m is the Haar measure of the compact metric group G. The quadratic form Q is again positive definite, and it is invariant under G. Fix an a ∈ SE such that Q(a) = 0. Renormalizing Q if necessary, we can assume that Q(a) = 1. As G operates transitively, for every x ∈ SE , there exists U ∈ G such that x = Ua; then Q(x) = Q(Ua) = Q(a) = 1. Hence, Q(x) = x2 for every x ∈ E, and the conclusion ensues. Two important properties of the Haar measure are now presented: Proposition III.2 1) The Haar measure is invariant under symmetry: m(−A) = m(A) for every Borel set A of G. 2) For every open non-empty subset  of G, m() > 0. Proof 1) If we set * m(A) = m(−A), then * m is a translation-invariant probability, and hence * m = m. 2) As G is compact, there exist elements x1 , . . . , xn ∈ G such that G = n j=1 ( + xj ), and since m( + xj ) = m(), we obtain 1 = m(G)  n m().

III.2 Convolution The convolution product μ ∗ ν of two (complex) measures μ and ν on G is defined as the image measure of the product measure μ⊗ν by the sum mapping

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(x, y) → x + y. In other words, for every continuous function f ∈ C(G), we have:  μ ∗ ν, f = f (x + y) dμ(x)dν(y) . G×G

Equipped with the convolution product, M(G) is a commutative Banach algebra, with a unit element: the Dirac mass δ0 at 0. We also have: μ ∗ ν  μ ν. The space L1 (G) is an ideal of M(G): if f ∈ L1 (G), then the identification of f and the measure f .m lead, for the convolution product, to: f ∗ μ ∈ L1 (G) and  ( f ∗ μ)(x) = f (x − t) dμ(t) . G

In particular, if f , g ∈

then f ∗ g ∈ L1 (G), and:  ( f ∗ g)(x) = f (x − t)g(t) dm(t) . L1 (G),

G

Next: Proposition III.3 If 1  p  +∞, and 1/p + 1/q = 1, then f ∗ g ∈ C(G) if f ∈ Lp (G) and g ∈ Lq (G). Moreover f ∗ g∞  f p gq . This proposition is a consequence of the following lemma, which, for X = C(G), ensues from the uniform continuity of f , and for X = Lp (G), from the density of C(G) in Lp (G): Lemma III.4 For X = C(G) or Lp (G), 1  p < +∞, the mapping x ∈ G → fx ∈ X, where fx (t) = f (t − x), is continuous,. Note that this lemma also implies: Proposition III.5 If f ∈ C(G) and μ ∈ M(G), then f ∗ μ ∈ C(G), and f ∗ μ∞  f ∞ μ. Corollary III.6 For every Borel set A of G of measure m(A) > 0, the set A − A is a neighborhood of 0. Proof The indicator function 1IA of A is in L2 (G). Then, when we denote ˇ fˇ (x) =  f (−x), the  convolution  product 1IA ∗ 1IA is continuous. Since 1IA ∗ 1ˇIA (x) = m A ∩ (A + x) , we have 1IA ∗ 1ˇIA (0) = m(A) > 0. Thus   there V of 0 such that, for every x ∈ V, 1IA ∗ 1ˇIA (x) =  exists a neighborhood  m A ∩ (A + x) > 0; in particular, A ∩ (A + x) = ∅ for every x ∈ V, hence V ⊆ A − A. Here is a consequence (however, it can be obtained without Corollary III.6: if m({x0 }) = δ > 0, then m({x}) = δ for every x ∈ G, and thus

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δ card(F) = m(F)  1 for every finite subset F ⊆ G; hence G itself is finite and δ card(G) = 1): Corollary III.7 diffuse.

If the compact group G is infinite, the Haar measure is

Proof First recall that a positive measure μ (defined on a measure space (X, A) in which all the singletons are measurable) is said to be diffuse (also said to be continuous) if μ({x}) = 0 for every x ∈ X. The corollary is immediate, since, if there existed an x ∈ G such that m({x}) > 0, the preceding corollary would state that {x} − {x} = {0} is a neighborhood of 0, which is false when G is infinite. Remark A positive measure μ is also said to be diffuse if it does not have any atoms, where an atom is a measurable subset A with μ(A) = 0, for which μ(B) = 0 or μ(B) = μ(A) for every measurable subset B ⊆ A. We will see that if μ is a regular bounded positive measure on a compact space K, then these two definitions coincide. More precisely, we will show that if A is an atom for μ, then there exists a t0 ∈ A such that μ({t0 }) = μ(A) (hence, in particular, μ({t0 }) > 0). Indeed, by the regularity of the measure, there exists a closed subset A ⊆ A such that μ(A )  μ(A)/2. As A is an atom, then necessarily μ(A ) = μ(A). We can thus assume that A is itself closed. Now, the set of all the closed subsets B ⊆ A such that μ(B) = μ(A) is inductive for “descending” inclusion: if (Bα )α is a totally ordered family of closed subsets of A, then again we have

μ(B) = μ(A) for B = α Bα . Indeed, for any ε > 0, by the regularity of μ, there exists an open subset  containing B and satisfying μ()  μ(B) + ε.

Since B ⊆ , we have α (c ∩ Bα ) = c ∩ B = ∅; hence the compactness implies the existence of an α0 such that c ∩ Bα0 = ∅, i.e. Bα0 ⊆ . Hence μ(A) = μ(Bα0 )  μ()  μ(B) + ε. By Zorn’s lemma, we can find a closed minimal subset B0 contained in A such that μ(B0 ) = μ(A). It only remains to show that B0 contains only a single point t0 . For this, we show that B0 contains a point t0 such that μ(V ∩ B0 ) > 0 for every open neighborhood V of t0 . Then μ(V)  μ(V ∩ B0 ) = μ(A), since A is an atom; hence μ({t0 }) = inf{μ(V) ; V open neighborhood of t0 }  μ(A) ; thus μ({t0 }) = μ(A) by the regularity of the measure μ, and hence B0 = {t0 } by the minimality of B0 .

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Indeed, suppose, on the contrary, that for every t ∈ B0 there exists an open  neighborhood Vt of t such that μ(Vt ∩ B0 ) = 0. Then, since B0 ⊆ t∈B0 Vt , we  could find a finite number of points t1 , . . . , tn of B0 such that B0 = nk=1 Vtk = n k=1 Vtk ∩ B0 , and we would obtain μ(B0 ) = 0. This contradiction completes the proof. We also use the following proposition (the case p = +∞ was handled in Proposition III.3): Proposition III.8 If 1  p < +∞, and f ∈ Lp (G), then f ∗ g ∈ Lp (G) for g ∈ L1 (G), and f ∗ gp  f p g1 .

III.3 The Dual Group A character of G is defined as any continuous homomorphism: γ : G −→ U = {z ∈ C ; |z| = 1} . The set  of characters of G can be equipped with the operation defined by: (γ1 + γ2 )(x) = γ1 (x) γ2 (x) , which makes it an Abelian group, called the dual group of G, denoted  =  G. For compatibility of notation, we denote by 0 the constant character 1I, and by −γ the character γ −1 = γ . Remark 1 The dual group  =  G can be equipped with a natural topology: that inherited from C(G), since the elements of  are, in particular, continuous functions. However, since√the characters are functions of modulus 1, the condition γ − 1I∞ < 2 implies γ = 1I (if γ (t0 ) = 1, we can write γ (t0 ) = eiα0 ; the point eiα0 may be on the right half-circle; however for n inα0 is on the left half-circle, i.e. an n large enough, √ γ (nt0 ) = γ (t0 ) = e is thus the discrete topology. |γ (nt0 ) − 1|  2). This topology √ Given that γ1 − γ2 ∞  2 if γ1 = γ2 ,  is hence countable, since C(G) is separable (G is metrizable). Remark 2 Let L2 (G) be the space of square integrable functions for the Haar 2 measure;  is a subset $ of L (G) and it is an orthonormal system. In fact, it suffices to show that G γ (t) dm(t) = 0 if γ = 1I. However, in this case there exists t0 ∈ G such that γ (t0 ) = 1, and the translation invariance of the Haar measure implies:    γ (t) dm(t) = γ (t0 ) γ (t − t0 ) dm(t) = γ (t0 ) γ (u) dm(u) ; G

G

G

the integral is thus indeed null. We will see that, in fact,  is an orthonormal basis of L2 (G).

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Examples 1) For G = T = R/2π Z, it is well known that each character of T can be uniquely written as: γ (t) = eint for some n ∈ Z, so that  T can be identified with the additive group Z. ∗ 2) For the Cantor group G =  = {−1, 1}N , we have the following representation: Proposition III.9 functions.

The dual of the Cantor group consists of the Walsh

The Walsh functions are defined as follows: every ω ∈  can be written as ω = (ε1 (ω), ε2 (ω), . . .) , where εn (ω) ∈ {−1, 1} for any n  1. Even though this is not used here, note that the εn ’s, n  1, form an independent sequence of Bernoulli random variables, taking on the values −1 and +1 with probability 1/2, and hence are equivalent to the sequence of Rademacher functions. For every finite subset A of N∗ , the Walsh function of index A is:  wA (ω) = εn (ω) . n∈A

Note that w∅ = 1I and w{k} = εk . Proof of the proposition Clearly every wA is a character of . Conversely, . First, since ω2 = (1, 1, 1, . . .) for every ω ∈ , we have let γ ∈  =  γ (ω) = ±1. For any n  1, set: ωn0 = (1, . . . , 1, −1, 1, . . .) , where the −1 is positioned at the n-th rank; we can write: γ (ωn0 ) = (−1)αn , with αn = 0 or 1. As ωn0 −−→ 1I, the continuity of γ gives γ (ωn0 ) −−→ 1, and n→+∞ n→+∞ hence the set A = {n  1 ; αn = 1} is finite. Now define the character γn by: γn (ω) = γ (1, . . . , 1, εn (ω), 1, 1, . . .) , where εn (ω) is positioned at the n-th rank, and note that: – if εn (ω) = 1, then γn (ω) = γ (1, 1, . . . , 1, 1 . . .) = 1; – if εn (ω) = −1, then γn (ω) = γ (ωn0 ) = (−1)αn ; thus, if n ∈ A, then γn = εn .

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We can now conclude: indeed, if we set: ωn = (1, . . . , 1, εn (ω), 1, 1, . . .) , where εn (ω) is positioned at the n-th rank, then:   ω = lim ε1 (ω), ε2 (ω), . . . , εn (ω), 1, 1, . . . = lim ω1 ω2 . . . ωn . n→+∞

n→+∞

Hence, since γ (ωj ) = 1 if j ∈ A, and γ (ωj ) = εj (ω) if j ∈ A, we obtain: γ (ω) = lim γ (ω1 ω2 . . . ωn ) = lim γ (ω1 ) . . . γ (ωn ) n→+∞ n→+∞  εj (ω) = wA (ω) . = j∈A

Theorem III.10 The dual  =  G separates the points of G: if x = y in G, there exists γ ∈  such that γ (x) = γ (y). We postpone the proof of this theorem to the next subsection, as a few additional notions are required; it will be part of Theorem III.16. However, some important consequences are presented right away. A trigonometric polynomial is defined as any (finite) linear combination of characters. The set of such polynomials is denoted P(G). The Stone– Weierstrass theorem implies: Corollary III.11 The set P(G) of trigonometric polynomials is a dense subspace of C(G). Corollary III.12 The set P(G) is dense in Lp (G) for 1  p < +∞. Corollary III.13 The dual  =  G is an orthonormal basis of L2 (G).

III.4 The Fourier Transform For μ ∈ M(G), the Fourier transform of μ is defined by:   γ (−t) dμ(t) = γ (t) dμ(t) , γ ∈ .  μ(γ ) = G

G

We have  μ ∈ ∞ (), and | μ(γ )|  μ. In particular, if f ∈ L1 (G), then:    f (γ ) = f (t)γ (−t) dm(t) = f (t)γ (t) dm(t) , G

and | f (γ )|  f 1 .

G

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 Fubini’s theorem easily shows that μ ∗ ν(γ )=  μ(γ ) ν(γ ) for μ, ν ∈ M(G); hence every γ ∈  defines a character χγ ∈ X M(G) : χγ (μ) =  μ(γ ) ,  1  and, by restriction, a character χγ ∈ X L (G) : χγ ( f ) =  f (γ ). Remarkably, this is the way to obtain all the characters of L1 (G).     Theorem III.14 X L1 (G) can be identified with : for every χ ∈ X L1 (G) , there exists γ ∈  such that χ ( f ) =  f (γ ) for any f ∈ L1 (G). Corollary III.15 Every character χ of L1 (G) can be extended to a character of M(G).   However, X M(G) is much more complicated, and in fact still not well understood (see Host–Méla–Parreau). Proof of Theorem III.14 A slight difficulty arises from the lack of a unit for the algebra L1 (G). However the sub-algebra A = L1 (G) ⊕1 Cδ0 of M(G) does possess a unit, and every character χ of L1 (G) can be extended to a character χ * of A by setting, for f ∈ L1 (G) and λ ∈ C: χ *( f + λδ0 ) = χ ( f ) + λ . We have seen that χ * is continuous on A and has norm is continuous  11. Hence ∗ χ ∞ 1 on L (G), and has norm  1. In particular, χ ∈ L (G) = L (G). Hence there exists γ ∈ L∞ (G) satisfying γ ∞  1 and  χ( f ) = f (t) γ (t) dm(t) . G

We now exploit χ ( f ∗ g) = χ ( f )χ (g). Since, for every g ∈ L1 (G),    χ ( f ∗ g) = f (x − t)γ (x) dm(x) g(t) dm(t) G

and

G

  χ ( f )χ (g) =

 f (x)γ (x) dm(x) g(t)γ (t) dm(t) ,

G

G

we obtain, for almost all t ∈ G:   (∗) f (x − t)γ (x) dm(x) = f (x)γ (x)γ (t) dm(x) , G

or:

G



 f (u)γ (u + t) dm(u) = G

f (u)γ (u)γ (t) dm(u) . G

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As this is true for every f ∈ L1 (G), necessarily: γ (u + t) = γ (u)γ (t) for almost all t ∈ G and almost all u ∈ G. Next, note that the relation (∗) can also be written: χ (τ−t f ) = χ ( f )γ (t) . Hence, if we choose a function f0 ∈ L1 (G) such that χ ( f0 ) = 0, the continuity of the mapping t ∈ G → τ−t f0 ∈ L1 (G), together with the continuity of χ , implies that γ is almost everywhere equal to the continuous function t → 1 χ ( f0 ) χ (τ−t f0 ). We can thus assume that γ is in fact continuous. Then we obtain γ (u + t) = γ (u) γ (t) for every u, t ∈ G. It remains to show that |γ (t)| = 1. In fact we already know that γ ∞  1, i.e. |γ (t)|  1 for any t, and as:    1    1,  |γ (t) =  γ (−t)  we indeed obtain |γ (t)| = 1. We can now conclude: Theorem III.16 1) The Fourier transform is injective on L1 (G). 2) The dual group  =  G separates the points of G. Proof 1) We must show that if f ∈ L1 (G) and f = 0, then there exists γ ∈  such that  f (γ ) = 0. This comes down to showing that there exists χ ∈ X L1 (G) such that  χ ( f ) = 0. To that purpose, it suffices to show that there exists χ ∈ X M(G) such that χ ( f ) = 0. For this, we construct a positive linear functional on the involutive Banach algebra with unit element M(G). The involution of M(G) is defined, for μ ∈ M(G), as * μ(S) = μ(−S), for every Borel set S of G, or:   ϕ(t) d* μ(t) = ϕ(−t) dμ(t) , G

G

for ϕ ∈ C(G). If f ∈ L1 (G), then * f (t) = f (−t) for t ∈ G, and hence * f ∈ L1 (G). For ϕ ∈ C(G) and μ ∈ M(G), we know that ϕ ∗ μ ∈ C(G); we can thus define Lϕ : M(G) → C by: Lϕ (μ) = (* ϕ ∗ ϕ ∗ μ)(0) .

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Since   Lϕ (μ ∗ * μ) = (μ ∗ ϕ) ∗ (μ ∗ ϕ) (0) = μ ∗ ϕ22  0 , Lϕ is positive. Moreover, for every μ = 0, there exists ϕ ∈ C(G) such that μ) = 0. Indeed, if Lϕ (μ ∗ * μ) = 0, then μ ∗ ϕ2 = 0; hence μ ∗ ϕ Lϕ (μ ∗ * is null almost everywhere. However, μ ∗ ϕ is continuous, and thus it must be null everywhere; in particular:  ϕ(−t) dμ(t) . 0 = (μ ∗ ϕ)(0) = G

This is possible for every ϕ ∈ C(G) only if μ = 0. Thus, by the Bochner– Weil–Raikov theorem, there exists χ ∈ X M(G) such that χ (μ ∗ * μ) = 0. As χ (μ ∗ * μ) = χ (μ)χ (* μ), we indeed have χ (μ) = 0. 2) If x, y ∈ G are distinct, there exists ϕ ∈ C(G) such that ϕ(x) = 1 and ϕ(y) = 0. Then: (τ−x ϕ)(0) = ϕ(x) = 1

and

(τ−y ϕ)(0) = ϕ(y) = 0 ;

hence τ−x ϕ = τ−y ϕ, and there exists γ ∈  such that τ −x ϕ(γ ) = τ −y ϕ(γ ), ϕ (γ ) = γ (y) ϕ (γ ), which implies γ (x) = γ (y). i.e. γ (x)   Remark In 1) , we have in fact shown that separates the points of  X M(G)  M(G). However not all the elements of X M(G) are represented by elements   of  (in fact χ ∈ X M(G) can be written χ (μ) =  μ(γ ) if and only if χ|L1 (G) = 0). Hence we cannot conclude directly that the Fourier transform is injective on M(G). This is nonetheless true: Corollary III.17

The Fourier transform is injective on M(G).

Proof If  μ(γ ) = 0, for every γ ∈ , then μ, p = 0 for every trigonometric polynomial p ∈ P(G). Hence μ = 0 since P(G) is dense in C(G). Note a consequence of the density of P(G) in L1 (G): Proposition III.18 (The Riemann–Lebesgue Lemma)  f ∈ c0 ().

If f ∈ L1 (G), then

III.5 Approximate Identities When G is infinite, L1 (G) does not possess a unit, since, if e were such an element, we would have  e(γ ) = 1 for all γ ∈ , in contradiction of the Riemann–Lebesgue lemma. Nonetheless, we will see that there exist approximate identities (also known as approximations of the identity).

376

Banach Algebras. Compact Abelian Groups

First note that in the classical case G = T, the Fejér kernel Fn is defined as  follows: if Dn (t) = |k|n eikt is the Dirichlet kernel, set: 1 Dk (t) . n+1 n

Fn (t) =

k=0

Then: Fn (t) =

    |k| sin(n + 1)t/2 2 1 1− eikt = ; n+1 n+1 sin t/2

|k|n

hence Fn is a trigonometric polynomial satisfying: a) Fn (t)  0 ; b) F  n 1 = 1 ; dt c) Fn (t) −−→ 0 for 0 < δ < π . 2π n→+∞ |t|δ In the general case, we have the following result (recall that G is assumed metrizable): Proposition III.19 Let (Vn )n1 be a decreasing basis of neighborhoods of 0 in G. There exist trigonometric polynomials Kn ∈ P(G) such that: a) Kn  0 ; b) K  n 1 = 1 ; Kn (t) dm(t) −−→ 0 for every neighborhood V of 0 in G. c) Vc

n→+∞

An approximate identity is defined as any sequence of trigonometric polynomials satisfying the three conditions of the proposition. Proof There exist neighborhoods Wn of 0 such that Wn −Wn ⊆ Vn . As 1IWn ∈ L2 (G), the function: 1 1IW ∗ 1I(−Wn ) [m(Wn )]2 n  is continuous on G. Moreover ϕn 1 = ϕn dm = 1 and supp(ϕn ) ⊆ ϕn =

G

Wn − Wn ⊆ Vn . By density, there exists Pn ∈ P(G) such that: Pn − ϕn ∞ 

1 · nϕn ∞

1 2  0, and Qn − ϕn ∞  . The nϕn ∞ nϕn ∞ trigonometric polynomial Kn = Qn /Qn 1 suits our purpose. In fact, it only

Then Qn = Re Pn +

III Compact Abelian Groups

377

remains to verify c) . If V is a neighborhood of 0, there exists N  1 such that Vn ⊆ V for n  N. Then, since supp(ϕn ) ⊆ Vn ⊆ V:   0 Kn (t) dm(t) = [Kn (t) − ϕn (t)] dm(t) Vc

Vc

 m(V c )Kn − ϕn ∞ 

8 −−→ 0 . n n→+∞

The terminology “approximate identity” comes from the following property: Theorem III.20 Let (Kn )n1 be an approximate identity, and let X = Lp (G), 1  p < +∞ or X = C(G). Then, for every f ∈ X: f ∗ Kn − f X −−→ 0 . n→+∞

Proof a) Note first that, for every γ ∈ , we have: 6n (γ ) −−→ 1 . K n→+∞

Indeed: 6n (γ ) = K



 Kn (t)γ (−t) dm(t) +

V

Vc

Kn (t)γ (−t) dm(t) .

As γ is continuous, for any ε > 0, there exists a neighborhood V of 0 such that |γ (−t) − 1|  ε for t ∈ V. The third condition in the definition of approximate identities states that:  Kn (t)γ (−t) dm(t) −−→ 0 . n→+∞

Vc

Hence:

    6 |Kn (γ ) − 1| =  Kn (t)[γ (−t) − 1] dm(t) G   ε Kn (t) dm(t) + 2 Kn (t) dm(t) V Vc  ε+2 Kn (t) dm(t) −−→ 0 . n→+∞

Vc

b) It follows that, for every trigonometric polynomial P:  Kn (t)P(t) dm(t) −−→ P(0) , G

n→+∞

and thus, by density of P(G) in C(G) (since Kn 1 = 1):  Kn (t)ϕ(t) dm(t) −−→ ϕ(0) G

for every ϕ ∈ C(G).

n→+∞

378

Banach Algebras. Compact Abelian Groups

c) Now let f ∈ C(G). We know that the mapping t ∈ G → τt f ∈ C(G) is continuous. Hence ϕ : t → τt f − f ∞ is continuous, and, by writing: |(Kn ∗ f )(x) − f (x)|      =  Kn (t)f (x − t) dm(t) − Kn (t)f (x) dm(t) G G   Kn (t)|τt f (x) − f (x)| dm(t) , G

we obtain:

 Kn ∗ f − f ∞ 

G

Kn (t)ϕ(t) dm(t) −−→ ϕ(0) = 0 . n→+∞

d) For g ∈ it now suffices to use the density of C(G) in Lp (G): for any ε > 0, there exists f ∈ C(G) such that g − f p  ε; then: Lp (G),

Kn ∗ g − gp  Kn ∗ g − Kn ∗ f p + Kn ∗ f − f p + f − gp  Kn 1 g − f p + Kn ∗ f − f ∞ + f − gp  2f − gp + Kn ∗ f − f ∞ .

Corollary III.21

w∗

If f ∈ L∞ (G) and μ ∈ M(G), then Kn ∗ f −−→ f and n→+∞

w∗

Kn ∗ μ −−→ μ. n→+∞

Corollary III.22 χγ  = 1.

If γ ∈  and χγ is the character f ∈ L1 (G) →  f (γ ), then

Example We have already encountered the Fejér kernel. We also present an ∗ example in the Cantor group  = {−1, 1}N . For this, we need what are known as the Riesz products: n    1 + εj (ω) = wA (ω) . Kn (ω) = j=1

A⊆{1,...,n}

We have: a) Kn (ω)  0;  n n   b) Kn (ω) dω = E(Kn ) = E(1 + εj ) = 1 = 1; 

j=1

j=1

c) with Vn = {ω ∈  ; ε1 (ω) = · · · = εn (ω) = 1}, then, if ω ∈ Vn , there exists j  n such that εj (ω) = −1, and hence Kn (ω) = 0; thus supp Kn ⊆ Vn . Another type of approximate identity is often very useful: the de la ValléePoussin kernel. Instead of having positive trigonometric polynomials, it is their

III Compact Abelian Groups

379

Fourier transforms that are positive. Moreover, there exists such a polynomial whose Fourier transform is equal to 1 on a prescribed finite subset. Theorem III.23 For every finite subset E ⊆ , and for any ε > 0, there exists a trigonometric polynomial V ∈ P(G) such that: a) b) c)

 V  0;  V (γ ) = 1 if γ ∈ E; V1  1 + ε.

Examples 1) For G = T, and E = [−N, N], the classical de la Vallée-Poussin kernel is: VN =

2N−1 1 Dn , N n=N

where Dn is the Dirichlet kernel of order n. We have VN 1  3 by noting that if Fn is the Fejér kernel of order n, then: VN = 2F2N−1 − FN . To obtain a norm less than or equal to 1 + ε, we modify this a bit: we set, for M > N,   |n| int int 1− e . VN,M (t) = e + M |n|N

N K0 large enough. Then: |F| = |H|(2K + 1)l (thanks to the uniqueness of the representation), and since: E − F ⊆ H ⊕ {n1 σ1 + · · · + nl σl ; |n1 |, . . . , |nl |  K + K0 } , we obtain: |E − F|  |H|[2(K + K0 ) + 1]l , and hence: |E − F|  |F|



2(K + K0 ) + 1 2K + 1

l −−→ 1 . K→+∞

Corollary III.24 Let μ ∈ M(G) be a measure on G, and let Cμ : L1 (G) −→ L1 (G) be the convolution operator defined by Cμ ( f ) = f ∗ μ. Then Cμ  = μ. Proof The inequality Cμ   μ is trivial. For the reverse inequality, set ε > 0 and let P be a trigonometric polynomial satisfying P∞ = 1 and:

III Compact Abelian Groups

381

    μ  (1 + ε) P(−x) dμ(x) = (1 + ε)|(P ∗ μ)(0)| . G

Let V be a de la Vallée-Poussin kernel such that  V (γ ) = 1 if γ is in the spectrum of P (i.e.  P(γ ) = 0), and V1  1 + ε. As P = P ∗ V, we obtain: μ  (1 + ε)|(μ ∗ P ∗ V)(0)| = (1 + ε)|[(μ ∗ V) ∗ P](0)|  (1 + ε)μ ∗ V1 P∞  (1 + ε)Cμ  V1  (1 + ε)2 Cμ  . Remark

The same proof shows that the convolution operator Cμ : C(G) −→ C(G)

is also of norm μ; in this case it suffices to write: μ  (1 + ε)μ ∗ P∞ V1 .

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Notation Index for Volume 1

A(D), 213 α (N), 240 BX , 2 c0 , 47 Cf , 243 Cq (X), 181 dX , 193 d(X, Y), 172 (εn )n1 , 26 E(X), 13 E( . | A0 ), 127 E, 126 EB (X), 35 γ2 (u), 193 2 (X, Y), 193 = G, 370 (gn )n1 , 26 h∗ (θ ), 279 H 1 (D), 213 H f , 278 H p (D), 296 H p (T), 296 KG , 217 K(X), 185 Ku , 91 K(H), 191

L0 , 15 L0 (E), 117 Lp (E), 126 Lψ , 32 2,1 , 201 2,∞ , 202 p , 47 nr , 192 Lp , 222 L(H), 191 M([0, 1]), 111 M(H p , H q ), 321 N (m, σ 2 ), 160  . 1,∞ , 281 [[ . ]]R , 246  . ψ , 32  . 2 , 33 ω0 , 11 O(n), 165 ϕ ! 0, 167 X , 19 πp (T), 211 p (X, Y), 212 P(G), 372 P , 232 P, 13 PX , 13 ψ " 0, 167 Rf , 277

413

414

σ (X), 14 σ (X, X ∗ ), 1 σ (X ∗ , X), 2 S( ), 232 Sr (H), 190 sp (μ), 232

Notation Index for Volume 1

wA , 144 w∗ , 2 P

Xn −−−→ X, 15 n→+∞

L

Xn −−−→ X, 18 n→+∞

a.s.

Tp (X), 180

Xn −−−→ X, 15 n→+∞

U(n), 165

X U , 176 X s , 24

w, 1  ∗ ∗ w∗ - +∞ n=1 x (en ) en , 69

(yi )p ≺ (xj )jq , 195

Author Index for Volume 1

Alaoglu, Leonidas, 2, 71 Albiac, Fernando, 182 Alfsen, Erik M., 12, 351 Alspach, Dale E., 305 Amir, Dan, 74, 204 Argyros, Spiros A., 111

Bretagnolle, Jean, xiv, 204 Brillhart, John, 259 Burkholder, Donald L., 91, 293, 309

Ball, Keith, 152 Banach, Stefan, xv, xxiv, 2, 48, 53, 63, 74, 75, 103, 204, 352 Barbe, Philippe, xvi Barthe, Franck, 288 Beauzamy, Bernard, xiv, 116 Beck, Anatole, xiv, 205 Beckner, William, 288 Bennett, Colin, 287 Bennett, Grahame, 75, 308 Benyamini, Yoav, xiv, 74, 157 Bergh, Jöran, 308 Bessaga, Czesław, xv, xvii, 63, 75, 83, 95, 110, 111 Billard, Pierre, xxviii, 151 Billingsley, Patrick, 42, 164 Biró, András, 44 Bochner, Salomon, xviii, 316 Boclé, Jean, 310 Bojanov, Borislav, 262 Bonami, Aline, 152 Borell, Christer, 146, 152 Borkar, Vivek S., 121, 151 Bourbaki, Nicolas, 2, 7, 364 Bourgain, Jean, xiv, xxiv, xxvi, xxvii, 75, 157, 232, 259, 309, 325, 350 Brascamp, Herm Jan, 288 Brassens, Georges, 314

Cambern, Michael, 74, 204 Carleman, Torsten, xviii, 203 Carleson, Lennart, 90, 315 Carlitz, Leonard, 259 Casazza, Peter G., 76, 116 Caveny, James, 321 Chatterji, Srishti D., 39 Choquet, Gustave, 351, 352 Christensen, Jens Peter Reus, 350 Cobos, Fernando, 205 Costé, Alain, 322 Cotlar, Mischa, 289 Cramér, Harald, xiv Creekmore, James, 205 Dacunha-Castelle, Didier, xiv, 204 Daugavet, Igor Karlovich, 308 Davie, Alexander M., xiv, xxiv, 48, 259 Day, Mahlon Marsh, xiv, 63, 75 de la Vallée-Poussin, Charles-Jean, 262 de Leeuw, Karel, 203 Debs, Gabriel, xxi, 326, 328, 350 Déchamps (see Déchamps-Gondim), 326 Déchamps-Gondim, Myriam, 231, 246, 259, 260 Deville, Robert, xiv, 207, 350, 352 Diestel, Joseph, xiv, 42, 63, 110, 157, 182, 185, 258, 259, 315, 322, 350, 353 Dilworth, Stephen J., xxii, 352 Doob, Joseph L., xvi, 22, 39, 130, 288 Dor, Leonard E., xxi, 75, 323, 336, 350

415

416

Author Index for Volume 1

Drury, Stephen W., 251, 259 Dudley, Richard Mansfield, xxv, xxviii, 203 Dunford, Nelson, xiv, 75, 157, 308, 316 Duren, Peter, 320, 322 Dvoretzky, Aryeh, 84, 227 Eberlein, William F., 2 Edwards, Robert E., 231 Enflo, Per, xxiv, 48, 74 Erdös, Paul, xiv Fabian, Marián, 1 Fack, Thierry, 191, 205 Fakhoury, Hicham, 353 Farahat, Jean, 350 Faraut, Jacques, 204 Farmaki, Vassiliki, 111 Favorov, Sergeij Yu., 152 Felouzis, Vaggelis, 111 Ferenczi, Valentin, 102, 114 Fernique, Xavier, xx, xxv, xxviii, 203, 255, 259 Fetter, Helga, 353 Figiel, Tadeusz, 74, 111, 115, 116 Fortet, Robert, xiv Frankiewicz, Riszard, 111 Fréchet, Maurice, 1 Fremlin, David H., 350 Gamboa de Buen, Berta, 353 Garling, David J. H., 259 Gasparis, Ioannis, 111 Gelbaum, Bernard R., 63, 75 Gelfand, Israel M., 274 Ghoussoub, Nassif, 75, 157, 352 Giesy, Daniel P., 205 Giné, Evarist, 204 Girardi, Maria, xxii, 352 Godefroy, Gilles, xiv, xv, xxi, 81, 113, 157, 207, 310, 313, 315, 323, 350–352, 354, 355, 367 Goldstine, Herman Heine, 2 Goodman, Victor, 75 Gordon, Yehoram, xxiii, 259 Gowers, W. Timothy, xvii, 75, 83, 101–104, 111, 275, 353 Graham, Colin C., 231, 314 Gross, Leonard, 152 Grothendieck, Alexandre, xxiv, 217, 218, 220, 258 Guédon, Olivier, xv, 152

Guerre-Delabrière, Sylvie, 309, 350 Gundy, Richard F., 309 Haagerup, Uffe, 152, 259 Habala, Petr, 1, 12 Hájek, Petr, 1, 12 Hagler, James, xxii, 113, 352, 353 Halmos, Paul R., 166 Hardy, Godfrey Harold, 185, 321 Harzallah, Khélifa, 204 Havin, Victor, 260 Haydon, Richard, 352 Helson, Henry, 259 Hoffmann-Jørgensen, Jørgen, xiv, 151, 154, 204 Host, Bernard, 314, 373 James, Robert C., xvi, xvii, 72, 75, 79, 81, 83, 96–98, 101, 111, 192, 205, 315, 353 Janicka, Liliana, 352 Jarchow, Hans, xiv, 182, 185, 258, 259 John, Fritz, 259 Johnson, William B., 69, 74–76, 81, 103, 115, 116 Jöricke, Burglind, 260 Junge, Marius, 42 Kadeˇc, Mikhail Iosifovich, 259 Kadets, Vladimir M., 308 Kahane, Jean-Pierre, 42, 139, 143, 151, 152, 185, 203, 240, 259 Kalton, Nigel J., 76, 81, 102, 113, 182, 260, 323, 352, 355 Kashin Boris S., 231 Katznelson, Yitzhak, xxviii, 169, 203 Khintchine, Aleksandr Yakovlevich, 30 Kisliakov, Serguei V., 259 Knaust, Helmut, 113 Koldobsky, Alexander, 204 Kolmogorov, Andre˘ı Nikolaevich, xvi, 17, 25, 36, 130, 280 Komlós, János, 315 Komorowski, Ryszard A., 103, 111 König, Hermann, 259 Kre˘ın, Mark Grigorievich, 6 Krickeberg, Klaus, 41 Krivine, Jean-Louis, xiv, 10, 12, 192, 204, 259 Kwapie´n, Stanisław, 111, 152, 154, 205, 232, 259, 261, 262, 308 Lacey, H. Elton, 75, 190 Lapresté, Jean-Thierry, 116

Author Index for Volume 1

Lasser, Rupert, 279, 280 Latała, Rafał, xviii, 144, 152 Ledoux, Michel, xiv, xvi, 121, 151, 153, 192, 204 Lefèvre, Pascal, 237, 260 Le Page, Raoul, 204 Lévy, Paul, xiv, 17, 20, 26, 27, 125, 126, 129, 132, 151, 166 Lewis, Daniel Ralph, xxvi, 259 Li, Daniel, 81, 113, 260, 323, 367 Lieb, Elliott H., 288 Lindahl, Lars-Åke, 231 Lindenstrauss, Joram, xiv, xviii, xix, xxiv, 48, 63, 69, 74–76, 78, 81, 82, 110, 112, 115, 151, 157, 204, 205, 217, 218, 221, 230, 258, 271, 310, 350, 353 Lions, Jacques-Louis, 308 Littlewood, John Edensor, 321 Loève, Michel, 42, 151 Löfström, Jörgen, 308 Lohman, Robert H., 353 Lonke, Yossi, 204 Loomis, Lynn H., 362 Lopez, Jorge M., 231 Lukacs, Eugene, 203 Lust-Piquard, Françoise, 205, 322, 354 Malliavin, Paul, 305 Marcus, Michael B., 192, 204 Maurey, Bernard, xiv, xxiii, xxvi, xxix, 75, 82, 90, 102–104, 111, 151, 152, 157, 182, 185, 188, 192, 204, 205, 214, 260, 275, 308, 309, 314, 315, 318, 323, 352 Mazur, Stanisław, 63, 74 McGehee, O. Carruth, 231 Megginson, Robert E., 1, 75, 315 Méla, Jean-François, 373 Meyer, Yves, 310 Milman, David Pinhusovich, 6 Milman, Vitali D., xiv, xxiii, xxiv, 82, 185, 192, 232, 259, 275 Milyutin, Alekse˘ı Alekseevich, 75 Mokobodzki, Gabriel, 350 Montesinos-Santalucía, Vicente, 1 Morrison, Terry J., 350 Morse, Anthony P., 75 Mourier, Édith, xiv Musial, Kazimierz, 352 Nahoum, Albert, 308 Naidenov, Nikola, 262

417

Nazarov, Fedor L., 152, 203, 205 Neveu, Jacques, 14, 37, 42, 203 Newman, Charles M., 75 Odell, Edward Wilfred, 82, 111, 113, 305 Oleszkiewicz, Krzysztof, 144, 152 Olevski˘ı, Alexander M., 310 Orlicz, Władysław, xx, 84, 110, 266, 284, 308 Ørno, Peter, 308 Pajor, Alain, xxi, xxv, 326, 336, 338, 350 Paley, Raymond E.A.C., xviii, xxi, 23, 203, 221, 267, 293, 309 Parreau, François, 314, 373 Parthasaraty, Kalyanapuram Rangachari, 151 Peetre, Jaak, 308 Pelant, Jan, 1 Pelczar, Anna Maria, 111 Pełczy´nski, Aleksander, xv, xvii, xix, xxii, 63, 67, 75, 76, 79, 82, 83, 95, 110–113, 205, 217, 218, 221, 232, 258–260, 274, 296, 308, 310, 326, 343 Pettis, Billy James, 110, 156, 157, 316 Pfitzner, Hermann, 113, 114 Phelps, Robert R., 308, 351 Phillips, Ralph S., 81 Pietsch, Albrecht, xiv, xix, 205, 258 Pisier, Gilles, xiv, xv, xx, xxiii, xxvi–xxix, 42, 43, 80, 151, 152, 182, 185, 188, 191, 192, 204, 205, 220, 224, 232, 246, 248, 250, 258–260, 309 Pitt, Harry Raymond, 78 Podkorytov, Anatoliy N., 152 Poulsen, Frode, 231 Prokhorov, Yuri Vasilyevich, 123 Queffélec, Hervé, xiii, 260 Queneau, Raymond, 7 Rainwater, John, 308 Read, Charles J., 76 Rényi, Alfréd, 42 Revuz, Daniel, xvi, 169 Ricard, Éric, 314 Rider, Daniel, 246, 259 Riemann, Bernhard, 84, 227 Riesz, Frédéric (Frigyes), 235 Riesz, Marcel, 53, 279, 308 Rodríguez-Piazza, Luis, xv, 260 Rogalski, Marc, 350 Rogers, C. Ambrose, 84, 227

418

Author Index for Volume 1

Rosenthal, Haskell Paul, xiv, xviii, xxvi, 69, 74–76, 110, 204, 205, 237, 326, 335, 348, 350, 352 Ross, Kenneth A., 231 Rudin, Walter, xiii, xv, xxvi, 1, 169, 231, 239, 259, 279, 297, 310 Ryll-Nardzewski, Czesław, 111 Saakyan, Artur Artushovitch, 231 Saint-Raymond, Jean, 367 Salem, Raphaël, xvi, xxviii, 34, 43, 259 Samorodnitsky, Gennady, 204 Saphar, Pierre David, 113 Schachermayer, Walter, 75, 157, 352 Schauder, Juliusz Paweł, 52, 54, 74 Schechtman, Gideon, xxiii, xxiv, 185, 192, 323 Schlumprecht, Thomas, 82, 102, 111 Schönberg, Mario M., xviii, 167, 204 Schwartz, Jacob T., xiv Semadeni, Zbigniew, 75 Shapiro, Harold S., 239 Shapiro, Joel, 310 Sharpley, Robert, 287 Shiryaev, Albert N., 42 Shura, Thaddeus J., 116 Shvidkoy, Roman V., 308 Sidon, Simon, 235 Sierpi´nski, Wacław, 331 Simons, Stephen, 315 Singer, Ivan, 74, 112 Sirotkin, Gleb G., 308 Skorohod, Anatoli Vladimirovitch, 164 Šmulian, Vitold L., 2 Snobar, M. G., 259 Sobczyk, Andrew, 80 Steˇckin (Stechkin), Serge˘ı Borisovich, 259 Stegall, Charles, 173, 204, 352, 353 Steinhaus, Hugo Dyonizy, 53, 308 Stroock, Daniel W., 42, 151 Szankowski, Andrzej, xxiv, 103 Szarek, Stanisław J., xxiv, 76, 152, 259 Szlenk, Wiesław, 78, 79, 315

Talagrand, Michel, xiv, xxvi, 110, 121, 151, 153, 181, 192, 204, 350–352 Taqqu, Murad S., 204 Taylor, Angus E., 316 Thorin, G. Olof, 308 Tomaszewski, Bogusław, 152 Tomczak-Jaegermann, Nicole, xiv, 82, 103, 191, 204, 205, 258, 325 Tonge, Andrew, xiv, 182, 185, 258, 259 Tsirelson, Boris S., 102, 115 Turán, Paul, 44 Turpin, Philippe, 153 Tzafriri, Lior, xxiv, 48, 63, 69, 75, 76, 110–112, 115, 116, 151, 218, 271, 310, 350 Uhl, J. Jerry, Jr., 157, 310, 322 Ullrich, David C., 42, 152 van Dulst, Dick, 352 Varopoulos, Nicholas Th., 232, 259 Veech, William A., 80 Weil, André, 364 Weissler, Fred B., 152 Wenzel, Jörg, xiv, 205 Werner, Dirk, 308 Whitley, Robert J., 5 Wojtaszczyk, Przemysław, 74, 76, 90, 221, 296, 308, 309, 315, 318, 320 Wright, E. Maitland, 185 Xu, Quanhua, 42, 309 Zafran, Misha, xxviii Zinn, Joel, 204 Zippin, Mordecay, 68, 69, 74, 75, 80, 81, 230 Zizler, Václav, xiv, 1, 12, 207, 350, 352 Zolotarev, Vladimir M., 204 Zuily, Claude, xiii Zygmund, Antoni, xvi, xxviii, 23, 34, 43, 279

Subject Index for Volume 1

σ -algebra, 13 complete, 13 Angle between two subspaces, 104 Approximate identity, 376 Atom of a measure, 270, 369 Banach algebra involutive, 362 Banach space, 1 B-convex, 205 containing nr ’s uniformly, 192 containing another Banach space, 94 crudely finitely representable, 173 distortable, 82 finitely representable, 173 hereditarily indecomposable, 102 HI, 102 K-convex, 185 L-summand in its bidual, 356 local theory, 173 of cotype q, 180 of type p, 180 UMD, 309 universal, 56 weakly sequentially complete (w.s.c.), 97 Banach spaces classical, 94 quasi-isometric, 172 Banach-valued random variable, 117 BAP, 76 Barycenter calculus, 351 of a probability measure, 351

Bases equivalent, 59 Basis, 46 block, 62 boundedly complete, 70 canonical of p , 48 of c0 , 48 constant of a basis, 49 Haar, 52 monotone, 49 natural of p , 48 of c0 , 48 normalized, 47 of a Banach space, 46 Schauder, 46 of C([0, 1]), 54 shrinking, 69 summing basis of c, 59 unconditional, 90 unconditional constant, 91 Block basic sequence, 62 C ∗ -algebra, 363 Character of a commutative algebra, 361 of a compact Abelian group, 370 Characteristic function of a Banach-valued r.v., 125 of a real r.v., 19 Class Schatten, 190 Complete (weakly sequentially), 97 Concatenation, 329

419

420

Subject Index for Volume 1

Condition Blaschke, 298 mesh, 240 Conditional expectation of a Banach-valued r.v., 127 of a real r.v., 35 Constant K-convexity, 185 cotype constant of an operator, 200 cotype-q constant of a Banach space, 181 Grothendieck, 217 of a basis, 49 of unconditionality, 91 Sidon, 232 type constant of an operator, 200 type-p constant of a Banach space, 180 Convergence almost sure of Banach-valued r.v.’s, 120 of sequences of real r.v.’s, 15 in distribution of Banach-valued r.v.’s, 122 of scalar r.v.’s, 18 in law of scalar r.v.’s, 18 in probability of Banach-valued r.v.’s, 119 of sequences of real r.v.’s, 15 weak, 123 r-convexity, 189 Convolution, 367 Cotype non-trivial, 182 of a Banach space, 180 Decomposition Doob, 40 into finite-dimensional subspaces (FDD), 76 Krickeberg, 41 Decoupling of real r.v.’s, 24 Distance Banach–Mazur, 172 Distribution of a real r.v., 13 Dual group, 370 Element in a C ∗ -algebra Hermitian, 363 unitary, 364 of first class (in the bidual of a Banach space), 341

Event space, 13 Events, 13 Expectation of a Banach-valued r.v., 126 of a real r.v., 13 Extreme face, 6 Family of elements dominated by another, 195 subordinate to another, 195 summable, 84 FDD, 76 Filter, 7 base, 7 cluster point, 9 coarser than another, 7 convergent, 8 finer than another, 7 Fréchet, 7 limit of a filter, 9 limit of a function along a filter, 9 section, 7 Filtering family, 10 convergence, 10 Filtering sub-family, 10 Filtration, 36 Formula integration by parts, 32 Fubinization principle, 26 Function affine first class, 343 conjugate, 278 of positive type, 167 Orlicz, 32 satisfying the barycenter calculus, 351 square, 189 square function of a martingale, 288 Functions Rademacher, 275 Walsh, 144, 371 Gaussian standard, 26, 160 Gaussian random variable standard complex, 164 Gaussian vectors rotation invariance of standard Gaussian vectors, 165 Group Cantor, 365

Subject Index for Volume 1

compact Abelian, 364 topological Abelian, 364 torus, 364 unit circle, 364 Identity generalized parallelogram, 178 Independence of families of events, 14 of real r.v.’s, 14 Inequalities Khintchine, 30 Kahane, 139 of truncation, 27 Inequality Bernstein, 240 Doob, 37 Grothendieck, 220 Hardy, 318 Kolmogorov, 21 Lax, 263 Malik, 263 Orlicz–Jensen, 33 Paley–Zygmund, 23 Paul Lévy maximal inequality, 129 Schaake–Van der Corput, 263 Invariance under rotation of Gaussian vectors, 165 K-convexity, 185 Kernel de la Vallée-Poussin, 378 Dirichlet, 53, 376 Fejér, 376 of negative type, 167 of positive type, 167 Poisson, 278 Law Kolmogorov’s zero–one law, 17 strong law of large numbers, 44 Lemma Borel–Cantelli, 14 du Bois–Reymond, 88 Riemann–Lebesgue, 375 Sierpi´nski, 331 Vitali covering, 313 Limit lower limit of a sequence of events, 14 upper limit of a sequence of events, 14

Linear functional coordinate, 49 positive, 362 Local reflexivity, 173 Martingale, 36 Banach-valued, 128 closed, 128 Burkholder transform, 289 closed, 37 square function, 288 Matrix covariance, 160 Hilbert, 319 Walsh, 238 Measure atom, 369 continuous, 369 diffuse, 369 doubling, 312 Haar, 364 Method Pełczy´nski decomposition, 68 Metric Hamming, 144 Multiplier, 264, 321 Norm 2-concave, 190 Luxemburg, 32 octahedral, 352 p-summing, 212 r-convex, 189 weak-2 , 195 Operator completely continuous, 316 Dunford–Pettis, 216, 316 factorizable through a Hilbert space, 193 of cotype q, 200 of type p, 200 p-summing, 211 (q, p)-summing, 211 representable, 155 strictly singular, 78 unconditionally convergent, 111 weak type (p, p), 279 Ordinal countable, 11 finite, 11

421

422

Subject Index for Volume 1

Ordinal (cont.) first uncountable, 11 infinite, 11 limit, 11 Polynomials random, 242 Rudin–Shapiro, 239 trigonometric, 372 Principle of contraction, 92 qualitative, 133 quantitative, 136 of local reflexivity, 173 Paul Lévy symmetry principle, 129 Probability, 13 Probability law of a real r.v., 13 Process increasing, 39 Product convolution, 367 Riesz, 235, 314, 378 Projection Riesz, 277 Property approximation bounded, 76 Banach–Saks, 315 Daugavet, 308 Dunford–Pettis, 353 extension, 229, 344 ideal, 212 Orlicz, 182 PIf ,∞ , 354 Radon–Nikodým (RNP), 155 Schur, 65 (u) of Pełczy´nski, 112 (V) of Pełczy´nski, 113 (V ∗ ) of Pełczy´nski, 113 Random variable complex standard Gaussian, 164 convergence in distribution of scalar r.v.’s, 18 convergence in law of scalar r.v.’s, 18 p-stable, 166 real Gaussian, 159 standard Gaussian, 160 symmetric in the complex sense, 137

tight, 118 with values in a Banach space, 117 Random variable (Banach-valued) characteristic function, 125 conditional expectation, 127 convergence almost sure, 120 convergence in distribution, 122 convergence in probability, 119 expectation, 126 symmetric, 129 Random variables (Banach-valued) independent, 127 Real r.v., 13 centered, 13 conditional expectation, 35 convergence a.s., 15 in probability, 15 distribution, 13 expectation, 13 independence, 14 probability law, 13 standard Gaussian, 26 symmetrization, 24 variance, 13 Resolvent, 358 Schatten class, 190 Selectors, 325 Separable, 327 Sequence basic, 59 strongly summing, 110 Bernoulli, 26 block basic, 62 of i.i.d. random variables, 26 of random variables symmetric, 129 Rademacher, 26 standard Gaussian, 26 uniformly integrable, 267 w-Cauchy, 97 weakly Cauchy, 97 Series unconditionally convergent, 83 weakly unconditionally Cauchy, 86 w.u.C., 86 Set Cantor, 56 dissociate, 314 equicontinuous, 268

Subject Index for Volume 1

Hadamard, 235 of words, 235, 314 Riesz, 310 Rosenthal, 237 Shapiro, 310 Sidon, 232 uniformly absolutely continuous, 268 uniformly integrable, 267 uniformly tight, 124 Space Gaussian, 206 Hagler JH, 353 James, 79 James function space, 353 James tree space, 353 Lorentz 2,1 , 201 Lp , 222 Orlicz, 32 Polish, 327 Tsirelson, 115 weak-2 , 202 Spectral radius, 359 Spectrum of a measure, 232 of an element of a Banach algebra, 358 Stopping time, 21 Sub-Gaussian, 33 Submartingale, 36 Subset (of a discrete Abelian group) nicely placed, 310 Subsets, Boolean independent, 334 Subspace of L1 , nicely placed, 310 Symmetrization of a real r.v., 24 System Franklin, 315 Haar, 52 Schauder, 54 Theorem Alaoglu–Bourbaki, 2 Banach–Dieudonné, 12 Banach–Mazur, 56 Banach–Saks, 317 Bernstein, 265 Bessaga–Pełczy´nski c0 theorem, 94 equivalence theorem, 60 selection theorem, 62 Bochner, 169 Bochner–Weil–Raikov, 362 boundedness, 34

Carleman, 201 Doob, 38 Doob vectorial, 128 Dor, 323 Drury, 247 Dunford–Pettis, 273 Dvoretzky–Rogers, 227 Eberlein–Šmulian, 2 factorization through a Hilbert space, 195 final form, 198 Fernique integrability of Gaussian vectors, 255 Frédéric and Marcel Riesz, 300 Gelfand–Mazur, 359 Goldstine, 2 Gowers dichotomy, 102 Grothendieck, 221 dual form, 224 Hausdorff–Young, 287 James, 96–98, 315 John, 229 Kadeˇc–Snobar, 229 Kolmogorov, 280 converse, 23 series of independent real r.v., 22 Komlós, 315 Krein–Milman, 6 Kwapie´n, 199 Lyapounov, 294 Marcel Riesz, 279 Marcinkiewicz, 280 Maurey–Pisier, 192 Odell–Rosenthal, 341, 350 Orlicz, 284 Orlicz–Pettis, 86 Paley, 201, 301 Paul Lévy, 27 continuity, 20 equivalence theorem, 130 Phillips, 81 Pietsch factorization, 213 Pietsch–Pełczy´nski, 216 Pitt, 78 Prokhorov, 124 vectorial form, 153 Rider, 247 Riesz factorization, 297 Riesz–Thorin, 286 Rosenthal’s 1 , 326

423

424

Subject Index for Volume 1

Theorem (cont.) Rosenthal–Bourgain–Fremlin–Talagrand, 327 Schönberg, 168 Schur, 65 Sobczyk, 80 Szlenk, 315 three-lines, 285 three-series, 25 Tychonov, 9 vectorial central limit theorem, 162 Vitali–Hahn–Saks, 269 Topological space dense-in-itself, 347 Topology balls, 352 Bohr, 310 w, 1 w∗ , 2 weak, 1 weak∗ , 1 Torus, 364

Transform Burkholder, 289 Fourier, 372 Gelfand, 363 Hilbert, 278 Type non-trivial, 182 of a Banach space, 180 Rademacher type p, 191 stable type p, 191 Type on a Banach space, 352 Ultrafilter, 7 trivial, 8 Ultrapower of a Banach space, 176 Ultraproduct of Banach spaces, 176 Unit circle, 364 Variance of a real r.v., 13 Vector, Gaussian, 160 Weakly sequentially complete, 97 Words of length  n, 235

Notation Index for Volume 2

A2,ϕ2 , 257 A2,ϕq , 287 ∗ , 37 α1∗ , . . . , αm A(X), 52 A(Y, X), 52 βN, 52 C as , 245 d2,A , 275 dA , 275 dA , 269 d(X), 25 F (X), 52 F (Y, X), 52 γ (z), 4 γ (u), 43 γN , 104 J(d), 80 K(ε), 30, 105 Kg (X), 163

K(X), 52 K(Y, X), 52 L2 , 197 λ(E, X), 41 λ(X), 41 L(X), 52 L(Y, X), 52 M2,2 , 255 N2,A , 276 NA , 276 NA , 269 n(E), 269 N(ε), 29 N(ε) = N(T, d, ε), 80 N(K, ε), 272  . K ◦ , 103 [[ . ]], 245 ψ, 93 ψA , 197 σX , 25 SN , 103 X ω , 73

425

Author Index for Volume 2

Aldous, David J., 177 Alspach, Dale E., 177, 186 Arkhipov, Gennady I., 229 Asmar, Nakhlé, 178 Azuma, Kazuoki, 48 Bachelis, Gregory F., 151, 178 Ball, Keith, 123 Banach, Stefan, xv, xxiv, 51 Barbe, Philippe, xvi Beauzamy, Bernard, xiv, 46 Beck, Anatole, xiv Bennett, Colin, 252 Benyamini, Yoav, xiv, 45, 46 Bernstein, Serge˘ı Natanoviˇc, 78 Bessaga, Czesław, xv, xvii Billard, Pierre, xxviii, 245 Bochner, Salomon, xviii Bohr, Harald, 233 Bonami, Aline, 150 Bourgain, Jean, xiv, xxiv, xxvi, xxvii, 36, 146, 159, 160, 176, 177, 179, 184, 186, 187, 193, 194, 197, 202, 204, 206, 217, 218, 223, 228–230, 261, 268, 271 Brascamp, Herm Jan, 123 Bretagnolle, Jean, xiv Bukhvalov, Alexander V., 177 Burkholder, Donald L., 217 Carleman, Torsten, xviii Casazza, Peter G., 68–70 ˇ Cervonenkis, Alexei Ya., 98 Chevet, Simone, 26 Cramér, Harald, xiv, 193

Dacunha-Castelle, Didier, xiv Davie, Alexander M., xiv, xxiv, 52, 59, 69, 71 Davis, William J., 46 Day, Mahlon Marsh, xiv Dean, David W., 46 Debs, Gabriel, xxi Déchamps-Gondim, Myriam, 206 Deville, Robert, xiv Diestel, Joseph, xiv Dilworth, Stephen J., xxii, 123 Doob, Joseph L., xvi Dor, Leonard E., xxi Drury, Stephen W., 193 Dudley, Richard Mansfield, xxv, xxviii, 45, 85, 122 Dunford, Nelson, xiv Duren, Peter, 124, 283 Dvoretzky, Aryeh, 19, 20, 46 Ebenstein, Samuel E., 151, 178 Elton, John Hancock, 123 Enflo, Per, xxiv, 52, 59, 69 Erdös, Paul, xiv, 193 Fathi, Albert, 122 Fernique, Xavier, xx, xxv, xxviii, 123, 245 Figiel, Tadeusz, 34, 46, 69, 163, 179 Fortet, Robert, xiv Girardi, Maria, xxii Gluskin, Efim D., 46 Godefroy, Gilles, xiv, xv, xxi, 69, 70, 177, 178, 185, 186 Gordon, Yehoram, xxiii, 1, 17, 27, 28, 39, 46 Gowers, W. Timothy, xvii

426

Author Index for Volume 2

Grothendieck, Alexandre, xxiv, 52, 53, 57–59, 69–71 Guédon, Olivier, xv, 19, 46 Guerre (see Guerre-Delabrière), 177 Guerre-Delabrière, Sylvie, 46, 177 Hagler, James, xxii Halász, Gábor, 286 Harcharras, Asma, 178 Hardin, Clyde D., Jr., 186 Hardy, Godfrey Harold, 252 Hare, Kathryn E., 188 Harmand, Peter, 178 Havin, Victor, 178 Hayman, Walter Kurt, 284 Hewitt, Edwin, 177 Hoffmann-Jørgensen, Jørgen, xiv, 178, 236, 286 Ibragimov, Ildar, 45 Itô, Kiyosi, 286 James, Robert C., xvi, xvii Jarchow, Hans, xiv John, Fritz, 22, 36, 46 Johnson, William B., 68, 69, 177, 179, 182 Kadeˇc, Mikhail Iosifovich, 130, 177, 178 Kahane, Jean-Pierre, 17, 45, 78, 122, 123, 229, 245, 263, 283, 286 Kalton, Nigel J., 68–70, 178, 185, 186 Katznelson, Yitzhak, xxviii, 193, 217, 262–264, 268, 287 Kolmogorov, Andre˘ı Nikolaevich, xvi, 284 Konyagin, Sergei V., 218, 229 Krivine, Jean-Louis, xiv, 46, 177 Lapresté, Jean-Thierry, 177 Larsen, Ronald, 257 Latała, Rafał, xviii Ledoux, Michel, xiv, xvi, 122, 179, 237 Lefèvre, Pascal, 234, 286, 287 Leindler, László, 123 Lévy, Mireille, 177 Lévy, Paul, xiv, 1, 25 Lewis, Daniel Ralph, xxvi, 127, 168, 179 Li, Daniel, 69, 70, 178, 185, 186, 286 Lieb, Elliott H., 123 Lifshits, Mikhail, 122 Lindenstrauss, Joram, xiv, xviii, xix, xxiv, 34, 45, 46, 60, 69, 70, 179

427

Lorentz, George Gunther, 179 Lozanovski, Grigorij Yakovlevich, 177 Lusky, Wolfgang, 186 Malliavin, Paul, 193, 194, 261 Malliavin-Brameret, Marie-Paule, 194, 261 Marcinkiewicz, Józef, 282, 284 Marcus, Michael B., 122, 123, 234, 286, 288 Mattila, Pertti, 45 Maurey, Bernard, xiv, xxiii, xxvi, xxix, 1, 2, 17, 45, 127, 177, 178, 249, 268, 270–272 Meyer, Mathieu, 46 Meyer, Yves, 150 Milman, Vitali D., xiv, xxiii, xxiv, 1, 19, 34, 36, 46, 123, 125, 126, 159, 179, 268, 271 Montgomery-Smith, Stephen, 178 Mooney, Michael C., 178 Mourier, Édith, xiv Neuwirth, Stefan, 70, 178 Nikishin, Evgenii Mikhailovich, 178 Nisio, Makiko, 286 Odell, Edward Wilfred, 177, 182 Orlicz, Władysław, xx Oskolkov, Konstantin I., 229 Pajor, Alain, xxi, xxv, 46, 114, 123, 230 Paley, Raymond E.A.C., xviii, xxi Papadopoulos, Stavros F., 230 Patterson, Joseph P., 123 Pedersen, Thomas Vils, 287 Pełczy´nski, Aleksander, xv, xvii, xix, xxii, 69, 130, 177, 178 Pfitzner, Hermann, 178 Pietsch, Albrecht, xiv, xix Pisier, Gilles, xiv, xv, xx, xxiii, xxvi–xxix, 1, 2, 8, 19, 21, 45, 46, 69, 70, 77, 122, 123, 125, 137, 146, 153, 158, 160, 177–179, 193, 194, 196, 197, 203, 223, 228, 234, 249, 261–264, 268, 271, 286–288 Plotkin, Alexander I., 186 Pólya, George, 229 Prékopa, András, 123 Preston, Christopher, 122 Prignot, Patrick, 287 Queffélec, Hervé, xiii, 46, 178, 259, 286 Rademacher, Hans, 5 Raynaud, Yves, 177

428

Author Index for Volume 2

Read, Charles J., 68, 69 Rényi, Alfréd, 193 Revuz, Daniel, xvi Rider, Daniel, 195 Robbins, Herbert, 213 Rodríguez-Piazza, Luis, xv, 178, 197, 230, 271, 281, 286 Rogers, C. Ambrose, 20 Rosenthal, Haskell Paul, xiv, xviii, xxvi, 69, 127, 150, 178, 181, 184, 186 Rudin, Walter, xiii, xv, xxvi, 5, 145, 146, 178, 179, 186, 187, 284, 286 Saffari, Bahman, 259 Salem, Raphaël, xvi, xxviii Saphar, Pierre David, 69, 70 Sauer, Norbert, 98 Schechtman, Gideon, xxiii, xxiv, 19, 36, 46, 48, 177, 179 Schönberg, Mario M., xviii Schwartz, Jacob T., xiv Sharpley, Robert, 252 Shelah, Saharon, 98 Shepp, Larry A., 122 Simon, Leon, 45 Singer, Ivan, 46 Sledd, William T., 124 Slepian, David, 45 Sudakov, Vladimir N., 45

Szankowski, Andrzej, xxiv, 46, 60, 69, 70 Szarek, Stanisław J., xxiv, 36, 69 Szegö, Gábor, 229 Talagrand, Michel, xiv, xxvi, 122, 123, 127, 158, 176, 177, 179, 181, 187, 237 Tomczak-Jaegermann, Nicole, xiv, 126, 179 Tonge, Andrew, xiv Tsirelson, Boris S., 45 Tzafriri, Lior, xxiv, 46, 60, 69, 70 Vapnik, Vladimir N., 98 Virot, Bernard, 217, 229 Wenzel, Jörg, xiv Werner, Dirk, 178 Werner, Wend, 178 Willis, George, 69 Wojtaszczyk, Przemysław, 47, 179 Wolfson, Haim J., 126 Yosida, Kôsaku, 177 Yurinski˘ı, Vadim V., 48 Zafran, Misha, xxviii, 263, 287 Zippin, Mordecay, 69 Zizler, Václav, xiv Zuily, Claude, xiii Zygmund, Antoni, xvi, xxviii, 282, 284

Subject Index for Volume 2

relatives, 15 strangers, 15

Algebra Pisier, 264 Banach algebra homogeneous, 263 semi-simple, 263 strongly homogeneous, 263 Banach space containing n1 ’s uniformly, 135 K-convex, 158 stable, 177 Basis Auerbach, 24 Brownian motion, 76 rapid points, 122 slow points, 122

Density combinatorial, 96 Diameter arithmetic diameter of a subset of a discrete Abelian group, 269 of a Banach space, 269 Dimension Gaussian, 25

Chaos Wiener, 191 Class analytic Lipschitz, 284 Nevanlinna, 283 Zygmund analytic, 283 Condition Tauberian, 240 Zygmund, 283 Conjecture Bloch–Nevanlinna, 283 Katznelson dichotomy, 263 Constant K-convexity, 158 of an FDD, 182 Couples of type max, 16 of type min, 16

Families of subsets Boolean independent, 105 Function entropy metric, 80 non-decreasing rearrangement, 93, 250

ε-net, 29 Entropy, 29 entropy integral, 80 entropy metric function, 80

Gaussian vector Gaussian dimension, 25 weak moment, 25 Inequality Azuma, 48 Brunn–Minkowski, 101 Chernov, 98 Marcus–Pisier, 249 Maurey–Pisier concentration of measure, 5 deviation, 4

429

430

Subject Index for Volume 2

Inequality (cont.) Prékopa–Leindler, 101 Urysohn, 103 Yurinski˘ı, 48 Integral entropy, 80 Law of the iterated logarithm for Brownian motion, 79 Lemma Dvoretzky–Rogers, 20 Auerbach, 23 Bourgain–Szarek, 36 Hardy, 252 Kadeˇc–Pełczy´nski, 130 Lewis, 21, 23 Sauer, 98 Slepian, 14 Slepian–Sudakov, 12 subsequence splitting lemma, 181 Method of selectors, 193 Kuhn–Tucker, 22 Rudin averaging, 187 Metric L2 of a process, 80 Minoration Sudakov, 14 Moment weak (of a Gaussian vector), 25 Multiplier, 255 Net, 29 Operator factorizable strongly through Lq , 151 Phenomenon concentration of measure, 1 Polynomials Hermite, 191 Problem of cosines, 229 of H. Bohr, 233 Procedure summation, 239 Process bounded version, 88 continuous version, 238

Gaussian, 72 metric L2 , 80 stationary, 85 sub-Gaussian, 80 trajectory, 72 version, 73 Property (π ), 68 approximation, 51 AP, 51 BAP, 51 BAPcomm , 68 bounded, 51 bounded commuting, 68 CAP, 69 compact, 69 MAP, 51 metric, 51 UMAP, 70 strong Schur property, 181 Quasi-independence, 194 Random variable q-stable, 143 Rearrangement non-decreasing rearrangement of a function, 93 Relation in a subset of Z, 198 length, 198 Selectors, 193 Sequence complex standard Gaussian, 92 Rudin–Shapiro, 48 Series A-bounded, 239 A-convergent, 239 Set (q), 145 almost sure p-Sidon, 286 p-Rider, 286 stationary, 287 Subset dense for a set of subsets, 96 Subset (of a Banach space) norming, 235 Subset (of a discrete Abelian group) quasi-independent, 194 Subspace of L1 , nicely placed, 177

Subject Index for Volume 2

Theorem Bachelis–Ebenstein, 151 Billard, 245 Bourgain–Milman, 268 Brunn–Minkowski, 101 Bukhvalov–Lozanovski, 177 Comparison (of Gaussian vectors), 11 Drury, 196 Dudley, abstract, 81 Dvoretzky, 24 Gaussian version, 25 Dvoretzky–Rogers, 20 Elton, 114 Elton–Pajor, 106 equivalence (for the convergence of random Fourier series), 243 Fernique minoration, 88 Gordon, 15 Itô–Nisio for C(K), 238 Itô–Nisio, abstract version, 235 Kadeˇc–Pełczy´nski, 128 reformulation, 128 Lewis, 23

431

Lindenstrauss–Tzafriri, 40 Marcinkiewicz–Zygmund–Kahane, 240 Marcus–Pisier, 248 Marcus–Shepp, 73 Maurey factorization theorem, 151 Milman–Schechtman, 36 Pisier, 214, 261 Rademacher, 5 Rider, 215 Rosenthal, 150 Rudin, 146 transfer, 145 Salem–Zygmund, 259 Sudakov minoration, 14 Tauberian, 240 Trajectory, of a process, 72 Transform Cramér, 99 Legendre, 99 Young, 99 Version bounded version of a process, 88 of a process, 73