Series in Banach Spaces

Operator Theory Advances and Applications Vol. 94

Editor I. Gohberg

Editorial Office: T. Kailath (Stanford) School of Mathematical H.G. Kaper (Argonne) Sciences S.T. Kuroda (Tokyo) Tel Aviv University P. Lancaster (Calgary) Ramat Aviv, Israel L.E. Lerer (Haifa) E. Meister (Darmstadt) Editorial Board: B. Mityagin (Columbus) J. Arazy (Haifa) V.V. Peller (Manhattan, Kansas) A. Atzmon (Tel Aviv) J.D. Pincus (Stony Brook) J.A. Ball (Blackburg) M. Rosenblum (Charlottesville) A. Ben-Artzi (Tel Aviv) J. Rovnyak (Charlottesville) H. Bercovici (Bloomington) D.E. Sarason (Berkeley) A. Bottcher (Chemnitz) H. Upmeier (Marburg) L. de Branges (West Lafayette) S. M. Verduyn-Lunel (Amsterdam) K. Clancey (Athens, USA) D. Voiculescu (Berkeley) L.A. Coburn (Buffalo) H. Widom (Santa Cruz) K. R. Davidson (Waterloo, Ontario) D. Xia (Nashville) R. G. Douglas (Stony Brook) D. Yafaev (Rennes) H. Dym (Rehovot) A. Dynin (Columbus) P.A. Fillmore (Halifax) Honorary and Advisory C. Foias (Bloomington) Editorial Board: P.A. Fuhrmann (Beer Sheva) P.R. Halmos (Santa Clara) S. Goldberg (College Park) T. Kato (Berkeley) B. Gramsch (Mainz) P.O. Lax (New York) G. Heinig (Chemnitz) M.S. Livsic (Beer Sheva) J.A. Helton (La Jolla) R. Phillips (Stanford) M.A. Kaashoek (Amsterdam) B. Sz.-Nagy (Szeged) Series in Banach Spaces

Conditional and Unconditional Convergence

Mikhail I. Kadets Vladimir M. Kadets

Translated from the Russian by Andrei lacob

Birkhauser Verlag Basel . Boston . Berlin Authors

M.l. Kadets and V.M. Kadets Pr. Pravdy 5, apt. 26 Kharkov 310022 Ukraine

1991 Mathematics Subject Classification 46B 15, 46B20, 46010, 42C20, 52A20, IOE05 , 60B 12

A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA

Deutsche Bibliothek Cataloging-in-Publication Data Kadec, Michail I.: Series in Banach spaces: conditional and unconditional convergence I Mikhail I. Kadets ; Vladimir M. Kadets. Trans!. from the Russ. by Andrei Iacob. - Basel; Boston; Berlin: Birkhauser, 1997 (Operator theory; Vo!. 94) ISBN-13: 978-3-0348-9942-0 e-ISBN-13: 978-3-0348-9196-7 DOl: 10.1007/978-3-0348-9196-7 NE: Kadec, Vladimir M.:; OT

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Printed on acid-free paper produced from chlorine-free pulp. TCF 00 Cover design: Heinz Hiltbrunner, Basel

ISBN-13: 978-3-0348-9942-0

987654321 CONTENTS

Introduction ...... vii Notations ...... 1

Chapter 1. Background Material §l. Numerical Series. Riemann's Theorem...... 5 §2. Main Definitions. Elementary Properties of Vector Series ...... 7 §3. Preliminary Material on Rearrangements of Series of Elements of a Banach Space ...... 9

Chapter 2. Series in a Finite-Dimensional Space §l. Steinitz's Theorem on the Sum Range of a Series ...... 13 §2. The Dvoretzky-Hanani Theorem on Perfectly Divergent Series ...... 21 §3. Pecherskii's Theorem ...... 23

Chapter 3. Conditional Convergence in an Infinite- Dimensional Space §l. Basic Counterexamples ...... 29 §2. A Series Whose Sum Range Consists of Two Points ...... 32 §3. Chobanyan's Theorem ...... 36 §4. The Khinchin Inequalities and the Theorem of M. I. Kadets on Conditionally Convergent Series in Lp ...... 39 Chapter 4. Unconditionally Convergent Series §l. The Dvoretzky-Rogers Theorem...... 45 §2. Orlicz's Theorem on Unconditionally Convergent Series in Lp Spaces ...... 49 §3. Absolutely Summing Operators. Grothendieck's Theorem 52

Chapter 5. Orlicz's Theorem and the Structure of Finite-Dimensional Subspaces §l. Finite Representability ...... 59 §2. The space Co, C-Convexity, and Orlicz's Theorem...... 62 §3. Survey on Results on Type and Cotype ...... 67

v vi CONTENTS

Chapter 6. Some Results from the General Theory of Banach Spaces §1. Frechet Differentiability of Convex Functions ...... 71 §2. Dvoretzky's Theorem ...... 73 §3. Basic Sequences ...... 79 §4. Some Applications to Conditionally Convergent Series 82 Chapter 7. Steinitz's Theorem and B-Convexity §1. Conditionally Convergent Series in Spaces with Infratype ...... 87 §2. A Technique for Transferring Examples with Nonlinear Sum Range to Arbitrary Infinite-Dimensional Banach Spaces. . . 93 §3. Series in Spaces That Are Not B-Convex ...... 97 Chapter 8. Rearrangements of Series in Topological Vector Spaces §1. Weak and Strong Sum Range ...... 101 §2. Rearrangements of Series of Functions ...... 106 §3. Banaszczyk's Theorem on Series in Metrizable Nuclear Spaces ...... 110 Appendix. The Limit Set of the Riemann Integral Sums of a Vector-Valued Function §1. Functions Valued in a B-Convex Space ...... 120 §2. The Example of Nakamura and Amemiya ...... 122 §3. Separability of the Space and the Structure of I(f) ...... 127 §4. Connection with the Weak Topology ...... 131 Comments to the Exercises ...... 141 References...... 149 Index ...... 155 INTRODUCTION

Series of scalars, vectors, or functions are among the fundamental objects of mathematical analysis. When the arrangement of the terms is fixed, investigating a series amounts to investigating the sequence of its partial sums. In this case the theory of series is a part of the theory of sequences, which deals with their convergence, asymptotic behavior, etc. The specific character of the theory of series manifests itself when one considers rearrangements (permutations) of the terms of a series, which brings combinatorial considerations into the problems studied. The phenomenon that a numerical series can change its sum when the order of its terms is changed is one of the most impressive facts encountered in a university analysis course. The present book is devoted precisely to this aspect of the theory of series whose terms are elements of Banach (as well as other topological linear) spaces. The exposition focuses on two complementary problems. The first is to char• acterize those series in a given space that remain convergent (and have the same sum) for any rearrangement of their terms; such series are usually called uncon• ditionally convergent. The second problem is , when a series converges only for certain rearrangements of its terms (in other words, converges conditionally), to describe its sum range, i.e., the set of sums of all its convergent rearrangements. Recall that for series of real numbers both problems are settled by a well• known theorem of Riemann (1867): a series converges unconditionally if and only if it converges absolutely; if the series converges conditionally, then its sum range is the whole real line. Difficultiet> arit>e when one considers more general series, for instance, se• ries in a finite-dimensional space, or even series of complex numbers. To be more precise, the answer to the first problem is readily obtained, as before: in any finite-dimensional vector space the commutativity law (invariance under rearrange• ments) holds only for absolutely convergent series. By contrast, the solution to the second problem-the description of the set of elements to which a series can con• verge for a suitable rearrangement of its terms-requires considerably more efforts. For series of complex numbers such a description was given by P. Levy in 1905 [50], while for case of series in finite-dimensional spaces it was provided by E. Steinitz [87]. The study of rearrangements of series in infinite-dimensional spaces was initi• ated by W. Orlicz [63 ,64] and subsequently developed by many mathematicians in

vii viii INTRODUCTION

different countries, and continues to be an active field of research. However, until now this theme found almost no exposition in textbooks and monographs. Several years ago we undertook the task of partially filling this gap by writ• ing a textbook Rearrangements of Series in Banach Spaces [43], intended for stu• dents at Tartu University. However, the beautiful results of D. V. Pecherskii, S. A. Chobanyan, W. Banaszczyk, and others, which appeared after that textbook was published, changed in essential manner the state of the art of the theory of series, have led to the development of new approaches, and in some instances even to a new point of view on the subject. For this reason we have decided to address again, but on a more modern level, the same topics, the result being the present book. One of our goals was to bring the topics under consideration and the theory of Banach spaces to the attention of young mathematicians. This explains the large number of exercises, among which are elementary as well as very difficult ones, and the fact that numerous open problems are pointed out. With the same goal in mind we included an Appendix titled The limit set of the Riemann integral sums of a vector-valued function , which, although does not belong directly to the main subject of the book, is closely related to it through the methods used in proofs and contains readily formulated open problems. We hope that our efforts will prove fruitful for beginner mathematicians as well for specialists in functional analysis and related areas of mathematics. We deal here mostly with general series, but beyond the topics covered in our exposition there are many other situations where the idea of considering rear• rangements of series has interesting applications: basic expansions (the theory of unconditional bases [12], [53], [83], [101]) , orthogonal systems of functions, Fourier analysis, etc. As a sample we wish to mention here the following brilliant theorem of Sz. Gy. Revsz [100J: for every 27f-periodic continuous function f it is possible to rearrange its Fourier series in such a way that some subsequence of the sequence of partial sums of the rearranged series will converge to f uniformly. For the understanding of the material presented in the book familiarity with standard courses of mathematical analysis and linear algebra and also with the elementary notions and results of the theory of Banach spaces is assumed (the content of the books [45], [55J or [78] is more than sufficient). The junior of the authors has used the book to teach a special course at Khar'kov University. Discussions with the participants in the course were ex• tremely useful in preparing the final version of the manuscript. Our work was supported in part by the AMS fSU Aid Grants and, in the final stage, by ISSEP Grants SPU 061025 and APU 06140. We wish to express our gratitude to W. Banaszczyk, I. S. Belov, W. A. Beyer, E. A. Gorin, V. S. Grinberg, and especially S. A. Chobanyan for useful discussions, to I. Halperin and T. Ando for kindly sending us supplementary bibliography on rearrangement of series, and to D. L. Dun, the spouse of the senior and the mother of the junior of the authors, who took upon herself the worries of everyday life-• without the freedom she thus provided we would have never decided to undertake the work whose results are offered now to the reader.