Sequences and series in banach spaces, joseph diestel

Graduate Texts in Mathematics 92 Editorial Board F W Gehring P R Halmos (Managing Editor) C C Moore Graduate Texts in Mathematics I TAKEUTIIZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFFER Topological Vector Spaces HILTON/STAMMBACH A Course in Homological Algebra MACLANE Categories for the Working Mathematician HUGHEs/PIPER Projective Planes SERRE A Course in Arithmetic T AKEUTIIZARING Axiometic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory 10 COHEN A Course in Simple Homotopy Theory 11 CONWAY Functions of One Complex Variable 2nd ed 12 BEALS Advanced Mathematical Analysis 13 ANDERSON/FuLLER Rings and Categories of Modules 14 GOLUBITSKy/GUILLEMIN Stable Mappings and Their Singularities IS BERBERIAN Lectures in Functional Analysis and Operator Theory 16 WINTER The Structure of Fields 17 ROSENBLATT Random Processes 2nd ed 18 HALMos Measure Theory 19 HALMOS A Hilbert Space Problem Book 2nd ed., revised 20 HUSEMOLLER Fibre Bundles 2nd ed 21 HUMPHREYS Linear Algebraic Groups 22 BARNEs/MACK An Algebraic Introduction to Mathematical Logic 23 GREUB Linear Algebra 4th ed 24 HOLMES Geometric Functional Analysis and its Applications 25 HEWITT/STROMBERG Real and Abstract Analysis 26 MANES Algebraic Theories 27 KELLEY General Topology 28 ZARISKIISAMUEL Commutative Algebra Vol I 29 ZARISKIISAMUEL Commutative Algebra Vol II 30 JACOBSON Lectures in Abstract Algebra I: Basic Concepts 31 JACOBSON Lectures in Abstract Algebra II: Linear Algebra 32 JACOBSON Lectures in Abstract Algebra III: Theory of Fields and Galois Theory 33 HIRSCH Differential Topology 34 SPITZER Principles of Random Walk 2nd ed 35 WERMER Banach Algebras and Several Complex Variables 2nd ed 36 KELLEy/NAMIOKA et al Linear Topological Spaces 37 MONK Mathematical Logic 38 GRAUERT/FRITZSCHE Several Complex Variables 39 ARVESON An Invitation to C*-Algebras 40 KEMENY/SNELUKNAPP Denumerable Markov Chains 2nd ed 41 APOSTOL Modular Functions and Dirichlet Series in Number Theory 42 SERRE Linear Representations of Finite Groups 43 GILLMAN/JERISOl':/ Rings of Continuous Functions 44 KENDIG Elementary Algebraic Geometry 45 LOEVE Probability Theory I 4th ed 46 LOEVE Probability Theory II 4th ed 47 MOISE Geometric Topology in Dimensions and continued after Index Joseph Diestel Sequences and Series in Banach Spaces Springer-Verlag New York Berlin Heidelberg Tokyo World PublishlI~g Corporation,Be;jing,China Joseph Diestel Department of Math Sciences Kent State University Kent, OH 44242 U.S.A Editorial Board P R Halmos F W Gehring Managing Editor Department of Mathematics Indiana University Bloomington, IN 47405 U.S.A Department of Mathematics University of Michigan Ann Arbor, Michigan 48104 U.S.A C C Moore Department of Mathematics University of California Berkeley, CA 94720 U.S.A Library of Congress Cataloging in Publication Data Diestel, Joseph, 1943Sequences and series in Banach spaces (Graduate texts in mathematics; 92) Includes bibliographies and index Banach spaces Sequences (Mathematics) Series I Title II Series QA322.2.D53 1984 515.7'32 83-6795 C> 1984 by Springer-Verlag New York, Inc Softcover reprint of the hardcover 1st edition 1984 All rights reserved No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, U.S.A Typeset by Science Typographers, Medford, New York For distribution and sale in the People' s Republic of ChUla unly fUll tE ~ A tUHll1jl :t IT ISBN-13: 978-1-4612-9734-5 DOl: 10.1007/978-1-4612-5200-9 e-ISBN-13: 978-1-4612-5200-9 Preface This volume presents answers to some natural questions of a general analytic character that arise in the theory of Banach spaces I believe that altogether too many of the results presented herein are unknown to the active abstract analysts, and this is not as it should be Banach space theory has much to offer the practitioners of analysis; unfortunately, some of the general principles that motivate the theory and make accessible many of its stunning achievements are couched in the technical jargon of the area, thereby making it unapproachable to one unwilling to spend considerable time and effort in deciphering the jargon With this in mind, I have concentrated on presenting what I believe are basic phenomena in Banach spaces that any analyst can appreciate, enjoy, and perhaps even use The topics covered have at least one serious omission: the beautiful and powerful theory of type and cotype To be quite frank, I could not say what I wanted to say about this subject without increasing the length of the text by at least 75 percent Even then, the words would not have done as much good as the advice to seek out the rich Seminaire Maurey-Schwartz lecture notes, wherein the theory's development can be traced from its conception Again, the treasured volumes of Lindenstrauss and Tzafriri also present much of the theory of type and cotype and are must reading for those really interested in Banach space theory Notation is standard; the style is informal Naturally, the editors have cleaned up my act considerably, and I wish to express my thanks for their efforts in my behalf I wish to express particular gratitude to the staff of Springer-Verlag, whose encouragement and aid were so instrumental in bringing this volume to fruition Of course, there are many mathematicians who have played a role in shaping my ideas and prejudices about this subject matter All that appears here has been the subject of seminars at many universities; at each I have received considerable feedback, all of which is reflected in this volume, be it in the obvious fashion of an improved proof or the intangible softening of a viewpoint Particular gratitude goes to my colleagues at Kent State University and at University College, Dublin, viii Preface who have listened so patiently to sermons on the topics of this volume Special among these are Richard Aron, Tom Barton, Phil Boland, Jeff Connor, Joe Creekmore, Sean Dineen, Paddy Dowlong, Maurice Kennedy, Mark Lit:cting co's presence · the Elton-Odell proof that each infinite dimensional Banach space contains a (L + E)-separated sequence of norm-one elements · exercIses notes and remarks bIbliography 248 XIV The Elton-Odell (l + f)-Separation Theorem if Ilbll = and b is of the form I b = tJa "i J (-1) , +.1 x m, = (J ( /oJ x ml - x m2 + + x m, ) ' where ml < m < < m,.la//1-11