In a contemporary course in mathematical analysis, the concept of series arises as a natural generalization of the concept of a sum over finitely many elements, and the simplest properties of finite sums carry over to infinite series. Standing as an exception among these properties is the commutative law, for the sum of a series can change as a result of a rearrangement of its terms. This raises two central questions: for which series is the commutative law valid, and just how can a series change upon rearrangement of its terms? Both questions have been answered for all finite-dimensional spaces, but the study of rearrangements of a series in an infinite-dimensional space continues to this day. In recent years, a close connection has been discovered between the theory of series and the so-called finite properties of Banach spaces, making it possible to create a unified theory from the numerous separate results. This book is the first attempt at such a unified exposition. This book would be an ideal textbook for advanced courses, for it requires background only at the level of standard courses in mathematical analysis and linear algebra and some familiarity with elementary concepts and results in the theory of Banach spaces. The authors present the more advanced results with full proofs, and they have included a large number of exercises of varying difficulty. A separate section in the last chapter is devoted to a detailed survey of open questions. The book should prove useful and interesting both to beginning mathematicians and to specialists in functional analysis.
Author(s): V. M. Kadets and M. I. Kadets
Series: Translations of Mathematical Monographs, Vol. 86
Publisher: American Mathematical Society
Year: 1991
Language: English
Pages: C+iv+123+B
Cover
Rearrangements of Series in Banach Spaces
Copyright ©1991 by the American Mathematical Society
ISBN 0-8218-4546-2
QA295.K2813 1991 512'.55-dc20
LCCN 91-6522 CIP
Contents
Introduction
CHAPTER 1 General Information
§1.1. Numerical series. The Riemann theorem
§1.2. Basic definitions. Elementary properties of vector series
§ 1.3. Preliminary information about rearrangements of series of elements of a Banach space
CHAPTER 2 Conditionally Convergent Series
§2.1. The domain of sums of a series in a finite-dimensional normed space
§2.2. Conditional convergence in an infinite-dimensional space. General results
§2.3. Conditionally convergent series in the spaces Lp
CHAPTER 3 Unconditionally Convergent Series
§3.1. The Dvoretzky-Rogers theorem
§3.2. The Orlicz theorem on unconditionally convergent series in the spaces Lp
§3.3. Absolutely summing operators
CHAPTER 4 Some Results in the General Theoryof Banach Spaces
§4.1. Finite representability
§4.2. Frechet differentiability of convex functions
§4.3. The Dvoretzky theorem
§4.4. Basic sequences
§4.5. Some applications to conditionally convergent series
CHAPTER 5 M-cotype and the Orlicz Theorem
§5.1. Unconditionally convergent series in the space Co
§5.2. C-convexity and the Orlicz theorem
§5.3. Summary of results on type and cotype
CHAPTER 6 The Steinitz Theorem and B-Convexity
§6.1. Conditionally convergent series in spaces with an infratype
§6.2. Series in spaces that are not B-convex
§6.3. The Chobanyan inequality
§6.4. Survey of unsolved problems in the theory of series
Comments on the Exercises
Chapter 1
Chapter 2
Chapter 3, Chapter 4
Chapter 5
Chapter 6
Bibliography
Subject Index
Back Cover
