**Translations of Mathematical Monographs**

1991;
123 pp;
Hardcover

MSC: Primary 46;

**Print ISBN: 978-0-8218-4546-2
Product Code: MMONO/86**

List Price: $66.00

AMS Member Price: $52.80

MAA Member Price: $59.40

**Electronic ISBN: 978-1-4704-4498-3
Product Code: MMONO/86.E**

List Price: $62.00

AMS Member Price: $49.60

MAA Member Price: $55.80

# Rearrangements of Series in Banach Spaces

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*V. M. Kadets; M. I. Kadets*

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.

#### Table of Contents

# Table of Contents

## Rearrangements of Series in Banach Spaces

- Cover Cover11
- Title page i2
- Contents iii4
- Introduction 16
- Chapter 1. General information 38
- Chapter 2. Conditionally convergent series 1318
- Chapter 3. Unconditionally convergent series 3338
- Chapter 4. Some results in the general theory of Banach spaces 4954
- Chapter 5. 𝑀-cotype and the Orlicz theorem 7378
- Chapter 6. The Steinitz theorem and 𝐵-convexity 8590
- Comments to the Exercises 113118
- Bibliography 119124
- Subject Index 123128
- Back Cover Back Cover1130