Hardcover ISBN:  9780821845462 
Product Code:  MMONO/86 
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AMS Member Price:  $132.00 
eBook ISBN:  9781470444983 
Product Code:  MMONO/86.E 
List Price:  $155.00 
MAA Member Price:  $139.50 
AMS Member Price:  $124.00 
Hardcover ISBN:  9780821845462 
eBook: ISBN:  9781470444983 
Product Code:  MMONO/86.B 
List Price:  $320.00 $242.50 
MAA Member Price:  $288.00 $218.25 
AMS Member Price:  $256.00 $194.00 
Hardcover ISBN:  9780821845462 
Product Code:  MMONO/86 
List Price:  $165.00 
MAA Member Price:  $148.50 
AMS Member Price:  $132.00 
eBook ISBN:  9781470444983 
Product Code:  MMONO/86.E 
List Price:  $155.00 
MAA Member Price:  $139.50 
AMS Member Price:  $124.00 
Hardcover ISBN:  9780821845462 
eBook ISBN:  9781470444983 
Product Code:  MMONO/86.B 
List Price:  $320.00 $242.50 
MAA Member Price:  $288.00 $218.25 
AMS Member Price:  $256.00 $194.00 

Book DetailsTranslations of Mathematical MonographsVolume: 86; 1991; 123 ppMSC: Primary 46
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 finitedimensional spaces, but the study of rearrangements of a series in an infinitedimensional space continues to this day.
In recent years, a close connection has been discovered between the theory of series and the socalled 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

Chapters

Introduction

Chapter 1. General information

Chapter 2. Conditionally convergent series

Chapter 3. Unconditionally convergent series

Chapter 4. Some results in the general theory of Banach spaces

Chapter 5. $M$cotype and the Orlicz theorem

Chapter 6. The Steinitz theorem and $B$convexity


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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 finitedimensional spaces, but the study of rearrangements of a series in an infinitedimensional space continues to this day.
In recent years, a close connection has been discovered between the theory of series and the socalled 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.

Chapters

Introduction

Chapter 1. General information

Chapter 2. Conditionally convergent series

Chapter 3. Unconditionally convergent series

Chapter 4. Some results in the general theory of Banach spaces

Chapter 5. $M$cotype and the Orlicz theorem

Chapter 6. The Steinitz theorem and $B$convexity