The beautiful Riemann theorem states that a series can change its sum after permutation of the terms. Many brilliant mathematicians, among them P. Levy, E. Steinitz and J. Marcinkiewicz considered such effects for series in various spaces. In 1988, the authors published the book Rearrangements of Series in Banach Spaces. Interest in the subject has surged since then. In the past few years many of the problems described in that book - problems which had challenged mathematicians for decades - have in the meantime been solved. This changed the whole picture significantly. In the present book, the contemporary situation from the classical theorems up to new fundamental results, including those found by the authors, is presented. Complete proofs are given for all non-standard facts. The text contains many exercises and unsolved problems as well as an appendix about the similar problems in vector-valued Riemann integration. The book will be of use to graduate students and mathe- maticians interested in functional analysis.
Author(s): M. I. Kadets
Publisher: Birkhauser
Year: 1997
Language: English
Pages: 168
Cover page......Page 1
Title page......Page 3
Contents......Page 5
Introduction......Page 7
Notations......Page 9
§1. Numerical Series. Hiemann's Theorem......Page 13
§2. Main Definitions. Elementary Propcrties of Vector Series......Page 15
§3. Preliminary Material on Rearrangements of Series of Elements of a Banach Space......Page 17
§1. Steinitz's Theorem on the Sum Range of a Series......Page 21
§2. The Dvoretzky-Hanani Theorem on Perfectly Divergent Series......Page 29
§3. Pecherskii's Theorem......Page 31
§1. Basic Counterexamples......Page 37
§2. A Series Whose Sum Range Consists of Two Points......Page 40
§3. Chobanyan's Theorem......Page 44
§4. The Khinchin Inequalities and the Theorem of M. I. Kadcts on Conditionally Convergent Series in L_p......Page 47
§1. The Dvorctzky-Rogers Theorem......Page 53
§2. Orlicz's Theorem on Unconditionally Convergent Series in L_p Spaces......Page 57
§3. Absolutely Summing Operators. Grothendieck's Theorem......Page 60
§1. Finite Representability......Page 67
§2. The space c_0, C-Convcxity, and Orlicz's Theorem......Page 70
§3. Survey on Results on Type and Cotype......Page 75
§1. Fréchet Differentiability of Convex Functions......Page 79
§2. Dvorctzky's Theorem......Page 81
§3. Basic Sequences......Page 87
§4. Some Applications to Conditionally Convergent Series......Page 90
§1. Conditionally Convergent Series in Spaces with Infratype......Page 95
§2. A Technique for Transferring Examples with Nonlinear Sum Range to Arbitrary Infinite-Dimensional Banach Spaces......Page 101
§3. Series in Spaces That Are Not B-Convex......Page 105
§1. Weak and Strong Sum Range......Page 109
§2. Rearrangements of Series of Functions......Page 114
§3. Banaszczyk's Theorem on Series in Metrizable Nuclear Spaces......Page 118
Appendix. The Limit Set of the Riemann Integral Sums of a Vector-Valued Function......Page 127
§1. Functions Vailled in a B-Convex Space......Page 128
§2. The Example of Nakamura and Amemiya......Page 130
§3. Separability of the Space and the Structure of I(f)......Page 135
§4. Connection with tile Weak Topology......Page 139
Comments to the Exercises......Page 149
References......Page 157
Index......Page 163
