Ebook: Series in Banach Spaces: Conditional and Unconditional Convergence

Series of scalars, vectors, or functions are among the fundamental objects of mathematical analysis. When the arrangement of the terms is fixed, investigating a series amounts to investigating the sequence of its partial sums. In this case the theory of series is a part of the theory of sequences, which deals with their convergence, asymptotic behavior, etc. The specific character of the theory of series manifests itself when one considers rearrangements (permutations) of the terms of a series, which brings combinatorial considerations into the problems studied. The phenomenon that a numerical series can change its sum when the order of its terms is changed is one of the most impressive facts encountered in a university analysis course. The present book is devoted precisely to this aspect of the theory of series whose terms are elements of Banach (as well as other topological linear) spaces. The exposition focuses on two complementary problems. The first is to char­ acterize those series in a given space that remain convergent (and have the same sum) for any rearrangement of their terms; such series are usually called uncon­ ditionally convergent. The second problem is, when a series converges only for certain rearrangements of its terms (in other words, converges conditionally), to describe its sum range, i.e., the set of sums of all its convergent rearrangements.

---

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.

---

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.
Content:
Front Matter....Pages i-viii
Notations....Pages 1-3
Background Material....Pages 5-12
Series in a Finite-Dimensional Space....Pages 13-27
Conditional Convergence in an Infinite-Dimensional Space....Pages 29-43
Unconditionally Convergent Series....Pages 45-57
Orlicz’s Theorem and The Structure of Finite-Dimensional Subspaces....Pages 59-70
Some Results from the General Theory of Banach Spaces....Pages 71-86
Steinitz’s Theorem and B-Convexity....Pages 87-100
Rearrangements of Series in Topological Vector Spaces....Pages 101-117
Back Matter....Pages 119-159

---

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.
Content:
Front Matter....Pages i-viii
Notations....Pages 1-3
Background Material....Pages 5-12
Series in a Finite-Dimensional Space....Pages 13-27
Conditional Convergence in an Infinite-Dimensional Space....Pages 29-43
Unconditionally Convergent Series....Pages 45-57
Orlicz’s Theorem and The Structure of Finite-Dimensional Subspaces....Pages 59-70
Some Results from the General Theory of Banach Spaces....Pages 71-86
Steinitz’s Theorem and B-Convexity....Pages 87-100
Rearrangements of Series in Topological Vector Spaces....Pages 101-117
Back Matter....Pages 119-159
....