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Series in Banach Spaces : Conditional and Unconditional Convergence - Mikhail I. Kadets

Series in Banach Spaces

Conditional and Unconditional Convergence

Hardcover Published: 20th March 1997
ISBN: 9783764354015
Number Of Pages: 159

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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.

Introduction
Notations
Background Material
Series in a Finite-Dimensional Space
Conditional Convergence in an Infinite-Dimensional Space
Unconditionally Convergent Series
Orlicz's Theorem and the Structure of Finite-Dimensional Subspaces
Some Results from the General Theory of Banach Spaces
Steinitz's Theorem and B-Convexity
Rearrangements of Series in Topological Vector Spaces
The Limit Set of the Riemann Integral Sums of a Vector-Valued Function
Comments to the Exercises
References
Index
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783764354015
ISBN-10: 3764354011
Series: Operator Theory: Advances and Applications
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 159
Published: 20th March 1997
Publisher: Birkhauser Verlag AG
Country of Publication: CH
Dimensions (cm): 23.4 x 15.6  x 1.27
Weight (kg): 0.93