Author	Kadets, Mikhail I. author
Title	Series in Banach Spaces [electronic resource] : Conditional and Unconditional Convergence / by Mikhail I. Kadets, Vladimir M. Kadets
Imprint	Basel : Birkhรคuser Basel, 1997
Connect to	http://dx.doi.org/10.1007/978-3-0348-9196-7
Descript	VIII, 159 p. online resource

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 charยญ 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 unconยญ 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

Notations -- 1. Background Material -- ยง1. Numerical Series. Riemannโ{128}{153}s Theorem -- ยง2. Main Definitions. Elementary Properties of Vector Series -- ยง3. Preliminary Material on Rearrangements of Series of Elements of a Banach Space -- 2. Series in a Finite-Dimensional Space -- ยง1. Steinitzโ{128}{153}s Theorem on the Sum Range of a Series -- ยง2. The Dvoretzky-Hanani Theorem on Perfectly Divergent Series -- ยง3. Pecherskiiโ{128}{153}s Theorem -- 3. Conditional Convergence in an Infinite-Dimensional Space -- ยง1. Basic Counterexamples -- ยง2. A Series Whose Sum Range Consists of Two Points -- ยง3. Chobanyanโ{128}{153}s Theorem -- ยง4. The Khinchin Inequalities and the Theorem of M. I. Kadets on Conditionally Convergent Series in Lp -- 4. Unconditionally Convergent Series -- ยง1. The Dvoretzky-Rogers Theorem -- ยง2. Orliczโ{128}{153}s Theorem on Unconditionally Convergent Series in LpSpaces -- ยง3. Absolutely Summing Operators. Grothendieckโ{128}{153}s Theorem -- 5. Orliczโ{128}{153}s Theorem and the Structure of Finite-Dimensional Subspaces -- ยง1. Finite Representability -- ยง2. The space c0, C-Convexity, and Orliczโ{128}{153}s Theorem -- ยง3. Survey on Results on Type and Cotype -- 6. Some Results from the General Theory of Banach Spaces -- ยง1. Frรฉchet Differentiability of Convex Functions -- ยง2. Dvoretzkyโ{128}{153}s Theorem -- ยง3. Basic Sequences -- ยง4. Some Applications to Conditionally Convergent Series -- 7. Steinitzโ{128}{153}s Theorem and B-Convexity -- ยง1. Conditionally Convergent Series in Spaces with Infratype -- ยง2. A Technique for Transferring Examples with Nonlinear Sum Range to Arbitrary Infinite-Dimensional Banach Spaces -- ยง3. Series in Spaces That Are Not B-Convex -- 8. Rearrangements of Series in Topological Vector Spaces -- ยง1. Weak and Strong Sum Range -- ยง2. Rearrangements of Series of Functions -- ยง3. Banaszczykโ{128}{153}s Theorem on Series in Metrizable Nuclear Spaces -- Appendix. The Limit Set of the Riemann Integral Sums of a Vector-Valued Function -- ยง2. The Example of Nakamura and Amemiya -- ยง4. Connection with the Weak Topology -- Comments to the Exercises -- References