Series in Banach Spaces: Conditional and Unconditional Convergence(94 Operator Theory: Advances and Applications)

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.

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