Series in Banach Spaces
Conditional and Unconditional Convergence
Published on 20. March 1997
Book
Hardback
VIII, 159 pages
978-3-7643-5401-5 (ISBN)
Description
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.
Reviews / Votes
"The material in the book is always started with motivation. There are plenty of exercises, with hints for solution. The list of references is complete and up-to-day. This is an amazing book on doctoral level, written by experienced authors, who contributed a lot to the subject presented in the book." (J.Musielak, zbMATH, 0876.46009, 1997)
More details
Series
Edition
1997
Language
English
Place of publication
Basel
Switzerland
Publishing group
Springer Basel
Target group
Professional and scholarly
Research
Illustrations
VIII, 159 p.
Dimensions
Height: 241 mm
Width: 160 mm
Thickness: 15 mm
Weight
430 gr
ISBN-13
978-3-7643-5401-5 (9783764354015)
DOI
10.1007/978-3-0348-9196-7
Schweitzer Classification
Other editions
Additional editions
Book
11/2011
1st Edition
Birkhäuser
€106.99
Shipment within 10-15 days
Person
Vladimir Kadets has authored two monographs and more than 100 articles in peer-reviewed journals, mainly in Banach space theory: sequences and series, bases, vector-valued measures and integration, measurable multi-functions and selectors, isomorphic and isometric structures of Banach spaces, operator theory. In 2005 he received the State Award of Ukraine in Science and Technology to honour his research. The present book reflects the author's teaching experience in the field, spanning over more than 20 years.
Content
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.