Descriptive Topology in Selected Topics of Functional Analysis

ISBN-13: 9781461405283 / Angielski / Twarda / 2011 / 496 str.

Jerzy K. Kol; Wies Aw Kubi; Manuel L. Pez-Pellicer

Opis

"Descriptive Topology in Selected Topics of Functional Analysis" is a collection of recent developments in the field of descriptive topology, specifically focused on the classes of infinite-dimensional topological vector spaces that appear in functional analysis. Such spaces include Frechet spaces, (LF)-spaces and their duals, and the space of continuous real-valued functions C(X) on a completely regular Hausdorff space X, to name a few. These vector spaces appear in functional analysis in distribution theory, differential equations, complex analysis, and various other analytical settings. This monograph provides new insights into the connections between the topological properties of linear function spaces and their applications in functional analysis.

Recenzje

From the reviews:

"The book summarizes many results which describe various aspects of the structure of topological vector spaces ... . Most of the chosen material is relatively recent and appears in book form for the first time. ... Together with a rich list of references the book can help in finding and understanding many finer questions concerning various aspects of the structure of subclasses of the class of topological vector spaces." (Petr Holický, Zentralblatt MATH, Vol. 1231, 2012)

"The authors collect a large amount of material displaying connections between general topology and functional analysis with regard to properties of infinite-dimensional topological vector spaces. ... It is a fine piece of scholarship with well-organized, readable proofs. ... It is an encyclopedia of a subject meant for researchers and advanced graduate students likely interested in research for an advanced degree, and/or adding more knowledge to a course in which they are enrolled." (E. Beckenstein, Mathematical Reviews, February, 2013)

Spis treści

Preface.- 1. Overview.- 2. Elementary Facts about Baire and Baire-Type Spaces.- 3. K-analytic and quasi-Suslin Spaces.- 4. Web-Compact Spaces and Angelic Theorems.- 5. Strongly Web-Compact Spaces and a Closed Graph Theorem.- 6. Weakly Analytic Spaces.- 7. K-analytic Baire Spaces.- 8. A Three-Space Property for Analytic Spaces.- 9. K-analytic and Analytic Spaces C_p(X).- 10. Precompact sets in (LM)-Spaces and Dual Metric Spaces.- 11. Metrizability of Compact Sets in the Class G.- 12. Weakly Realcompact Locally Convex Spaces.- 13. Corson’s Propery (C) and tightness.- 14. Fréchet-Urysohn Spaces and Groups.- 15. Sequential Properties in the Class G.- 16. Tightness and Distinguished Fréchet Spaces.- 17. Banach Spaces with Many Projections.- 18. Spaces of Continuous Functions Over Compact Lines.- 19. Compact Spaces Generated by Retractions.- 20. Complementably Universival Banach Space.- Index.

Blurb

A large mathematical community throughout the world actively works in functional analysis and uses profound techniques from topology. As the first monograph to approach the topic of topological vector spaces from the perspective of descriptive topology, this work provides also new insights into the connections between the topological properties of linear function spaces and their role in functional analysis. 

Descriptive Topology in Selected Topics of Functional Analysis is a self-contained volume that applies recent developments and classical results in descriptive topology to study the classes of infinite-dimensional topological vector spaces that appear in functional analysis. Such spaces include Fréchet spaces, LF-spaces and their duals, and the space of continuous real-valued functions C(X) on a completely regular Hausdorff space X, to name a few. These vector spaces appear in distribution theory, differential equations, complex analysis, and various other areas of functional analysis.

Written by three experts in the field, this book is a treasure trove for researchers and graduate students studying the interplay among the areas of point-set and descriptive topology, modern analysis, set theory, topological vector spaces and Banach spaces, and continuous function spaces.  Moreover, it will serve as a reference for present and future work done in this area and could serve as a valuable supplement to advanced graduate courses in functional analysis, set-theoretic topology, or the theory of function spaces.