Banach Spaces and Descriptive Set Theory: Selected Topics

ISBN-13: 9783642121524 / Angielski / Miękka / 2010 / 168 str.

Pandelis Dodos

Opis

These notes are devoted to the study of some classical problems in the Geometry of Banach spaces. The novelty lies in the fact that their solution relies heavily on techniques coming from Descriptive Set Theory. Thecentralthemeisuniversalityproblems.Inparticular, thetextprovides an exposition of the methods developed recently in order to treat questions of the following type: (Q) LetC be a class of separable Banach spaces such that every space X in the classC has a certain property, say property (P). When can we ?nd a separable Banach space Y which has property (P) and contains an isomorphic copy of every member ofC? We will consider quite classical properties of Banach spaces, such as "- ing re?exive," "having separable dual," "not containing an isomorphic copy of c," "being non-universal," etc. 0 It turns out that a positive answer to problem (Q), for any of the above mentioned properties, is possible if (and essentially only if) the classC is "simple." The "simplicity" ofC is measured in set theoretic terms. Precisely, if the classC is analytic in a natural "coding" of separable Banach spaces, then we can indeed ?nd a separable space Y which is universal for the class C and satis?es the requirements imposed above.

Recenzje

From the reviews:

"The book under review ... is mainly focused on the author's work on characterizations of the existence of universal elements in subclasses of separable Banach spaces ... . The book collects the most important of these results into a self-contained framework that clarifies the ideas under the successive improvements between each paper and the following ones. All this makes the book a mandatory reference for anyone interested in universality in Banach spaces." (Matias Raja, Mathematical Reviews, Issue 2011 j)

"The author uses descriptive set theory to prove results on the structure of Banach spaces. ... this book may be useful for people interested in Banach space theory or/and descriptive set theory. It is very well written and contains a lot of results and techniques from these two theories, and thus may serve as a reference book." (Daniel Li, Zentralblatt MATH, Vol. 1215, 2011)

Spis treści

Basic Concepts.- The Standard Borel Space of All Separable Banach Spaces.- The ?2 Baire Sum.- Amalgamated Spaces.- Zippin’s Embedding Theorem.- The Bourgain–Pisier Construction.- Strongly Bounded Classes of Banach Spaces.

Blurb

This volume deals with problems in the structure theory of separable infinite-dimensional Banach spaces, with a central focus on universality problems. This topic goes back to the beginnings of the field and appears in Banach's classical monograph. The novelty of the approach lies in the fact that the answers to a number of basic questions are based on techniques from Descriptive Set Theory. Although the book is oriented on proofs of several structural theorems, in the main text readers will also find a detailed exposition of numerous “intermediate” results which are interesting in their own right and have proven to be useful in other areas of Functional Analysis. Moreover, several well-known results in the geometry of Banach spaces are presented from a modern perspective.