Banach Spaces and Descriptive Set Theory: Selected Topics

Pandelis Dodos

ISBN 10: 3642121527 / ISBN 13: 9783642121524

Published by Berlin Springer Berlin Heidelberg Springer Mai 2010, 2010

About this Item

This item is printed on demand - it takes 3-4 days longer - Neuware -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. 168 pp. Englisch.

Seller Inventory # 9783642121524

Bibliographic Details

• Title: Banach Spaces and Descriptive Set Theory: Selected Topics
• Publisher: Berlin Springer Berlin Heidelberg Springer Mai 2010
• Publication Date: 2010
• Language: English
• ISBN 10: 3642121527
• ISBN 13: 9783642121524
• Binding: Taschenbuch
• Condition: Neu
• Dimensions: 235W x 155H x 10D millimeters 235 millimeters width by 155 millimeters height by 10 millimeters depth
• Weight: 286 grams

About this title

Synopsis

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.

From the Back Cover

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.