Banach Spaces and Descriptive Set Theory: Selected Topics, Issue 1993

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Springer Science & Business Media, May 10, 2010 - Mathematics - 161 pages
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.
 

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Contents

Selected Topics 1 Basic Concepts
1
Selected Topics 2 The Standard Borel Space of All Separable Banach Spaces
9
Selected Topics 3 The 2 Baire Sum
37
Selected Topics 4 Amalgamated Spaces
57
Selected Topics 5 Zippins Embedding Theorem
71
Selected Topics 6 The BourgainPisier Construction
89
Selected Topics 7 Strongly Bounded Classes of Banach Spaces
115
Selected Topics A Rank Theory
127
Selected Topics B Banach Space Theory
137
Selected Topics C The KuratowskiTarski Algorithm
149
Selected Topics D Open Problems
151
Selected Topics References
155
Selected Topics Index
159
Selected Topics Series
163
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