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

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159 | |

Selected Topics Series | 163 |

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analytic subset assume B-tree Banach Space Theory basic sequence bijection block subspace Borel map Borel subset bounded linear operator Bourgain canonical triple claim is proved containing l1 contains an isomorphic contradiction convex Corollary countable ordinal define deﬁned Deﬁnition denote dense Descriptive Set Theory exists fact ﬁnite finite-dimensional ﬁrst following are satisfied Hence Il}-rank isometric embedding isomorphic copy isomorphic embedding Kechris Lemma limsup map f metrizable space Moreover non-empty norm normalized Schauder basis Notice open subset pairwise incomparable parameterized Polish space proof is completed Proof of Claim proof of Theorem Proposition Ramsey's Theorem recursively REFL result satisﬁed Schauder basis Schauder tree basis Sect segment separable Banach space separable dual sequence yn shrinking Schauder basis standard Borel space strongly bounded transfinite induction vector weakly null weakly X-singular X G SB X-singular subspace xt)teT Y G A