Banach Spaces and Descriptive Set Theory: Selected Topics
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 no
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1993
Pandelis Dodos
Banach Spaces and Descriptive Set Theory: Selected Topics
123
Pandelis Dodos Department of Mathematics University of Athens Panepistimiopolis 157 84 Athens Greece [email protected]
ISBN: 978-3-642-12152-4 e-ISBN: 978-3-642-12153-1 DOI: 10.1007/978-3-642-12153-1 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2010924170 Mathematics Subject Classification (2000): 46B03, 46B15, 46B07, 46B70, 46M40, 03E15, 03E75, 05D10 c Springer-Verlag Berlin Heidelberg 2010 ° This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publisher Services Printed on acid-free paper springer.com
Preface
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. The central theme is universality problems. In particular, the text provides an exposition of the methods developed recently in order to treat questions of the following type: (Q) Let C be a class of separable Banach spaces such that every space X in the class C has a certain property, say property (P). When can we find a separable Banach space Y which has property (P) and contains an isomorphic copy of every member of C? We will consider quite classical properties of Banach spaces, such as “being reflexive,” “having separable dual,” “not containing an isomorphic copy of c0 ,” “being non-universal,” etc. 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 class C is “simple.” The “simplicity” of C is measured in set theoretic terms. Precisely, if the class C is analytic in a natural “coding” of separable Banach spaces, then we can indeed find a separable space Y which is universal for the class C and satisfies the requirements imposed above. The text is addressed to both Functional Analysts and Set Theorists. We have tried to follow the terminology and notation employed by these two groups of researchers. Concerning Banach Space Theory, we follow the conventions adopted in the monograph of Linden
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