Three-space Problems in Banach Space Theory
This book on Banach space theory focuses on what have been called three-space problems. It contains a fairly complete description of ideas, methods, results and counterexamples. It can be considered self-contained, beyond a course in functional analysis a
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1667
Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo
Jesus M.P. Castillo Manuel Gonzalez
Three-space Problems in Banach Space Theory
Springer
Authors Jesus M. F. Castillo Departamento de Maternaticas Universidad de Extremadura Avda, de Elvas sIn E-06071 Badajoz, Spain E-mail: [email protected]
Manuel Gonzalez Departamento de Matematicas Universidad de Cantabria Avda. de los Castros sIn E-39071 Santander, Spain E-mail: [email protected]
LIbrary of Congress Cataloglng-ln-Publlcatlon Data
Castfll0. Jesus M. F. Three-space problems In Banach space theory I Jesus M.F. CastIllo. Manuel Gonzalez. p. em. -- (Lecture notes tn mathematIcs 1667) Includes blblfographtcal references (p. I and fndex. ISBN 3-540-63344-8 (softcover : alk. paper) 1. Banach spaces. 2. Ultraproducts. 3. Dual1ty theory (Mathematics) I. Gonzalez. Manuel. II. TItle. III. SerIes: Lecture notes fn mathematIcs (Sprfnger-Verlagl ; 1667. QA3.L28 no. 1667 [QA332.21 510 s--dc21 [515'.7321
Mathematics Subject Classification (1991): 46B03, 46B20, 46B08, 46B 10 ISSN 0075-8434 ISBN 3-540-63344-8 Springer-Verlag Berlin Heidelberg New York 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, re-use of illustrations, recitation, broadcasting, reproduction on microfilms 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- Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1997 Printed in Germany
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Foreword A three-space problem, in the Banach space setting, has the form: Let P be a Banach space property, let Y be a subspace of X and let XI Y be the corresponding quotient space. Is it true that X has property P when Y and XI Y have property P ? If the answer is positive then P is said to be a three-space property. We shall often shorten "three-space" to 3SP (see below). Three-space questions face the general problem of the structure of arbitrary subspaces and quotients of Banach spaces. Nowadays, it is clear that only Hilbert spaces can be labelled as simple: all subspaces are complemented Hilbert spaces and all quotients are Hilbert spaces. Other spaces may contain (and usually do) bizarre uncomplemented subspaces andlor quotients. The point of view of 3SP problems is to consider the structure of a Banach sp
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