A Nonlinear Transfer Technique for Renorming
Abstract topological tools from generalized metric spaces are applied in this volume to the construction of locally uniformly rotund norms on Banach spaces. The book offers new techniques for renorming problems, all of them based on a network analysis for
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Aníbal Moltó José Orihuela Stanimir Troyanski Manuel Valdivia
A Nonlinear Transfer Technique for Renorming
1951
123
Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris
1951
Aníbal Moltó · José Orihuela Stanimir Troyanski · Manuel Valdivia
A Nonlinear Transfer Technique for Renorming
ABC
Aníbal Moltó Manuel Valdivia
José Orihuela Stanimir Troyanski
Departamento de Análisis Matemático Facultad de Matemáticas Universidad de Valencia Dr. Moliner 50 46100 Burjasot, Valencia Spain [email protected] [email protected]
Departamento de Matemáticas Universidad de Murcia Campus Espinardo 30100 Murcia Spain http://webs.um.es/joseori [email protected] [email protected]
ISBN: 978-3-540-85030-4 e-ISBN: 978-3-540-85031-1 DOI: 10.1007/978-3-540-85031-1 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008932180 Mathematics Subject Classification (2000): 46B03, 46B20, 46B26, 46T20, 54D20, 54E18, 54E25, 54E35 c 2009 Springer-Verlag Berlin Heidelberg ° 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 Publishing Services Printed on acid-free paper 987654321 springer.com
Preface
Banach spaces are objects with a linear structure so linear maps have been considered the natural tool for transferring good norms from one Banach space to another. It is well known that a Banach space X admits an equivalent strictly convex (rotund) norm if there is a bounded linear one-to-one operator T : X → Y where Y has such a norm. For example, J. Lindenstrauss proved that in any reflexive space X there is such an operator T : X → c0 (Γ) for some set Γ. F. Dashiell and J. Lindenstrauss gave an example of a strictly convex renormable space without such an operator into c0 (Γ) for any Γ. For that reason we are searching for a non linear transfer technique. We consider here locally uniformly rotund (LUR) norms, a property adding to strict convexity the coincidence of the weak and the norm topologies on the unit sphere. For these norms a class of non linear maps was not only more powerful but even more natural for this purpose, as evinced by the solution of an old open problem due to Kadec using this class of non linear
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