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Topological Properties of the Set of Norm-Attaining Linear Functionals

Published online by Cambridge University Press:  20 November 2018

Gabriel Debs ,
Gilles Godefroy  and
Jean Saint Raymond
Gabriel Debs
Affiliation:
Equipe d'Analyse Universite Paris VI Boîte 186 4, Place Jussieu 75252-Paris Cedex 05 France
Gilles Godefroy
Affiliation:
Equipe d'Analyse Universite Paris VI Boîte 186 4, Place Jussieu 75252-Paris Cedex 05 France
Jean Saint Raymond
Affiliation:
Equipe d'Analyse Universite Paris VI Boîte 186 4, Place Jussieu 75252-Paris Cedex 05 France
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Abstract

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If X is a separable non-reflexive Banach space, then the set NA of all norm-attaining elements of X* is not a w*-Gδ subset of X*. However if the norm of X is locally uniformly rotund, then the set of norm attaining elements of norm one is w*-Gδ. There exist separable spaces such that NA is a norm-Borel set of arbitrarily high class. If X is separable and non-reflexive, there exists an equivalent Gâteaux-smooth norm on X such that the set of all Gâteaux-derivatives is not norm-Borel.

Keywords

46B2004A15
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 2 , 01 April 1995 , pp. 318 - 329
Copyright
Copyright © Canadian Mathematical Society 1995

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