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Separable Reduction and Supporting Properties of Fréchet-Like Normals in Banach Spaces

Published online by Cambridge University Press:  20 November 2018

Marián Fabian  and
Boris S. Mordukhovich
Marián Fabian
Affiliation:
Mathematical Institute, Czech Republic Academy of Sciences, Žitná 25, 115 67 Prague 1, Czech Republic
Boris S. Mordukhovich
Affiliation:
Department of Mathematics, Wayne State University, Detroit, MI 48202, USA
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Abstract

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We develop a method of separable reduction for Fréchet-like normals and $\epsilon$-normals to arbitrary sets in general Banach spaces. This method allows us to reduce certain problems involving such normals in nonseparable spaces to the separable case. It is particularly helpful in Asplund spaces where every separable subspace admits a Fréchet smooth renorm. As an applicaton of the separable reduction method in Asplund spaces, we provide a new direct proof of a nonconvex extension of the celebrated Bishop-Phelps density theorem. Moreover, in this way we establish new characterizations of Asplund spaces in terms of $\epsilon$-normals.

Keywords

49J5258C2046B20nonsmooth analysisBanach spacesseparable reductionFréchet-like normals and subdifferentialssupporting propertiesAsplund spaces
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 51 , Issue 1 , 01 February 1999 , pp. 26 - 48
Copyright
Copyright © Canadian Mathematical Society 1999

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