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

Published:1999-02-01
Printed: Feb 1999
• Marián Fabian
• Boris S. Mordukhovich
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## Abstract

We develop a method of separable reduction for Fr\'{e}chet-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 Fr\'{e}chet 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: nonsmooth analysis, Banach spaces, separable reduction, Fréchet-like normals and subdifferentials, supporting properties, Asplund spaces
 MSC Classifications: 49J52 - Nonsmooth analysis [See also 46G05, 58C50, 90C56] 58C20 - Differentiation theory (Gateaux, Frechet, etc.) [See also 26Exx, 46G05] 46B20 - Geometry and structure of normed linear spaces