On the structure of Banach spaces with Mazur's intersection property
type: Item
instance of: scholarly article
title: On the structure of Banach spaces with Mazur's intersection property
zbMATH Open document ID: 0726.46008
DOI: 10.1007/BF01445220
published in: Mathematische Annalen
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164879
review text: We establish structural properties for Banach spaces where every bounded closed convex set is the intersection of closed balls and for those with dual space where every bounded \(weak^*\) closed convex set is the intersection of dual balls. It is shown that if a Banach space has both properties then the duality mapping is a homeomorphism between certain residual subsets of the unit spheres of the space and its dual, the points of these sets being simultaneously points of Fréchet differentiability of the norm and strongly exposed points of the unit ball. Among spaces with the second property is the special class where every point of the unit sphere is a denting point of the closed unit ball. For Banach spaces isomorphic to a space of this class, and this includes all the weakly compactly generated spaces, it is shown that every continuous convex function on an open convex subset of the dual is Fréchet differentiable on a dense \(G_{\delta}\) subset of its domain provided the set of points where the function has a weak\({}^*\) continuous subgradient is residual in its domain.
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46B10
zbMATH DE Number: 4198798
zbMATH Keywords: Banach spaces where every bounded closed convex set is the intersection of closed balls
zbMATH Keywords: dual space where every bounded \(weak^ *\) closed convex set is the intersection of dual balls
zbMATH Keywords: duality mapping
zbMATH Keywords: points of Fréchet differentiability of the norm
zbMATH Keywords: strongly exposed points of the unit ball
zbMATH Keywords: denting point
zbMATH Keywords: \(weak^ *\) continuous subgradient
author: Petar S. Kenderov
author: John Giles
reviewed by: Petar S. Kenderov
MaRDI profile type: Publication
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OpenAlex ID: W2059926230
description: scientific article; zbMATH DE number 4198798