On non-fragmentability of Banach spaces
type: Item
instance of: scholarly article
title: On non-fragmentability of Banach spaces
zbMATH Open document ID: 0937.46012
DOI: 10.1007/BF02841549
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 2000-05-23T00:00:00Z
review text: The paper is concerned with fragmentability in Banach spaces, a notion due to \textit{J. E. Jayne} and \textit{C. Rogers} [J. Funct. Anal. 56, 279-299 (1984; Zbl 0581.28007)], and which has many interesting connections with other topics in Banach space theory as locally uniformly convex and Kadec renormings, generic differentiability of convex functions, Radon-Nikodým property. \textit{R. Haydon} and \textit{V. Zizler} [Can. Math. Bull. 32, No. 1, 122-128 (1989; Zbl 0694.46011)], have constructed a closed linear subspace \(X\) of \(\ell^\infty\) which is a tree-complete sublattice (see loc. cit.) of the lattice \(\ell^\infty\), contains the space \(c_0\) and does not contain any isomorphic copy of \(\ell^\infty\). The aim of the present paper is to exhibit new properties of the space \(X\), namely, the space \((X/c_0,\) weak) is not fragmented by any metric and, as a consequence, \(X/c_0\) admits no equivalent strictly convex renorming.
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46B22
Mathematics Subject Classification ID: 46B03
zbMATH DE Number: 1334255
zbMATH Keywords: fragmentability in Banach spaces
zbMATH Keywords: Kadec renormings
zbMATH Keywords: generic differentiability of convex functions
zbMATH Keywords: Radon-Nikodým property
zbMATH Keywords: tree-complete sublattice
zbMATH Keywords: strictly convex renorming
author: Alireza Kamel Mirmostafaee
reviewed by: Stefan Cobzaş
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1334255