Banach spaces not antiproximinal in their second dual
type: Item
instance of: scholarly article
title: Banach spaces not antiproximinal in their second dual
zbMATH Open document ID: 0731.41035
DOI: 10.1016/0021-9045(91)90069-M
author: Mu-San Wang
author: Jian-Hua Wang
published in: Journal of Approximation Theory
publication date: 1991-01-01T00:00:00Z
review text: A subspace M of a Banach space X is said to be antiproximinal in X if the only vectors in X which have closest proximinants from M are elements of M. In [\textit{K. R. Davidson}, J. Approximation Theory 47, 203-213 (1986; Zbl 0595.41034)], the author claims that if a Banach space X has the projection approximation property, then X has an equivalent norm \(| \cdot |\) under which it is antiproximinal in \(X^{**}\). The authors point out a gap in Davidson's argument and show that, in fact, \((\ell_ 1,| \cdot |)\) is not antiproximinal in its second dual.
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 4208757
zbMATH Keywords: proximal subspaces
zbMATH Keywords: projection approximation
zbMATH Keywords: antiproximinal
reviewed by: Richard M. Aron
MaRDI profile type: Publication
cites work: Banach spaces antiproximinal in their biduals
cites work: Q4104377
cites work: Q3966611
full work available at URL: 0021-9045(91)90069-m
OpenAlex ID: W2024151100
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description: scientific article; zbMATH DE number 4208757