Banach spaces antiproximinal in their biduals
type: Item
instance of: scholarly article
title: Banach spaces antiproximinal in their biduals
zbMATH Open document ID: 0595.41034
DOI: 10.1016/0021-9045(86)90028-6
author: Kenneth R. Davidson
published in: Journal of Approximation Theory
publication date: 1986-01-01T00:00:00Z
review text: A subspace X of a Banach space Y is proximinal (antiproximinal) if every vector \(y\in Y\setminus X\) has a (has no) closest approximant in X. Most classical Banach spaces are proximinal in their second duals \(X^{**}\). It is shown that if X has a Schauder basis, then X can be renormed so that it is anti-proximinal in \(X^{**}\). Also, X can be renormed so that the compact operators \({\mathcal K}(X)\) are not proximinal in \({\mathcal B}({\mathcal H})\). (It cannot be antiproximinal.)
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 3957889
zbMATH Keywords: proximinal
zbMATH Keywords: antiproximinal
zbMATH Keywords: Banach spaces
zbMATH Keywords: Schauder basis
MaRDI profile type: Publication
full work available at URL: 0021-9045(86)90028-6
OpenAlex ID: W2010616164
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description: scientific article; zbMATH DE number 3957889