Proximinality in \(L_p(\mu,X)\)
type: Item
instance of: scholarly article
title: Proximinality in \(L_p(\mu,X)\)
zbMATH Open document ID: 0914.41020
author: José Mendoza
published in: Journal of Approximation Theory
publication date: 1998-10-27T00:00:00Z
review text: Let \( X \) be a Banach space, let \( p\in [1,\infty] \) and let \( L_p(\mu,X) \) be the Banach space of \(X\)-valued Bochner \(p\)-integrable functions defined on a positive complete \(\sigma\)-finite measure space \( (\Omega,\Sigma,\mu)\). The author gives an example of a pair of Banach spaces \( (X,Y), Y\subseteq X\), with \( Y \) proximinal in \( X \) such that \( L_p(\mu,Y) \) is not proximinal in \( L_p(\mu,X)\). The pair \( (X,Y) \) was also used in other context by \textit{R. Holmes} and \textit{B. Kripke} [Mich. Math. J. 15, 225-248 (1968; Zbl 0177.16201)]. On the other hand the author shows that for separable proximinal subspaces \( Y \) of \( X, L_p(\mu,Y) \) is proximinal in \( L_p(\mu,X)\). A number of known results are derived from this last proposition.
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 1197085
zbMATH Keywords: best approximation in \(L_p(\mu
zbMATH Keywords: X)\) spaces
zbMATH Keywords: proximinal subspaces
reviewed by: Ioan Şerb
MaRDI profile type: Publication
full work available at URL: jath.1997.3163
OpenAlex ID: W2199161675
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cites work: Q4148368
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DOI: 10.1006/JATH.1997.3163
description: scientific article; zbMATH DE number 1197085