Proximinal subspaces of \(C(\mathcal Q)\) of finite codimension
type: Item
instance of: scholarly article
title: Proximinal subspaces of \(C(\mathcal Q)\) of finite codimension
zbMATH Open document ID: 0965.41012
published in: Journal of Approximation Theory
publication date: 2000-03-02T00:00:00Z
review text: The structure of a proximinal subspace \(G\) of \(C(\mathcal Q)\) of codimension \(n\), \(\mathcal Q\) being a compact Hausdorff space, is studied in terms of the geometry of the range of the vector measure \(\nu=(\nu_1, \dots, \nu_n)\), where \(\{\nu_1, \dots, \nu_n\}\) is a basis for the annihilator \(G^\perp\). In particular, if \(\nu\) is non-atomic, \(G\) is proximinal if and only if for every \(P\in \text{Ext}\,R(v)\) there exists a clopen subset \(C\) of \(\bigcup_{i=1}^n S(\nu_i)\) such that \(\nu(C)=P\). It is shown that the annihilator of a proximinal finitely complemented subspace of \(C(\mathcal Q)\) is a finite dimensional subspace of \(C(\mathcal Q)\). A set \(G\subset C(\mathcal Q)\) of codimension 2 is proximinal if and only if every extreme point of the range of the measure \((\nu_1,\nu_2)\) is the image under \((\nu_1,\nu_2)\) of a clopen subset of \(S(\nu_1)\cup S(\nu_2)\).
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 46E99
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 1389111
zbMATH Keywords: proximinal space
zbMATH Keywords: vector measure
zbMATH Keywords: clopen set
author: Francesca Centrone
author: Anna Martellotti
reviewed by: Thomas V. Tonev
MaRDI profile type: Publication
full work available at URL: jath.1999.3361
OpenAlex ID: W2027366718
cites work: Q3879482
cites work: Geometric properties of the range of two-dimensional quasi-measures with respect to the Radon-Nikodým property
cites work: Best approximation in \(L^ p(I,X)\), \(0<p<1\)
cites work: Q3802019
cites work: Q5674504
cites work: Q3216840
cites work: Proximinal Subspaces of Finite Codimesion in General Normed Linear Spaces
cites work: Best approximation in \(L^ p(\mu,X)\). II
cites work: Best Approximation in L 1 (I, X)
cites work: Some best-approximation theorems in tensor-product spaces
cites work: Q3791393
cites work: Q4317513
cites work: Cebysev subspaces of finite codimension in \(C(X)\)
cites work: Q5595759
cites work: Q5734734
cites work: Q3905894
cites work: Subspaces of finite codimension: The existence of elements of best approximation
cites work: The existence of elements of best approximation in the complex space C(Q)
DOI: 10.1006/JATH.1999.3361
description: scientific article; zbMATH DE number 1389111