Chebyshev subspaces of spaces of abstract affine functions
type: Item
instance of: scholarly article
title: Chebyshev subspaces of spaces of abstract affine functions
zbMATH Open document ID: 0562.46023
DOI: 10.1007/BF01139555
author: G. M. Ustinov
published in: Mathematical Notes
publication date: 1984-01-01T00:00:00Z
review text: The author investigates finite dimensional subspaces of Chebyshev rank s, \(s>0\), in the space of affine continuous functions defined on a compact Choquet simplex and taking values in an arbitrary Banach space. The characterization of those subspaces, obtained in the paper can be viewed as generalization of the results by \textit{S. I. Zukhovitskij} and \textit{S. B. Stechkin}, Dokl. Akad. Nauk SSSR, 106, 773-776 (1956; Zbl 0075.117), which characterize finite-dimensional Chebyshev subspaces \((s=0)\) in the space of continuous functions from a compact Hausdorff space to a finite- dimensional, strictly convex Banach space. As an application of the results in the paper, the author obtains a proof of the known criterion due to \textit{G. Sh. Rubinshtejn}, saying that a subspace \(E\subset C(Q)\), dim E\(=N<\infty\) has Chebyshev rank \(0\leq S\leq N-1\) if and only if any \((S+1)\) linearly independent functions in E have at most N-S-1 common zeros. [Dokl. Akad. Nauk SSSR, 102, 451-454 (1955; Zbl 0070.109)]. (Here, Q is a compact Hausdorff space and C(Q) is a Banach space of continuous real-valued functions on Q with the usual norm.)
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 46A55
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 43A17
zbMATH DE Number: 3895669
zbMATH Keywords: Chebyshev rank s
zbMATH Keywords: space of affine continuous functions defined on a compact Choquet simplex and taking values in an arbitrary Banach space
zbMATH Keywords: finite-dimensional Chebyshev subspaces
reviewed by: Dmitry Khavinson
MaRDI profile type: Publication
cites work: Spaces of Affine Continuous Functions on Simplexes
cites work: Q3237473
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description: scientific article; zbMATH DE number 3895669