Finite-codimensional Chebyshev subspaces in the complex space \(C(Q)\)
type: Item
instance of: scholarly article
title: Finite-codimensional Chebyshev subspaces in the complex space \(C(Q)\)
zbMATH Open document ID: 0936.41019
DOI: 10.1007/BF02355903
author: L. P. Vlasov
published in: Mathematical Notes
publication date: 2000-05-14T00:00:00Z
review text: Let \(C_\mathbb{R}(Q)\) and \(C_\mathbb{C}(Q)\) be the real and complex spaces of continuous functions on a compact Hausdorff space \(Q\). The paper is concerned with the existence and characterisation of Chebyshev subspaces \(L\) of finite codimension in \(C_\mathbb{C}(Q)\). The paper continues the author's work in [Mat. Zametki 40, No. 5, 627-634 (1986; Zbl 0655.41031)] and establishes analogues of earlier results for the spaces \(C_\mathbb{R}(Q)\) due to \textit{A. L. Garkavi} [Mat. Sb., N. Ser. 74(116), No. 2, 209-217 (1967; Zbl 0186.44802)] and \textit{A. L. Brown} [Chebyshev subspaces of finite codimension in spaces of continuous functions, J. Austral. Math. Soc., Ser. A 26, No. 1, 99-109 (1978; Zbl 0386.41026)]. Theorem 2 presents a characterisation of those \(L\) of finite codimension \(n\) in either \(C_\mathbb{R}(Q)\) or \(C_\mathbb{C}(Q)\) which are Chebyshev; it is in terms of the annihilator \(L^\perp\) of \(L\) in the space \(C(Q)^*\) of real or complex measures on \(Q\), and includes the necessary condition that \(\mu,\nu\in L^\perp\{0\}\) are mutually absolutely continuous on the derived set \(Q'\). Further statements and proofs involve a detailed analysis of the Radon-Nykodym derivatives. The results include Theorem 7 which asserts that if \(C_\mathbb{R}(Q)\) contains a Chebyshev subspace of finite codimension \(\geq 2\) then \(C_\mathbb{C}(Q)\) contains Chebyshev subspaces of every finite codimension, and Theorem 8 which extends to \(C_\mathbb{C}(Q)\) Garkavi's characterisation of those metrizable \(Q\) for which \(C_\mathbb{R}(Q)\) contains Chebyshev subspaces of finite codimension \(\geq 2\).
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 46B04
Mathematics Subject Classification ID: 46E27
Mathematics Subject Classification ID: 54C35
zbMATH DE Number: 1257019
zbMATH Keywords: Chebyshev subspace
zbMATH Keywords: continuous function
zbMATH Keywords: finite codimension
reviewed by: Aldric L. Brown
MaRDI profile type: Publication
cites work: Cebysev subspaces of finite codimension in \(C(X)\)
cites work: Q5550654
cites work: Q5580584
cites work: Q4167103
cites work: Q5595759
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full work available at URL: https://doi.org/10.1007/bf02355903
OpenAlex ID: W2052523975
description: scientific article; zbMATH DE number 1257019