Finite-codimensional Chebyshev subspaces in the complex space \(C(Q)\)
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 \(\geq 2\).
scientific article; zbMATH DE number 1257019