Chebyshev subspaces of spaces of abstract affine functions
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.)
space of affine continuous functions defined on a compact Choquet simplex and taking values in an arbitrary Banach space
scientific article; zbMATH DE number 3895669
scientific article