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Best constrained approximation in Banach spaces. (English) Zbl 1312.41039

Summary: In this article, for a real Banach space \(X\) and a closed subspace \(Y\), we consider aspects of proximinality, its stronger variants and the notion of centrability, Chebyshev centers, for a class of subspaces that are relatively constrained in a Banach space, in the sense that for \(x\notin Y\), \(Y\) is constrained in \(\mathrm{span}x,Y\). We show that if \(X\), under the canonical embedding has the strong-\(1\frac12\)-ball property in its bidual, then the same is true of \(Y\). We also give applications of these results to proximinality in spaces of Bochner integrable functions. We consider a class of Banach spaces for which a formula due to P. W. Smith and J. D. Ward [Proc. Am. Math. Soc. 48, 165–172 (1975; Zbl 0296.41022)] on relative Chebyshev centers and radius is valid. We show that any locally constrained subspace of the Grothendieck \(G\)-space is a \(G\)-space.

MSC:

41A50 Best approximation, Chebyshev systems
46B20 Geometry and structure of normed linear spaces
46E40 Spaces of vector- and operator-valued functions

Citations:

Zbl 0296.41022
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References:

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