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A theorem on best approximations. (English) Zbl 0635.41022

A theorem on the existence of best approximation for an approximatively compact subset of a normed space is proved. The result herein contains a recent result of Prolla. In a recent paper, Prolla proved the following theorem: Theorem 1: Let M be a nonempty compact and convex subset of a normed space E and \(g: M\to M\) be a continuous, almost affine and an onto mapping. Then for each continuous mapping \(f: M\to E\) there exists an \(x\in M\) satisfying (1) \(\| g(x)-f(x)\| =d(f(x),M)\) where \(d(f(x),M)=\inf \{\| f(x)-m\|:m\in M\}\). The purpose of this paper is to investigate result (1) when the subset M in Theorem 1 is an approximativelopriate to their result.
Reviewer: R.Artzy

MSC:

41A50 Best approximation, Chebyshev systems
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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References:

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