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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) $\parallel g\left(x\right)-f\left(x\right)\parallel =d\left(f\left(x\right),M\right)$ where $d\left(f\left(x\right),M\right)=inf\left\{\parallel f\left(x\right)-m\parallel :m\in M\right\}$. 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
##### Keywords:
approximatively compact subset