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Zbl 0635.41022
Sehgal, V.M.; Singh, S.P.
A theorem on best approximations.
(English)
[J] Numer. Funct. Anal. Optimization 10, No.1-2, 181-184 (1989). ISSN 0163-0563; ISSN 1532-2467/e

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) $\Vert g(x)-f(x)\Vert =d(f(x),M)$ where $d(f(x),M)=\inf \{\Vert f(x)-m\Vert:m\in M\}$. The purpose of this paper is to investigate result (1) when the subset M in Theorem 1 is an approximativelopriate to their result.
[R.Artzy]
MSC 2000:
*41A50 Best approximation
41A65 Abstract approximation theory

Keywords: approximatively compact subset

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