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Characterizations of elements of best approximation in non-Archimedean normed spaces. (English) Zbl 0623.41035
The author gives some characterizations of elements of best approximation in a non-Archimedean normed space. Thus, for example, he proves that if G is a linear subspace of a spherically complete (every nest of closed spheres has a non-empty intersection) non-Archimedean normed linear space X, $$x\in X\setminus G$$ and $$g_ 0\in G$$, then $$g_ 0$$ is an element of best approximation to x in G if and only if there exists $$f\in X^*$$ such that (i) $$f(g)=0$$, (ii) $$| f(x-g_ 0)| =\| x-g_ 0\|$$ and (iii) $$| f(x-g)| \leq \| x-g\|$$, for every $$g\in G$$.
Reviewer: C.G.Lascarides
MSC:
 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 41A50 Best approximation, Chebyshev systems 46S10 Functional analysis over fields other than $$\mathbb{R}$$ or $$\mathbb{C}$$ or the quaternions; non-Archimedean functional analysis
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