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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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