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Convergence theorems for best approximations in a nonreflexive Banach space. (English) Zbl 0924.41021

Let \(X\) be a Banach space, \(A\subset X\), \(x\in X\) and \(P_A(x):=\{y_0\in A: d(x,A)=\inf\{\| x-y\|: y\in A\} = \| x-y_0\|\}\). Denote by \({\mathcal C}\) the family of all proximinal subsets of \(X\) (a set is called proximinal if \(P_A(x)\neq\emptyset\), for every \(x\in X\)).
The author studies continuity properties of the application: (1) \((x,A)\to P_A(x)\), \(x\in X\), \(A\in {\mathcal A}\), with respect to Wijsman strong convergence in a nonreflexive Banach space.
If \(X\) is a reflexive Banach space then Mosco convergence implies Wijsman strong convergence (Proposition 2.1) so that the obtained results are extensions to the nonreflexive case of known results in reflexive Banach spaces with respect to Mosco convergence (e.g. M. Tsukada, J. Approximation Theory 40, 301-309 (1984; Zbl 0545.41042)]; N. S. Papageorgiou and D. A. Kandilakis, J. Approximation Theory 49, 41-54 (1987; Zbl 0619.41033)].

MSC:

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