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Continuity of metric projection. (English. Russian original) Zbl 0586.41026
Math. Notes 37, 114-119 (1985); translation from Mat. Zametki 37, No. 2, 200-211 (1985).
Continuing his previous investigation, the author gives new geometric characterizations of the continuity of the metric projection \(P_ M\), for all M in a given class \({\mathfrak M}\) of subsets of a Banach space X (for example, \({\mathfrak M}\) may be the class of all weakly compact convex subsets of X, the class of all boundedly weakly compact convex subsets of X, etc.). The main result of the paper is Theorem 2: For a Banach space X the following conditions are equivalent: 1) \(X\in (A_{\partial})\cap (RBR)\); 2) \(P_ M\) is Hausdorff continuous, \(M\in {\mathfrak M}\); 3) \(P_ M\) is Hausdorff lower semicontinuous, \(M\in {\mathfrak M}\); 4) \(P_ M\) is lower semicontinuous, \(M\in {\mathfrak M}\). Condition 1) is given in terms involving faces of the unit sphere S of X and weak and norm convergence of sequences in S.
Reviewer: S.Cobzaş

MSC:
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
46B20 Geometry and structure of normed linear spaces
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