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The representations and continuity of the metric projections on two classes of half-spaces in Banach spaces. (English) Zbl 1474.46041

Summary: We show a necessary and sufficient condition for the existence of metric projection on a class of half-space \(K_{x_0^*, c} = \{x \in X : x^*(x) \leq c \}\) in Banach space. Two representations of metric projections \(P_{K_{x_0^*, c}}\) and \(P_{K_{x_0, c}}\) are given, respectively, where \(K_{x_0, c}\) stands for dual half-space of \(K_{x_0^*, c}\) in dual space \(X^*\). By these representations, a series of continuity results of the metric projections \(P_{K_{x_0^*, c}}\) and \(P_{K_{x_0, c}}\) are given. We also provide the characterization that a metric projection is a linear bounded operator.

MSC:

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