The representations and continuity of the metric projections on two classes of half-spaces in Banach spaces (Q1725243)

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.