The Representations and Continuity of the Metric Projections on Two Classes of Half-Spaces in Banach Spaces

Abstract

We show a necessary and sufficient condition for the existence of metric projection on a class of half-space $K_{x_0^{*},c}=\{x\in X:x^{*}(x)\le c\}$ 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.