Zur Stetigkeit der mengenwertigen metrischen Projektion in endlichdimensionalen Räumen

Abstract

Let X be a real normed linear space, X^* its dual, V a linear subspace of X and S(V^⊥) the unit sphere in the orthogonal space$v^ \bot : = \{ x* \in \chi * :x*(v) = O\forall v \in V\} .$ In this note we prove in the case of finite-dimensional X the following sufficient condition for the continuity of the set-valued metric projection$P_V (x): = \{ v_O \in V:\parallel x - v_O \parallel \leqslant \parallel x - v\parallel \forall v \in V\}$ in terms of the mapping$T(\begin{array}{*{20}c} o \\ x \\ \end{array} *): = \{ x \in ^o x:\parallel x\parallel \leqslant 1$ and$x*(x) = \parallel x*\parallel \}$: If the restriction of T to S(V^⊥) is lower semi-continuous then P_V is lower semi-continuous.