Continuous selections for the metric projection onC 1

Abstract

C1(K) is the space of real continuous functions onK endowed with the usualL1,-norm where\(K = \overline {\operatorname{int} K}\) is compact inRm · U is a finite-dimensional subspace ofC1,(K). The metric projection ofC1,(K) ontoU contains a continuous selection with respect toL1, -convergence if and only ifU is a unicity (Chebyshev) space forC1,(K). Furthermore, ifK is connected andU is not a unicity space forC1,(K), then there is no continuous selection with respect toL∞-convergence. An example is given of aU and a disconnectedK with no continuous selection with respect toL1-convergence, but many continuous selections with respect toL∞-convergence.