CONTINUOUS SELECTIONS FOR METRIC PROJECTIONS

Abstract

This chapter highlights the continuous selections for metric projections. It presents the problem of existence of continuous selections for metric projections. The problem of characterizing the spaces that admit continuous selections for metric projections from among the finite dimensional subspaces of C(X), X compact, has been posed by Lazar-Morris-Wulbert. A solution of this problem is given provided that X = [a, b], a real compact interval. Furthermore, some of the interesting properties of the weak Chebyshev subspaces of C[a, b] is shown, as the theory of these subspaces is of fundamental importance to prove our characterization theorem. In addition, characterization of the spaces G that admit continuous selections from among the finite dimensional subspaces of C[a, b] is given. There exists a continuous selection for PG if and only if the following conditions are satisfied: (1) G is weak Chebyshev, (2) No g ∈ G, g ≠ 0, vanishes on more than one interval, and (3) the numbers of the boundary zeros of the elements g ∈ G are bounded in a certain sense.