A VECTOR-VALUED VERSION OF RUSSO-DYE THEOREM

In this paper, we will study the extremal structure of the unit ball of U(M, X), the space of uniformly continuous and bounded functions, from a not necessarily compact metric space M into a normed space X. Concretely, if X is uniformly convex and dim X >= 2, where dim X denotes the dimension of X as a real vector space, it is proved that every element y in U(M, X), with parallel to y parallel to < 1, is a convex combination of a finite number of extreme points of the unit ball. As a result, the unit ball of U(M, X) coincides with the closed-convex hull of its extreme points.