Extreme points in spaces of continuous functions

Abstract

We study the A-property for the space <£( T, X) of continuous and bounded functions from a topological space T into a strictly convex Banach space X. We prove that the A-property for <t(T, X) is equivalent to an extension property for continuous functions of the pair (T, X). We show also that, when X has even dimension, the A-property is equivalent to the fact that the unit ball of <£( T, X) is the convex hull of its extreme points and that this last property is true if X is infinite dimensional. As a result we get that the identity mapping on the unit ball of an infinite-dimensional strictly convex Banach space can be expressed as the average of four retractions of the unit ball onto the unit sphere. Given a Banach space X, B(X) denotes its closed unit ball, S(X) the unit sphere of X, and ext.s(A') the set of extreme points of B(X). Along this paper we will consider only real Banach spaces. If X is a Banach space whose unit ball has some extreme point, we define the X-function of X by