On the extremal structure of the unit balls of Banach spaces of weakly continuous functions and their duals

Abstract

A sufficient and then a necessary condition are given for a function to be an extreme point of the unit ball of the Banach space C ( K , ( X , w ) ) C(K,(X,w)) of continuous functions, under the supremum norm, from a compact Hausdorff topological space K K into a Banach space X X equipped with its weak topology w w . Strongly extreme points of the unit ball of C ( K , ( X , w ) ) C(K,(X,w)) are characterized as the norm-one functions that are uniformly strongly extreme point valued on a dense subset of K K . It is shown that a variety of stronger types of extreme points (e.g. denting points) never exist in the unit ball of C ( K , ( X , w ) ) C(K,(X,w)) . Lastly, some naturally arising and previously known extreme points of the unit ball of C ( K , ( X , w ) ) ∗ C(K,(X,w))^{*} are shown to actually be strongly exposed points.