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

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