Krein-Milman's Extreme Point Theorem and Weak Topology on Hyperspace

Abstract

Let $\operatorname{WCC}(X)$ be the collection of all non-empty, weakly compact, convex subsets of a Banach space $X$ endowed with the Hausdorff metric $h$. Weak topology $\mathcal{T}_{w}$ will be defined on $\operatorname{WCC}(X)$. We shall prove that every weakly compact ($\mathcal{T}_{w}$-compact) convex subset $\mathcal{K} \subset (\operatorname{WCC}(X), \mathcal{T}_{w})$ has an extreme point. We also show that there exists strongly bounded ($h$-bounded), closed ($h$-closed) convex subsets which are not weakly closed (i.e., not $\mathcal{T}_{w}$-closed).