Exercise 18, § 7

In a Hausdorff locally convex space E, let $(K_\alpha)_{\alpha \in I}$ be a decreasing directed family of convex sets that are compact and non-empty. For all $\alpha \in I$, denote the set of extremal points of $K_\alpha$ by $A_\alpha$, and by $F_\alpha$ the closure of the union of the $A_\beta$ for $\beta \geq \alpha$, so that $(F_\alpha)$ is a decreasing directed family of compact sets. Let A be the intersection (non-empty) of the $F_\alpha$, and K the intersection (non-empty) of the $K_\alpha$. Show that K is the closed convex envelope of A. (If f is a continuous linear form on E, and $x_\alpha$ a point of $F_\alpha$ where f attains its maximum in $F_\alpha$, show that $f(y) \leq f(x_\alpha)$ for all $y \in K$; then take a cluster point of the family $(x_\alpha)$ following the filter of sections of I.)