Exercise 30, § 7

Let $E$ be a Hausdorff locally convex space and $A$ a cap of $C$, a closed convex set in $E$. Show that if $a \in A$ is an extremal point of $A$ then the facet $F$ of $a$ in $C$ (II, p. 87, exerc. 3) is of dimension $\leq 1$ (use exerc. 26 of II, p. 91). Deduce that $F \cap A$ is a cap of $C$.