Theorem 1 (Krein-Milman)

— In a Hausdorff locally convex space E, every compact convex set A is the closed convex envelope of the set of its extremal points.

For, let C be the closed convex envelope of the set of extremal points of A; clearly $C \subset A$. To see that $A \subset C$, it is sufficient to prove that, if $u$ is an affine linear function, continuous in E and if $u(x) \geq 0$ in C then also $u(x) \geq 0$ in A (II, p. 39, cor. 4); but this follows from prop. 1 applied to $-u$.