Krein-Milman theorem

Theorem.

Let $X$ be a locally convex topological vector space, and let $K\mathrm{\subset }X$be a compact convex subset (http://planetmath.org/ConvexSet). Then $K$ is the closed convex hull of its extreme points.

The closed convex hull above is defined as the intersection of all closed convex subsets of $X$ that contain $K$. This turns out to be the same as the closure of the convex hull in a topological vector space.