Corollary 2

— In a Hausdorff locally convex space, every compact convex set $A$ is the intersection of the closed half-spaces which contain it and which are determined by support hyperplanes of $A$.

For, let $x_0 \notin A$; $\{ x_0 \}$ is closed, therefore there exists a closed hyperplane $H$ which separates $x_0$ strictly from $A$ (prop. 4); let $f(x) = \alpha$ be an equation of $H$ ($f$ a continuous linear form) and suppose that $f(x) > \alpha$ for all $x \in A$. If we put $\gamma = \inf_{x \in A} f(x)$, the half-space defined by $f(x) \geq \gamma$ contains $A$, is determined by the support hyperplane of equation $f(x) = \gamma$, and does not contain $x_0$; whence the corollary.

It is possible that a closed convex set that is not compact and has no interior point, in a locally convex space, does not have any closed support hyperplane (II, p. 86, exerc. 18 : cf. also V, p. 71, exerc. 11).