Corollary 4

Let $C$ be a closed convex set in a locally convex space $E$. A subset $A$ of $E$ is contained in $C$, if, and only if, for every real valued continuous affine function $u$ in $E$ such that $u(x) \geqslant 0$ for all $x$ in $C$, we have $u(y) \geqslant 0$ for all $y$ in $A$.

The condition is obviously necessary. Conversely we show that it is sufficient; if a point $x \in A$ is not contained in $C$, there exists a closed hyperplane of equation $f(z) = \alpha$ separating $x$ strictly from $C$; if we suppose for example that $f(x) < \alpha$, then the continuous affine function $u = f - \alpha$ contradicts the hypotheses.