Exercise 6, § 6

Let F, G be two vector spaces in separating duality, and A a weakly compact convex set in F. Let C be a convex cone with vertex 0, that is weakly closed in G. Suppose that, for all $y \in C$, there exists $x \in A$ such that $\langle x, y \rangle \geq 0$. Show that there exists $x_0 \in A$ such that $\langle x_0, y \rangle \geq 0$ for all $y \in C$ (apply prop. 4 of II, p. 38, to A and $C^\circ$).