Exercise 13, § 1

Let $F$ and $G$ be two vector spaces over $\mathbf{R}$ in separating duality. a) Let $A$ be a convex set in $F$, not containing the origin, and compact for the topology $\sigma(F, G)$; let $C$ be the convex cone with vertex 0 generated by $A$. Show that the polar cone $C^\circ$ in $G$ has an interior point for the topology $\tau(G, F)$: b) Conversely, let $C$ be a proper convex cone with vertex 0, closed for $\sigma(F, G)$ and having an interior point for the topology $\tau(F, G)$. Show that in $G$ there exists a hyperplane $H$, closed for $\sigma(G, F)$, not containing the origin, such that $H \cap C^\circ$ is compact for $\sigma(G, F)$ and $(H \cap C^\circ) \cup \{0\}$ generates $C^\circ$.