Exercise 16, § 2

Let A, B be two closed convex subsets of a Hausdorff topological vector space E. Suppose further that B is locally compact and that $C_A \cap C_B = \{0\}$. Show that $A - B$ is closed in E. (Let $b \in B$, and W be a closed neighbourhood of 0 in E such that $B \cap (b + W)$ is compact. Let $c \in A - B$; for every neighbourhood V of 0 in E consider the set $M_V$ of those $y \in B$ such that $A \cap (c + y + V) \neq \varnothing$. Consider two cases according as to whether there exists a V for which $M_V$ is relatively compact, or there does not exist such a V; in the second case, consider the filter base formed from the sets $P_{V,n} = M_V \cap (b + nW)$ where V varies in the set of closed neighbourhood of 0 in E and $n$ varies in $\mathbf{N}$; form the cone with vertex $b$ generated by $P_{V,n}$ and its intersection with the frontier of $b + W$).