Exercise 39, § 2

Let E be a finite dimensional Hausdorff topological vector space. a) Let $\mathfrak{F}(E)$ be the family of closed non-empty sets of E, carrying the uniform structure deduced from the uniform structure of E by the procedure of GT, II, § 1, exerc. 5, a). Show that, the set $\mathfrak{C}(E)$ of non-empty closed convex sets of E, is closed in the space $\mathfrak{F}(E)$. Deduce that if K is a compact set in E, the set of non-empty closed convex sets in E that are contained in K, is a compact set in $\mathfrak{C}(E)$ ($cf.$ GT, § 4, exerc. 11). b) Let $\mathfrak{K}_0(E)$ be the set of compact convex subsets of E that contain 0 as an interior point. For every set $A \in \mathfrak{K}_0(E)$, let $p_A$ be the gauge of A (II, p. 20). Show that $A \mapsto p_A$ is an isomorphism of the uniform subspace $\mathfrak{K}_0(E)$ of $\mathfrak{C}(E)$ on a subspace of the space $\mathscr{C}_c(E; \mathbf{R})$ of continuous real valued functions in E, carrying the uniform structure of compact convergence (GT, X, § 1.6).