Exercise 16, § 6

Let $F$ be a product of lines and $C$ a closed convex set in $F$. *a) Show that there exists $x_0 \in F$, two sets $I$ and $J$ and a topological isomorphism $u$ of $F$ on $\mathbf{R}^I \times \mathbf{R}^J$ such that $u(x_0 + C)$ is of the form $\mathbf{R}^I \times A$, where $A$ is a closed convex set of $\mathbf{R}_+^J$. (Note that we have $F = G^*$ where $F$ has the topology $\sigma(G^*, G)$; consider the polar $C^\circ$ of $C$ in $G$, the vector subspace of $G$ generated by $C^\circ$ and a complement of this subspace.) *b) If $C$ does not contain any affine line, the mapping $(x, y) \mapsto x + y$ of $C \times C$ in $F$ is proper. *c) Suppose that $C$ is a cone with vertex 0 and that the uniform structure induced on $C$ by that of F is metrisable. Then if the sets I and J, the point $x_0$ and the mapping $u$ satisfy the conditions of a) then I is enumerable and there exists an enumerable subset H of J such that the restriction of the canonical projection $p : \mathbf{R}^I \times \mathbf{R}^J \to \mathbf{R}^I \times \mathbf{R}^H$ to $u(x_0 + C)$ is an isomorphism of the uniform subspace $u(x_0 + C)$ of $\mathbf{R}^I \times \mathbf{R}^J$ on the uniform subspace $p(u(x_0 + C))$ of $\mathbf{R}^I \times \mathbf{R}^H$.