Proposition 5

Let C be a proper convex cone with vertex 0 in E, a Hausdorff weak space; suppose that C is complete for the uniform structure induced by that of E, and that there is an enumerable fundamental system of neighbourhoods of 0 in C. Then C is the union of its caps and is the closed convex envelope of the union of its extremal generators.

The second statement follows from the first and from cor. 2 above. Using prop. 11 of II, p. 52 reduces the proposition to the case when $E = \mathbf{R}^l$ and $C \subset \mathbf{R}_+^l$. For all $\alpha \in I$, denote the projection $pr_\alpha$ in E by $f_\alpha$; then $f_\alpha$ is a continuous linear form. On the other hand let $(V_n)_{n \in \mathbf{N}}$ be an enumerable fundamental system of neighbourhoods of 0 in C. By the definition of the topology of E, for each $n \in \mathbf{N}$, there exists a finite subset $J_n$ of I and a number $\varepsilon_n > 0$ such that $V_n$ contains the set $W_n$ of the $x \in C$ such that $f_\alpha(x) \leq \varepsilon_n$ for all $\alpha \in J_n$; put $J = \bigcup_{n \in \mathbf{N}} J_n$. Let $y \neq 0$ be a point of C, and p be the function $\sum_{\alpha \in J} \lambda_\alpha (f_\alpha|C)$ where the $\lambda_\alpha > 0$ are chosen so that $p(y) = 1$; this is possible, since if $f_\alpha(y) = 0$ for all $\alpha \in J$, then $y \in V_n$ for all $n$, which implies that $y = 0$, and this is contrary to hypothesis. Now we remark that for all $\alpha \in I$, the function $f_\alpha|C$ is continuous at the point 0, therefore there is an $n \in \mathbf{N}$, such that $f_\alpha$ is bounded in a $W_n$, therefore bounded above in C by a linear combination of a finite number of functions $f_\beta|C$, where $\beta \in J$. It follows that if A in the set of $x \in C$ such that $p(x) \leq 1$, then $f_\alpha$ is bounded in A for all $\alpha \in I$. As p is lower semi-continuous in C, it follows that A is closed and non-empty in C and therefore is compact. Since it is clear that p verifies the conditions (i) and (ii) of prop. 4 of II, p. 58, we see that A is a cap in C and contains y.