Proposition 6

Let E be a Hausdorff locally convex space and K a convex compact set in E which does not contain 0. Then the smallest pointed cone C of vertex 0 which contains $K$ is a proper convex cone in $E$ and is a locally compact and complete subspace of $E$; also, there exists a closed hyperplane $H$ in $E$ that does not contain 0 and is such that $H$ meets all the half-lines originating at 0 contained in $C$ and such that $H \cap C$ is compact. Further, if $D$ is the half-space containing 0 determined by $H$, a closed hyperplane with these properties, then $C \cap D$ is a cap of $C$ and $C$ is the union of the $\lambda(C \cap D)$ for $\lambda > 0$.

By prop. 4 of II, p. 38, there exists a closed hyperplane $H$ which separates 0 strictly from $K$. Now, the convex envelope $A$ of the union of $\{0\}$ and of $K$ is compact (II, p. 14, prop. 15) and is the union of the $\lambda K$ with $0 \leq \lambda \leq 1$. As 0 and $K$ are strictly on opposite sides of $H$, for every $x \in K$ there exists $\lambda$ such that $0 < \lambda < 1$ and $\lambda x \in H$. As $C$ is the union of the $\lambda A$ for $\lambda \geq 1$, we see that $H$ meets every half-line originating at 0 contained in $C$ and that $H \cap A = H \cap C$ is compact. Further, $C$ is also the union of the $\lambda(H \cap C)$ for $\lambda \geq 0$; let $C_n$ be the union of the $\lambda(H \cap C)$ for $0 \leq \lambda \leq n$. Clearly $C_n$ is the convex envelope of the union of $\{0\}$ and of $n(H \cap C)$, therefore it is compact. Also, for all $x \in E$, there is a closed neighbourhood $V$ of $x$ in $E$ and an integer $n$ such that $V \cap C \subset C_n$; in fact, if $H$ is defined by the equation $f(z) = \alpha$, where $\alpha > 0$, it is sufficient to take for $V$ the closed half-space determined by $nH$ and containing 0, where $n$ is so large that $n\alpha > f(x)$. This shows that $C$ is locally compact (taking $x \in C$), and that it is closed in $E$. We can also consider $K$ as a subset of the completion $\hat{E}$, therefore $C$ is also closed in $\hat{E}$ and therefore complete.

Given a cone $C$ and a closed hyperplane $H$ in a Hausdorff topological vector space $E$, such that $H$ does not contain the vertex $s$ of $C$ and $C$ is the smallest cone with vertex $s$ containing $H \cap C$, then we call the intersection $H \cap C$ a « sole » of the cone $C$. Prop. 6 shows that in a Hausdorff locally convex space $E$, the smallest cone of vertex 0, containing a compact convex set $K$ to which 0 does not belong, is a cone of compact sole, and that every convex cone having a compact sole $S$, is locally compact and complete.