Proposition 14

In a topological vector space E, the closure of a convex set (resp. of a convex cone) is a convex set (resp. a convex cone with the same vertex).

For, let A be a convex set; the mapping $(x, y) \mapsto \lambda x + (1 - \lambda) y$, where $0 < \lambda < 1$, is continuous in $E \times E$ and maps $A \times A$ in A; thus (GT, I, § 2.1, th. 1) it maps $\overline{A} \times \overline{A}$ in $\overline{A}$, which shows that $\overline{A}$ is convex. Similarly, if C is a convex cone with vertex 0 then $\overline{C} + \overline{C} \subset \overline{C}$ and $\lambda \overline{C} \subset \overline{C}$ for all $\lambda > 0$.