Proposition 3

— Let E and F be two locally convex spaces, F being assumed Hausdorff, and let $\mathfrak{S}$ be a family of bounded subsets of E. If the union A of the sets of $\mathfrak{S}$ is total in E, then the space $\mathscr{L}_\mathfrak{S}(E; F)$ is Hausdorff.

Let $u_0$ be a non-zero element of $\mathscr{L}(E; F)$; since $u_0$ is continuous and A is total in E, there exists an $x_0$ in A such that $u_0(x_0) \neq 0$. Since F is Hausdorff, there exists a neighbourhood V of 0 in F such that $u_0(x_0) \notin V$. Let $M \in \mathfrak{S}$ be such that $x_0 \in M$. Then the set U of all $u \in \mathscr{L}(E; F)$ such that $u(M) \subset V$ is a neighbourhood of 0 in $\mathscr{L}(E; F)$, and we have $u_0 \notin U$, hence $\mathscr{L}(E; F)$ is Hausdorff.

In particular, the following topologies on $\mathscr{L}(E; F)$ are Hausdorff whenever F is Hausdorff : simple convergence, compact convergence, precompact or compact convex, and bounded convergence.