Proposition 1

Let E be a topological vector space and F a separated topological vector space.

a) Every compact linear mapping from E into F is continuous ;

b) Every continuous linear mapping of finite rank from E into F is compact;

c) The set $\mathscr{L}^c(E; F)$ of compact linear mappings from E into F is a vector subspace of $\mathscr{L}(E; F)$.

Let us prove a). Let $v: E\rightarrow F$ be a compact linear mapping and U a neighbourhood of 0 in F. Choose a neighbourhood V of 0 in E such that $v(V)$ is relatively compact in F. Then $v(V)$ is bounded and there exists therefore a real number $\lambda >0$ such that the set $v(\lambda V)$ is contained in U; consequently, $v$ is continuous.

Let $u: E\rightarrow F$ be a continuous linear mapping of finite rank. Since the image $u(E)$ of $u$ is separated and of finite dimension, there exists a compact neighbourhood A of 0 in $u(E)$. The set $V =\overset{-1}{u}(A)$ is a neighbourhood of 0 in E and we have $u(V)\subset A$, hence the linear mapping $u$ is compact. This proves assertion b).

Let $u_1$ and $u_2$ be compact linear mappings of E into F. Let $V_1$ and $V_2$ be neighbourhoods of 0 in E such that the sets $u_1(V_1)$ and $u_2(V_2)$ are relatively compact in F. Put $V = V_1\cap V_2$ and $u=u_1+u_2$. Then $u(V)$ is contained in $u_1(V_1) +u_2(V_2)$, hence is relatively compact in F, and the linear mapping $u$ is compact. It is immediate that $\mathscr{L}^c(E; F)$ is stable under homothety. This proves that $\mathscr{L}^c(E; F)$ is a vector subspace of $\mathscr{L}(E; F)$, whence assertion c).