Proposition 4

Let E be a topological vector space, $\widehat{E}$ its separated completion and $j$ the canonical mapping of E into $\widehat{E}$. Let $u$ be a compact linear mapping of E into a separated topological vector space F. Then there exists a unique compact linear mapping $v$ of $\widehat{E}$ into F such that $u=v\circ j$.

Identify F with a topological vector subspace of $\widehat{F}$ and denote by $\widehat{u}:\widehat{E}\rightarrow \widehat{F}$ the unique continuous linear mapping such that $\widehat{u}\circ j$ coincides with $u$ on E. Since $u$ is compact, there exist a neighbourhood V of 0 in E and a compact subset A of F such that $u(V)\subset A$. We have $\widehat{u}(j(V))\subset A$, whence $\widehat{u}(j(V))\subset A$. Now $j(V)$ is a neighbourhood of 0 in $\widehat{E}$ (TG, III, p. 24, Prop. 7). It follows that the image of $\widehat{u}$ is contained in F and the continuous linear mapping $v:\widehat{E}\rightarrow F$ deduced from $\widehat{u}$ by passing to the subspace is compact. Since $j(E)$ is dense in $\widehat{E}$, the mapping $v$ is the unique continuous linear mapping of $\widehat{E}$ into F such that $u=v\circ j$.