Proposition 10

— Let E be a locally convex space, M a vector subspace of E such that every point of E is in the closure of a bounded subset of M. Then every continuous linear mapping f from M into a Hausdorff quasi-complete locally convex space F uniquely extends to a continuous linear mapping from E into F. The hypothesis implies that M is dense in E, hence f extends uniquely to a continuous linear mapping $\hat{f}$ from E into the completion $\hat{F}$ of F (GT, III, § 3, No. 4, corollary). But every $x \in F$ lies in the closure of a bounded subset B of M; hence $\hat{f}(x)$ is in the closure of $f(B)$ in $\hat{F}$. But $f(B)$ is bounded in F, hence its closure in F is complete, and coincides with its closure in $\hat{F}$. This proves that $\hat{f}(x) \in F$.