Let $f : X \to Y$ be a continuous map of complete topological vector spaces. Suppose that $A \subseteq X$ and $B \subseteq Y$ are proper, dense linear subspaces, and that the restriction map $f : A \to B$ is an isomorphism. Is the original map an isomorphism?
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Robertson & Robertson: Topological Vector Spaces. Chapter VI (Completeness), Proposition 6 and Corollary 1.
If $E$ and $F$ are separated l.c. spaces, $t:E\to F$ continuous, linear, then there is a unique continuous linear extension $\hat t: \hat E \to \hat F$. Here $\hat E$ is the completition.
If further $t$ is an isimorphism onto $t(E)$, then $\hat t$ is an isomorphism of $\hat E$ onto $\hat t(\hat E)$.