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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closed as unclear what you're asking by Tom LaGatta, Yemon Choi, Daniel Moskovich, Todd Trimble♦, Qiaochu Yuan Jul 2 '13 at 7:48Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question. 


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 $\hat t(\hat E)$. 

