Isomorphism on dense subset

Question

I am wondering if the following could be done. I want to show two Banach spaces $X$ and $Y$ are isomorphic. If $A$ is dense in $X$, and $B$ is dense in $Y$, is it sufficient to show there is an isomorphism $S : A\to B$ to deduce (extending in the obvious way) that $X$ and $Y$ are isomorphic?

Answer 1

If $S$ is a linear isometry then it maps Cauchy sequences to Cauchy sequences, and so does its inverse. Every element $x\in X$ is the limit of a sequence in $S,$ and the difference of any two such sequences converges to 0. Define $S(x)$ as the limit of the image of such a sequence, then it is not difficult to see that this definition does not depend on the choice of the sequence. Moreover when $S^{-1}$ is extended to $Y$ in the same way, it is easily shown to be the two-sided inverse of the extended $S.$ Finally, the extended $S$ is norm preserving by continuity of the norm.