Let $X$ be a normed vector space and $Y$ be a Banach space such that $X$ is continuously embedded into $Y$ (this will be denoted by $X\hookrightarrow Y$ in the sequel).
Is it always possible to find a Banach space $\tilde{X}$ such that $X\hookrightarrow \tilde{X}\hookrightarrow Y$, $X$ is dense in $\tilde{X}$ and, for all $x\in X$, $\left\Vert x\right\Vert_X=\left\Vert x\right\Vert_{\tilde{X}}$ ?
In other words, is it possible to find a completion of $X$ inside $Y$ ? I guess the answer is no in general, but I was unable to find a counterexample. Perhaps this is a very classical fact.
Playing with Hamel bases you get that on every infinite dimensional Banach space $(Y,\|\cdot\|_Y)$ there is a discontinuous linear functional $f$. Then define $X=Y$ and $\|x\|_X=\|x\|_Y + |f(x)|$. Then the extension of the identical map $X\hookrightarrow Y$ to the completion is not injective since otherwise it were bijective and hence an isomorphism be the open mapping theorem which would imply the continuity of $f$.