Is completeness over a normed space invariant under linear homeomorphisms?

If $X$ and $Y$ are normed spaces and $T : X \to Y$ is a linear homeomorphism then both $T$ and $T^{-1}$ are uniformly continuous, and hence $X$ is complete if and only if $Y$ is complete.

Theorem: Let $X$ and $Y$ be metric spaces and let $f: X \to Y$ be a uniformly continuous homeomorphism. If $Y$ is complete then $X$ is complete.

Proof sketch: Let $x_n$ be a Cauchy sequence in $X$. By uniform continuity $f(x_n)$ is a Cauchy sequence in $Y$. By completeness, $f(x_n)$ has a limit $y$. Let $x = f^{-1}(y)$, then by continuity of $f$, $x$ is a limit of $x_n$. $\square$

Corollary: The affirmative answer to the question follows from the above theorem and the fact that every linear homeomorphism and its inverse are uniformly continuous.