Q:

Functional analysis, help to show a short result

The following problem is from the theory of compact operators:
Suppose $X,Y$ are normed spaces and $T:X\to Y$ is linear. Show that if $T$ is compact and invertible  then $\mbox{dim}(X)$ and $\mbox{dim}(Y)$ are finite.
I have started like this:

Using Riesz's Lemma, $\mbox{dim}(X)$ is finite if and only if $\overline{B}(0,1)$ is compact.

If $T$ is compact then we have $\overline{T(B(0,1))}$ is compact, because   $\overline{B}(0,1)$ is bounded and a compact operator transforms a bounded set in a relatively compact set.
Now how do we put $\overline{T(B(0,1))}$ in relation with $\overline{B}(0,1)$ using Riesz's Lemma?

A:

If $T$ is invertible then the image of the open unit ball in $X$ is open in $Y$ (it is the preimage of the ball under $T^{-1}$). If $Y$ is infinite-dimensional, the closure of this image cannot be compact (by Riesz's lemma, as you mentioned). Therefore $Y$ must be finite-dimensional. From there $X$ clearly must be finite-dimensional as well.

