Functional analysis, help to show a short result

Question

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:

1. 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?

Answer 1

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.

Answer 2

As continuous functions map compact sets into compact sets, composition of a bounded operator with a compact operator is compact. So, if $T$ is compact and invertible, so are $I_X=T^{-1}T$, $I_Y=TT^{-1}$. Now if $I_X$ is compact, it maps the unit ball into a compact set, itself. So the unit ball of $X$ is compact, which implies that $X$ is finite-dimensional. Similarly, $Y$ is finite-dimensional.