I have a problem answering this exercise that seems to me to be an application of the Open Application Theorem.
"Let $E, F$ Banach spaces and $T \in B(E, F)$ surjective, such that $T(B(0, r))$ is contained in a compact set for all $r> 0$. Show that $\dim E < \infty$."
Perhaps the result below (which I already know how to prove) will help in the resolution, but I can't make the adjustment.
"Let $N$ be a normed space with $\dim N = \infty$. Show that any subset that contains a non-empty open is not compact."
Any idea? Thanks in advance.
This is false. Let $E$ be any real Banach space and $T$ be a non-zero continuous linear functional on it. Then $T$ is surjective. The image of any bounded subset if $E$ is bounded in $\mathbb R$ and hence it is contained in a compact set. But $E$ need not be finite dimensional.