Let $X,Y$ be Banach Spaces, and let $T\in K(X,Y)$ be a compact operator from $X$ to $Y$. I have to prove that $T(X)$ is closed in Y if, and only if, $\dim(T(X))<\infty$.
Can anybody help me with this proof, please? There is surely some property I haven't thought about, but I'm getting really weird right now... Thank you!