Refrence for closedness of image of a compact operator

Question

Let $\mathcal{H}$ be a Hilbert space, and let $T\in K(\mathcal{H})$ be a compact operator. There exists a theorem in the following way: "$T(\mathcal{H})$ is closed in $\mathcal{H}$ if, and only if, $\dim(T(\mathcal{H}))<\infty$."

Can anybody give me a reference for this theorem, please?

Answer 1

This is an easy conseqeuence of Open Mapping Theorem. If $T(H)$ is closed we can think of $T$ as a surjective operator from $H$ to $T(H)$. By Open Mapping Theorem it is an open map. Hence the image of the closed unit ball contains some open ball around $0$ in $H$. Since $T$ is compact this open ball is relatively compact and this makes $T(H)$ finite dimensional.

Answer 2

P.R. Halmos: Hilbert Space Problem Book, Problem 141:

Every closed subspace included in the range of a compact operator is finite- dimensional.