I want to prove the following:
Suppose $\mathcal{H}_1$ and $\mathcal{H_2}$ are Hilbert spaces and let $T: \mathcal{D} \rightarrow \mathcal{H}_2$ be a closed operator, where $\mathcal{D} \subset \mathcal{H}_1$ denotes its domain. For any relatively compact subset $C$ of $\mathcal{D}$ we have that $T(C)$ is closed.
Any input is really appreciated.
Thanks, Giacomo