Let $T$ be a continuous linear operator on Hilbert space $\mathcal{H}$. Prove that $T$ is compact if and only if every closed subspace contained in the range of $T$ is finite dimensional.
For the left side implies the right side, it is a consequence of open-mapping theorem. But on the other hand I don’t know how to deal with. I tried to use finite rank operator to approach $T$, but field. Any hint or references? Thanks!
By the polar decomposition one may assume that $T\geq0$.
In this case, for every $n$, let $f_n$ be the characteristic function of $[1/n,\infty)$. The function $$ g_n(t) = f_n(t)/t, $$ is then bounded and $f_n(t)=tg_n(t)$. Therefore $f_n(T)=Tg_n(T)$, so we see that the range of $f_n(T)$ is contained in the range of $T$. Since $f_n(T)$ is a projection, its range is closed, so by hypothesis we deduce that $f_n(T)$ has finite rank.
Observing that $Tf_n(T)$ converges to $T$, we conclude that $T$ is compact.