I am asked to prove the following:
For a compact self-adjoint operator T on a Hilbert Space H, show that: $$ \operatorname{Im}(T) \mbox{ is closed} \iff \dim(\operatorname{Im}(T)) < \infty$$
I know that for self adjoint operators you can write them with the spectral representation. However, I am still missing a specific property about self adjoint operators that I don't know to show this. Can anyone give me a hint?