Let $K$ be a compact Hermitian operator on a Hilbert space $H$. How to show that if $K$ has infinite rank then the range of $K$ is not closed in $H$?
My thought is that suppose range $K$ is closed in $H$ and then we need to prove that $K$ has finite rank. So far I think that spectral theorem can be applied. I want to find contradiction to compactness, but I still have no idea. Thank you for your help.