Let $E$ and $F$ be Hilbert spaces. For, $T \in B(E,F)$, show that the following are equivalent:
$(i)$ $T$ is compact
$(ii)$ $T^*$ is compact
$(iii)$ There exists a sequence of finite rank operators $\{T_n\}$ such that $||T_n-T|| \to 0$
I was able to show the equivalence relation between $(i)$ and $(ii)$. But couldn't show the equivalence relation between $(iii)$ with the others.
Please give me some hints on it!