I am having serious trouble understanding the proof that an operator is compact.
This is the original question I asked and the proof is included very helpfully in the answer.
When showing the operator $T$ is compact the main criteria mentioned is "show $T$ is the norm limit of a sequence $T_n$ of finite rank operators".
What does this statement mean?
Do we have a finite rank operator?
Apologies if I am not asking the right question because I am not sure where to begin.
In the proof, why do we have to show $$||T-T_k || \to 0$$
Is it because this shows that $T$ is the limit of a sequence of finie rank operators? So $T_k$ are the finite rank operators and $T$ is the limit?