I was wondering about something related to compact operators. If we have a compact operator $T:X \mapsto Y$ and a bounded sequence $(x_n)n$, then we know that there is a convergent subsequence $(Tx_{n_k})$.
My question is: When can we deduce from this that there is a $w \in X$ such that $\lim_{k \rightarrow \infty}(Tx_{n_k})=Tw$? Is there a theorem that deals with this case?