Let $\;H\;$ be a Hilbert space and consider $\;T_n:H\rightarrow H \;$ a sequence of compact operators. It is known that if $\; \vert \vert T_n - T \vert \vert \to 0\;$ as $\;n\to \infty\;$ then $\;T\;$ is also compact.
Substituting the above convergence by $\;T_nx \to Tx\; \forall x\in H\;$ the statement isn't valid.
I'm trying to find a counterexample to confirm this but I lack good ideas. Any help would be valuable.
Thanks in advance!!