# Can you have a weakly convergent sequence of unbounded linear operators? (Example)

Is it at all feasible to have a weakly convergent sequence of unbounded linear operators? If so, what is a concrete example of a sequence of not necessarily bounded linear operators who converge in the weak operator topology?

Is it in fact the case that being a sequence of operators convergent in the weak operator topology necessitates that one in fact has a sequence of bounded linear operators?

Take any unbounded operator $$T$$, and let $$T_n=\tfrac1n\,T$$. Then, for any $$x$$ in the domain of $$T$$, you have $$\|T_nx\|=\tfrac1n\|Tx\|\to0.$$ So $$T_n\to0$$ in the strong operator topology.