Suppose I have a linear operator T from a Hilbert space H to itself, and T maps every weak convergent sequence to a weak convergent sequence. Show that T is continuous.
I feel that this statement will not be true for general Banach spaces. Weak convergence in a Hilbert space means the inner product of Xn and u converges to the inner product of X and u for any u in H. But I don't see how it helps me in proving the claim. And I think closed graph theorem will not help either, since the assumption is about strong convergence.
Any help is appreciated.