Convergent Operator, weakly convergent sequence => weakly convergent? Suppose we have a Hilbert space $X$, a weakly convergent sequence $u_k\rightharpoonup u$ and a convergent operator $T_k \rightarrow T$ in the norm of $\mathcal{L}(X)$ (bounded, linear operators).
Is the assertion $T_k u_k \rightharpoonup Tu$ correct?
Thanks in advance!

ok suppose $f\in{X^*}$ then $foT\in{X^*}$  since $u_k$ is weakly convergent we have 
$foT_k u_k \rightharpoonup foTu$
