# Weakly convergent subsequence under continuous operator

Suppose we have two Hilbert spaces $H_1,~ H_2$, a linear continuous operator $T:H_1 \to H_2$ and a weakly convergent sequence $u_k\rightharpoonup u$ in $H_1$.

Is $Tu_k \rightharpoonup Tu$ in $H_2$ true? If yes, could you give me a short sketch of the proof?