Let $H,G$ be Hilbert spaces. Assume that $T: H \rightarrow G$ is a continuous linear transformation. I want to show that if $(x_n)$ is a weakly convergent sequence in $H$, then $(Tx_n)$ is a weakly convergent sequence in G.
Here is what I've done so far:
By definition, we have $\langle x_n, v\rangle \rightarrow \langle x,v \rangle$, for all $v \in H$, and some $x \in H$.
Since $G$ is continuous, we have that $T(x_n) \rightarrow T(x)$ in G.
Hence, $\langle T(x_n),v \rangle \rightarrow \langle T(x),v \rangle$.
I'm not sure if this is correct, as I the only things I've used are definition of weak convergence, and continuity of $T$.
I'm new to weak convergence. If the proof is wrong, could you please let me know which part is wrong, and how should I fix it?