Weak convergence of Hilbert Schmidt operators Let $\{T_i\}$ be a sequence of positive operators in $B_2(H)$ converging to $T$ with respect to the weak topology on $B_2(H)$ (space of Hilbert Schmidt operators on $H$). Does $\{T_ix\}$ converges to $Tx$  in norm (or weakly) for each $x\in H$?

I know that $T$ must be positive since positive operators form a convex set and convex weakly closed sets are norm closed.
Fix $x,y\in H$. Then the functional $T\mapsto \langle Tx,y\rangle$ is continuous.
Then the weak convergence $T_i\rightharpoonup T$ implies
$$
\langle T_i x,y\rangle \to \langle Tx,y\rangle \quad \forall x,y\in H,
$$
which is equivalent to
$$
T_ix \rightharpoonup Tx \quad \forall x\in H.
$$
Fix $x,y\in H$. Then the functional $T\mapsto \langle Tx,y\rangle$ is continuous.
Then the weak convergence $T_i\rightharpoonup T$ implies
$$
\langle T_i x,y\rangle \to \langle Tx,y\rangle \quad \forall x,y\in H,
$$
which is equivalent to
$$
T_ix \rightharpoonup Tx \quad \forall x\in H.
$$