weak operator convergence on dense subsets implies weak operator convergence

Question

Let $X$ be a Banach space, $E_1$ be a dense subset of $X$ and $E_2$ be a weak* dense subset of the dual $X^*$.

If $(T_j)$ is a bounded net of bounded linear operators acting on $X$ and if $T$ is a bounded linear operator such that $$ \langle T_j(x),y \rangle \to \langle T(x), y\rangle $$ for any $x \in E_1$ and any $y \in E_2$ then it seems to me the net $(T_j)$ converges to $T$ for the weak operator topology of $B(X)$.

1. Can you confirm this ? Note $E_2$ is weak* dense and not a norm dense subset of $X^*$.

2. Do you know a reference (at least of the case where $E_2$ is norm dense)? Or it is possible to deduce this fact from results of the literature ?

Answer 1

This is not true. You can take $X=\ell_1$ with dual $\ell_\infty$ which contains the weak$^*$-dense subspace $E_2=c_0$. For $E_1=X$ and $T_n:\ell_1\to\ell_1$, $x\mapsto x_1e_n$ (where $e_n$ is the unit sequence wit $1$ in the $n$th spot and zeros else) you get $\langle T_n(x),y\rangle = x_1y_n\to 0$ for every $y\in E_2=c_0$ but it does not converge for all $y\in\ell_\infty$.