I am working on a problem from Reed and Simon, which states:
Suppose $\{A_\alpha\}$ and $\{B_\alpha\}$, $\alpha \in I$, are nets. Let $A_\alpha^* \to A^*$ and $B_\alpha \to B$ in the Strong Operator Topology. Prove that $A_\alpha B_\alpha \to AB$ in the Weak Operator Topology.
Note that these operators are defined on a Hilbert space.
Clearly, SOT convergence implies WOT convergence, so we most certainly have that \begin{align*} \lim_\beta \lim_\alpha \langle A_\alpha(B_\beta x), y \rangle = \langle A(Bx), y \rangle = \lim_\alpha \lim_\beta \langle A_\alpha(B_\beta x), y \rangle. \end{align*}
The problem is this does not give us convergence in a single $\alpha$ limit, and I'm not aware of any theorems that might help. I've tried playing around with the inner product representation of the SOT convergence of $A_\alpha^*$ and $B_\alpha$, but so far to no avail. Any help/hints would be appreciated.