Continuity in the weak operator topology implies continuity in the strong op. top.?

Question

Let $\mathcal{H}$ be a Hilbert space. I would like to show:

A $\phi : \mathcal{B(H)}\to\mathbb{C}$ linear functional is continous in the strong operator-topology iff is continous in the weak operator-topology.

The SOT $\Rightarrow$ WOT implication is clear, but the reverse is not.

Answer 1

$\phi$ is WOT-continous iff for every WOT-convergent net $A_n\to A\in\mathcal{B(H)}$ we have $\phi(A_n)\to\phi(A)$. If you replace "WOT" with "SOT" in that statement, you make the assumption stronger, so the statement becomes weaker.