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

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.

• The question now makes sense, but now you have it the wrong way around. The trivial direction is "WOT$\implies$ SOT" Nov 27, 2016 at 12:53
• $\phi$ is WOT-continous iff for every WOT-convergent sequence $A_n\to A\in B(H)$ 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. Nov 27, 2016 at 12:59
• Yes, I had realised and deleted my comment. Nov 27, 2016 at 13:00
• Also, since WOT and SOT are not metrizisable, it is not enough to check simply that the implication holds on the level of sequential continuity. Nov 27, 2016 at 14:40
• just replace "sequence" with "net". Also: I would be interested in the "SOT$\implies$ WOT" proof, as that is the non-trivial one. Nov 27, 2016 at 15:34

$\phi$ 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.
• @s.harp well, I wasn't thinking too much either. I though "hey, $\mathbb{C}$ is metric, so sequences are enough"... which is not true. Nov 27, 2016 at 16:15