It was mentioned in the comments to https://math.stackexchange.com/questions/517369/comparison-of-strong-operator-and-weak-topologies-on-bh that continuous linear functionals on $\mathfrak{B}(\mathbb{H})$ are the same in the strong operator topology as in the weak operator topology. It doesn't seem obvious and I couldn't find it proved. Can someone give an idea of a proof or a reference to it?

Does the same apply to $\sigma$-strong and $\sigma$-weak (weak*) topologies, can continuous linear functionals be identified with trace class operators in both cases?