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?



Continuous linear functionals in strong operator and $\sigma$-strong topologies
The duals under the strong and the weak topologies can be identified with the finite rank operators, under the ultrastrong and the ultraweak with the trace class operators. This can be found in the classic "von Neumann algebras" by Dixmier.
Can you make precise what you mean by the $\sigma$-strong and $\sigma$-weak topologies?


