Proposition 37.1.4.label Let $H$ be a complex Hilbert space, then:

1. (1)

   The dual of $B(H)$ with respect to its strong and weak operator topologies is $H \otimes H$.

2. (2)

   Every bounded subset of $B(H)$ is relatively compact in the weak operator topology.

3. (3)

   The composition map $(S, T) \mapsto ST$ is separately continuous in the strong and weak operator topologies.

4. (4)

   The composition map $(S, T) \mapsto ST$ is left-hypocontinuous with respect to the strong operator topology and strong-operator bounded subsets of $B(H)$.