Proposition 37.1.3.label Let $H$ be a complex Hilbert space, then:
- (1)
The dual of $B(H)$ with respect to its strong and weak operator topologies is $H \otimes H$.
- (2)
Every bounded subset of $B(H)$ is relatively compact in the weak operator topology.
- (3)
The composition map $(S, T) \mapsto ST$ is separately continuous in the strong and weak operator topologies.
- (4)
For any bounded subset $B \subset B(H)$, the composition map $(S, T) \mapsto ST$ restricted to $B \times B(H)$ is continuous in the strong operator topology.
- (5)
The adjoint map $T \mapsto T^{*}$ is continuous in the weak operator topology and the ultraweak topology.
Proof. (2): By Proposition 37.1.2.
(4): Let $\angles{S_\alpha}_{\alpha \in A}\subset B$, $\angles{T_\alpha}_{\alpha \in A}\subset B(H)$, and $(S, T) \in B \times B(H)$ such that $S_{\alpha} \to S$ and $T_{\alpha} \to T$ in the strong operator topology. Since $\{S_{\alpha}| \alpha \in A\}$ is equicontinuous, for any $x \in H$,
$\square$
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