Definition 37.1.2 (Ultrastrong Topology).label Let $H$ be a complex Hilbert space. For each $x = \seq{x_n}\in L^{2}(\natp; H)$, let
\[\Phi_{x}: B(H) \to L^{2}(\natp; H) \quad (\Phi_{x}T)_{n} = Tx_{n}\]
then the ultrastrong/$\sigma$-strong topology on $B(H)$ is the topology generated by the maps $\bracsn{\Phi_x|x \in L^2(\natp; H)}$.
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