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In Stratila and Zsido, as well as some other sources, the ultraweak topology on $B(H)$ is taken to be the smallest topology for which every element in the closure of the span in $B(H)$ of the elements $\omega_{\xi, \eta}$ where $\omega_{\xi, \eta}(x)=<\xi, x\eta>$ is continuous. Another popular definition is found in Dixmier Chapter 3, or just as well here: p.2 of this notes

A third popular definition can be found here.

Can someone help me establish the three are equivalent? Thanks.

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A linear functional of the form $x \mapsto \sum_{i = 1}^\infty \langle \xi_i, x \eta_i \rangle$ where $\sum_{i = 1}^\infty \| \xi_i \|^2 < \infty$ and $\sum_{i = 1}^\infty \| \eta_i \|^2 < \infty$ can be written more succinctly as $x \mapsto {\text Tr}(Tx)$ where $T$ is the trace class operator $T\zeta = \sum_{i = 1}^\infty \langle \xi_i, \zeta \rangle \eta_i$. (or something close to this formula)

Since the trace class operators from the predual of $B(H)$ it follows that the second and third definitions are equivalent.

Since every such linear functional above can be approximated by a finite sum (or equivalently since the finite rank operators are dense (in trace norm) in the trace class operators) it follows that the first and second/third definitions are equivalent.

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    No problem, glad I could help.2012-07-16