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.