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There is a quotation below (in the book "C*-algebras and Finite-Dimensional Approximations")

Remark 2.1.3. It follows from Sakai's predual uniqueness theorem that when checking point-ultra weak convergence of bounded nets, it always suffices to check convergence on certain vector functionals. That is, if $N\subset B(H)$ is any faithful normal representation and $\Omega\subset H$ is any set of vectors whose linear span is dense in $H$, then $\psi_{n} \circ\phi_{n} \rightarrow \theta$ in the point-ultraweak topology if and only if $$\langle \psi_{n} \circ\phi_{n} (a)v, w \rangle \rightarrow \langle\theta(a)v, w\rangle$$ for all $a\in A$ and $v, w\in \Omega$.

My question is:

1. What does the bold words above mean?

2. Does the author want to describle the the point-ultraweak topology is equivalent to point-weak topology on the bounded nets?

3. How to explain the $\Omega$ that the author take?

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Yes, they are just saying that the ultraweak topology agrees with the weak operator topology on bounded sets. So you can test convergence on functionals of the form $x\mapsto\langle x\xi,\eta\rangle$, $\xi,\eta\in H$.

And if you have a set $\Omega$ with dense span in $H$, one can check that if $\langle x_j v,w\rangle\to0$ for all $v,w\in \Omega$, then $\langle x_j\xi,\eta\rangle\to0$ for all $\xi,\eta\in H$.

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