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?
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 $\xi,\eta\in H$.