I have a couple questions about how we define the strong operator topology on $\mathscr{B} (H)$ that I'm hoping someone can help me with.
First, I thought that the strong operator topology was the initial topology with respect to the family of maps $F_v$, $v\in H$, defined by $T\mapsto \|Tv\|$ (I got this definition from Analysis Now by Pedersen), but I also see online (such as on the wikipedia page for the strong operator topology) it being defined as the initial topology with respect to the maps $T\mapsto Tv$. The first implies the second, but I don't see why they would be equivalent?
Second, I'm trying to understand exactly why defining the subbasic open sets of $\mathscr{B} (H)$ as being of the form $$\mathcal{U}_{T_0,v,\epsilon}= \{T\in\mathscr{B}(H):\|Tv-T_0v\|<\epsilon\}$$ gives us the initial topology for the family of maps $F_v$.
I want to show that sets of the above form give us the weakest topology making every $F_v$ continuous, so it seems that the strategy should be to show that, given a basic open subset $B_\epsilon(x)$ of $\mathbb{R}$, $F_v^{-1}(B_\epsilon(x))$ is of the form above, so
$$F_v^{-1}(B_\epsilon(x)) = \{T\in\mathscr{B}(H): |\|Tv\|-x|<\epsilon\} \subseteq \{T\in\mathscr{B}(H): \|Tv-x\|<\epsilon\}$$
However, I don't see how to related this back to sets of the same form as $\mathcal{U}_{T_0,v,\epsilon}$. Would I just pick an operator $T_0$ so that $T_0v=x$? And even then, how can one get equality for the sets?
Many thanks for any help.
EDIT: It's been pointed out that I made a mistake when I said Pedersen defines the SOT as the initial topology given by the maps $F_v$. It actually the topology induced by these maps. If that's the case, is it the same as the initial topology defined by the maps $T\mapsto Tv$ like wikipedia says?