Added basic topology facts about B(H).

src/op/index.tex

@@ -3,5 +3,6 @@
 
 \input{./banach/index.tex}
 \input{./c-star/index.tex}
+\input{./vn/index.tex}
 \input{./example/index.tex}
 \input{./notation.tex}

src/op/vn/topologies.tex

@@ -3,7 +3,7 @@
 
 Let $H$ be a complex Hilbert space. Thanks to its self-duality, there is a natural dual pairing
 \[
-B(H) \times (H \otimes H) \quad \dpn{T, \phi \otimes x}{B(H)} =  \dpn{Tx, \phi}{H}
+B(H) \times (H \otimes H) \to \complex \quad \dpn{T, \phi \otimes x}{B(H)} =  \dpn{Tx, \phi}{H}
 \]
 
 Depending on the topology placed on $H \otimes H$, and the corresponding completion, a handful of different topologies arise on $B(H)$. In fact, the above duality produces a predual for $B(H)$, being the trace class operators: 
@@ -42,7 +42,7 @@ Now, a few facts about the more familiar operator topologies:
 \begin{proof}
     (2): By \autoref{proposition:bh-ultraweak-bounded}. 
 
-    (4): Let $\angles{S_\alpha}_{\alpha \in A} \subset B$, $\angles{T_\alpha}_{\alpha \in A} \subset H$, and $S, T \in H$ such that $S_\alpha \to S$ and $T_\alpha \to T$ in the strong operator topology. Since $\{S_\alpha| \alpha \in A\}$ is equicontinuous, for any $x \in H$,
+    (4): Let $\angles{S_\alpha}_{\alpha \in A} \subset B$, $\angles{T_\alpha}_{\alpha \in A} \subset B(H)$, and $(S, T) \in B \times B(H)$ such that $S_\alpha \to S$ and $T_\alpha \to T$ in the strong operator topology. Since $\{S_\alpha| \alpha \in A\}$ is equicontinuous, for any $x \in H$,
     \[
     \lim_{\alpha \in A} S_\alpha T_\alpha x = \lim_{\alpha \in A}S_\alpha Tx = \lim_{\alpha \in A}STx
     \]