Slight adjustments.

src/op/example/bounded.tex

@@ -6,6 +6,22 @@
     Let $H$ be a complex Hilbert space, then $B(H) = L(H; H)$ is the algebra of all bounded linear operators on $H$. 
 \end{definition}
 
+\begin{lemma}
+\label{lemma:unitary-ots}
+    Let $H$ be a complex Hilbert space, then the strong and weak operator topologies on $U(B(H))$ coincide. 
+\end{lemma}
+\begin{proof}
+    Let $S, T \in U(B(H))$ and $\xi \in H$, then
+    \begin{align*}
+        \norm{(S - T)\xi}_H^2 &= \dpn{(S - T)\xi, (S - T)\xi}{H} \\
+        &= \norm{S\xi}_H^2 + \norm{T\xi}_H^2 - 2\text{Re}\angles{S\xi, T\xi}_H \\
+        &= 2\norm{\xi}_H^2  - 2\text{Re}\angles{S\xi, T\xi}_H
+    \end{align*}
+
+    which tends to $0$ as $S \to T$ in the weak operator topology. 
+    
+\end{proof}
+
 
 % 1. Every non-trivial ideal of B(H) contains the finite-rank operators.
 % 2. If H is separable, then the only non-trivial closed idela of B(H) are the compact operators.