Let $G$ be a locally compact group. A unitary representation on a Hilbert space $H$ is group homomorphism $u:G\to U(B(H))$ with $u$ is continuous with respect to weak operator topology. ($U(B(H))$ denotes the unitaries in $B(H)$).
Question: How can 'continuous with respect to weak operator topology' characterized in terms of nets? Is it as follows: A net $(g_i)_{i\in I}\subseteq G$ and $g\in G$ such that $g_i\to g$ in $G$, then $\lvert \langle u_{g_i}(h),k \rangle \rvert \to \lvert \langle u_{g}(h),k \rangle \rvert$ for all $h,k\in H$? Thank you
You're close, a net $(A_i)_i$ in $B(H)$ converges to some $A \in B(H)$ in the weak operator topology if and only if $$ \lvert \langle (A_i-A)h, k \rangle \rvert \to 0 $$ for all $h,k \in H$. So more concretely in your case, for any net $(g_i)_{i\in I}\subseteq G$ that converges to some $g\in G$, we have $\lvert \langle (u(g_i)-u(g))(h),k \rangle \rvert \to 0$ for all $h,k\in H$.