A unitary representation of a locally compact topological group$G$ on a Hilbert space$H$ is a representation of $G$ as an abstract group on $H$ by unitary linear operators, i.e. the homomorphism of groups $\rho : G\to U(H)\subset End H$ which is continuous with respect to the topology on $U(H)$induced by the strong operator topology on $End H$. In other words, $\rho: G\to U(H)$ is a map satisfying $\rho(gh) = \rho(g)\rho(h)$, $\rho(g)^{-1} = \rho(g^{-1}) = \rho(g)^*$ for all $g,h\in G$ and for all $x\in H$, the function $g\mapsto \rho(g)(x)$ on $G$ is norm continuous.