unitary representation

Let $G$ be a topological group. A unitary representation of $G$is a pair $(\pi ,H)$ where $H$ is a Hilbert space and$\pi :G\to U(H)$ is a homomorphism such thatthe mapping of $G\times H\to H$ that sends $(g,v)$ to $\pi (g)v$is continuous. Here $U(H)$ denotes the set of unitary operatorsof $H$.The group $G$ is said to act unitarily on $H$ or sometimes,$G$ is said to act by unitary representation on $H$.