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 that the 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 operators of $H$ The group $G$ is said to act unitarily on $H$ or sometimes, $G$ is said to act by unitary representation on $H$