Definition 3

Let G be a topological group and E a Hilbert space over K. A unitary representation of G in E is a continuous linear representation $\varrho$ of G in E such that, for every $g$ in G, the endomorphism $\varrho (g)$ of E is a unitary endomorphism (TVS, V, p. 40) of E.

In other words, a unitary representation is an isometric representation in a Hilbert space. In particular, a unitary representation is bounded.

The trivial representation of a topological group in a Hilbert space is unitary. Every subrepresentation of a unitary representation is unitary. The restriction of a unitary representation to a subgroup is unitary.