Corollary

Let $\varrho$ be a unitary representation of a topological group G in a Hilbert space E. The following conditions are equivalent:

(i) The representation $\varrho$ is irreducible;

(ii) The contragredient representation $\breve{\varrho}$ is irreducible;

(iii) The conjugate representation $\overline{\varrho}$ is irreducible.

This results from Proposition 2, and from the remark preceding it concerning the subrepresentations of E.