Exercise 4, § 9

Let G be a Lie group, H a compact subgroup of $G,\rho$ a unitary representation of H on a real Hilbert space V. Prove that there exists a real Hilbert space W, a unitary representation $\sigma$ of G on W and an isometric injection $u: V\rightarrow W$ with closed image, such that $u(\rho (h)v) =\sigma (h)u(v)$ for $h\in H$, $v\in V$ (same method as for Exerc. 3).