Exercise 12, § 2

Let G be a real Lie group and $\varrho$ a unitary representation of G in a Hilbert space E. An element $x\in E$ is said to be a vector of class $C^{\infty}$ if the mapping of G into E defined by $g\mapsto \varrho (g)x$ is of class $C^{\infty}$. a) Let $\pi$ be a unitary representation of G in F and $u: E\rightarrow F$ a G-morphism. For every vector $x\in E$ of class $C^{\infty}$, the vector $u(x)$ is of class $C^{\infty}$ in F.

We denote by $C^{\infty}(\varrho )$ the space of vectors of class $C^{\infty}$ in E. By question a), every G-morphism of $\varrho$ into $\pi$ defines, by passing to the subspaces, a linear mapping of $C^{\infty}(\varrho )$ into $C^{\infty}(\pi )$.

b) The space $C^{\infty}(\varrho )$ is dense in E.