Remark 1

Let $\varrho$ be a unitary representation of H in a complex Hilbert space F and let $u:\pi \rightarrow \varrho$ be an H-morphism. For every function $f\in \mathscr{K}_{\overline{\pi}}$(G), let $v(f)$ denote the function $g\mapsto u(f(g))$ from G into F; it belongs to $\mathscr{K}_{\overline{\varrho}}(G)$ and satisfies $N_p(v(f))\leqslant \|u\|N_p(f)$. The linear mapping from $\mathscr{K}_{\overline{\pi}}(G)$ into $\mathscr{K}_{\overline{\varrho}}(G)$ which associates $v(f)$ with $f$ therefore extends to a continuous H-morphism from Ind$^G_H(\pi )$ into Ind$^G_H(\varrho )$ which is denoted by Ind$^G_H(u)$. We have Ind$^G_H(1_{\pi}) = 1_{Ind^G_H(\pi)}$. Let $\sigma$ be a unitary representation of H and let $v:\varrho \rightarrow \sigma$ be an H-morphism; we have Ind$^G_H(v\circ u) =$ Ind$^G_H(v)\circ$ Ind$^G_H(u)$.

In other words, the construction which associates Ind$^G_H(\pi )$ with $\pi$ and Ind$^G_H(u)$ with $u$ is a functor from the category of unitary representations of H into that of unitary representations of G (cf. CAT, I, § 2, in preparation).