Lemma 7

The mapping $g\mapsto \varrho (g)$ is a unitary representation of G in E.

By Lemma 4 of V, p. 380, it is enough to prove that for every $i$ in I and every $x$ in $E_i$, the mapping $g\mapsto \varrho (g)x$ is continuous at the identity element $e$ of G. This mapping is the composition of the continuous mapping $g\mapsto \varrho_i(g)x_i$ and the canonical injection of $E_i$ into E. It is therefore continuous.

The representation $\varrho$ is called the Hilbert sum of the unitary representations $(\varrho_i)_{i\in I}$; it is denoted by $\varrho =\bigoplus_{i\in I}\varrho_i$.

Let G and H be topological groups. Let $\varrho_1$ (resp. $\varrho_2$) be a unitary representation of G (resp. H) in a Hilbert space $E_1$ (resp. $E_2$). Let $E = E_1\widehat{\otimes}_2E_2$ be the Hilbert tensor product space of $E_1$ and $E_2$ (EVT, V, p. 28, Def. 1). For $(g, h)\in G\times H$, let $\varrho (g, h)$ be the continuous endomorphism $\varrho_1(g)\widehat{\otimes}_2\varrho_2(h)$ of E (EVT, V, p. 28) ; it will be denoted simply by $\varrho_1(g)\otimes \varrho_2(h)$ when there is no danger of ambiguity.