Exercise 1, § 2

Let $(G_\ell)_{\ell \in I}$ be a family of locally compact groups, all but finitely many of them compact. Let $U_\ell$ be a continuous linear representation of $G_\ell$ in a locally convex space $E_\ell$. For every $s = (s_\ell) \in G = \prod_\ell G_\ell$, let $U(s)$ be the endomorphism $(x_\ell) \mapsto (U_\ell(s)x_\ell)$ of $E = \prod_\ell E_\ell$. Show that $U$ is a continuous linear representation of $G$ in $E$. Let $E'$ be the topological direct sum of the $E_\ell$. Let $V(s)$ be the restriction of $U(s)$ to $E'$. Show that $V$ is a continuous linear representation of $G$ in $E'$.

$^1$ Translation of saillant (TVS, II, §2, No. 4).