Corollary 1

Let $G$ be a Lie group, $E_1, \ldots, E_{n+1}$ complete normable spaces and $\pi_1, \ldots, \pi_{n+1}$ analytic linear representations of $G$ on $E_1, \ldots, E_{n+1}$. Let

the complete normable space of continuous multilinear mappings of $E_1 \times \cdots \times E_n$ into $E_{n+1}$ (General Topology, Chapter X, § 3, no. 2). For all $g \in G$, let $\pi(g)$ be the automorphism of $E$ defined by

Then $\pi$ is an analytic linear representation of $G$ on $E$ and

for all $a \in L(G)$, $u \in E$, $x_1 \in E_1, \ldots, x_n \in E_n$.

Every element $(A_1, \ldots, A_{n+1})$ of $\mathscr{L}(E_1) \times \cdots \times \mathscr{L}(E_{n+1})$ defines a continuous endomorphism $\theta(A_1, \ldots, A_{n+1})$ of $E$ by the formula

The mapping $\theta$ of $\mathscr{L}(E_1) \times \cdots \times \mathscr{L}(E_{n+1})$ into $\mathscr{L}(E)$ is continuous and multilinear. Then, for all $g \in G$, and hence $\pi$ is analytic. We apply Proposition 41 to the mapping of $E_1 \times \cdots \times E_n \times E$ into $E_{n+1}$. Then and hence

When the $E_i$ are finite-dimensional, the representation $L(\pi)$ of $L(G)$ is derived from the representations $L(\pi_1), \ldots, L(\pi_{n+1})$ under the procedure of Chapter I, § 3, Proposition 3.