Corollary 3

Let $G$ be a Lie group and $\pi_1, \ldots, \pi_n$ analytic linear representations of $G$ on finite-dimensional vector spaces $E_1, \ldots, E_n$. Then the representation $\pi_1 \otimes \cdots \otimes \pi_n$ of $G$ (Appendix) is analytic and $L(\pi_1 \otimes \cdots \otimes \pi_n)$ is the tensor product of $L(\pi_1), \ldots, L(\pi_n)$.

The mapping $(A_1, \ldots, A_n) \mapsto A_1 \otimes \cdots \otimes A_n$ of $\mathscr{L}(E_1) \times \cdots \times \mathscr{L}(E_n)$ into $\mathscr{L}(E_1 \otimes \cdots \otimes E_n)$ is multilinear, whence the fact that $\pi$ is analytic. Consider the mapping $(x_1, \ldots, x_n) \mapsto x_1 \otimes \cdots \otimes x_n$ of $E_1 \times \cdots \times E_n$ into

By Proposition 41, we see that for all $a \in L(G)$, $x_i \in E_i$ for $1 \leq i \leq n$. Hence $L(\pi)$ is the tensor product of the $L(\pi_i)$.