Corollary 2

Let $G, E, \pi$ be as in Proposition 39. Let $F$ be a subset of $E$. The set of $g \in G$ such that $\pi(g)x = x$ for all $x \in F$ is a Lie subgroup of $G$ and its Lie algebra is the set of $a \in L(G)$ such that $(L(\pi)a)(x) = 0$ for all $x \in F$.

We apply Proposition 39 with $E_2 = \{0\}$ and $E_1$ the vector subspace of $E$ generated by $F$.

Let $\pi_1, \pi_2, \ldots, \pi_n$ be analytic linear representations of $G$. Clearly the direct sum $\pi$ of the $\pi_i$ (Algebra, Chapter VIII, § 13, no. 1) is an analytic linear representation of $G$ and $L(\pi)$ is the direct sum of $L(\pi_1), L(\pi_2), \ldots, L(\pi_n)$ (Chapter I, § 3, no. 1).