Example

(1) Let G be a Lie group. For all $x \in G$, $\operatorname{Int}(x)$ is an automorphism of the Lie group G.

(2) Let G be a Lie group. Let $G^\vee$ denote the opposite group to G, with the same manifold structure as G. It is immediate that $G^\vee$ is a Lie group (called the opposite Lie group to G) and that the mapping $g \mapsto g^{-1}$ is an isomorphism of the Lie group G onto the Lie group $G^\vee$.

(3) Let G be a Lie group and E a complete normable space. An analytic linear representation of G on E (or simply a linear representation of G on E when no confusion can arise) is a morphism of the Lie group G into the Lie group $\mathbf{GL}(E)$, in other words an analytic mapping $\pi$ of G into $\mathbf{GL}(E)$ such that $\pi(gg') = \pi(g)\pi(g')$ for $g, g'$ in G. Suppose that E admits a finite basis $(e_1, e_2, \ldots, e_n)$ over K; let $(e_1^*, e_2^*, \ldots, e_n^*)$ be the dual basis; let $\rho$ be a homomorphism of the group G into the group $\mathbf{GL}(E)$; then the following conditions are equivalent: (i) $\rho$ is an analytic linear representation; (ii) for all $x \in E$ and $x' \in E'$, the function $g \mapsto \langle \rho(g)x, x' \rangle$ on G is analytic; (iii) for all i and j, the function $g \mapsto \langle \rho(g)e_i, e_j^* \rangle$ on G is analytic. For the implications (i) $\Rightarrow$ (ii) $\Rightarrow$ (iii) are clear. On the other hand, the functions $u \mapsto \langle ue_i, e_j^* \rangle$ form a coordinate system on $\mathscr{L}(E)$; hence their restrictions to $\mathbf{GL}(E)$ form a coordinate system on $\mathbf{GL}(E)$, whence the implication (iii) $\Rightarrow$ (i).

Let G be a real Lie group, E a real complete normable space and $\rho$ a homomorphism of the group G into the group $\mathbf{GL}(E)$. We shall see in § 8, Theorem 1 that, if $\rho$ is continuous (when $\mathbf{GL}(E)$ has the topology derived from the norm on $\mathbf{L}(E)$), then $\rho$ is analytic. But note that this notion of continuity is different from that considered in Integration, Chapter VIII, § 2, Definition 1 (iii) (Exercise 1).

(4) Let G be a real Lie group and E a complex complete normable space. An analytic linear representation of G on E is a morphism of G into the underlying real Lie group of $\mathbf{GL}(E)$.