Example

Let $E$ be a normable space, $G = \mathbf{GL}(E)$, $T$ the translation group of $E$ and $A$ the permutation group of $E$ generated by $G$ and $T$. The group $A$ is algebraically the semi-direct product of $G$ by $T$. (If $E$ is finite-dimensional, $A$ is the affine group of $E$, cf. Algebra, Chapter II, § 9, no. 4). Let $\sigma$ be the identity linear representation of $G$ on $E$ and $S$ the external semi-direct product of $G$ by $E$ relative to $\sigma$. For all $x \in E$, let $t_x$ be the translation of $E$ defined by $x$. The mapping $(x, u) \mapsto t_x \circ u$ is isomorphism $\Phi$ of the group $S$ onto the group $A$. The mapping $(x, u) \mapsto \sigma(u)x = u(x)$ of $E \times \mathscr{L}(E)$ into $E$ is continuous and bilinear and hence analytic; its restriction to $E \times G$ is therefore analytic. Thus the group $S$, with the product manifold structure of $E$ and $G$, is a Lie group. We transport this structure to $A$ by means of $\Phi$. Then $A$ becomes a Lie group, the internal semi-direct product of $G$ by $T$ as a Lie group.