Lemma 3

Let L be a complete normed Lie algebra over K. Let G be the set of $x \in L$ such that $\|x\| < \lambda$. Let $H : G \times G \to G$ be the Hausdorff function of L (Chapter II, § 8, no. 3). (i) With the law of composition H, G is a Lie group in which 0 is identity element and $-x$ the inverse of x for all $x \in G$. (ii) Let $\phi$ be the identity mapping of G into L. The differential of $\phi$ at 0 is an isomorphism of the normable Lie algebra $L(G)$ onto L. (iii) For all $\mu \in \mathbf{R}_+^*$, let $G_\mu$ be the set of $x \in L$ such that $\|x\| < \mu$. Then the $G_\mu$, for $\mu < \lambda$, form a fundamental system of open and closed neighbourhoods of 0 and are subgroups of G. Assertions (i) and (iii) follow from Chapter II, § 8, no. 3, Proposition 3 and (ii) can be proved as in Lemma 2.

G is called the Lie group defined by L.