Lemma 2

Let L be a complete normed Lie algebra over $\mathbf{R}$ or $\mathbf{C}$. Let G be the set of $x \in L$ such that $\|x\| < \frac{1}{3} \log \frac{3}{2}$. Let $\theta$ be the mapping $x \mapsto -x$ of G into G. Let H be the restriction to $G \times G$ of the Hausdorff function of L (Chapter II, § 7, no. 2). (i) $(G, 0, \theta, H)$ is a Lie group germ. (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. (i) follows from Chapter II, § 7, no. 2. As $\phi$ is a chart on G, the differential $\psi$ of $\phi$ at 0 is an isomorphism of normable spaces. On the other hand, the expansion as an integral series $H = \sum_{i,j \geq 0} H_{ij}$ of the mapping H is such that $H_{11}(x, y) = \frac{1}{2}[x, y]$. By § 3, Proposition 24, for all $a, b$ in $L(G)$,

which proves (ii).

G is called the Lie group germ defined by L.

Suppose that K is ultrametric. Let $p$ be the characteristic of the residue field of K. If $p \neq 0$, let $\lambda = |p|^{1/(p-1)}$; if $p = 0$, let $\lambda = 1$.