Example

(1) Let G be the additive group of a complete normable space E. The canonical isomorphism of L(G) onto E satisfies condition (i) of Theorem 4 and is therefore an exponential mapping of G.

(2) Let A be a complete normed unital associative algebra. Let A* be the Lie group consisting of the invertible elements of A. We identify L(A*) with A (§ 3, no. 9, Corollary to Proposition 33). If K = \mathbf{R} or \mathbf{C}, we know that the mapping exp of A into A* defined in Chapter II, § 7, no. 3 satisfies condition (i) of Theorem 4 and hence is an exponential mapping. Now let K be 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 $\lambda = 1$. Let U be the set of $x \in A$ such that $\|x\| < \lambda$. We know (Chapter II, § 8, no. 4) that the mapping exp of U into A* satisfies condition (i) of Theorem 4 and hence is an exponential mapping. Note that U is an additive subgroup of A.

This example explains the terminology adopted in Definition 1.

Let G be a Lie group germ and $\phi$ an exponential mapping of G. Then $\phi$ is étale at 0 and hence there exists an open neighbourhood U of 0 in L(G) such that $\phi(U)$ is open in G and $\phi|U$ is an isomorphism of the analytic manifold U onto the analytic manifold $\phi(U)$.

A canonical chart (of the first species) on G is a chart $\psi$ on the analytic manifold G whose inverse mapping is an exponential mapping. If further G is finite-dimensional and a basis of L(G) is chosen, the coordinate system defined by $\psi$ and this basis in the domain of $\psi$ is called a canonical coordinate system (of the first species).