Proposition 2

The homomorphism $f_t : L(I) \to g$ such that $f_t(T_i) = t_i$ (§ 2, no. 4) can be extended by continuity to one and only one continuous homomorphism $\hat{f}_t$ of $\hat{L}(I)$ into $g$.

There exists $\alpha > 0$ such that $t_i \in g_\alpha$ for all $i \in I$; hence $f_t(L^\nu(I)) \subset g_{|\nu| \wedge \alpha}$ for all $\nu$ (Lemma 1), which implies the continuity of $\hat{f}_t$.

If $u \in \hat{L}(I)$, we write $u((t_i)) = \hat{f}_t(u)$. In particular, taking $g = \hat{L}(I)$, $u = u((T_i))$; in the general case, $u((t_i))$ is called the result of substituting the $t_i$ for the $T_i$ in the Lie formal power series $u((T_i))$. If $u = \sum_{\nu \in \mathbf{N}^{(I)}} u_\nu$, where $u_\nu \in L^\nu(X)$, the family $(u_\nu((t_i)))_{\nu \in \mathbf{N}^{(I)}}$ is summable and

Let $\sigma$ be a continuous homomorphism of $g$ into a complete Hausdorff filtered Lie algebra $g'$ such that $g' = \bigcup_{\alpha > 0} g'_\alpha$. For every finite family $t = (t_i)_{i \in I}$ of elements of $g$ and all $u \in \hat{L}(I)$,

for the homomorphism $\sigma \circ \hat{f}_t$ is continuous and maps $T_i$ to $\sigma(t_i)$ for $i \in I$.

Let $u = (u_j)_{j \in J}$ be a finite family of elements of $\hat{L}(I)$ and let $v \in \hat{L}(J)$; by substituting the $u_j$ for the $T_j$ in $v$, we obtain an element $w = v((u_j)_{j \in J})$ of $\hat{L}(I)$ denoted by $v \circ u$. Then

for every finite family $t = (t_i)_{i \in I}$ of elements of $g$, as is seen by operating with the continuous homomorphism $\hat{f}_t$ on the equation $w = v((u_j)_{j \in J})$.

Let $u = \sum_{\nu \in \mathbf{N}^{I}} u_\nu \in \hat{L}(I)$, where $u_\nu \in L^\nu(I)$. The mapping $\tilde{u}$ $(t_i) \mapsto u((t_i))$ of $g^I$ into $g$ is continuous: for in each of the open sets $g_\alpha$ with $\alpha > 0$ the family of $\tilde{u}_\alpha$ is uniformly summable and it suffices to prove that each $\tilde{u}_\nu$ is continuous, which is immediate by induction on $|\nu|$.

† A Lie formal power series is not in general a formal power series in the sense of Algebra, Chapter IV, § 6.