Definition 1

A normed Lie algebra over K is a Lie algebra with a norm such that

for all $x, y$ in $g$.

Throughout the rest of this paragraph, g denotes a complete normed Lie algebra.

For every finite set I we define as in § 7, no. 1 a continuous Lie algebra homomorphism $u \mapsto \tilde{u}$ of $\hat{L}(I)$ into $\hat{P}(g^I; g)$. We see as in § 7 that if $u = \sum_v u_v$, with $u_v \in L^v(I)$ for $v \in \mathbf{N}^I$, then $\tilde{u} = \sum_v \tilde{u}_v$, where $\tilde{u}_v$ is the polynomial mapping $(t_i)_{i \in I} \mapsto u_v((t_i))$ defined in § 2, no. 4. The composition formula (2) of § 7, no. 1, remains valid.