Lemma 1

Tag 0AA8, permanent. Cite it as [Bourbaki, Tag 0AA8].

Lie Groups and Lie Algebras, chapter III, § 3. Passage from a Lie group to its Lie algebra

Let X and $X'$ be complete normable spaces, $X_0$ an open neighbourhood of 0 in X and f an analytic mapping of $X_0$ into $X'$ such that $f(0) = 0$ . Let $f = f_1 + f_2 + f_3 + \cdots$ be the expansion of f as an integral series about 0, where $f_i$ is a homogeneous continuous-polynomial of degree i on X with values in $X'$ . Let t be an element of TS $^2(X)$ , considered as a point distribution on X with support contained in \{0\}. Let $t' = f_*(t) \in TS(X')$ . The homogeneous component of $t'$ of degree 1 is $\langle f_2, t \rangle$ .

Let $t'_1$ be this component. Then, for every continuous linear mapping u of $X'$ into a polynormed space,

u(t'_1) = \langle t', u \rangle \quad \text{because } u \text{ is continuous and linear}

= \langle t, u \circ f \rangle \quad (\text{Diff. \& Anal. Man., R, 13.2.3})

= \langle t, u \circ f_2 \rangle \quad \text{because } t \in TS^2(X)

= u(\langle t, f_2 \rangle) \quad (\text{Diff. \& Anal. Man., R, 13.2.2}),

whence the lemma.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Proposition 24, § 3

Cites

Nothing.