Proposition 17

Tag 0A9V, permanent. Cite it as [Bourbaki, Tag 0A9V].

Let $t \in \mathscr{T}^{(s)}(G)$ , $t' \in \mathscr{T}^{(s')}(X)$ and $f : X \to F$ a function of class $C^r$ with $s + s' \leq r$ . Then

\langle t', t * f \rangle = \langle t^\vee * t', f \rangle.

\begin{align*} \langle t', t * f \rangle &= \langle t', x \mapsto \langle t, g \mapsto f(g^{-1}x) \rangle \rangle \quad \text{by (4)} \\ &= \langle t \otimes t', (g, x) \mapsto f(g^{-1}x) \rangle \quad (\text{Diff. \& Anal. Man., R, 13.4.4}) \\ &= \langle t^\vee \otimes t', (g, x) \mapsto f(gx) \rangle \quad (\text{Diff. \& Anal. Man., R, 13.2.3}) \\ &= \langle t^\vee * t', f \rangle. \end{align*}

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

Cited by

Nothing.