Corollary 1

Tag 01FF, permanent. Cite it as [Bourbaki, Tag 01FF].

Lie Groups and Lie Algebras, chapter IX, § 6. Integration on compact Lie groups

Let F be a Banach space, $\rho : H\rightarrow \mathbf{G}\mathbf{L}(F)$ an analytic linear representation, $f\in \mathscr{C}^r(X; F)$ . For $x\in X$ , put

f^{\sharp}(x) =\int_H\rho (h).f(h^{-1}x)dh

Then $f^{\sharp}$ is a morphism of class $C^r$ from X to F, compatible with the operations of H; for $x\in X$ , we have (with $\tau_h$ denoting the automorphism $x \rightarrow hx$ of X)

d_xf^{\sharp}=\int_H(\rho (h)\circ d_{h^{-1}x}f\circ T_x(\tau_{h^{-1}}))dh\in \mathscr{L}(T_x(X); F) \tag{14}

The first assertion follows from the proposition applied to the bundle $X\times$ F, equipped with the law of operation $(h; (x, f)) \rightarrow (hx, \rho (h).f)$ . The second follows from Integration, Chap. III, §3, no. 2, Prop. 2, by applying to the vector integral $f^{\sharp}$ the homomorphism $d_x:\mathscr{C}^r(X; F)\rightarrow \mathscr{L}(T_x(X); F)$ which is continuous by the definition of the topology of compact $C^r$ -convergence.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Theorem 2, § 6

Cites

Integration, Chap. III, §3, no. 2, Prop. 2 not in this corpus: INT III