Corollary 1

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

Lie Groups and Lie Algebras, chapter IX, § 9. Compact Lie groups operating on manifolds

Assume that X is compact. There exists an analytic linear representation $\rho$ of G on a finite dimensional vector space E and an embedding $\varphi : X\rightarrow E$ such that $\varphi (gx) =\rho (g)\varphi (x)$ for $g\in G, x\in X$ .

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

Corollary 2, § 9

Cites

Nothing.