Theorem 1

Let G be a compact Lie group, $(g, x) \rightarrow gx$ a law of left operation of class $C^r$ of G on X, and A a compact subset of X. There exists an analytic linear representation $\rho$ of G on a finite dimensional vector space E, a morphism $\varphi : X\rightarrow E$ of class $C^r$, compatible with the operations of G, and an open neighbourhood U of A, stable under G, such that the restriction of $\varphi$ to U is an embedding.

Replacing A by the compact set GA, we are reduced to the case in which A is stable under G.

Let $E_0$ be a finite dimensional vector space such that there exists an element of $\mathscr{C}^r(X;E_0)$ that is an embedding in the neighbourhood of A (no. 1, Prop. 4); the set $\mathscr{P}$ of morphisms having this property is a non-empty open subset of $\mathscr{C}^r(X;E_0)$ (no. 1, Prop. 3). Consider the continuous linear representation of the compact group G on the space $\mathscr{C}^r(X;E_0) ($§6, no. 4, Lemma 4). By the Peter-Weyl theorem (Spectral Theory, in preparation), the union of the finite dimensional subspaces stable under G is dense in $\mathscr{C}^r(X;E_0)$; hence, there exists an element $\varphi_0$ of $\mathscr{P}$ such that the maps $x \rightarrow \varphi_0(gx)$, for $g\in G$, generate a finite dimensional vector subspace $E_1$ of $\mathscr{C}^r(X;E_0)$, which is clearly stable under the operation of G.

Take E to be the space Hom$_{\mathbf{R}}(E_1,E_0),\rho$ to be the representation of G on E induced by the action on $E_1$, and $\varphi : X\rightarrow E$ to be the map that associates to $x\in X$ the linear map $\psi \rightarrow \psi (x)$ from $E_1$ to $E_0$. This is a morphism of class $C^r$; for $x\in X,g\in G,\psi \in E_1$, we have (denoting by $\tau (g)$ the automorphism $x \rightarrow gx$ of X):

Let $\alpha :$ Hom$_{\mathbf{R}}(E_1,E_0)\rightarrow E_0$ be the linear map $u \rightarrow u(\varphi_0)$; we have $\alpha \circ \varphi =\varphi_0$, so $\varphi$ is an embedding in the neighbourhood of A because $\varphi_0$ is one. Hence, there exists an open neighbourhood U of A such that the restriction of $\varphi$ to U is an embedding; we can choose U stable under G by Lemma 3, hence the theorem.