Corollary 3

Assume that X is paracompact. There exists a real Hilbert space E, a continuous unitary representation $^{10}\rho$ of G on E and an embedding $\varphi : X\rightarrow E$ of class $C^r$ such that $\varphi (gx) =\rho (g)\varphi (x)$ for all $g\in G$ and all $x\in X$.

The space $X/G$ is locally compact (General Topology, Chap. III, §4, no. 5, Prop. 11). Its connected components are the images of the connected components of X, which are countable at infinity (General Topology, Chap. I, §9, no. 10, Th. 5); thus, they are themselves countable at infinity, which implies that $X/G$ is paracompact (loc. cit.). Hence, there exists a locally finite covering $(U'_{\alpha})_{\alpha\in I}$ of $X/G$ by relatively compact open sets, and a covering $(V'_{\alpha})_{\alpha\in I}$ such that $\overline{V}'_{\alpha}\subset U'_{\alpha}$ for all $\alpha \in I ($General Topology, Chap. IX, §4, no. 3, Cor. 1 of Th. 3); taking the inverse image, we obtain two locally finite coverings $(U_{\alpha})_{\alpha\in I}$ and $(V_{\alpha})_{\alpha\in I}$ of X by relatively compact open sets stable under G, such that $\overline{V}_{\alpha}\subset U_{\alpha}$ for all $\alpha \in I$.

For all $\alpha \in I$, there exists a representation $\rho_{\alpha}$ of G on a finite dimensional real vector space $E_{\alpha}$ and a morphism $\varphi_{\alpha}\in \mathscr{C}^r(X;E_{\alpha})$, compatible with the operations of G, whose restriction to $U_{\alpha}$ is an embedding (Th. 1). For $\alpha \in I$, let $a_{\alpha}$ be a numerical function of class $C^r$ on X, equal to 1 on $V_{\alpha}$ and to 0 outside $U_{\alpha}($Differentiable and Analytic Manifolds, Results, 5.3.6). Put $b_{\alpha}(x) =\int_Ga_{\alpha}(gx)dg$ for $x\in X$. The function $b_{\alpha}$ is of class $C^r$, invariant under G (§6, no. 4, Cor. 2), equal to 1 on $V_{\alpha}$ and to 0 outside $U_{\alpha}$. Give each $E_{\alpha}$ a Hilbert scalar product invariant under G (§1, no. 1), and $\mathbf{R}$ its canonical Hilbert structure; let E be the Hilbert sum of the family $(E_{\alpha}\oplus \mathbf{R})_{\alpha\in I}$, and let $\rho$ be the representation of G on E induced by the $\rho_{\alpha}$ and the trivial action of G on $\mathbf{R}$. For $x\in X$, put $\varphi (x) = (b_{\alpha}(x)\varphi_{\alpha}(x), b_{\alpha}(x))_{\alpha\in I}$. Then $\varphi$ is a morphism of class $C^r$ from X to E, compatible with the operations of G; we verify as in the proof of Prop. 4 (no. 1) that $\varphi$ is an embedding, which implies the corollary.

$^{10}$That is (Spectral Theory, in preparation) a continuous linear representation (Integration, Chap. VIII, §2, no. 1) such that the operator $\rho (g)$ is unitary for all