Corollary 2

Let H be a closed subgroup of G. There exists an analytic linear representation of G on a finite dimensional vector space E and a point $v\in E$ with fixer H.

Apply Cor. 1 to the canonical operation of G on the compact manifold $G/H$. This gives an analytic linear representation $\rho : G\rightarrow \mathbf{G}\mathbf{L}(E)$ and an embedding $\varphi : G/H\rightarrow E$ such that $\varphi (gx) =\rho (g)\varphi (x),g\in G, x\in G/H$. Let $\overline{e}\in G/H$ be the class of $e\in G$, and $v=\varphi (\overline{e})$ its image. For all $g\in G$, we have $\rho (g)v=v\Leftarrow \Rightarrow \varphi (g\overline{e}) =\varphi (\overline{e})\Leftarrow \Rightarrow g\overline{e}= \overline{e}\Leftarrow \Rightarrow g\in H$.