Exercise 7, § 1

Let E be a complete normable space over $\mathbf{R}$ and F a closed vector subspace of E such that there exists no bicontinuous linear bijection of E onto $F \times (E/F)$.‡ Let X be the analytic manifold $\mathbf{R} \times E \times F$. Let G be the

† For more details, see R. S. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. of Math., 72 (1961), pp. 295–323. ‡ For an example of such an ordered pair (E, F), cf. for example A. Douady, Le problème des modules pour les sous-espaces analytiques compacts d’un espace analytique donné, Ann. Inst. Fourier, 16 (1966), p. 16.

additive group of F, which operates on X by the mapping

of $G \times X$ into X. For all $x \in X$, let $R_x$ be the G-orbit of x, which is a quasi-submanifold of X. Then condition (a) of Proposition 10 is satisfied, but there exists no ordered pair $(Y, \pi)$ with the following properties: Y is an analytic manifold, $\pi$ is a morphism of X into Y and, for all $x \in X$, is surjective with kernel $T_x(R_x)$.

(Suppose that $(Y, \pi)$ exists. Let H be the tangent space to Y at $\pi(0, 0, 0)$. Then H is isomorphic to $\mathbf{R} \times F \times (E/F)$. On the other hand H is isomorphic to $T_{\pi(x)}Y$ for x sufficiently close to $(0, 0, 0)$ and is hence isomorphic to $\mathbf{R} \times E$.)