This is joint work with Ivar Ekeland. The Nash-Moser theorem allows to solve a functional equation F(x)=y of unknown x in a "scale" of Banach spaces, assuming that F(0)=0, that y is small and that near 0 the differential DF has a right inverse which loses derivatives. The classical proof uses a Newton iteration scheme, which converges when F is of class C^2 and y is very small. In contrast, we only assume that F is continuous and has a Gâteau first differential, which is right-invertible with loss of derivatives. Of course we assume that y is small, but our smallness assumption is less demanding. We solve the functional equation thanks to an iteration scheme. Each step of the scheme consists in solving a Galerkin approximation of the equation, using Ekeland's variational principle.