Alberto Richsfeld: An introduction to the Nash–Moser theorem

The Nash–Moser theorem generalizes the inverse function theorem to maps between Fréchet spaces. This generalization is particularly useful for solving equations of the form $P(f)=0$ where $P$ is a nonlinear differential operator whose linearization exhibits loss of derivatives, a situation in which the classical inverse function theorem fails.

We begin by reviewing Newton’s method and explaining the phenomenon of loss of derivatives. We then show how Newton’s method can be adapted to the Fréchet-space setting, leading to the Nash–Moser theorem. We conclude by discussing the role of the Nash–Moser theorem in stability problems in general relativity.