The Nash–Moser theorem, attributed to mathematicians John Forbes Nash and Jürgen Moser, is a generalization of the inverse function theorem on Banach spaces to a class of "tame" Fréchet spaces. In contrast to the Banach space case, in which the invertibility of the derivative at a point is sufficient for a map to be locally invertible, the Nash–Moser theorem requires the derivative to be invertible in a neighborhood. The theorem is widely used to prove local uniqueness for non-linear partial differential equations in spaces of smooth functions.
While Nash (1956) originated the theorem as a step in his proof of the Nash embedding theorem, Moser (1966a, 1966b) showed that Nash's methods could be successfully applied to solve problems on periodic orbits in celestial mechanics.
- Moser, Jürgen (1966a), A rapidly convergent iteration method and non-linear partial differential equations. I, Ann. Scuola Norm. Sup. Pisa (3) 20: 265–315, MR 0199523
- Moser, Jürgen (1966b), A rapidly convergent iteration method and non-linear partial differential equations. II, Ann. Scuola Norm. Sup. Pisa (3) 20: 499–535, MR 0206461
- Nash, John (1956), The imbedding problem for Riemannian manifolds, Annals of Mathematics 63 (1): 20–63, doi:10.2307/1969989, JSTOR 1969989, MR 0075639.