Nash–Moser theorem

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.

References

• Hamilton, Richard S. (1982). "The inverse function theorem of Nash and Moser" (PDF-12MB). Bulletin of the American Mathematical Society 7 (1): 65–222. doi:10.1090/S0273-0979-1982-15004-2. MR0656198. http://www.ams.org/bull/1982-07-01/S0273-0979-1982-15004-2. . (A detailed exposition of the Nash–Moser theorem and its mathematical background.)

• Moser, Jürgen (1966a), "A rapidly convergent iteration method and non-linear partial differential equations. I", Ann. Scuola Norm. Sup. Pisa (3) 20: 265–315, MR0199523, http://www.numdam.org/item?id=ASNSP_1966_3_20_2_265_0 
• Moser, Jürgen (1966b), "A rapidly convergent iteration method and non-linear partial differential equations. II", Ann. Scuola Norm. Sup. Pisa (3) 20: 499–535, MR0206461, http://www.numdam.org/item?id=ASNSP_1966_3_20_3_499_0 
• Nash, John (1956), "The imbedding problem for Riemannian manifolds", Annals of Mathematics 63 (1): 20–63, doi:10.2307/1969989, JSTOR 1969989, MR0075639 .