# Nash–Moser theorem

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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.

## Introduction

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.

## History

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.

## Formal statement

The formal statement of the theorem is as follows:[1]

Let $F$ and $G$ be tame Frechet spaces and let $P:U\subseteq F\rightarrow G$ be a smooth tame map. Suppose that the equation for the derivative $DP(f)h = k$ has a unique solution $h=VP(f)k$ for all $f \in U$ and all $k$, and that the family of inverses $VP: U \times G\rightarrow F$ is a smooth tame map. Then P is locally invertible, and each local inverse $P^{-1}$ is a smooth tame map.

## References

1. ^ 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. MR 0656198.. (A detailed exposition of the Nash–Moser theorem and its mathematical background.)