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. 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.