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In the mathematical field of analysis, the Nash–Moser theorem, discovered by mathematician John Forbes Nash and named for him and Jürgen Moser, is a generalization of the inverse function theorem on Banach spaces to settings when the required solution mapping for the linearized problem is not bounded.
For functions of a single variable, the theorem states that if is a continuously differentiable function with nonzero derivative at the point ; then is injective (or bijective onto the image) in a neighborhood of , the inverse is continuously differentiable near = (), and the derivative of the inverse function at is the reciprocal of the derivative of at : ′ = ′ = ′ (()).
Nash's proof of the C k - case was later extrapolated into the h-principle and Nash–Moser implicit function theorem. A simpler proof of the second Nash embedding theorem was obtained by Günther (1989) who reduced the set of nonlinear partial differential equations to an elliptic system, to which the contraction mapping theorem could be ...
A 1969 theorem of David Henderson states that every infinite-dimensional, separable, ... "The inverse function theorem of Nash and Moser". Bull. Amer. Math. Soc.
Due to Jürgen Moser's extension of Nash's ideas for application to other problems (notably in celestial mechanics), the resulting implicit function theorem is known as the Nash–Moser theorem. It has been extended and generalized by a number of other authors, among them Gromov, Richard Hamilton , Lars Hörmander , Jacob Schwartz , and Eduard ...
With some additional constraints on the Fréchet spaces and functions involved, there is an analog of the inverse function theorem called the Nash–Moser inverse function theorem, having wide applications in nonlinear analysis and differential geometry.
Such theorems are now known as Nash–Moser theorems. [ 27 ] In 1982, Hamilton published his formulation of Nash's reasoning, casting the theorem into the setting of tame Fréchet spaces ; Nash's fundamental use of restricting the Fourier transform to regularize functions was abstracted by Hamilton to the setting of exponentially decreasing ...
The former is the more common definition in areas of nonlinear analysis where the function spaces involved are not necessarily Banach spaces. For instance, differentiation in Fréchet spaces has applications such as the Nash–Moser inverse function theorem in which the function spaces of interest often consist of smooth functions on a manifold.