Наш-Мозер теоремасы: кері функция теоремасының жалпылануы

Generalization of the inverse function theorem
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.

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 existence for non linear partial differential equations in spaces of smooth functions. It is particularly useful when the inverse to the derivative "loses" derivatives, and therefore the Banach space implicit function theorem cannot be used.

The Nash–Moser theorem traces back to , who proved the theorem in the special case of the isometric embedding problem. It is clear from his paper that his method can be generalized. , for instance, showed that Nash's methods could be successfully applied to solve problems on periodic orbits in celestial mechanics in the KAM theory. However, it has proven quite difficult to find a suitable general formulation; there is, to date, no all encompassing version; various versions due to Gromov, Hamilton, Hörmander, Saint Raymond, Schwartz, and Sergeraert are given in the references below. That of Hamilton's, quoted below, is particularly widely cited.