An Inverse Function Theorem in Frechet Spaces

TL;DR

This paper introduces a new inverse function theorem for Frechet spaces that broadens applicability by weakening differentiability requirements and avoids Newton iteration, using Lebesgue's dominated convergence and Ekeland's variational principle.

Contribution

It presents a novel inverse function theorem in Frechet spaces that does not depend on classical differentiability or Newton iteration, expanding the scope of inverse function results.

Findings

The theorem applies to maps with minimal differentiability assumptions.

It generalizes the classical Nash-Moser inverse function theorem.

The proof employs Lebesgue's dominated convergence theorem and Ekeland's variational principle.

Abstract

I present an inverse function theorem for differentiable maps between Frechet spaces which contains the classical theorem of Nash and Moser as a particular case. In contrast to the latter, the proof does not rely on the Newton iteration procedure, but on Lebesgue's dominated convergence theorem and Ekeland's variational principle. As a consequence, the assumptions are substantially weakened: the map F to be inverted is not required to be C^2, or even C^1, or even Frechet-differentiable.