An inverse function theorem converse

An inverse function theorem in Fréchet spaces

I present an inverse function theorem for differentiable maps between Fréchet 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: th...

On the Inverse Function Theorem

Weil Converse Theorem

Hecke generalized this equivalence, showing that an integral form has an associated Dirichlet series which can be analytically continued to C and satisfies a functional equation. Conversely, Weil showed that, if a Dirichlet series satisfies certain functional equations, then it must be associated to some integral form. Our goal in this paper is to describe this work. In the first three sections...

An Inverse Spectral Theorem

We prove a substantial extension of an inverse spectral theorem of Ambarzumyan, and show that it can be applied to arbitrary compact Riemannian manifolds, compact quantum graphs and finite combinatorial graphs, subject to the imposition of Neumann (or Kirchhoff) boundary conditions.

An Inverse Function Theorem for Metrically Regular Mappings

We prove that if a mapping F : X → → Y , where X and Y are Banach spaces, is metrically regular at x̄ for ȳ and its inverse F−1 is convex and closed valued locally around (x̄, ȳ), then for any function G : X → Y with lipG(x̄) · regF (x̄ | ȳ)) < 1, the mapping (F + G)−1 has a continuous local selection x(·) around (x̄, ȳ + G(x̄)) which is also calm.