On Yamamuro's inverse and implicit function theorems in terms of calibrations

TL;DR

This paper investigates conditions under which certain smooth maps on the space of smooth functions satisfy Yamamuro's inverse function theorem, revealing that only affine functions meet the criteria, and emphasizes testing inverse theorems with simple maps.

Contribution

The paper demonstrates that the map x ↦ φ ∘ x satisfies Yamamuro's inverse function theorem only when φ is affine, highlighting the importance of testing inverse theorems with simple examples.

Findings

The map satisfies Yamamuro's theorem only if φ is affine.

Simple maps can effectively test the applicability of inverse theorems.

The paper discusses the limitations of inverse theorems in infinite-dimensional spaces.

Abstract

For the Frechet space E=C^{\infty}(S^1) and for a smooth \phi: R to R, we prove that the associated map E to E given by x mapsto\phi\circ x satisfies the continuous B\Gamma--differentiability condition in Yamamuro's inverse function theorem only if \phi is affine. Via more complicated examples, we also generally discuss the importance of testing the applicability of proposed inverse and implicit function theorems by this kind of simple maps.