The Convenient Setting for non-Quasianalytic Denjoy--Carleman Differentiable Mappings

TL;DR

This paper establishes that for certain non-quasianalytic Denjoy--Carleman classes, the class of $C^M$-mappings is stable under composition and forms a cartesian closed category, with applications to diffeomorphism groups.

Contribution

It proves that $C^M$-mappings are characterized by their action on $C^M$-curves and that these mappings form a cartesian closed category in the non-quasianalytic setting.

Findings

$C^M$-mappings are characterized by their action on $C^M$-curves.

The category of $C^M$-mappings is cartesian closed.

The group of $C^M$-diffeomorphisms forms a $C^M$-Lie group.

Abstract

For Denjoy--Carleman differential function classes $C^M$ where the weight sequence $M=(M_k)$ is logarithmically convex, stable under derivations, and non-quasianalytic of moderate growth, we prove the following: A mapping is $C^M$ if it maps $C^M$-curves to $C^M$-curves. The category of $C^M$-mappings is cartesian closed in the sense that $C^M(E,C^M(F,G))\cong C^M(E\x F, G)$ for convenient vector spaces. Applications to manifolds of mappings are given: The group of $C^M$-diffeomorphisms is a $C^M$-Lie group but not better.