The Convenient Setting for Denjoy--Carleman Differentiable Mappings of Beurling and Roumieu Type

TL;DR

This paper establishes that Denjoy--Carleman classes of Beurling and Roumieu types form a convenient setting under certain conditions, enabling cartesian closed categories and applications to diffeomorphism groups.

Contribution

It proves that these classes admit a cartesian closed structure in the convenient setting, extending the functional analytic framework for Denjoy--Carleman differentiable mappings.

Findings

Categories are cartesian closed for log-convex, moderate growth weight sequences.

The group of diffeomorphisms forms a regular Lie group within these classes.

Applications include manifolds of mappings and Lie group structures.

Abstract

We prove in a uniform way that all Denjoy--Carleman differentiable function classes of Beurling type $C^{(M)}$ and of Roumieu type $C^{\{M\}}$, admit a convenient setting if the weight sequence $M=(M_k)$ is log-convex and of moderate growth: For $\mathcal C$ denoting either $C^{(M)}$ or $C^{\{M\}}$, the category of $\mathcal C$-mappings is cartesian closed in the sense that $\mathcal C(E,\mathcal C(F,G))\cong \mathcal C(E\times F, G)$ for convenient vector spaces. Applications to manifolds of mappings are given: The group of $\mathcal C$-diffeomorphisms is a regular $\mathcal C$-Lie group if $\mathcal C \supseteq C^\omega$, but not better.