%0 Journal Article
%T The Convenient Setting for Denjoy--Carleman Differentiable Mappings of Beurling and Roumieu Type
%A Andreas Kriegl
%A Peter W. Michor
%A Armin Rainer
%J Mathematics
%D 2011
%I arXiv
%R 10.1007/s13163-014-0167-1
%X 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.
%U http://arxiv.org/abs/1111.1819v3