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

Andreas Kriegl; Peter W. Michor; Armin Rainer

doi:10.1016/j.jfa.2009.03.003.x

paper

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

arXiv:0804.2995 · doi:10.1016/j.jfa.2009.03.003.x

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.

LaTeX, 29 pages, Some misprints corrected. Again some misprints corrected