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

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

2011-11-08

arxiv.org/abs/1111.1819v1

Mathematics

Functional Analysis

40 pages. arXiv admin note: substantial text overlap with arXiv:0909.5632

Scientific paper

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 $\mathcal C(E,\mathcal C(F,G))\cong \mathcal C(E\x F, G)$ 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 $\mathcal C \supseteq C^\om$, but not better.

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