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

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.

41 pages; numbering changed in order to make it consistent with the published version; to appear in Rev. Mat. Complut. arXiv admin note: text overlap with arXiv:0909.5632

References in corpus (5)

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• On quasianalytic local rings
• Denjoy-Carleman differentiable perturbation of polynomials and unbounded operators
• Perturbation theory for normal operators
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