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The convenient setting for Denjoy–Carleman differentiable mappings of Beurling and Roumieu type

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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.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090, Wien, Austria

    Andreas Kriegl, Peter W. Michor & Armin Rainer

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  1. Andreas Kriegl
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  2. Peter W. Michor
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  3. Armin Rainer
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Correspondence to Peter W. Michor.

Additional information

AK was supported by FWF-Project P 23028-N13; PM by FWF-Project P 21030-N13; AR by FWF-Project P 22218-N13.

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Kriegl, A., Michor, P.W. & Rainer, A. The convenient setting for Denjoy–Carleman differentiable mappings of Beurling and Roumieu type. Rev Mat Complut 28, 549–597 (2015). https://doi.org/10.1007/s13163-014-0167-1

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