The Convenient Setting for Denjoy–carleman Differentiable Mappings of Beurling and Roumieu Type

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 = (Mk) is log-convex and of moderate growth: For C denoting either C(M) or C{M}, the category of C-mappings is cartesian closed in the sense that C(E, C(F,G)) ∼= C(E × F,G) for convenient vector spaces. Applications to manifolds of mappings are given: The group of C-diffeomorphisms is a regular C-Lie group if C ⊇ Cω , but not better.