The convenient setting for non-quasianalytic Denjoy–Carleman differentiable mappings

For Denjoy–Carleman differentiable function classes CM where the weight sequence M = (Mk) is logarithmically convex, stable under derivations, and non-quasianalytic of moderate growth, we prove the following: A mapping is CM if it maps CM-curves to CM-curves. The category of CM-mappings is cartesian closed in the sense that CM(E,CM(F,G))∼= CM(E × F,G) for convenient vector spaces. Applications to manifolds of mappings are given: The group of CM-diffeomorphisms is a CM-Lie group but not better. © 2009 Elsevier Inc. All rights reserved