The convenient setting for quasianalytic DenjoyCarleman differentiable mappings

Abstract. For quasianalytic Denjoy–Carleman differentiable function classes CQ where the weight sequence Q = (Qk) is log-convex, stable under deriva-tions, of moderate growth and also an L-intersection (see (1.6)), we prove the following: The category of CQ-mappings is cartesian closed in the sense that CQ(E,CQ(F,G)) ∼ = CQ(E × F,G) for convenient vector spaces. Applications to manifolds of mappings are given: The group of CQ-diffeomorphisms is a regular CQ-Lie group but not better. Classes of Denjoy-Carleman differentiable functions are in general situated be-tween real analytic functions and smooth functions. They are described by growth conditions on the derivatives. Quasianalytic classes are those where infinite Taylor expansion is an injective mapping. That a class of mappings S admits a convenient setting means essentially that we can extend the class to mappings between admissible infinite dimensional spaces E,F,... so that S(E,F) is again admissible and we have S(E × F,G) canonically S-diffeomorphic to S(E,S(F,G)) (the exponential law). Usually this comes han