Armin Rainer

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.

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

PERTURBATION THEORY FOR NORMAL OPERATORS

C^{(M)}

Equivalence of stability properties for ultradifferentiable function classes

C(M) and of Roumieu type

A new proof of Bronshtein’s theorem

C^{\{M\}}

An exotic zoo of diffeomorphism groups on \(\mathbb {R}^n\)

C{M}, admit a convenient setting if the weight sequence