No abstract

Abstract. We clarify the question whether for a smooth curve of polynomials one can choose the roots smoothly and related questions. Applications to perturbation theory of operators are given.

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

We present here "the" cartesian closed theory for real analytic mappings. It is based on the concept of real analytic curves in locally convex vector spaces. A mapping is real analytic, if it maps smooth curves to smooth curves and real analytic curves to real analytic curves. Under mild completeness conditions the second requirement can be replaced by: real analytic along affine lines. Enclosed and necessary is a careful study of locally convex topologies on spaces of real analytic mappings.As an application we also present the theory of manifolds of real analytic mappings: the group of real analytic diffeomorphisms of a compact real analytic manifold is a real analytic Lie group.1991 Mathematics Subject Classification. 26E15, 58C20, 58B10, 58D05, 58D15, 26E05, 26E20, 46F15.