Keywords: Convenient vector space, diierentiation in innnite-dimensions, holo-morphic mappings in innnite-dimensions, real analytic mappings in innnite dimensions , partitions of unity, smoothly real compact spaces, innnite dimensional man-ifolds, regular innnite dimensional Lie groups, evolution operator, manifolds of mappings, diieomorphism groups,… (More)
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.
We investigate the rudiments of Riemannian geometry on orbit spaces M/G for isometric proper actions of Lie groups on Riemannian manifolds. Minimal geodesic arcs are length minimising curves in the metric space M/G and they can hit strata which are more singular only at the end points. This is phrased as convexity result. The geodesic spray, viewed as a… (More)
We show that one can lift locally real analytic curves from the orbit space of a compact Lie group representation, and that one can lift smooth curves even globally, but under an assumption.
Smooth, real analytic and holomorphic mappings deened on non-open subsets of innnite dimensional vector spaces are treated. 0. Introduction In this paper we will generalize the concept of diierentiable maps f : E X ! F deened on open subsets to such on more general subsets of innnite dimensional vector spaces. We will refer to the theories for open domains… (More)
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:… (More)
Regular Lie groups are infinite dimensional Lie groups with the property that smooth curves in the Lie algebra integrate to smooth curves in the group in a smooth way (an 'evolution operator' exists). Up to now all known smooth Lie groups are regular. We show in this paper that regular Lie groups allow to push surprisingly far the geometry of principal… (More)
We show that the roots of any smooth curve of polynomials with real roots only can be parametrized twice differentiable (but not better). In  we claimed that there exists a smooth curve of polynomials of degree 3 for which no C 1-parametrization of the roots exists. Unfortunately there was an error in the calculation of b 3 and we have been informed by… (More)
The convenient setting for smooth mappings, holomorphic mappings, and real analytic mappings in infinite dimension is sketched. Infinite dimensional manifolds are discussed with special emphasis on smooth partitions of unity and tangent vectors as derivations. Manifolds of mappings and diffeomorphisms are treated. Finally the differential structure on the… (More)