zbMATH — the first resource for mathematics

Smooth negligibility of compact sets in infinite-dimensional Banach spaces, with applications. (English) Zbl 0917.46007
Summary: We give a method of constructing $$C^p$$ diffeomorphisms (with $$p\in\mathbb{N}\cup \{\infty\})$$ between an infinite-dimensional Banach space $$X$$ and $$X\setminus A$$, where $$A$$ is either a compact set or an infinite-codimensional subspace, provided that $$X$$ has a (not necessarily equivalent) $$C^p$$ smooth norm. As an application we give a complete smooth classification of the convex bodies of every Banach space. In particular, we prove that every smooth convex body containing no linear subspaces in an infinite-dimensional Banach space is diffeomorphic to a half-space. Other applications concern, e.g., Garay’s phenomena for ODE’s in Banach spaces, and the existence of periodic diffeomorphisms without fixed points on Banach spaces.

MSC:
 46B20 Geometry and structure of normed linear spaces 58B99 Infinite-dimensional manifolds
Full Text: