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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
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