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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.

46B20 Geometry and structure of normed linear spaces
58B99 Infinite-dimensional manifolds
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