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The Failure of Rolle's Theorem in Infinite-Dimensional Banach Spaces

✍ Scribed by Daniel Azagra; Mar Jiménez-Sevilla

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
171 KB
Volume
182
Category
Article
ISSN
0022-1236

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✦ Synopsis

We prove the following new characterization of C p (Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space X has a C p smooth (Lipschitz) bump function if and only if it has another C p smooth (Lipschitz) bump function f such that its derivative does not vanish at any point in the interior of the support of f (that is, f does not satisfy Rolle's theorem). Moreover, the support of this bump can be assumed to be a smooth starlike body. The ``twisted tube'' method we use in the proof is interesting in itself, as it provides other useful characterizations of C p smoothness related to the existence of a certain kind of deleting diffeomorphisms, as well as to the failure of Brouwer's fixed point theorem even for smooth self-mappings of starlike bodies in all infinite-dimensional spaces.

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