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Azagra Rueda, Daniel and Jiménez Sevilla, María del Mar
(2001)
*The Failure of Rolle's Theorem in Infinite-Dimensional Banach Spaces.*
Journal of Functional Analysis , 182
(1).
pp. 207-226.
ISSN 0022-1236

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Official URL: http://www.sciencedirect.com/science/journal/00221236

## Abstract

We prove the following new characterization of Cp Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space X has a Cp smooth (Lipschitz)

bump function if and only if it has another Cp 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 Cp 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.

Item Type: | Article |
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Uncontrolled Keywords: | Negligibility; Rolle theorem; Smooth norm; Brouwer fixed point theorem; Bump |

Subjects: | Sciences > Mathematics > Functional analysis and Operator theory |

ID Code: | 14493 |

Deposited On: | 01 Feb 2012 11:35 |

Last Modified: | 25 Jun 2013 15:11 |

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