Azagra Rueda, Daniel and Gómez Gil, Javier and Jaramillo Aguado, Jesús Ángel
(1997)
*Rolle’s Theorem and Negligibility of Points in Infinite Dimensional Banach Spaces.*
Journal of Mathematical Analysis and Applications, 213
(2).
pp. 487-495.
ISSN 0022-247X

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Official URL: http://www.sciencedirect.com/science/journal/0022247X

## Abstract

In this note we prove that if a differentiable function oscillates between y« and « on the boundary of the unit ball then there exists a point in the interior of the ball in which the differential of the function has norm equal or less than« . This kind of approximate Rolle’s theorem is interesting because an exact Rolle’s theorem does not hold in many infinite dimensional Banach spaces. A characterization of those spaces in which Rolle’s theorem does not hold is given within a large class of Banach spaces. This question is closely related to the existence of C1 diffeomorphisms between a Banach space X and X _ _04 which are the identity out of a ball, and we prove that such diffeomorphisms exist for every C1 smooth Banach space which can be linearly injected into a Banach space whose dual norm is locally uniformly rotund (LUR).

Item Type: | Article |
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Uncontrolled Keywords: | Rolle’s theorem in infinite-dimensional Banach spaces; Approximate Rolle’s theorem; Continuous norm whose dual norm is locally uniformly rotund; C1 bump function |

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

ID Code: | 14492 |

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Deposited On: | 01 Feb 2012 09:41 |

Last Modified: | 28 Jan 2016 16:01 |

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