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Rolle’s Theorem and Negligibility of Points in Infinite Dimensional Banach Spaces

Azagra Rueda, Daniel and Gómez Gíl, 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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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
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:14 May 2013 14:36

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