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

✍ Scribed by D Azagra; J Gómez; J.A Jaramillo

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
198 KB
Volume
213
Category
Article
ISSN
0022-247X

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✦ Synopsis

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 C 1 Ä 4 diffeomorphisms between a Banach space X and X _ 0 which are the identity out of a ball, and we prove that such diffeomorphisms exist for every C 1 smooth Banach space which can be linearly injected into a Banach space whose dual norm Ž . is locally uniformly rotund LUR .

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