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oai:docta.ucm.es:20.500.14352/571282023-08-26T06:40:00Zcom_20.500.14352_14col_20.500.14352_15

Rolle’s Theorem and Negligibility of Points in Infinite Dimensional Banach Spaces Azagra Rueda, Daniel Gómez Gil, Javier Jaramillo Aguado, Jesús Ángel 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). 2023-06-20T16:49:12Z 2023-06-20T16:49:12Z 1997-09-15 journal article 0022-247X 10.1006/jmaa.1997.5552 https://hdl.handle.net/20.500.14352/57128 http://www.sciencedirect.com/science/journal/0022247X eng open access Elsevier