%0 Journal Article
%A Azagra Rueda, Daniel
%A Gómez Gil, Javier
%A Jaramillo Aguado, Jesús Ángel
%T Rolle’s Theorem and Negligibility of Points in Infinite Dimensional Banach Spaces
%D 1997
%@ 0022-247X
%U https://hdl.handle.net/20.500.14352/57128
%X 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).
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