Rolle's theorem and negligibility of points in infinite dimensional Banach spaces

[摘要] In this note we prove that if a differentiable function oscillates between -epsilon and epsilon 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 epsilon. 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 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 is locally uniformly rotund (LUR). (C) 1997 Academic Press.

[发布日期] 1997-09-15