Complutense University Library

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


Azagra Rueda, Daniel and Gómez Gil, 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

[img] PDF

Official URL:


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

1. C. Bessaga, Every infinite-dimensional Hilbert space is diffeomorphic with its unit sphere, Bull. Acad. PoZon. Sci. Sér. Sci. Math. 14 (1966), 27-31.

2. C. Bessaga and A. Pe1czynski, Selected tapies in infinite-dimensional topology, in "Monografie Matematyczne," PWN, Warsaw, 1975.

3. R Deville, G. Godefroy, and V. Zizler, Smoothness and renormings in Banach spaces, in "Pitman Monographs and Surveys in Pure and Applied Mathematics," VoL 64, Longman, Harlow, 1993.

4. T. Dobrowolski, Smooth and R-analytic negligibility of subsets and extension of homeomorphism in Banach spaces, Studia Math. 65 (1979), 115-139.

5. 1. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. (N S.) 1, No. 3 (1979),443-474.

6. M. Fabian, P. Hájek, and J Vanderwerff, On smooth variational principies in Banach spaces, 1. Math. Anal. Appl. 197 (1996), 153-172.

7. J Bés and J Ferrera, private communication.

8. J Ferrer, Rolle's theorem fails in 12 , Amer. Math. Month1y 103, No. 2 (1996), 161-165.

9. R R Phelps, Convex functions, monotone operators and differentiability, in "Lecture Notes in Mathematics," VoL 1364, Springer-Verlag, BerlinjNew York, 1993.

10. S. A. Shkarin, On Rolle's theorem in infinite-dimensional Banach spaces, Mat. Zametki 51, No. 3 (1992), 128-136.

Deposited On:01 Feb 2012 09:41
Last Modified:28 Jan 2016 16:01

Repository Staff Only: item control page