The Failure of Rolle's Theorem in Infinite-Dimensional Banach Spaces

Journal of Functional Analysis

Volumn 182, Issue 1, 2001, Pages 207-226

  Azagra, Daniel ^a,b   Jiménez Sevilla, Mar ^a,b  

^a UNIVERSIDAD COMPLUTENSE DE MADRID   (Spain)

^b SORBONNE UNIVERSITÉ   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000208587     PISSN: 00221236     EISSN: None     Source Type: Journal    
DOI: 10.1006/jfan.2000.3709     Document Type: Article

References (33)

1. Azagra D. Diffeomorphisms between spheres and hyperplanes in infinite-dimensional Banach spaces. Studia Math. 125:1997;179-186.
2. D. Azagra, and, M. Cepedello, Smooth Lipschitz retractions of starlike bodies onto their boundaries in infinite-dimensional Banach spaces, Bull. London Math. Soc, in press.
3. Azagra D., Deville R. Subdifferential Rolle's and mean value inequality theorems. Bull. Austral. Math. Soc. 56:1997;319-329.
4. D. Azagra, and, R. Deville, James theorem fails for starlike bodies, J. Funct. Anal, in press.
5. Azagra D., Dobrowolski T. Smooth negligibility of compact sets in infinite-dimensional Banach spaces, with applications. Math. Ann. 312:1998;445-463.
6. Azagra D., Gómez J., Jaramillo J. A. Rolle's theorem and negligibility of points in infinite-dimensional Banach spaces. J. Math. Anal. Appl. 213:1997;487-495.
7. Azagra D. Smooth Negligibility and Subdifferential Calculus in Banach Spaces, with Applications. 1997;Universidad Complutense de Madrid.
8. Benyamini Y., Lindenstrauss J. Geometrical Nonlinear Functional Analysis, Vol. I. Amer. Math. Soc. Colloq. Publ. 48:2000;Amer. Math. Soc. Providence.
9. Benyamini Y., Sternfeld Y. Spheres in infinite-dimensional normed spaces are Lipschitz contractible. Proc. Amer. Math. Soc. 88:1983;439-445.
10. 0, private communication.
11. Bessaga C. Every infinite-dimensional Hilbert space is diffeomorphic with its unit sphere. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 14:1966;27-31.
12. Bessaga C. Interplay between infinite-dimensional topology and functional analysis: Mappings defined by explicit formulas and their applications. Topology Proc. 19:1994.
13. Bessaga C., Pelczynski A. Selected Topics in Infinite-Dimensional Topology. 1975;Monografie Matematyczne, Warszawa.
14. Cartan H. Calcul différentiel. 1967;Hermann, Paris.
15. Day M. M. Normed Linear Spaces. 1973;Springer-Verlag, Berlin.
16. Deville R. A mean value theorem for the non-differentiable mappings. Serdica Math. J. 21:1995;59-66.
17. Deville R., Godefroy G., Zizler V. Smoothness and Renormings in Banach Spaces. Pitman Monogr. Surveys Pure Appl. Math. 64:1993;Longman, Harlow.
18. Diestel J. Sequences and Series in Banach Spaces. 1984;Springer-Verlag, New York.
19. Dobrowolski T. Smooth and R-analytic negligibility of subsets and extension of homeomorphism in Banach spaces. Studia Math. 65:1979;115-139.
20. Dobrowolski T. Every infinite-dimensional Hilbert space is real-analytically isomorphic with its unit sphere. J. Funct. Anal. 134:1995;350-362.
21. 2. Amer. Math. Monthly. 103:1996;161-165.
22. Godefroy G. Some remarks on subdifferential calculus. Rev. Mat. Complut. 11:1998;269-279.
23. Goebel K. On the minimal displacement of points under Lipschitzian mappings. Pacific J. Math. 45:1973;151-163.
24. Goebel K., Kirk W. A. Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics. 28:1990;Cambridge Univ. Press, Cambridge.
25. Goebel K., Kirk W. A. A fixed point theorem for transformations whose iterates have uniform Lipschitz constant. Studia Math. 47:1973;135-140.
26. 0. Israel J. Math. 104:1998;17-27.
27. Haydon R. G. A counterexample to several questions about scattered compact spaces. Bull. London Math. Soc. 22:1990;261-268.
28. Kakutani S. Topological properties of the unit sphere of a Hilbert Space. Proc. Imp. Acad. Tokyo. 19:1943;269-271.
29. Klee V. L. Convex bodies and periodic homeomorphisms in Hilbert space. Trans. Amer. Math. Soc. 74:1953;10-43.
30. Klee V. L. Some topological properties of convex sets. Trans. Amer. Math. Soc. 78:1955;30-45.
31. Lin P. K., Sternfeld Y. Convex sets with the Lipschitz fixed point property are compact. Proc. Amer. Math. Soc. 93:1985;633-639.
32. Nowak B. On the Lipschitzian retraction of the unit ball in infinite-dimensional Banach spaces onto its boundary. Bull. Acad. Polon. Sci. 27:1979;861-864.
33. Shkarin S. A. On Rolle's theorem in infinite-dimensional Banach spaces. Mat. Zametki. 51:1992;128-136.