Publication: The Failure of Rolle's Theorem in Infinite-Dimensional Banach Spaces
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2001-05-10
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Elsevier
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Abstract
We prove the following new characterization of Cp Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space X has a Cp smooth (Lipschitz)
bump function if and only if it has another Cp smooth (Lipschitz) bump function f such that its derivative does not vanish at any point in the interior of the support of f (that is, f does not satisfy Rolle's theorem). Moreover, the support of this bump can be assumed to be a smooth starlike body. The ``twisted tube'' method we use in the proof is interesting in itself, as it provides other useful characterizations of Cp smoothness related to the existence of a certain kind of deleting diffeomorphisms, as well as to the failure of Brouwer's fixed point theorem even for smooth self-mappings of starlike bodies in all infinite-dimensional spaces.
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1. D. Azagra, 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 anto their boundaries in infinite-dimensional Banach spaces, Bull. London Math. Soc. , in press.
3. D. Azagra and R. Deville, Subdifferential Rolle's and mean value inequality theorems, Bull. Austral. Math. Soco 56 (1997), 319-329.
4. D. Azagra and R. Deville, James theorem fai1s for starlike bodies, J. FWlct. Anal., in press.
5. D. Azagra and T. Dobrowo1ski, Smooth negligibi1ity of compact sets in infinite-dimensional Banach spaces, with applications, Math. Ann. 312 (1998), 445-463.
6. D. Azagra, J. Gómez, and J. A. Jaramillo, Rolle's theorem and negligibi1ity of points in infinite-dimensional Banach spaces, J. Math. Anal. Appl. 213 (1997), 487-495.
7. D. Azagra, "Smooth Negligibility and Subdifferential Caleulus in Banach Spaces, with Applications," doctoral dissertation, Universidad Complutense de Madrid, 1997.
8. Y. Benyamini and J. Lindenstrauss, "Geometrical Nonlinear Functional Analysis, Vol. 1,"Amer. Math. Soc. Colloq. Publ., Vol. 48, Amer. Math. Soc., Providenee, 2000.
9. Y. Benyamini and Y. Sternfe1d, Spheres in infinite-dimensional normed spaces are Lipschitz contractib1e, Proc. Amer. Math. Soco 88 (1983), 439-445.
10. J. Bes and J. Ferrera, Rolle's theorem fai1s in t 2 and co , private communication.
11. C. Bessaga, Every infinite-dimensional Hi1bert space is diffeomorphic with its unit sphere, Bull. Acad. PoZon. Sd. Sér. Sd. Math. Astr. Phys. 14 (1966), 27-31.
12. C. Bessaga, Interp1ay between infinite-dimensional topo1ogy and functional analysis: Mappings defined by explicit formulas and their applications, TopoZogy Proc. 19 (1994).
13. C. Bessaga and A Pe1czynski, "Se1eeted Topics in Infinite-Dimensional Topo1ogy," Monografie Matematyezne, Warszawa, 1975.
14. H. Cartan, "Caleul différentie1," Hennann, Paris, 1967.
15. M. M. Day, "Nonned Linear Spaces," third ed., Springer-Verlag, Berlin, 1973.
16. R. Deville, A mean value theorem for the non-differentiab1e mappings, Serdica Math. J. 21 (1995), 59-66.
17. R. Deville, G. Godefroy, and V. Ziz1er, "Smoothness and Renormings in Banach Spaces," Vol. 64, Pitman Monogr. Surveys Pure App1. Math., Vol. 64, Longman, Har1ow, 1993
18. J. Diestel, "Sequences and Series in Banach Spaces," Springer-Verlag, New York, 1984.
19. T. Dobrowolski, Smooth and R-ana1ytic negligibility of subsets and extension of homeomorphism in Banach spaces, Studia Math. 65 (1979), 115-139.
20. T. Dobrowolski, Every infinite-dimensional Hilbert space is real-ana1ytically isomorphic with its unit sphere, J. FWlct. Anal. 134 (1995), 350-362.
21. 1. Ferrer, Rolle's theorem faiIs in t¡, Amer. Math. Monthly 103 (1996), 161-165.
22. G. Godefroy, Sorne remarks on subdifferential calculus, Rev. Mat. Complut. 11 (1998), 269-279.
23. K. Goebel, On the minimal displacement of points under lipschitzian mappings, Pacific J. Math. 45 (1973), 151-163.
24. K. Goebel and W. A. Kirk, "Tapies in Metric Fixed Point Theory," Cambridge Studies in Advanced Mathematics, Vol. 28, Cambridge Univ. Press, Cambridge, UK, 1990.
25. K. Goebel and W. A. Kirk, A fixed point theorem for transformations whose iterates have uniform Lipschitz constant, Studia Math. 47 (1973),135-140.
26. P. Hájek, Smooth functions on co, Israel J. Math. 104 (1998), 17-27.
27. R. G. Haydon, A counterexample to several questions about scattered compact spaces, Bull. London Math. Soc. 22 (1990), 261-268.
28. S. Kakutani, Topological properties of the unit sphere of a Hilbert Space, Proc. Imp. Awd. Tokyo 19 (1943), 269-271.
29. V. L. Klee, Convex bodies and periodic homeomorphisms in Hilbert space, Trans. Amer. Math. Soc. 74 (1953), 10-43.
30. V. L. Klee, Some topological properties of convex sets, Trans. Amer. Math. Soc. 78 (1955), 30-45.
31. P. K. Lin and Y. Sternfeld, Convex sets with the Lipschitz fixed point property are compact, Proc. Amer. Math. Soc. 93 (1985), 633-639.
32. B. Nowak, On the Lipschitzian retraction of the unit ball in infinite-dimensional Banach spaces onto its boundary, Bull. Acad. Polon. Sd. 27 (1979),861-864.
33. S. A. Shkarin, On Rolle's theorem in infinite-dimensional Banach spaces, Mat. Zametki 51 (1992), 128-136.