Azagra Rueda, Daniel and Deville, Robert
(2001)
*James' Theorem Fails for Starlike Bodies.*
Journal of Functional Analysis , 180
(2).
pp. 328-346.
ISSN 0022-1236

PDF
5MB |

Official URL: http://www.sciencedirect.com/science/journal/00221236

## Abstract

Starlike bodies are interesting in nonlinear functional analysis because they are strongly related to bump function sand to n-homogeneous polynomials on Banach spaces, and their geometrical proper ties are thus worth studying. In this paper we deal wit the question whether James' theorem on the characterization of reflexivity holds for (smooth) starlike bodies, and we establish that a feeble form of this result is trivially true for starlike bodies in nonreflexive Banach spaces, but a reasonable strong version of James' theorem for starlike bodies is never true, even in the smooth case. We also study the related question as to how large the set of gradients of a bump function can be, and among other results we obtain the following new characterization of smoothness in Banach spaces: a Banach space X has a C-1 Lipschitz bump function if and only if there exists another C-1 smooth Lipschitz bump function whose set of gradients contains the unit ball of the dual space X*. This result might also be relevant to the problem of finding an Asplund space with no smooth bump functions.

Item Type: | Article |
---|---|

Uncontrolled Keywords: | Starlike body; Convex body; James' theorem; Characterization of reflexivity |

Subjects: | Sciences > Mathematics > Functional analysis and Operator theory |

ID Code: | 14494 |

References: | 1. D. Azagra, Diffeomorphisms between spheres and hyperplanes in infinite-dimensional Banach spaces, Studia Math. 125 (1997), 179-186. 2. D. Azagra and T. Dobrowolski, Smooth negligibility of compact sets in infinite-dimensional Banach spaces, with applications, Math. Ann. 312 (1998), 445-463. 3. D. Azagra, 1. Gómez, and 1. A. Jaramillo, Rolle's theorem and negligibility of points in infinite-dimensional Banach spaces, J. Math. Anal. Appl. 213 (1997), 487-495. 4. D. Azagra and M. Jiménez-Sevilla, Rolle's theorem is either false or trivial in infinitedimensional Banach spaces, preprint, 2000. 5. D. Azagra, "Smooth Negligibility and Subdifferential Calculus in Banach Spaces, with Applications," doctoral dissertation, Universidad Complutense de Madrid, 1997. 6. S. M. Bates, On smooth, nonlinear surjeetions of Banaeh spaees, Israel J. Math. 100 (1997), 209-220. 7. H. Cartan, "Caleul différentiel," Hennann, Paris, 1967. 8. G. Debs, G. Godefroy, and 1. Saint Raymond, Topologieal properties of the set ofnonnattaining linear funetions, Canad. J. Math. 47 (1995), 318-329. 9. R. Deville, G. Godefroy, and V. Zizler, "Smoothness and Renonnings in Banaeh Spaees," Pitman Monographies and Surveys in Pure and Applied Mathematies, Vol. 64, Pitman, London, 1993. 10. P. Hájek, Smooth funetions on co , Israel J. Math. 104 (1998), 17-27. 11. R. C. James, Weakly eompaet sets, Trans. Amer. Math. Soco 113 (1964), 129-140. 12. M. Jiménez-Sevilla and 1. P. Moreno, A note on nonn attaining funetions, Froc. Amer. Malh. So,. 126 (1998), 1989-1997. 13. E. B. Leaeh and 1. H. M. Whitfield, Differentiable funetions and rough norms on Banaeh spaees, Froc. Amer. Math. Soco (1972), 120-126. 14. S. A. Shkarin, On Rolle's theorem in infinite-dimensional Banaeh spaees, transl. from Mal. Z 51 (1992), 128-136. |

Deposited On: | 01 Feb 2012 09:32 |

Last Modified: | 01 Feb 2012 09:32 |

Repository Staff Only: item control page