Publication: A note on norm attaining functionals
Loading...
Official URL
Full text at PDC
Publication Date
1998-07
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
American Mathematical Society
Citations
Exportar
Abstract
We are concerned in this paper with the density of functionals which do not attain their norms in Banach spaces. Some earlier results given for separable spaces are extended to the nonseparable case. We obtain that a Banach space X is reflexive if and only if it satisfies any of the following properties: (i) X admits a norm |· | with the Mazur Intersection Property and the set N A|· | of all norm attaining functionals of X* contains an open set, (ii) the set N A|· | 1 of all norm one elements of N A|· | contains a (relative) weak* open set of the unit sphere, (iii) X* has C* PCP and N A|· | 1 contains a (relative) weak open set of the unit sphere, (iv) X is WCG, X* has CPCP and N A|· | 1 contains a (relative) weak open set of the unit sphere. Finally, if X is separable, then X is reflexive if and only if N A|· | 1 contains a (relative) weak open set of the unit sphere.
Description
UCM subjects
Unesco subjects
Keywords
Citation
M. D. Acosta and M. R. Galan, New characterizations of the reflexivity in terms of the set of norm attaining functionals, preprint.
Richard D. Bourgin, Geometric aspects of convex sets with the Radon-Nikodým property, vol. 993, Lecture Notes in Mathematics, Springer-Verlag, 1983, 57-59.
Dongjian Chen and Bor-Luh Lin, Ball separation properties in Banach spaces, preprint.
G. Debs, G. Godefroy and J. Saint Raymond, Topological properties of the set of norm attaining linear functionals, Canad. J. Math.47(2), (1995), 318-329.
R. Deville, G. Godefroy, D. E. G. Hare and V. Zizler, Differentiability of convex functions and the convex point of continuity property in Banach spaces, Isr. J. Math.59(2), 1987, 245-255.
R. Deville, G. Godefroy, and V. Zizler, Smoothness and renormings in Banach spaces, vol. 64, Pitman Monograph and Surveys in Pure and Applied Mathematics, 1993.
P. G. Georgiev, On the residuality of the set of norms having Mazur's intersection property, Mathematica Balkanica5 (1991), 20-26.
J. R. Giles, D. A. Gregory, and B. Sims, Characterization of normed linear spaces with Mazur's intersection property, Bull. Austral. Math. Soc.18 (1978), 105-123.
G. Godefroy, Nicely smooth Banach spaces, Texas Functional Analysis Seminar, 1984-1985, Univ. of Texas Press, Austin, TX, 1985, pp. 117-124.
G. Godefroy, Boundaries of a convex set and interpolation sets, Math. Ann.277 (1987), 173-184.
B. V. Godun and S. L. Troyanski, Renorming Banach spaces with fundamental biorthogonal system, Contemporary Math.144 (1993), 119-126.
Zhibao Hu and Bor-Luh Lin, Smoothness and the asymptotic-norming properties of Banach spaces, Bull. Austral. Math. Soc.45 (1992), 285-296.
D. E. G. Hare, A dual characterization of Banach spaces with the convex point-of-continuity property, Canad. Math. Bull. Vol. 32274-280.
R. C. James, Reflexivity and the sup of linear functionals, Isr. J. Math.13 (1972), 289-300.
M. Jiménez Sevilla and J. P. Moreno, The Mazur intersection property and Asplund spaces, C. R. Acad. Sci. Paris Sér. I Math.321, 1219-1223, 1995.
M. Jimenez Sevilla and J. P. Moreno, Renorming Banach spaces with the Mazur Intersection Property, J. Funct. Anal.144 (1997), 486-504.
Bor-Luh Lin, Pei-Kee Lin and S. L. Troyanski, Characterizations of denting points, Proc. Amer. Math. Soc.102, (1988), 526-528.
S. Mazur, Uber schwache Konvergentz in den Raumen (LP) , Studia Math.4 (1933), 128- 133.
W. B. Moors, The relationship between Goldstine 's Theorem and the convex point of con- tinuity property, J. Math. Anal. Appl.188 (1994), 819-832.
J. P. Moreno, On the Weak* Mazur Intersection Property and Frechet differentiable norms on open dense sets, Bull. Sci. Math., to appear.
J. P. Moreno, Geometry of Banach spaces with (a, e)-property or (/3, e)-property, Rocky Mountain J. Math.27 (1997), 241-256. CMP 97:13
Y. I. Petunin and A. N. Plichko Some properties of the set of functionals which attain their supremum on the unit sphere, Ukrainian Math. Journal, 26 (1974), 85-88.
R. R. Phelps, Convex functions, monotone operators and differentiability, Lecture Notes in Math. 1364, Springer-Verlag, 1989; rev. ed., 1993.
S. Sullivan, Geometrical properties determined by the higher duals of a Banach space, Illinois J. of Math.21 (1977), 315-331.
S. L. Troyanski, On locally uniformly convex and differentiable norms in certain non sepa- rable Banach spaces, Studia Math.37 (1971), 173-180.