Rough and strongly rough norms on Banach spaces
HTML articles powered by AMS MathViewer
- by G. Godini PDF
- Proc. Amer. Math. Soc. 87 (1983), 239-245 Request permission
Abstract:
We give equivalent conditions for a norm on a Banach space $X$ to be rough using the sets $A(x)$ defined for each $x \in X$ by $A(x) = \{ {f \in {X^ * }|f(x) = || x ||,|| f || = 1\}}$. This enables us to obtain unitary characterizations for rough and strongly rough norms.References
- R. Anantharaman, T. Lewis, and J. H. M. Whitfield, Smoothability, strong smoothability and dentability in Banach spaces, Canad. Math. Bull. 24 (1981), no. 1, 59–68. MR 611210, DOI 10.4153/CMB-1981-009-9
- Béla Bollobás, An extension to the theorem of Bishop and Phelps, Bull. London Math. Soc. 2 (1970), 181–182. MR 267380, DOI 10.1112/blms/2.2.181
- Mahlon M. Day, Strict convexity and smoothness of normed spaces, Trans. Amer. Math. Soc. 78 (1955), 516–528. MR 67351, DOI 10.1090/S0002-9947-1955-0067351-1
- Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
- J. R. Giles, D. A. Gregory, and Brailey Sims, Characterisation of normed linear spaces with Mazur’s intersection property, Bull. Austral. Math. Soc. 18 (1978), no. 1, 105–123. MR 493266, DOI 10.1017/S0004972700007863
- James Hagler and Francis Sullivan, Smoothness and weak$^{\ast }$ sequential compactness, Proc. Amer. Math. Soc. 78 (1980), no. 4, 497–503. MR 556620, DOI 10.1090/S0002-9939-1980-0556620-4
- K. John and V. Zizler, On rough norms on Banach spaces, Comment. Math. Univ. Carolin. 19 (1978), no. 2, 335–349. MR 500126
- J. Kurzweil, On approximation in real Banach spaces, Studia Math. 14 (1954), 214–231 (1955). MR 68732, DOI 10.4064/sm-14-2-214-231
- E. B. Leach and J. H. M. Whitfield, Differentiable functions and rough norms on Banach spaces, Proc. Amer. Math. Soc. 33 (1972), 120–126. MR 293394, DOI 10.1090/S0002-9939-1972-0293394-4
- R. R. Phelps, A representation theorem for bounded convex sets, Proc. Amer. Math. Soc. 11 (1960), 976–983. MR 123172, DOI 10.1090/S0002-9939-1960-0123172-X
- Francis Sullivan, Dentability, smoothability and stronger properties in Banach spaces, Indiana Univ. Math. J. 26 (1977), no. 3, 545–553. MR 438088, DOI 10.1512/iumj.1977.26.26042
- S. L. Troyanski, On locally uniformly convex and differentiable norms in certain non-separable Banach spaces, Studia Math. 37 (1970/71), 173–180. MR 306873, DOI 10.4064/sm-37-2-173-180 J. H. M. Whitfield, Rough and strongly rough norms on Banach spaces, Abstracta of the Seventh Winter School Praha, 1979.
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 87 (1983), 239-245
- MSC: Primary 46B20
- DOI: https://doi.org/10.1090/S0002-9939-1983-0681828-X
- MathSciNet review: 681828