Keywords
duality maps
uniformly convex spaces
uniformly smooth spaces
modulus of convexity
modulus of smoothness
espacios de Banach
mapeos de dualidad
espacios uniformemente convexos
espacios uniformemente suaves
módulo de convexidad
módulo de suavidad
How to Cite
Abstract
Using a result of C. E. Chidume, we present a shorter and improved proof of a result of Xu and Roach on characterization of uniformly convex and uniformly smooth Banach spaces by the duality map and inequalities involving strictly increasing and strictly decreasing functions.
References
Beurling, A.; Livingston, A. E. (1962) “A thoerem on duality mappings on Banach spaces”, Arkiv Matematik 4: 405–411.
Cioranescu, I. (1990) Geometry of Banach spaces, duality mappings and nonlinear problems. “Mathematics and its Applications” 62, Kluwer Academic Publishers, London.
Chidume, C.E. (1998) Geometric properties of Banach spaces and nonlinear iteractions. I.C.T.P., Trieste.
Day, M.M. (1944) “Uniform convexity in factor and conjugate spaces”, Annals of Mathematics 45(2): 375–385.
Day, M.M. (1941) “Reflexive Banach spaces not isomorphic to uniform convex spaces”, Bull. Amer. Math. Soc. 47: 313–317.
Diestel, J. (1975) Geometry of Banach spaces. Lec. N. in Math. 485, Springer-Verlag, Berlin.
Lindenstrauss, J.; Tzafriri, L. (1979) Classical Banach Spaces, II. Springer-Verlag, Berlin.
Reich, S. (1981) “On the asymptotic behavior of nonlinear semigroups and the range of accretive operators”, J. Math. Anl. Appl. 79: 113–126.
Xu, Z.; Roach, G.F. (1991) “Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces”, J. Math. Anal. Appl. 157: 189–210.