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Uniform Kadec-Klee Lorentz Spaces Lw,1 and Uniformly Concave Functions

Published online by Cambridge University Press:  20 November 2018

S. J. Dilworth  and
C. J. Lennard
S. J. Dilworth
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A., e-mail:dilworth@math.scarolina.edu
C. J. Lennard
Affiliation:
Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A., e-mail:chris@lennext.math.pitt.edu
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Abstract

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We consider the notion of a uniformly concave function, using it to characterize those Lorentz spaces Lw,1 that have the weak-star uniform Kadec-Klee property as precisely those for which the antiderivative ϕ of w is uniformly concave; building on recent work of Dilworth and Hsu. We also derive a quite general sufficient condition for a twice-differentiable ϕ to be uniformly concave; and explore the extent to which this condition is necessary.

Keywords

Primary: 46Esecondary: 46BLorentz spacesuniform Kadec-Klee propertyweak-star convergenceuniformly concave functions
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 39 , Issue 3 , 01 September 1996 , pp. 266 - 274
Copyright
Copyright © Canadian Mathematical Society 1996

References

[Ak] Akimovic, B. A., On uniformly convex and uniformly smooth Orlicz spaces, Teor. Funktsiï Funktsional. Anal, i Prilozhen. 15(1972), 114–220.Google Scholar
[AI] Altshuler, Z., Uniform convexity in Lorentz sequence spaces, Israel J. Math. 20(1975), 260–274.Google Scholar
[BM] Brodski, M. S.ï and D. P. Mil'man, On the center of a convex set, Dokl. Akad. Nauk. SSSR(N.S.) 59(1948), 837–840.Google Scholar
[C] Carothers, N. L., Symmetric Structures in Lorentz Spaces, Ph.D. Diss., Ohio State U. 1982.Google Scholar
[CDL] Carothers, N. L., Dilworth, S. J. and Lennard, C. J., On a localization of the UKK property and the fixed point property in Lw\, Proc. Conf. Func. Anal., Harm. Anal, and Prob., Univ. Missouri-Columbia, 1994, to appear.Google Scholar
[CDLT] Carothers, N. L., Dilworth, S. J., Lennard, C. J. and Trautman, D. A., A fixed point property for the Lorentz space Lp,1(μ), Indiana Univ. Math. J. 40(1991) 345–352.Google Scholar
[DH] Dilworth, S. J. and Hsu, Y. P., The uniform Kadec-Klee property for the Lorentz spaces Lw,1 , J. Austral. Math. Soc, to appear.Google Scholar
[DDDLS1 Dodds, P. G., Dodds, T. K., Dowling, P. N. and Sukochev, F. A., A uniform Kadec-Klee property for symmetric operator spaces, Math. Proc. Camb. Philo. Soc, to appear.Google Scholar
[DS]D. van Dulst and Sims, B., Fixed points of nonexpansive mappings and Chebyshev centers in Banach spaces with norms of type (KK), (Banach Theory and its Applications, Proceedings Bucharest, Lecture Notes in Mathematics 991), Springer-Verlag(1983), 35–43.Google Scholar
[Ha] Halperin, I., Uniform convexity in function spaces, Duke Math. J. 21(1954), 195—204.Google Scholar
[HK] Hudzik, H. and Kamihska, A., Monotonicity properties of Lorentz spaces, 1994, preprint.Google Scholar
[Hu] Huff, R., Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math. 10( 1980) 743–749.Google Scholar
[Kal] Kamihska, A., On uniformly convex Orlicz spaces, Indag. Math. 44(1982), 27–36.Google Scholar
[Ka2] Kamihska, A., Uniformly convexity of generalized Orlicz spaces, Arch. Math. 56(1991), 181—188.Google Scholar
[Kil] Kirk, W. A., A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72(1965), 1004–1006.Google Scholar
[Ki2] Kirk, W. A., An abstract fixed point theorem for nonexpansive mappings, Proc. Amer. Math. Soc. 82(1981), 640–642.Google Scholar
[Lu] Luxemburg, W. A. J., Banach function spaces, Thesis, Delft 1955.Google Scholar
[S] Sedaev, A. A., The H-property in symmetric spaces, Teor. Funktsiï Funktsional. Anal, i Prilozhen. 11(1970), 67–80.Google Scholar