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The uniform Kadec-Klee property for the Lorentz spaces Lw,1

Part of: General convexity Topological linear spaces and related structures Linear function spaces and their duals Spaces with richer structures Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  09 April 2009

S. J. Dilworth  and
Yu-Ping Hsu
S. J. Dilworth
Affiliation:
Department of MathematicsUniversity of South CarolinaColumbia, SC 29208, USA
Yu-Ping Hsu
Affiliation:
Department of General StudiesNational Taiwan Ocean UniversityKeelung, TaiwanROC
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Abstract

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In this paper we show that the Lorentz space Lw, 1(0, ∞) has the weak-star uniform Kadec-Klee property if and only if inft>0 (w(αt)/w(t)) > 1 and supt>0(φ(αt) / φ(t))< 1 for all α ∈ (0, 1), where φ(t) = ∫t0 w(s) ds.

Keywords

Lorentz spaceuniform Kadec-Klee Propertyfixed point property.

MSC classification

Secondary: 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46B25: Classical Banach spaces in the general theory 54E40: Special maps on metric spaces 52A07: Convex sets in topological vector spaces 46A50: Compactness in topological linear spaces; angelic spaces, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 60 , Issue 1 , February 1996 , pp. 7 - 17
Copyright
Copyright © Australian Mathematical Society 1996

References

[1]Altshuler, Z., ‘Uniform convexity in Lorentz sequence spaces’, Israel J. Math. 20 (1975) 260–275.CrossRefGoogle Scholar
[2]Altshuler, Z., ‘The modulus of convexity in Lorentz and Orlicz sequences’, Notes in Banach spaces (Lacey, E., ed. University of Texas Press, 1980).Google Scholar
[3]Besbes, M., Dilworth, S. J., Dowling, P. N. and Lennard, C. J., ‘New convexity and fixed point properties in Hardy and Lebesgue-Bochner spaces’, J. Funct. Anal. 119 (1994) 340–357.CrossRefGoogle Scholar
[4]Carothers, N. L., ‘Rearrangement invariant subspaces of Lorentz function spaces’, Israel J. Math. 40 (1981) 217–228.CrossRefGoogle Scholar
[5]Carothers, N. L., Dilworth, S. J., Lennard, C. J. and Trautman, D. A., ‘A fixed point property for the Lorentz space L ρ,1 (μ)’, Indiana Univ. Math. J. 40 (1991) 345–352.CrossRefGoogle Scholar
[6]Carothers, N. L., Dilworth, S. J. and Trautman, D. A., ‘On the geometry of the unit spheres of the Lorentz spaces Lw,1’, Glasgow Math. J. 34 (1992) 21–25.CrossRefGoogle Scholar
[7]van Dulst, D. and Sims, B., ‘Fixed points of non-expansive mappings and Chebyshev centers in Banach spaces with norms of type (KK)’, in: Banach space theory and its applications, Lecture Notes in Mathematics 991 (Springer, Berlin, 1983) pp. 35–43.CrossRefGoogle Scholar
[8]Halperin, I. ‘Uniform convexity in function spaces’ Duke Math. J. 21 (1954) 195–204.CrossRefGoogle Scholar
[9]Hsu, Yu-Ping, ‘The lifting of the UKK property from E to CE’, Proc. Amer. Math. Soc., to appear.Google Scholar
[10]Huff, R., ‘Banach spaces which are nearly uniformly convex’, Rocky Mountain J. Math. 10 (1980) 743–749.CrossRefGoogle Scholar
[11]Kirk, W. A., ‘A fixed point theorem for mappings which do not increase distances’, Amer. Math. Monthly 72 (1965) 1004–1006.CrossRefGoogle Scholar
[12]Lennard, C. J., ‘A new convexity property that implies a fixed point property for L 1’, Studia Math. 100 (1991) 95–108.CrossRefGoogle Scholar
[13]Lennard, C. J., ‘1 is uniformly Kadec-Klee’, Proc. Amer. Math. Soc. 109 (1990) 71–77.Google Scholar
[14]Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces II: Function spaces (Springer-Verlag, Berlin-Heidelberg, 1979).CrossRefGoogle Scholar
[15]Lorentz, G. G., ‘Some new functional spaces’, Ann. of Math. 51 (1950) 37–55.CrossRefGoogle Scholar
[16]Sedaev, A. A., ‘The H-property in symmetric spaces’, Teor. Funktsii Funktsional. Anal. i Prilozhen. 11 (1970) 67–80 (in Russian).Google Scholar