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Banach spaces which admit a norm with the uniform Kadec-Klee property

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Several results are established about Banach spaces Ӿ which can be renormed to have the uniform Kadec-Klee property. It is proved that all such spaces have the complete continuity property. We show that the renorming property can be lifted from Ӿ to the Lebesgue-Bochner space $L_2(Ӿ)$ if and only if Ӿ is super-reflexive. A basis characterization of the renorming property for dual Banach spaces is given.
Słowa kluczowe
  • Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208, U.S.A.
  • Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208, U.S.A.
  • Institute of Mathematics, Bulgarian Academy of Sciences, 1090 Sofia, Bulgaria.
  • [1] M. Besbes, S. J. Dilworth, P. N. Dowling and C. J. Lennard, New convexity and fixed point properties in Hardy and Lebesgue-Bochner spaces, J. Funct. Anal. 119 (1994), 340-357.
  • [2] J. Bourgain, La propriété de Radon-Nikodým, Publ. Math. Univ. Pierre et Marie Curie 36 (1979).
  • [3] J. Bourgain, D. H. Fremlin and M. Talagrand, Pointwise compact sets of Baire-measurable functions, Amer. J. Math. 100 (1979), 845-886.
  • [4] J. Bourgain and H. P. Rosenthal, Martingales valued in certain subsets of $L_1$, Israel J. Math. 37 (1980), 54-75.
  • [5] N. L. Carothers, S. J. Dilworth, C. J. Lennard and D. A. Trautman, A fixed point property for the Lorentz space $L_{p,1}(μ)$, Indiana Univ. Math. J. 40 (1991), 345-352.
  • [6] J. Castillo and F. Sanchez, Weakly p-compact, p-Banach-Saks and super-reflexive Banach spaces, J. Math. Anal. Appl., to appear.
  • [7] J. Diestel, Sequences and Series in Banach Spaces, Springer, New York, 1983.
  • [8] J. Diestel and J. J. Uhl, Vector Measures, Amer. Math. Soc., 1977.
  • [9] S. J. Dilworth and Y.-P. Hsu, The uniform Kadec-Klee property for the Lorentz spaces $L_{w,1}$, J. Austral. Math. Soc., to appear.
  • [10] W. T. Gowers, A space not containing $c_0$, $ℓ_1$, or a reflexive subspace, preprint, 1992.
  • [11] M. Girardi and W. B. Johnson, The complete continuity property and finite-dimensional decompositions, Canad. Math. Bull., to appear.
  • [12] V. I. Gurariĭ and N. I. Gurariĭ, On bases in uniformly convex and uniformly smooth Banach spaces, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 210-215 (in Russian).
  • [13] Y.-P. Hsu, The lifting of the UKK property from E to $C_E$, Proc. Amer. Math. Soc., to appear.
  • [14] Z. Hu and B.-L. Lin, RNP and CPCP in Lebesgue-Bochner function spaces, Illinois J. Math. 37 (1993), 329-347.
  • [15] R. Huff, Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math. 10 (1980), 743-749.
  • [16] R. C. James, Some self-dual properties of normed linear spaces, in: Symposium on Infinite Dimensional Topology, Ann. of Math. Stud. 69, Princeton Univ. Press, 1972, 159-175.
  • [17] R. C. James, A non-reflexive Banach space that is uniformly non-octahedral, Israel J. Math. 18 (1974), 145-155.
  • [18] R. C. James, Non-reflexive spaces of type 2, ibid. 30 (1978), 1-13.
  • [19] R. C. James, KMP, RNP and PCP for Banach spaces, in: Contemp. Math. 85, Amer. Math. Soc., 1989, 281-317.
  • [20] R. C. James, Unconditional bases and the Radon-Nikodým property, Studia Math. 95 (1990), 255-262.
  • [21] W. B. Johnson, H. P. Rosenthal and M. Zippin, On bases, finite-dimensional decompositions, and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488-506.
  • [22] D. Kutzarova and T. Zachariades, On orthogonal convexity and related properties, preprint, 1992.
  • [23] G. Lancien, Applications de la théorie de l'indice en géometrié des espaces de Banach, Ph.D. thesis, Univ. Paris VI, 1992.
  • [24] G. Lancien, On uniformly convex and uniformly Kadec-Klee renormings, preprint.
  • [25] C. J. Lennard, $C_1$ is uniformly Kadec-Klee, Proc. Amer. Math. Soc. 109 (1990), 71-77.
  • [26] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces 1, Sequence Spaces, Springer, Berlin, 1977.
  • [27] E. W. Odell and H. P. Rosenthal, A double-dual characterization of Banach spaces containing $ℓ_1$, Israel J. Math. 22 (1978), 290-294.
  • [28] J. R. Partington, On nearly uniformly convex Banach spaces, Math. Proc. Cambridge Philos. Soc. 93 (1983), 127-129.
  • [29] A. Pełczyński, On the impossibility of embedding of the space L in certain Banach spaces, Colloq. Math. 8 (1961), 199-203.
  • [30] S. Prus, Finite-dimensional decompositions of Banach spaces with (p,q)-estimates, Dissertationes Math. 263 (1987).
  • [31] S. Prus, Nearly uniformly smooth Banach spaces, Boll. Un. Mat. Ital. 7 (1989), 507-521.
  • [32] H. P. Rosenthal, Weak* Polish Banach spaces, J. Funct. Anal. 76 (1988), 267-316.
  • [33] H. P. Rosenthal, On the structure of non-dentable closed bounded convex sets, Adv. in Math. 70 (1988), 1-58.
  • [34] B. Tsirelson, Not every Banach space contains an imbedding of $ℓ_p$ or $c_0$, Functional Anal. Appl. 8 (1974), 138-141.
  • [35] A. Wessel, Some results on Dunford-Pettis operators, strong regularity and the Radon-Nikodým property, Séminaire d'Analyse Fonctionnelle, Paris VI-VII, 1985-86, Publ. Math. Univ. Paris VII.
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