Abstract
We introduce and study the notion of a distance type, on a Banach space, defined by a nested sequence of convex sets. Among other things, we show that there always exist distance types that are not types in the classical sense. Then, we recover the notion of the flat nested sequence of Milman and Milman and show that distance types defined by flat nested sequences coincide with the bidual types of Farmaki. These results are applied to show that a flat nested sequence of convex sets is Wijsman convergent to the intersection of their weak*-closures in bidual space.
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References
Appell, J., De Pascale, E. and Zabrejko, P. P.: Some remarks on Banach limits, Atti.Sem.Mat. Fis.Univ.Modena 42(1994), 273–278.
Beer, G.: Topologies on Closed and Closed Convex Sets, Math. Appl. 268, Kluwer Acad. Publ., ordrecht, 1993.
Beer, G. and Borwein, J.: Mosco convergence and reflexivity, Proc.Amer.Math.Soc. 109 (1990), 427–436.
Borwein, J. M. and Lewis, A.: Convergence of decreasing sequences of convex sets in nonreflexive Banach spaces, Set-Valued Anal. 1(1993), 355–363.
Borwein, J. M. and Vanderwerff, J.: Further arguments for slice convergence in nonreflexive spaces, Set-Valued Anal. 2(1994), 529–544.
Farmaki, V.: c 0-subspaces and fourth-dual types, Proc.Amer.Math.Soc. 102(1988), 321–328.
Godefroy, G. and Kalton, N. J.: The ball topology and its applications, Contemp.Math. 85 (1989), 195–238.
Granero, A. S., Jiménez Sevilla,M. and Moreno, J. P.: Sequential continuity in the ball topology of a Banach space, Preprint, 1997; Indag.Math., to appear.
Lindenstrauss, J. and Tzafriri, L.: Classical Banach Spaces, I.Sequence Spaces, Springer, New York, 1977.
Maurey, B.: Types and l 1 subspaces, Longhorn Notes, The University of Texas, Functional Analysis Seminar (1982–83), 123–137.
Milman, D. P. and Milman, V. D.: The geometry of nested families with empty intersection. Structure of the unit sphere of a nonreflexive space, Mat.Sb. 66(1965), 109–118; English transl. Math Notes 85 (1969), 233–243.
Plans, A.: Orthogonality and convergence in Banach spaces, Communication presented at Functional Analysis and Applications (FAA 93), Gargnano del Garda, Italy, 1993.
Plans, A.: Una caracterización del espacio de Banach Fréchet diferenciable, en términos de la convergencia débil, Colloquium Seminario del Departamento de Análisis Matemático, Univ. Complutense, 1995.
Rosenthal, H.: Double dual types and the Maurey characterization of Banach spaces containing l 1, Longhorn Notes, The University of Texas, Functional Analysis Seminar (1983–84), 1–36.
Sims, B.: 'Ultra'-Techniques in Banach Space Theory, Queen's Papers in Pure Appl. Math. 60, Queen's Univ., Kingston, Ontario, Canada, 1982.
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Castillo, J.M.F., Papini, P.L. Distance Types in Banach Spaces. Set-Valued Analysis 7, 101–115 (1999). https://doi.org/10.1023/A:1008702707348
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DOI: https://doi.org/10.1023/A:1008702707348