Sequential continuity in the ball topology of a Banach space

Thumbnail Image
Full text at PDC
Publication Date
Advisors (or tutors)
Journal Title
Journal ISSN
Volume Title
Elsevier Science
Google Scholar
Research Projects
Organizational Units
Journal Issue
Consider the isometric property (P): the restriction to the unit ball of every bounded linear functional is sequentially continuous in the ball topology. We present in this paper a systematic study of this property, which is a sequential version of the well known ball generated property. A separable Banach space is Asplund if and only if there exists an equivalent norm with property (P). However, in nonseparable spaces this equivalence does not work. Examples of nonseparable Banach spaces with property (P) which are not Asplund are given. Also, we exhibit a nonseparable Asplund space, namely the space of continuous functions on the Kunen compact, admitting no equivalent norm with this property. We characterize reflexive spaces as those satisfying that every equivalent norm has property (P), thus improving a previous characterization involving the ball generated property. Finally, we investigate the relationships between property (P) and Grothendieck spaces.
UCM subjects
Unesco subjects
I. Balcar. B. and F. Franek Independent famthes in complete Boolean algebras. Trans. Amer.Math.Sot.274(2)6077618 (1982). Borwein, J.M. and J.D. Vanderwerff ~ Banach spaces that admit support sets. Proc. Amer. Math. Sot. 124 (3). 751-755 (1996). Cascales, B. and G. Godefroy Angehcity and the boundary problem. Mathematika 45. 105-112 (1998). Castillo. J.M.F. and Papnn, P.L. - DistanceTypes in Banach spaces (preprint). Chen, Dongjian, Zhibao Hu and Bor-Luh Lin - Ball intersection properties of Banach spaces. Bull. Austral. Math. Sot. 45, 333-342 (1992). Chen. Dongjian and Bor-Luh Lm ~ Ball topology on Banach spaces. Houston J. Math. 22. No. 4.821-833 (1996). Chen, Donglian and Bor-Luh Lin Ball separation properties m Banach spaces (preprint). Corson, H.H. and J. Lindenstrauss On weakly compact subsets of Banach spaces. Proc. Amer. Math. Sot. 17,407-412 (1966). Deville, R.. G. Godefroy and V. Zizler Smoothness and renormmgs in Banach spaces. Pitman Monograph and Surveys in Pure and Applied Mathematics, vol. 64 (1993). Diestel. J. - Sequences and series in Banach spaces. Graduate Text in Mathematics, vol. 92. Springer Verlag (1984). Diestel, J. and J.J. Uhl. Jr. - Vector measures. Math Surveys 15. Amer. Math. Sot.. Providence, R I. (1977) Finet. C. and W. Schachermayer -~ Equtvalent norms in separable Asplund spaces. Studia Math. 92. 275-283 (1989). Fremlin, D.H. - Consequences of Martm’s Axiom. Cambridge Univ. Press (1985). Finet, C. and G. Godefroy Biorthogonal systems and big quotient spaces Contemporary Math., vol. 85,87-l 10 (1989). Giles, J.R.. D.A. Gregory and B. Sims Charactertzation of normed linear spaces with Mazur’s mtersectton property.Bull. Austr Math. Sot. 18.471-476 (1978). Godefroy, G. and N.J. Kalton - The ball topology and its applications. Contemp. Math. 85. 1955237 (1989) Codefroy. G. and PD. Saphar - Duality in spaces of operators and smooth norms on Banach spaces, Illinois J. Math. 32, No. 4,6722695 (1988). Godun. B.V. and S.L. Troyanski Renorming Banach spaces with fundamental biorthogonal systems. Contemporary Math. 144. 1199126 (1993) Granero, AS. and H. Hudzik - The classical Banach spaces I',/h,P.r oc. Amer. Math Sot. 124, No. 12.3777-3787 (1996). Granero. A.S . M. Jimenez Sevtlla and J.P. Moreno Convex sets in Banach spaces and a problem of Rolewicz Studia Math. 129 (1). 19929 (1998). Hagler. J. - A counterexample to several question about Banach spaces. Studta Math. 60. 289-308 (1977). Haydon. R. - An unconditional result about Grothendieck spaces. Proc. Amer. Math. Sot 100. 511-516 (1987). Hu. Zhibao and Bor-Luh Lm Smoothness and the asymptotic-normmg properties of Banach spaces. Bull. Austral. Math. Sot. 45. 2855296 (1992). Jimenez Sevilla, M and J P. Moreno The Mazur intersection property and Asplund spaces. C R. Acad. Sci. Paris. S&e I, 321. 121991223 (1995). Jimenez Sevilla, M. and J.P. Moreno - Renormmg Banach spaces with the Mazur intersection property. J Funct.Anal. 144 (2). 4866504 (1997). Klee, V - Two renorming constructions related to a question of Anselone. Studia Mathematica, v. 33 1-7 31-242 (1969) Lacey. H.E. The Isometry Theory of Classtcal Banach Spaces. Springer-Verlag (1974). Meyer-Nieberg, P. ~ Banach Latttces. Springer-Verlag (1994). Negrepontts, S. ~ Banach Spaces and Topology. Handbook of Set-theorettc Topology. K. Kunen and J.E.Vaughan. North-Holland, 104551142 (1984). Plans. A. Una caracterizacion del espacio de Banach Frtchet diferenciable, en terminos de la convergencia debil. Colloquium, Seminario del Departamento de Anilisis Matematico. Fat. de Matematicas. Umv. Complutense (1995). Rosenthal, H.P - A characterization of Banach spaces contaming pt Proc. Nat. Acad. SC. USA, 71.2411-2413 (1974). Van Dulst, D. Reflexive and superreflexrve Banach spaces. Mathematisch Centrum, Mathematical CentreTracts 102. Amsterdam (1978). Walker, R.C. - The Stone-Tech Compactification. Springer-Verlag (1974). Watson. S. - A compact Hausdorff space without P-points m which GA sets have Interior. Proc. Amer. Math. Sot. 123.2575-2577 (1995).