Abstract
We show that every Banach space X of density smaller that the Baire category number admits a quotient with a long Schauder basis that can be taken of length ω1 if X is not separable. So, assuming that the Baire category number does not take its minimal possible value, a Banach space X is separable if and only if all biorthogonal systems of X are countable.
Similar content being viewed by others
References
Banach, S.: Théorie des opérations linéaires, Warszawa 1932
Benyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis. Amer. Math. Soc., Providence 2000
Bell, M., Ginsburg, J., Todorcevic, S.: Countable spread of exp Y and λY. Topology and its Appl. 14, 1–12 (1982)
Borwein, J.M., Vanderwerff, J.D.: Banach spaces that admit support sets. Proc. Amer. Math. Soc. 124, 751–755 (1996)
Davis, W.J., Johnson, W.B.: On the existence of fundamental and total bounded biorthogonal systems in Banach spaces. Studia Math. 45, 173–179 (1972)
Engelking, R.: General Topology, Heldermann Verlag Berlin 1989
Fabian, M., Habala, P., Hajek, P., Montesinos, V., Pelant, J., Zizler, V.: Functional Analysis and Infinite-Dimensional Geometry. CMS Books in Mathematics, Springer-Verlag, 2001
Finet, C., Godefroy, G.: Biorthogonal systems and big quotient spaces. Contemp. Math. 85, 275–283 (1989)
Foreman, M., Magidor, M., Shelah, S.: Martin's maximum, saturated ideals and non-regular ultrafilters. Ann. Math. 127, 1–47 (1988)
Fremlin, D.H.:Consequences of Martin's axioms. Cambridge Univ. Press 1984
Frontisi, J.: Lissité et dialité dans les espaces de Banach. Thése de Dectorat, Univ. Paris VI, 1996
Godefroy, G., Louveau, A.: Axioms of determinacy and biorthogonal systems. Israel J. Math. 67, 109–116 (1989)
Grodefroy, G., Talagrand, M.: Espaces de Banach representables. Israel J. Math. 41, 321–330 (1982)
Godun, B.V., Kadets, M.: Banach spaces without complete minimal systems. Funk. Analiz i Ego Pril. 14, 67–68 (1980)
Johnson, W.B.: No infinite dimensional P space admits a Markushevich basis. Proc. Amer. Math. Soc. 28, 467–468 (1970)
Johnson, W.B.: E-mail message of November 16, 2005
Johnson, W.B., Rosenthal, H.P.: On w*-basic sequences and their applications in the study of Banach spaces. Studia Math. 43, 77–92 (1972)
Koszmider, P.: A problem of Rolewics about Banach spaces that admit support sets, preprint 2004
Kunen, K.: Products of S-spaces, note of May 1975
Kunen, K.: On hereditarily Lindelof Banach spaces, note of July 1980
Kutzarova, D.: Convex sets containing only support points in Banach spaces with an uncountable minimal system. C.R. l'Acad. Bulg. Sci., 39, 13–14 (1986)
Lazar, A.J.: Points of support for closed convex sets. Illinois J. Math. 25, 302–305 (1981)
Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces, I, Springer-Verlag, Berlin, 1977
Mujica, J.: Separable quotients of Banach spaces. Revista Mat. Univ. Complut. 10, 299–330 (1997)
Montesinos, V.: Solution to a problem of S. Rolewicz. Studia Math. 81, 65–69 (1985)
Negrepontis, S.: Banach spaces and topology, Handbook of Set-Theoretic topology. In: Kunen, K., Vaughan, J.E., (eds), North-Holland, 1984, pp. 1045–1142
Pelczynski, A.: Some problems on basis in Banach and Fréchet space. Israel J. Math. 2, 132–138 (1964)
Plichko, A.N.: Fundamental biorthogonal systems and projective bases in Banach spaces. Math. Zametki 34, 473–476 (1983)
Plichko, A.N.: Banach spaces without fundamantal biorthogonal systems. Soviet Math. Dokl. 254(4), 978–801 (1980)
Rolewicz, S.: On convex sets containing only points of support. Comment. Math., Prace Math. Special issue 1, 279–281 (1978)
Singer, I.: Bases in Banach spaces II, Springer-Verlag, Berlin, 1981
Shelah, S.: Uncountable constructions for BA, e.g. groups and Banach spaces. Israel J. Math. 51, 273–297 (1985)
Todorcevic, S.: Forcing positive partition relations. Trans. Amer. Math. Soc. 280, 703–720 (1983)
Todorcevic, S.: Directed sets and cofinal types. Trans. Amer. Math. Soc. 290, 711–723 (1985)
Todorcevic, S.: Partition problems in topology, Amer. Math. Soc. Providence 1989
Todorcevic, S.: Irredundant sets in Boolean algebras. Trans. Amer. Math. Soc. 339, 35–44 (1990)
Todorcevic, S.: Some applications of S and L combinatorics. Ann. New York Acad. Sci.,705, 130–167 (1993)
Todorcevic, S.: Basis problems in combinatorial set theory, Proc. International Congress of Mathematicians Berlin 1998, Docum. Math. Extra Volume ICM 1998. II. 43–52
Todorcevic, S.: A generic function space (K) with no support set, in preparation
Wojtaszczyk, P.: A theorem on convex sets related to the abstract Pontriagin maximality principle. Bull. Acad. Polon Sci 21, 93–95 (1973)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Todorcevic, S. Biorthogonal systems and quotient spaces via baire category methods. Math. Ann. 335, 687–715 (2006). https://doi.org/10.1007/s00208-006-0762-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-006-0762-7