Abstract
It is shown that for every separable Banach space X with non-separable dual, the space \(X^{**}\) contains an unconditional family of size \(|X^{**}|\) . The proof is based on Ramsey Theory for trees and finite products of perfect sets of reals. Among its consequences, it is proved that every dual Banach space has a separable quotient.
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Apatsidis, D., Argyros, S.A., Kanellopoulos, V.: On the subspaces of JF and JT with non-separable dual (preprint)
Argyros, S.A., Arvanitakis, A.D., Tolias, A.G.: Saturated extensions, the attractors method and hereditarily James tree spaces. In: Castillo, J.M., Johnson, W.B. (eds.) Methods in Banach spaces. LMS Lecture Notes, 337, pp. 1–90
Argyros, S.A., Dodos, P., Kanellopoulos, V.: A classification of separable Rosenthal compacta and its applications. Dissertations Math (to appear)
Argyros S.A., Felouzis V., Kanellopoulos V. (2002). A proof of Halpern-Läuchli Partition Theorem. Eur. J. Comb. 23: 1–10
Argyros, S.A., Godefroy, G., Rosenthal, H.P.: Descriptive set theory and Banach spaces. In: Johnson, W.B., Lindenstrauss, J. (eds.) Handbook of the Geometry of Banach Spaces, vol. 2. Elsevier, Amsterdam (2003)
Argyros, S.A., Todorčević, S.: Ramsey methods in analysis. Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser, Verlag, Basel (2005)
Argyros, S.A., Tolias, A.: Methods in the theory of hereditarily indecomposable Banach spaces. Memoirs AMS 806 (2004)
Bessaga C., Pełczyński A. (1958). On bases and unconditional convergence of series in Banach spaces. Stud. Math. 17: 151–164
Blass A. (1981). A partition for perfect sets. Proc. AMS 82: 271–277
Carlson T.J. (1988). Some unifying principles in Ramsey theory. Discrete Math. 68: 117–169
Edgar G.A., Wheeler R.F. (1984). Topological properties of Banach spaces. Pacific J. Math. 115: 317–350
Ellentuck E. (1974). A new proof that analytic sets are Ramsey. J. Symb. Logic 39: 163–165
Farahat, J.: Espaces be Banach contenant \(\ell_1\) d’ apres H.P. Rosenthal. Seminaire Maurey-Schwartz, Ecole Polytechnique, pp. 1973–74
Godefroy G., Louveau A. (1989). Axioms of determinacy and biorthogonal systems. Israel J. Math. 67: 109–116
Godefroy G., Talagrand M. (1982). Espaces de Banach représentable. Israel J. Math. 41: 321–330
Gowers W.T. (2002). An infinite Ramsey theorem and some Banach-space dichotomies. Ann. Math. 156: 797–833
Gowers, W.T.: Ramsey methods in Banach spaces. In: Johnson, W.B., Lindenstrauss, J. (eds.) Handbook of the Geometry of Banach Spaces, vol. 2. Elsevier, Amsterdam (2003)
Gowers W.T., Maurey B. (1993). The unconditional basic sequence problem. J. AMS 6: 851–874
Hagler J.N., Johnson W.B. (1977). On Banach spaces whose dual balls are not weak* sequentially compact. Israel J. Math. 28: 325–330
Halpern J.D., Läuchli H. (1966). A partition theorem. Trans. AMS 124: 360–367
James R.C. (1974). A separable somewhat reflexive Banach space with non-separable dual. Bull. AMS 80: 738–743
Johnson W.B., Rosenthal H.P. (1972). On weak* basic sequences and their applications to the study of Banach spaces. Stud. Math. 43: 77–92
Kanellopoulos V. (2005). Ramsey families of subtrees of the dyadic tree. Trans. AMS 357: 3865–3886
Kechris, A.S.: Classical Descriptive Set Theory. Grad. Texts in Math. 156, Springer, Heidelberg (1995)
Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces I and II. Springer, Heidelberg (1996)
Louveau A., Shelah S., Veličković B. (1993). Borel partitions of infinite subtrees of a perfect tree. Ann. Pure Appl. Logic 63: 271–281
Maurey, B.: Types and \(\ell_1\) subspaces. Texas functional analysis seminar 1982–1983 (Austin, Tex.), 123-137, Longhorn Notes, University of Texas Press, Austin (1983)
Milliken K. (1981). A partition theorem for the infinite subtrees of a tree. Trans. AMS 263: 137–148
Miller A.W. (1989). Infinite combinatorics and definability. Ann. Pure Appl. Logic 41: 179–203
Milman V.D. (1970). Geometric theory of Banach spaces. I. Theory of basic and minimal systems (Russian). Uspehi Mat. Nauk 25: 113–174
Odell, E.: Applications of Ramsey theorems to Banach space theory. Notes in Banach spaces, 379–404, University of Texas Press, Austin (1980)
Odell E., Rosenthal H.P. (1975). A double-dual characterization of separable Banach spaces not containing \(\ell_1\). Israel J. Math. 20: 375–384
Pawlikowski J. (1990). Parametrized ellentuck theorem. Topol. Appl. 37: 65–73
Pełczyński A. (1968). On Banach spaces containing \(L_1(\mu)\). Stud. Math. 30: 231–246
Rosenthal H.P. (1974). A characterization of Banach spaces containing \(\ell_1\). Proc. Nat. Acad. Sci. USA 71: 2411–2413
Stegall C. (1975). The Radon-Nikodym property in conjugate Banach spaces. Trans. AMS 206: 213–223
Stern J. (1978). A Ramsey theorem for trees with an application to Banach spaces. Israel J. Math. 29: 179–188
Todorčević S. (1999). Compact subsets of the first Baire class. J. AMS 12: 1179–1212
Todorčević, S.: Introduction to Ramsey spaces (to appear)
Todorčević S. (2006). Biorthogonal systems and quotient spaces via Baire category methods. Math. Ann. 335: 687–715
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Argyros, S.A., Dodos, P. & Kanellopoulos, V. Unconditional families in Banach spaces. Math. Ann. 341, 15–38 (2008). https://doi.org/10.1007/s00208-007-0179-y
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DOI: https://doi.org/10.1007/s00208-007-0179-y