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A Ramsey Theorem with an Application to Sequences in Banach Spaces

Published online by Cambridge University Press:  20 November 2018

Robert Service
Robert Service*
Affiliation:
Department of Mathematics and Statistics, University of Helsinkie-mail: [email protected]
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Abstract

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The notion of a maximally conditional sequence is introduced for sequences in a Banach space. It is then proved using Ramsey theory that every basic sequence in a Banach space has a subsequence which is either an unconditional basic sequence or a maximally conditional sequence. An apparently novel, purely combinatorial lemma in the spirit of Galvin's theorem is used in the proof. An alternative proof of the dichotomy result for sequences in Banach spaces is also sketched, using the Galvin–Prikry theorem.

Keywords

46B1505D10Banach spacesRamsey theory
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 2 , 01 June 2012 , pp. 410 - 417
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Abad, J. L. and Todorcevic, S., Partial unconditionality of weakly null sequences. RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 100(2006), no. 1-2, 237–277.Google Scholar
[2] Argyros, S. A., Merkourakis, S., and Tsarpalias, A., Convex unconditionality and summability of weakly null sequences. Israel J. Math. 107(1998), 157–193. http://dx.doi.org/10.1007/BF02764008 Google Scholar
[3] Bessaga, C. and Pełczyński, A., A generalization of results of R. C. James concerning absolute bases in Banach spaces. Studia Math. 17(1958), 165–174.Google Scholar
[4] Bollobás, B., Combinatorics: Set Systems, Hypergraphs, Families of Vectors and Combinatorial Probability. Cambridge University Press, Cambridge, 1986.Google Scholar
[5] Dilworth, S. J., Odell, E., Schlumprecht, T., and Zsak, A., Partial unconditionality. arXiv:math/0510609v1.Google Scholar
[6] Elton, J., Weakly null normalized sequences in Banach spaces. Ph.D-Thesis, Yale University, New Haven, CT, 1978.Google Scholar
[7] Gowers, W. T., An infinite Ramsey theorem and some Banach-space dichotomie. Ann. of Math. 156(2002), no. 3, 797–833. http://dx.doi.org/10.2307/3597282 Google Scholar
[8] Gowers, W. T. and Maurey, B., The unconditional basic sequence problem. J. Amer. Math. Soc. 6(1993), no. 4, 851–874.Google Scholar
[9] Maurey, B., A note on Gowers’ dichotomy theorem. In: Convex Geometric Analysis. Math. Sci. Res. Inst. Publ. 34. Cambridge University Press, Cambridge, 1999, pp. 149–157.Google Scholar
[10] Maurey, B. and Rosenthal, H. P., Normalized weakly null sequence with no unconditional subsequence. Studa Math. 61(1977), no. 1, 77–98.Google Scholar
[11] Megginson, R. E., An Introduction to Banach Space Theory. Graduate Texts in Mathematics 183. Springer-Verlag, New York, 1998.Google Scholar
[12] Odell, E. On Schreier unconditional sequences. In: Banach Spaces. Contemp. Math. 144. American Mathematical Society, Providence, RI, 1993, pp. 197–201.Google Scholar
[13] Odell, E., On subspaces, asymptotic structure, and distortion of Banach spaces; connections with logic. In: Analysis and Logic. London Math. Soc. Lecture Note Ser. 262. Cambridge University Press, Cambridge, 2002, pp. 189–267.Google Scholar