Geodesic
A Ramsey Theorem with an Application to Sequences in Banach Spaces
Canadian mathematical bulletin, Tome 55 (2012) no. 2, pp. 410-417

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-2011-073-5
Mots-clés : 46B15, 05D10, Banach spaces, Ramsey theory 
Service, Robert. A Ramsey Theorem with an Application to Sequences in Banach Spaces. Canadian mathematical bulletin, Tome 55 (2012) no. 2, pp. 410-417. doi: 10.4153/CMB-2011-073-5
@article{10_4153_CMB_2011_073_5,
     author = {Service, Robert},
     title = {A {Ramsey} {Theorem} with an {Application} to {Sequences} in {Banach} {Spaces}},
     journal = {Canadian mathematical bulletin},
     pages = {410--417},
     year = {2012},
     volume = {55},
     number = {2},
     doi = {10.4153/CMB-2011-073-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-073-5/}
}
TY  - JOUR
AU  - Service, Robert
TI  - A Ramsey Theorem with an Application to Sequences in Banach Spaces
JO  - Canadian mathematical bulletin
PY  - 2012
SP  - 410
EP  - 417
VL  - 55
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-073-5/
DO  - 10.4153/CMB-2011-073-5
ID  - 10_4153_CMB_2011_073_5
ER  - 
%0 Journal Article
%A Service, Robert
%T A Ramsey Theorem with an Application to Sequences in Banach Spaces
%J Canadian mathematical bulletin
%D 2012
%P 410-417
%V 55
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-073-5/
%R 10.4153/CMB-2011-073-5
%F 10_4153_CMB_2011_073_5

[1] [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] [2] Argyros, S. A., Merkourakis, S., and Tsarpalias, A., Convex unconditionality and summability of weakly null sequences. Israel J. Math. 107(1998), 157–193. Google Scholar | DOI

[3] [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] [4] Bollobás, B., Combinatorics: Set Systems, Hypergraphs, Families of Vectors and Combinatorial Probability. Cambridge University Press, Cambridge, 1986. Google Scholar

[5] [5] Dilworth, S. J., Odell, E., Schlumprecht, T., and Zsak, A., Partial unconditionality. arXiv:math/0510609v1. Google Scholar

[6] [6] Elton, J., Weakly null normalized sequences in Banach spaces. Ph.D-Thesis, Yale University, New Haven, CT, 1978. Google Scholar

[7] [7] Gowers, W. T., An infinite Ramsey theorem and some Banach-space dichotomie. Ann. of Math. 156(2002), no. 3, 797–833. Google Scholar | DOI

[8] [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] [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] [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] [11] Megginson, R. E., An Introduction to Banach Space Theory. Graduate Texts in Mathematics 183. Springer-Verlag, New York, 1998. Google Scholar

[12] [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] [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

Cité par Sources :