Skip to main content
SpringerLink
Log in

Banach spaces and Ramsey Theory: some open problems

Espacios de Banach y Teoría de Ramsey: Ciertos problemas abiertos

  • Published:
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas Aims and scope Submit manuscript

Abstract

We discuss some open problems in the Geometry of Banach spaces having Ramsey-theoretic flavor. The problems are exposed together with well known results related to them.

Resumen

Presentamos ciertos problemas en la geometría de los espacios de Banach que tienen conexiones con la teoría de Ramsey. Los problemas son expuestos juntamente con resultados bien conocidos relacionados con ellos.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aharoni, I., (1974). Every separable metric space is Lipschitz equivalent to a subset of c 0, Israel J. Math., 19, 284–291.

    Article  MathSciNet  Google Scholar 

  2. Androulakis, G.; Odell, E.; Schlumprecht, Th. and Tomczak-Jaegermann, N., (2005). On the structure of the spreading models of a Banach space, Canad. J. Math., 57, 673–707.

    Article  MathSciNet  MATH  Google Scholar 

  3. Anisca, R., (2010). The ergodicity of weak Hilbert spaces, Proc. Amer. Math. Soc., 138, 1405–1413. DOI: 10.1090/S0002-9939-09-10164-8

    Article  MathSciNet  MATH  Google Scholar 

  4. Argyros, S. A.; Arvanitakis, A. D. and Tolias, A. G., (2006). Saturated extensions, the attractors method and hereditarily James tree spaces, in Methods in Banach spaces, Proceedings of the V Conference on Banach Spaces, Cáceres, Spain, 13–18 September 2004, J. M. Castillo and W. B. Johnson eds., LMS Lecture Notes, 337, 1–90, Cambridge Univ. Press. DOI: 10.1017/CBO9780511721366.003

  5. Argyros, S. A.; Dodos, P. and Kanellopoulos, V., (2008). A classification of separable Rosenthal compacta and its applications, Dissertationes Math., 449, 1–52. DOI: 10.4064/dm449-0-1

    Article  MathSciNet  Google Scholar 

  6. Argyros, S. A.; Dodos, P. and Kanellopoulos, V., (2008). Unconditional families in Banach spaces, Math. Ann., 341, 15–38. DOI: 10.1007/s00208-007-0179-y

    Article  MathSciNet  MATH  Google Scholar 

  7. Argyros, S. A.; Godefroy, G. and Rosenthal, H. P., (2003) Descriptive Set Theory and Banach spaces, in Handbook of the Geometry of Banach Spaces, Volume 2, W. B. Johnson and J. Lindenstrauss eds., 1007–1069, North-Holland. Elsevier DOI: 10.1016/S1874-5849(03)80030-X

    Chapter  Google Scholar 

  8. Argyros, S. A.; Lopez-Abad, J. and Todorcevic, S., (2005). A class of Banach spaces with few nonstrictly singular operators, J. Funct. Anal., 222, 306–384. DOI: 10.1016/j.jfa.2004.11.001

    Article  MathSciNet  MATH  Google Scholar 

  9. Argyros, S. A. and Todorcevic, S., (2005). Ramsey Methods in Analysis. Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser, Verlag, Basel.

    Google Scholar 

  10. Banach, S., (1932). Théorie des opérations linéaires. Warszawa.

  11. Benyamini, Y. and Lindenstrauss, J., (2000). Geometric Nonlinear Functional Analysis. AMS Colloquium Publications.

  12. Bessaga, C. and Pelczyński, A., (1960). Spaces of continuous functions IV, Studia Math., 19, 53–62.

    MathSciNet  MATH  Google Scholar 

  13. Bourgain, J., (1980). A result on operators on C[0, 1], J. Operator Theory, 3, 275–289.

    MathSciNet  MATH  Google Scholar 

  14. Bourgain, J. and Szarek, S. J., (1988). The Banach-Mazur distance to the cube and the Dvoretsky-Rogers factorization, Israel J. Math., 62, 169–180. DOI: 10.1007/BF02787120

    Article  MathSciNet  MATH  Google Scholar 

  15. Brech, C. and Koszmider, P., (2010). Thin very-tall compact scattered spaces which are hereditarily separable, Trans. Amer. Math. Soc., (to be published).

  16. Brunel, A. and Sucheston, L., (1974). On B convex Banach spaces, Math. Systems Th., 7, 294–299.

    Article  MathSciNet  MATH  Google Scholar 

  17. Brunel, A. and Sucheston, L., (1975). On J-convexity and some ergodic super properties of Banach spaces, Trans. Amer. Math. Soc., 204, 79–90. DOI: 10.2307/1997350.DOI: 10.1090/S0002-9947-1975-0380361-2

    MathSciNet  MATH  Google Scholar 

  18. Davis, W. J. and Johnson, W. B., (1972). On the existence of fundamental and total bounded biorthogonal systems in Banach spaces, Studia Math., 45, 173–179.

    MathSciNet  Google Scholar 

  19. Dilworth, S. J.; Kalton, N. J. and Kutzarova, D., (2003). On the existence of almost greedy bases in Banach spaces, Studia Math., 159, 1, 67–101. DOI: 10.4064/sm159-1-4

    Article  MathSciNet  MATH  Google Scholar 

  20. Dilworth, S. J.; Odell, E. and Sari, B., (2007). Lattice structures and spreading models, Israel J. Math., 161, 387–410. DOI: 10.1007/s11856-007-0084-9

    Article  MathSciNet  MATH  Google Scholar 

  21. Dilworth, S. J.; Odell, E.; Schlumprecth, Th. and Zsák, A., (2009). Partial unconditionality, Houston J. Math., 35, 1251–1311.

    MathSciNet  MATH  Google Scholar 

  22. Dodos, P., (2009). Operators whose dual has non-separable range, (preprint).

  23. Dodos, P.; Lopez-Abad, J. and Todorcevic, S., (2008). Unconditional basic sequences in spaces of large density, (preprint).

  24. Elton, J., (1978). Weakly null normalized sequences in Banach spaces. Ph. D. Thesis, Yale University.

  25. Elton, J., (1983). Sign-embeddings of l n1 , Trans. Amer. Math. Soc., 279, 1, 113–124. DOI: 10.2307/1999374. DOI: 10.1090/S0002-9947-1983-0704605-4

    MathSciNet  MATH  Google Scholar 

  26. Ferenczi, V.; Pelczar, A. M. and Rosendal, C., (2004). On a question of Haskell P. Rosenthal concerning a characterization of c 0 and l p , Bull. Lond. Math. Soc., 36, 396–406. DOI: 10.1112/S0024609303003047

    Article  MathSciNet  MATH  Google Scholar 

  27. Ferenczi, V. and Rosendal, C., Complexity and homogeneity in Banach spaces, in Banach Spaces and their Applications in Mathematics, B. Randrianantoanina and N. Randrianantoanina eds., Walter de Gruyter, Berlin, 83–110.

  28. Fremlin, D. H., (1984). Consequences of Martin’s axiom. Cambridge University Press.

  29. Fremlin, D. H., (2010). Problems; Version of 11.1.10. University of Essex, England. http://www.essex.ac.uk/ maths/staff/fremlin/problems.htm

    Google Scholar 

  30. Galvin, F. and Prikry, K., (1973). Borel Sets and Ramsey’s Theorem, J. Symbolic Logic, 38, 193–198. DOI: 10.2307/2272055

    Article  MathSciNet  MATH  Google Scholar 

  31. Gasparis, I., (2005). Operators on C[0, 1] preserving copies of asymptotic l1 spaces, Math. Ann., 333, 831–858. DOI: 10.1007/s00208-005-0702-y

    Article  MathSciNet  MATH  Google Scholar 

  32. Giannopoulos, A., (1995). A note on the Banach-Mazur distance to the cube, Oper. Theory Adv. Appl., 77, 67–73.

    MathSciNet  Google Scholar 

  33. Godefroy, G., (2009). Descriptive set theory and the geometry of Banach spaces, in Statistical Science and Interdisciplinary research — Vol. 8, Perspectives in Mathematical Sciences II, Pure Mathematics, N. S. N. Sastry, T. S. S. R. K. Rao, Mohan Delampady and B. Rajeev eds., 63–82. DOI: 10.1142/9789814273657 0004

  34. Godefroy, G.; Kalton, N. J. and Lancien, G., (2000). Subspaces of c 0(ℕ) and Lipschitz isomorphisms, Geom. Funct. Anal., 10, 798–820. DOI: 10.1007/PL00001638

    Article  MathSciNet  MATH  Google Scholar 

  35. Gowers, W. T., (1992). Lipschitz functions on classical spaces, European J. Combin., 13, 3, 141–151. DOI: 10.1016/0195-6698(92)90020-Z

    Article  MathSciNet  MATH  Google Scholar 

  36. Gowers, W. T., (1994). A space not containing c 0, l 1 or a reflexive subspace, Trans. Amer. Math. Soc., 344, 407–420.

    Article  MathSciNet  MATH  Google Scholar 

  37. Gowers, W. T., (2002). An infinite Ramsey theorem and some Banach space dichotomies, Ann. of Math. (2), 156, 797–833. DOI: 10.2307/3597282

    Article  MathSciNet  MATH  Google Scholar 

  38. Gowers, W. T., Ramsey methods in Banach spaces, (2003). in Handbook of the Geometry of Banach Spaces, Volume 2, W. B. Johnson and J. Lindenstrauss eds., Elsevier. DOI: 10.1016/S1874-5849(03)80031-1

  39. Hagler, J. N. and Johnson, W. B., (1977). On Banach spaces whose dual balls are not weak* sequentially compact, Israel J. Math., 28, 325–330. DOI: 10.1007/BF02760638

    Article  MathSciNet  MATH  Google Scholar 

  40. Halpern, J. D. and Läuchli, H., (1966). A partition theorem, Trans. Amer. Math. Soc., 124, 360–367. DOI: 10.2307/1994406.DOI: 10.1090/S0002-9947-1966-0200172-2

    Article  MATH  Google Scholar 

  41. James, R. C., (1972). Uniformly non-square Banach spaces, Ann. of Math. (2), 69, 3, 409–419. DOI: 10.2307/ 1970663

    Google Scholar 

  42. James, R. C., (1974). A separable somewhat reflexive Banach space with non-separable dual, Bull. Amer. Math. Soc. (N.S.), 80, 738–743.

    Article  MATH  Google Scholar 

  43. Johnson, W. B. and Rosenthal, H. P., (1972). On weak* basic sequences and their applications to the study of Banach spaces, Studia Math., 43, 77–92.

    MathSciNet  Google Scholar 

  44. Kalton, N. J., (2008). The nonlinear geometry of Banach spaces, Rev. Mat. Complut., 21, 7–60.

    MathSciNet  MATH  Google Scholar 

  45. Kalton, N. J. and Lancien, G., (2008). Best constants for Lipschitz embeddings of metric spaces into c 0, Fund. Math., 199, 249–272.

    Article  MathSciNet  MATH  Google Scholar 

  46. Kanellopoulos, V., (2005). Ramsey families of subtrees of the dyadic tree, Trans. Amer. Math. Soc., 357, 3865–3886.

    Article  MathSciNet  MATH  Google Scholar 

  47. Ketonen, J., (1974). Banach spaces and large cardinals, Fund. Math., 81, 291–303.

    MathSciNet  MATH  Google Scholar 

  48. Koszmider, P., (2009). On a problem of Rolewicz about Banach spaces that admit support sets, J. Funct. Anal., 257, 9, 2723–2741. DOI: 10.1016/j.jfa.2009.07.016

    Article  MathSciNet  MATH  Google Scholar 

  49. Kunen, K., (1975). Products of S-spaces, note of May 1975.

  50. Kunen, K., (1980). On hereditarily Lindelöf Banach spaces, note of July 1980.

  51. Lindenstrauss, J. and Tzafriri, L., (1977). Classical Banach spaces vol. I: Sequence spaces. Ergebnisse, 92, Springer.

  52. Lindenstrauss, J. and Wulbert, D. E., (1969). On the classification of the Banach spaces whose duals are L 1 spaces, J. Funct. Anal., 4, 332–349. DOI: 0.1016/0022-1236(69)90003-2

    Article  MathSciNet  MATH  Google Scholar 

  53. Lopez-Abad, J. and Todorcevic, S., (2006). Partial unconditionality of weakly null sequences, RACSAM Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat., 100, 237–277. http://www.rac.es/ficheros/doc/00226.pdf

    MathSciNet  MATH  Google Scholar 

  54. Lopez-Abad, J. and Todorcevic, S., (2010). Generic Banach spaces and generic simplexes, (preprint).

  55. Maurey, B. and Pisier, G., (1976). Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Studia Math., 58, 45–90.

    MathSciNet  MATH  Google Scholar 

  56. Maurey, B. and Rosenthal, H. P., (1977). Normalized weakly null sequence with no unconditional subsequence, Studia Math., 61, 77–98.

    MathSciNet  MATH  Google Scholar 

  57. Milman, V. D., (1971). Geometric theory of Banach spaces part II: Geometry of the unit sphere of a Banach space, Usp. Mat. Nauk, 26, 73–149.

    MathSciNet  Google Scholar 

  58. Mujica, J., (1997). Separable quotients of Banach spaces, Rev. Mat. Complut., 10, 299–330.

    MathSciNet  MATH  Google Scholar 

  59. Negrepontis, S., (1984). Banach spaces and topology, in Handbook of Set-Theoretic topology, K. Kunen and J. E. Vaughan eds., North-Holland, 1045–1142.

  60. Odell, E., (1980). Applications of Ramsey theorems to Banach space theory, in Notes in Banach spaces, H. Lacey ed., Longhorn Notes, University of Texas, Texas Functional Analysis Seminar, 379–404.

  61. Odell, E. and Schlumprecht, Th., (1994). The distortion problem, Acta Math., 173, 259–281. DOI: 10. 1007/BF02398436

    Article  MathSciNet  MATH  Google Scholar 

  62. Odell, E. and Schlumprecht, Th., (2003). Distortion and asymptotic structure, in Handbook of the Geometry of Banach spaces, Volume 2, W. B. Johnson and J. Lindenstrauss eds., Elsevier. DOI: 10.1016/ S1874-5849(03)80038-4

  63. Pelczyński, A., (1960). Projections in certain Banach spaces, Studia Math., 32, 209–228.

    Google Scholar 

  64. Pelczyński, A., (1964). Some problems on basis in Banach and Fréchet spaces, Israel J. Math., 2, 132–138.

    Article  MathSciNet  MATH  Google Scholar 

  65. Pelczyński, A., (1984). Structural theory of Banach spaces and its interplay with analysis and probability, in Proceedings of the ICM 1983, PWN-North Holland, 237–269.

  66. Pelczyński, A. and Semadeni, Z., (1959). Spaces of continuous functions III. Spaces C(Ω) for (Ω) without perfect subsets, Studia Math., 18, 211–222.

    MathSciNet  MATH  Google Scholar 

  67. Rolewicz, S., (1978). On convex sets containing only points of support, Comment. Math. Prace Math., Special issue 1, 279–281.

    MathSciNet  Google Scholar 

  68. Rosenthal, H. P., (1972). On factors of C([0, 1]) with non-separable dual, Israel J. Math., 13, 361–378. DOI: 10.1007/BF02762811

    Article  MathSciNet  Google Scholar 

  69. Rosenthal, H. P., (2003). The Banach spaces C(K), in Handbook of the Geometry of Banach Spaces, Volume 2, W. B. Johnson and J. Lindenstrauss eds., Elsevier. DOI: 10.1016/S1874-5849(03)80043-8

  70. Sauer, N., (1972), On the density of families of sets, J. Comb. Theory Ser. A, 13, 1, 145–147. DOI: 10.1016/ 0097-3165(72)90019-2

    Article  MathSciNet  MATH  Google Scholar 

  71. Shelah, S., (1972). A combinatorial problem: stability and order for models and theories in infinitary languages, Pacific J. Math., 41, 247–261.

    MathSciNet  MATH  Google Scholar 

  72. Shelah, S., (1985). Uncountable constructions for BA, e.g. groups and Banach spaces, Israel J. Math., 51, 273–297. DOI: 10.1007/BF02764721

    Article  MathSciNet  MATH  Google Scholar 

  73. Szankowski, A., (1978). Subspaces without the approximation property, Israel J. Math., 30, 123–129. DOI: 10.1007/BF02760833

    Article  MathSciNet  MATH  Google Scholar 

  74. Szarek, S. J., (1990). Spaces with large distance to l n∞ and random matrices, Amer. J. Math., 112, 899–942. DOI: 10.2307/2374731

    Article  MathSciNet  MATH  Google Scholar 

  75. Szarek, S. J. and Talagrand, M., (1989). An isomorphic version of the Sauer-Shelah lemma and the Banach-Mazur distance to the cube, in Geometric Aspects of Functional Analysis, Seminar 87–88, Lecture Notes in Mathematics, Springer, 1376, 105-112. DOI: 10.1007/BFb0090050

  76. Todorcevic, S., (1989). Partition Problems in Topology, Contemporary Mathematics, 84, American Mathematical Society, Providence.

  77. Todorcevic, S., (1999). Compact subsets of the first Baire class, J. Amer. Math. Soc., 12, 1179–1212.

    Article  MathSciNet  MATH  Google Scholar 

  78. Todorcevic, S., (2000). A dichotomy for P-ideals of countable sets, Fund. Math., 166, 251–267.

    MathSciNet  MATH  Google Scholar 

  79. Todorcevic, S., (2006). Biorthogonal systems and quotient spaces via Baire category methods, Math. Ann., 335. 687–715. DOI: 10.1007/s00208-006-0762-7

    Article  MathSciNet  MATH  Google Scholar 

  80. Todorcevic, S., (2007). Walks on ordinals and their characteristics, Progress inMathematics, 263, Birkhäuser.

  81. Todorcevic, S., (2010). Introduction to Ramsey Spaces. Annals Math. Studies, 174, Princeton Univ. Press.

  82. Todorcevic, S., (2009). Combinatorial Dichotomies in Set Theory, (preprint).

  83. Tsirelson, B. S., (1974). Not every Banach space contains an imbedding of l p or c 0, Funct. Anal. Appl., 8, 138–141. DOI: 10.1007/BF01078599

    Article  Google Scholar 

  84. Zippin, M., (1966). On perfectly homogeneous bases in Banach spaces, Israel J. Math., 4. 265–272. DOI: 10.1007/BF02771642

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Athens, Panepistimiopolis 157 84, Athens, Greece

    P. Dodos

  2. Instituto de Ciencias Matematicas (ICMAT), CSIC-UAM-UC3M-UCM, Madrid, Spain

    J. Lopez-Abad

  3. Université Denis Diderot - Paris 7, C.N.R.S., 2 place Jussieu - Case 7012, UMR 7056, 72521, Paris Cedex 05, France

    S. Todorcevic

  4. University of Toronto, M5S 2E4, Toronto, Canada

    S. Todorcevic

Authors
  1. P. Dodos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Lopez-Abad
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Todorcevic
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. Dodos.

Additional information

Dedicated to Professor Manuel Valdivia on the occasion of his 80th birthday

Submitted by Vicente Montesinos

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dodos, P., Lopez-Abad, J. & Todorcevic, S. Banach spaces and Ramsey Theory: some open problems. RACSAM 104, 435–450 (2010). https://doi.org/10.5052/RACSAM.2010.27

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.5052/RACSAM.2010.27

Keywords

Search

Navigation