Nonstandard analysis and the theory of Banach spaces

  • C. Ward Henson
  • L. C. MooreJr.
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 983)


Banach Space Banach Lattice Winning Strategy Nonstandard Analysis Finite Dimensional Subspace 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AHL]
    I. Aharoni and J. Lindenstrauss, Uniform equivalence between Banach spaces, Bull. Amer. Math. Soc. 84 (1978), 281–283.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [AML]
    D. Amir and J. Lindenstrauss, The structure of weakly compact sets in Banach spaces, Ann. of Math. 88 (1968), 35–46.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [BA]
    K. J. Barwise, Back and forth through infinitary logic, in Studies in Model Theory, M. Morley (ed.), Math. Assn. of America (Providence, 1974), 5–34.Google Scholar
  4. [BEL 1]
    S. Bellenot, Prevarieties and intertwined completeness of locally convex spaces, Math. Ann. 217 (1975), 59–67.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [BEL 2]
    _____, On nonstandard hulls of convex spaces, Canad. J. Math. 28 (1976), 141–147.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [BEN]
    Y. Benyamini, Separable G spaces are isomorphic to C(K) spaces, Israel J. Math. 14 (1973), 287–293.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [BNL]
    Y. Benyamini and J. Lindenstrauss, A predual of l 1 which is not isomorphic to a C(K) space, Israel J. Math. 13 (1972), 246–254.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [BER]
    S. Bernau, A unified approach to the principle of local reflexivity, in Notes in Banach Spaces, H. E. Lacy (ed.), Univ. Texas Press (Austin, 1980), 427–439.Google Scholar
  9. [BRR]
    A. Bernstein and A. Robinson, Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos, Pacific J. Math. 16 (1966), 421–431.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [BDCK]
    J. Bretagnolle, D. Dacunha-Castelle and J.-L. Krivine, Lois stables et espaces Lp, Ann. Inst. Henri Poincaré, Sect. B 2 (1966), 231–259.MathSciNetzbMATHGoogle Scholar
  11. [BRS 1]
    Brunel and Sucheston, On B-convex Banach spaces, Math. System Theory 7 (1974), 294–299.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [BRS 2]
    _____, on J-Convexity and ergodic super-properties of Banach spaces, Trans. Amer. Math. Soc. 204 (1975), 79–90.MathSciNetzbMATHGoogle Scholar
  13. [CON]
    J. Conroy, The finite dimensional Riesz subspace structure of Banach lattices and applications to the theory of non-standard hulls, Ph.D. Thesis, Duke University, 1976.Google Scholar
  14. [CNM]
    J. Conroy and L. C. Moore, Jr., Local reflexivity in Banach lattices, (unpublished manuscript).Google Scholar
  15. [CK]
    H. Corson and V. Klee, Topological classification of convex sets, Proc. Symp. Pure. Math. Vol. 7 (Convexity), Amer. Math. Soc. (Providence, 1963), 37–51.Google Scholar
  16. [CZM]
    D. Cozart and L. C. Moore, Jr., The nonstandard hull of a normed Riesz space, Duke Math. J. 41 (1974), 263–275.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [DCK 1]
    D. Dacunha-Castelle and J.-L. Krivine, Applications des ultraproduits a l'etude des espaces et des algèbres de Banach, Studia Math. 41 (1972), 315–334.MathSciNetzbMATHGoogle Scholar
  18. [DCK 2]
    _____, Ultraproduits d'espaces de Banach, Sém. Goulaouic-Schwartz, 1971–1972, Exposés IX, X.Google Scholar
  19. [DCK 3]
    _____, Sous-espaces de L1, Israel J. Math. 26 (1977), 320–351.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [DA]
    M. Davis, Applied Nonstandard Analysis, J. Wiley Pub. (New York, 1977).zbMATHGoogle Scholar
  21. [DO]
    H. P. Dowson, Spectral Theory of Linear Operators, Academic Press (New York, 1977).zbMATHGoogle Scholar
  22. [DS]
    N. Dunford and J. Schwartz, Linear Operators Part I, Interscience Publishers (New York, 1958).zbMATHGoogle Scholar
  23. [EH]
    A. Ehrenfeucht, An application of games to the completeness problem for formalized theories, Fund. Math. 49 (1961), 129–141.MathSciNetzbMATHGoogle Scholar
  24. [ELP]
    P. Enflo, J. Lindenstrauss and G. Pisier, On the "three space problem," Math. Scand. 36 (1975), 199–210.MathSciNetzbMATHGoogle Scholar
  25. [ER]
    P. Enflo and H. P. Rosenthal, Some results concerning Lp(μ) spaces, J. Functional Analysis 14 (1973), 325–348.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [FR]
    R. Fraissé, Sur quelques classifications des systèmes des relations, Publ. Sci. Univ. Alge. Ser. A 1 (1954), 35–82.Google Scholar
  27. [GU]
    V. I. Gurarii, Spaces of universal disposition, isotropic spaces and the Mazur problem on rotations of Banach spaces, Sibirski Math. Z. 7 (1966), 1002–1013.MathSciNetzbMATHGoogle Scholar
  28. [HEI 1]
    S. Heinrich, Finite representability and super-ideals of operators, Dissertationes Math. vol. 172 (1980).Google Scholar
  29. [HEI 2]
    _____, Finite representability of operators, in Proc. Int. Conf. on Operator Algebras, Ideals and their Applications in Theoretical Physics, Leipzig, 1977.Google Scholar
  30. [HEI 3]
    _____, Closed operator ideals and interpolation, J. Funct. Analysis 35 (1980), 397–411.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [HEI 4]
    _____, Ultraproducts in Banach space theory, J. Reine Angew. Math. 313 (1980), 72–104.MathSciNetzbMATHGoogle Scholar
  32. [HEI 5]
    _____, Ultraproducts of L1-predual spaces, Fund. Math. 113 (1981), 221–234.MathSciNetzbMATHGoogle Scholar
  33. [HEI 6]
    _____, The isomorphic problem of envelopes, to appear.Google Scholar
  34. [HH]
    S. Heinrich and C. W. Henson, Banach space model theory II, in preparation.Google Scholar
  35. [HHM]
    S. Heinrich, C. W. Henson and L. C. Moore, Jr., Elementary equivalence of L1-preduals, in preparation.Google Scholar
  36. [HMK]
    S. Heinrich and P. Mankiewicz, Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces, preprint (Institute of Math., Polish Acad. Sci. 1980).Google Scholar
  37. [HP]
    S. Heinrich and A. Pietsch, A characterization of (∞,p,q)-integral operators, Math. Nachr. 89 (1979), 197–202.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [HEN 1]
    C. W. Henson, The isomorphism property in nonstandard analysis and its use in the theory of Banach spaces, J. Symbolic Logic 39 (1974), 717–731.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [HEN 2]
    _____, When do two Banach spaces have isometrically isomorphic nonstandard hulls? Israel J. Math. 22 (1975), 57–67.MathSciNetCrossRefzbMATHGoogle Scholar
  40. [HEN 3]
    _____, Nonstandard hulls of Banach spaces, Israel J. Math. 25 (1976), 108–144.MathSciNetCrossRefzbMATHGoogle Scholar
  41. [HEN 4]
    _____, Ultraproducts of Banach spaces, in the Altgeld Book 1975–1976, The University of Illinois Functional Analysis Seminar.Google Scholar
  42. [HEN 5]
    _____, Banach space model theory I, in preparation.Google Scholar
  43. [HMR 1]
    C. W. Henson and L. C. Moore, Jr., The nonstandard theory of topological vector spaces, Trans. Amer. Math. Soc. 172 (1972), 405–435.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [HMR 2]
    _____, Nonstandard hulls of the classical Banach spaces, Duke Math. J. 41 (1974), 227–284.MathSciNetCrossRefzbMATHGoogle Scholar
  45. [HMR 3]
    _____, Subspaces of the nonstandard hull of a normed space, Trans. Amer. Math. Soc. 197 (1974), 131–143.MathSciNetCrossRefzbMATHGoogle Scholar
  46. [HMR 4]
    _____, The Banach spaces lp(n) for large p and n, to appear in Manuscripta Mathematica.Google Scholar
  47. [HRB]
    K. Hrbacek, Axiomatic foundations for nonstandard analysis, Fund. Math. 98 (1978), 1–19.MathSciNetzbMATHGoogle Scholar
  48. [JA]
    R. C. James, Characterizations of reflexivity, Studia Math. 23 (1963/64), 205–216.MathSciNetzbMATHGoogle Scholar
  49. [KR 1]
    J.-L. Krivine, Theorie des modèles et espaces Lp, C. R. Acad. Sci. Paris Ser. A 275 (1972), 1207–1210.MathSciNetzbMATHGoogle Scholar
  50. [KR 2]
    _____, Langages à valeurs réelles et applications, Fund. Math. 81 (1974), 213–253.MathSciNetzbMATHGoogle Scholar
  51. [KR 3]
    _____, Sous espaces de dimension finie des espaces de Banach réticulés, Ann. of Math. 104 (1976), 1–29.MathSciNetCrossRefzbMATHGoogle Scholar
  52. [KU]
    K. D. Kürsten, On some questions of A. Pietsch II, Teor. Funkcional. Anal. i Priloženia (Kharkov) 29 (1978), 61–73. (Russian)zbMATHGoogle Scholar
  53. [LE]
    H. Lemberg, Nouvelle demonstration d'un théorème de J.-L. Krivine sur la finie representation de lp dans un éspace de Banach, Israel J. Math. 39 (1981), 341–348.MathSciNetCrossRefzbMATHGoogle Scholar
  54. [LT 1]
    J. Lindenstrauss and L. Tzafriri, The uniform approximation property in Orlicz spaces, Israel J. Math. 23 (1976), 142–155.MathSciNetCrossRefzbMATHGoogle Scholar
  55. [LT 2]
    _____, Classical Banach Spaces I, Sequence Spaces, Ergebnisse der Math. und ihrer Grenzgebiete 92, Springer-Verlag (Heidelberg, 1977).CrossRefzbMATHGoogle Scholar
  56. [LT 3]
    _____, Classical Banach Spaces II, Function Spaces, Ergebnisse der Math. und ihrer Grenzgebiete 97, Springer-Verlag (Heidelberg, 1979).CrossRefzbMATHGoogle Scholar
  57. [LO]
    P. Loeb, Conversion from non-standard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113–122.MathSciNetCrossRefzbMATHGoogle Scholar
  58. [LUS]
    W. Lusky, The Gurarii spaces are unique, Arch. Math. (Basel) 27 (1976), 627–635.MathSciNetCrossRefzbMATHGoogle Scholar
  59. [LUX 1]
    W. A. J. Luxemburg, A General theory of monads, in Applications of Model Theory to Algebra, Analysis and Probability, W. A. J. Luxemburg (ed.), Holt, Rinehart and Winston (New York, 1969), 18–86.Google Scholar
  60. [LUX 2]
    _____, On some concurrent binary relations occurring in analysis, in Contributions to Non-Standard Analysis, W. A. J. Luxemburg and A. Robinson (eds.), North-Holland (Amsterdam, 1972), 85–100.CrossRefGoogle Scholar
  61. [LUX 3]
    _____, Notes on Banach function spaces XVIB, Proc. Acad. Sci. Amsterdam A68 (1965), 658–667.MathSciNetzbMATHGoogle Scholar
  62. [LXZ]
    W. A. J. Luxemburg and A. C. Zaanen, Riesz Spaces I, North-Holland (Amsterdam, 1971).zbMATHGoogle Scholar
  63. [MN]
    P. Meyer-Nieberg, Characterisierung einiger topologischer und ordnungstheoretischer Eigenschaften von Banachverbänden unit Hilfe disjunkter Folgen, Arch. Math. (Basel) 24 (1973), 640–647.MathSciNetCrossRefzbMATHGoogle Scholar
  64. [MO 1]
    L. C. Moore, Jr., Hyperfinite extensions of bounded operators on a separable Hilbert space, Trans. Amer. Math. Soc. 218 (1976), 285–295.MathSciNetCrossRefzbMATHGoogle Scholar
  65. [MO 2]
    _____, Hyperfinite-dimensional subspaces of the non-standard hull of c0, Proc. Amer. Math. Soc. 80 (1980), 597–603.MathSciNetzbMATHGoogle Scholar
  66. [MO 3]
    _____, Approximately finite-dimensional Banach spaces, J. Funct. Analysis 42 (1981), 1–11.MathSciNetCrossRefzbMATHGoogle Scholar
  67. [MO 4]
    _____, Unitary equivalence of nonstandard extensions of bounded operators on Hilbert space, preprint.Google Scholar
  68. [NE]
    E. Nelson, Internal set theory: A new approach to nonstandard analysis, Bull. Amer. Math. Soc. 83 (1977), 1165–1198.MathSciNetCrossRefzbMATHGoogle Scholar
  69. [PR]
    A. Pelczynski and H. P. Rosenthal, Localization techniques in Lp-spaces, Studia Math. 52 (1975), 263–289.MathSciNetzbMATHGoogle Scholar
  70. [PT]
    A. Pietsch, Ultraprodukte von Operatoren in Banachräumen, Math. Nachr. 61 (1974), 123–132.MathSciNetCrossRefzbMATHGoogle Scholar
  71. [RA]
    S. A. Rakov, C-convexity and the "three space problem," Dokl. Acad. Nauk. SSSR 228 (1976), 303–305.MathSciNetGoogle Scholar
  72. [RI 1]
    M. Ribe, On uniformly homeomorphic normed spaces, Ark. Math. 14 (1976), 237–244.MathSciNetCrossRefzbMATHGoogle Scholar
  73. [RI 2]
    _____, On uniformly homeomorphic normed spaces, II, Ark. Math. 16 (1978), 1–9.MathSciNetCrossRefzbMATHGoogle Scholar
  74. [ROB 1]
    A. Robinson, On generalized limits and linear functionals, Pacific J. Math. 14 (1964), 269–283.MathSciNetCrossRefzbMATHGoogle Scholar
  75. [ROB 2]
    _____, Non-Standard Analysis, North-Holland (Amsterdam, 1966).zbMATHGoogle Scholar
  76. [RZ]
    A. Robinson and E. Zakon, A set-theoretical characterization of enlargements, in Applications of Model Theory to Algebra, Analysis and Probability, W. A. J. Luxemburg (ed.), Holt, Rinehart and Winston (New York, 1969), 109–122.Google Scholar
  77. [ROS 1]
    H. P. Rosenthal, A characterization of Banach spaces containing l1, Proc. Nat. Acad. Sci. (USA), 71 (1974), 2411–2413.CrossRefzbMATHGoogle Scholar
  78. [ROS 2]
    _____, On a theorem of J. L. Krivine, J. Funct. Analysis 28 (1978), 197–225.CrossRefzbMATHGoogle Scholar
  79. [SI]
    B. Simon, Trace Ideals and Their Applications, London Math. Soc. Lecture Notes 35, Cambridge Univ. Press (Cambridge, 1979).zbMATHGoogle Scholar
  80. [STE 1]
    J. Stern, Sur certaines classes d'espaces de Banach caracterisees par des formules, C. R. Acad. Sci. Paris Ser. A 278 (1974), 525–528.MathSciNetzbMATHGoogle Scholar
  81. [STE 2]
    _____, Some applications of model theory in Banach space theory, Ann. Math. Logic 9 (1976), 49–121.MathSciNetCrossRefzbMATHGoogle Scholar
  82. [STE 3]
    _____, The problem of envelopes for Banach spaces, Israel J. Math. 24 (1976), 1–15.MathSciNetCrossRefzbMATHGoogle Scholar
  83. [STE 4]
    _____, Ultrapowers and local properties of Banach spaces, Trans. Amer. Math. Soc. 240 (1978), 231–252.MathSciNetCrossRefzbMATHGoogle Scholar
  84. [STL]
    K. D. Stroyan and W. A. J. Luxemburg, Introduction to the Theory of Infinitesimals, Academic Press (New York, 1976).zbMATHGoogle Scholar
  85. [ZI]
    M. Zippin, On perfectly homogeneous bases in Banach spaces, Israel J. Math. 4 (1966), 265–272.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • C. Ward Henson
    • 1
  • L. C. MooreJr.
    • 2
  1. 1.Department of MathematicsUniversity of Illinois at Urbana/ChampaignUrbana
  2. 2.Department of MathematicsDuke UniversityDurham

Personalised recommendations