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Beiträge zur Strukturtheorie der Grothendieck-Räume

Conference paper
Part of the Sitzungsberichte der Heidelberger Akademie der Wissenschaften book series (HD AKAD, volume 1985 / 1)

Zusammenfassung

Ein Banachraum E heißt Grothendieck-Raum, falls in E′ jede σ(E′, E)-konvergente Folge σ(E′, E′′)-konvergent ist. Man sagt dann auch, E besitzt die Grothendieck-Eigenschaft.

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Literatur

  1. 1.
    Aliprantis CD, Burkinshaw O (1978) Locally Solid Riesz Spaces. Academic Press, New York San Francisco LondonGoogle Scholar
  2. 2.
    Andô T (1961) Convergent sequences of finitely additive measures. Pacific J Math 11:395 – 404CrossRefGoogle Scholar
  3. 3.
    Beauzamy B (1982) Introduction to Banach spaces and their geometry. North-Holland, Amsterdam New York OxfordGoogle Scholar
  4. 4.
    Bessaga C, Pelczyński A (1958) On bases and unconditional convergence of series in Banach spaces. Stud Math 17:151 – 164Google Scholar
  5. 5.
    Bourgain J (1981) New classes of L p-spaces. Lecture Notes in Mathematics 889. Springer-Verlag, Berlin Heidelberg New YorkGoogle Scholar
  6. 6.
    Burkinshaw O (1974) Weak compactness in the order dual of a vector lattice. Trans Amer Math Soc 187:183 – 201CrossRefGoogle Scholar
  7. 7.
    Burkinshaw O, Dodds P (1976) Weak sequential compactness and completeness in Riesz spaces. Canad J Math 28:1332 – 1339CrossRefGoogle Scholar
  8. 8.
    Burkinshaw O, Dodds P (1977) Disjoint sequences, compactness and semireflexivity in locally convex Riesz spaces. Illinois J Math 21:759 – 775Google Scholar
  9. 9.
    Cartwright DI, Lotz HP (1977) Disjunkte Folgen in Banachverbänden und Kegelabsolutsummierende Abbildungen. Arch Math 28:525 – 532CrossRefGoogle Scholar
  10. 10.
    Cembranos P (1982) Algunas propiedades del espacio de Banach c 0 (E). Actualités mathématiques, Actes de 6e Congr. Group Math Expr Latine (Luxembourg 1981): 333 – 336Google Scholar
  11. 11.
    Dashiell FK (1981) Nonweakly compact operators from order-Cauchy complete C(S) lattices, with applications to Baire classes. Trans Amer Math Soc 266:397 – 413Google Scholar
  12. 12.
    Diestel J, Seifert CJ (1978) The Banaeh-Saks ideal, I. Operators acting on C(Ω). Com- mentationes Math, Tomus specialis in honorem Ladislai Orlicz, I., 109 – 118Google Scholar
  13. 13.
    Diestel J (1980) A survey of results related to the Dunford-Pettis property. Proc Conf on Integration, Topology and Geometry in Linear Spaces, Chapel Hill, 15 – 60. Contemporary Mathematics, Vol. 2, American Math Soc, Providence, Rhode IslandGoogle Scholar
  14. 14.
    Diestel J (1984) Sequences and series in Banach spaces. Springer-Verlag, New York Berlin Heidelberg TokyoCrossRefGoogle Scholar
  15. 15.
    Dodds PG (1975) Sequential convergence in the order duals of certain classes of Riesz spaces. Trans Amer Math Soc 203:391 – 403CrossRefGoogle Scholar
  16. 16.
    Duhoux M (1978) Mackey topologies on Riesz spaces and weak compactness. Math Z 158:199 – 209CrossRefGoogle Scholar
  17. 17.
    Dunford N, Schwartz JT (1958) Linear operators. Part I: General theory. Wiley, New YorkGoogle Scholar
  18. 18.
    Faires B (1976) On Vitali-Hahn-Saks-Nikodym type theorems. Ann Inst Fourier Univ Grenoble 26,4:99 – 114CrossRefGoogle Scholar
  19. 19.
    Faires BT (1978) Varieties and vector measures. Math Nachr 85:303 – 314CrossRefGoogle Scholar
  20. 20.
    Figiel T, Johnson WB, Tzafriri L (1975) On Banach lattices and spaces having local unconditional structure, with applications to Lorentz function spaces. J Approx Theory 13:395 – 412CrossRefGoogle Scholar
  21. 21.
    Figiel T, Ghoussoub N, Johnson WB (1981) On the structure of non-weakly compact operators on Banach lattices. Math Annalen 257:317 – 334CrossRefGoogle Scholar
  22. 22.
    Fremlin DH (1973) Topological Riesz spaces and measure theory. Cambridge University Press, LondonGoogle Scholar
  23. 23.
    Freniche Ibáñez FJ (1983) Teorema de Vitali-Hahn-Saks en algebras de Boole. Thesis, SevillaGoogle Scholar
  24. 24.
    Freniche Ibáñez FJ (1984) The Vitali-Hahn-Saks theorem for Boolean algebras with the subsequential interpolation property. Proc Amer Math Soc 92:362 – 366CrossRefGoogle Scholar
  25. 25.
    Gillman L, Jerison M (1976) Rings of continuous functions. Springer-Verlag, New York Heidelberg BerlinGoogle Scholar
  26. 26.
    Graves WH, Wheeler RF (1983) On the Grothendieck and Nikodym properties for algebras of Baire, Borel and universally measurable sets. Rocky Mountain J Math 13:333 – 353CrossRefGoogle Scholar
  27. 27.
    Grothendieck A (1953) Sur les applications linéaires faiblement compactes d’espaces du type C(K). Canad J Math 5:129 – 173CrossRefGoogle Scholar
  28. 28.
    Haydon R (1981) A non-reflexive Grothendieck space that does not contain l . Israel J Math 40:65 – 73CrossRefGoogle Scholar
  29. 29.
    Johnson WB, Zippin W (1973) Separable L 1 preduals are quotients of C(Δ). Israel J Math 16:198 – 202CrossRefGoogle Scholar
  30. 30.
    Johnson WB, Tzafriri L (1977) Some more Banach spaces which do not have local unconditional structure. Houston J Math 3:55 – 60Google Scholar
  31. 31.
    Kühn B (1977) Orthogonalkompakte Teilmengen topologischer Vektorverbände. Dissertation, DortmundGoogle Scholar
  32. 32.
    Kühn B (1979) Banachverbände mit ordnungsstetiger Dualnorm. Math Z 167: 271 – 277CrossRefGoogle Scholar
  33. 33.
    Kühn B (1980) Schwache Konvergenz in Banachverbänden. Arch Math 35:554 – 558CrossRefGoogle Scholar
  34. 34.
    Kuo T-H (1976) Weak compactness of operators on Grothendieck spaces. J Nat Chiao Tung Univ 2:133 – 138Google Scholar
  35. 35.
    Lacey HE (1974) The isometric theory of classical Banach spaces. Springer-Verlag, Berlin Heidelberg New YorkCrossRefGoogle Scholar
  36. 36.
    Leavelle TL (1983) The reciprocal Dunford-Pettis property. PreprintGoogle Scholar
  37. 37.
    Lindenstrauss J, Tzafriri L (1977) Classical Banach spaces I. Sequence spaces. Springer-Verlag, Berlin Heidelberg New YorkCrossRefGoogle Scholar
  38. 38.
    Lindenstrauss J, Tzafriri L (1979) Classical Banach spaces II. Function spaces. Springer-Verlag, Berlin Heidelberg New YorkGoogle Scholar
  39. 39.
    Lotz HP, Rosenthal HP (1978) Embeddings of C(Δ) and L1 [0,1] in Banach lattices. Israel J Math 31:169 – 179CrossRefGoogle Scholar
  40. 40.
    Meyer-Nieberg P (1973) Charakterisierung einiger topologischer und ordnungstheoretischer Eigenschaften von Banachverbänden mit Hilfe disjunkter Folgen. Arch Math 24:640 – 647CrossRefGoogle Scholar
  41. 41.
    Meyer-Nieberg P (1973) Zur schwachen Kompaktheit in Banachverbänden. Math Z 134:303 – 315CrossRefGoogle Scholar
  42. 42.
    Meyer-Niberg P (1974) Über Klassen schwach kompakter Operatoren in Banachverbänden. Math Z 138:145–159CrossRefGoogle Scholar
  43. 43.
    Meyer-Nieberg P (1978) Ein elementarer Beweis einer Charakterisierung von M-Räumen. Math Z 161:95 – 96CrossRefGoogle Scholar
  44. 44.
    Moltó A (1981) On the Vitali-Hahn-Saks theorem. Proc Royal Soc Edinburgh 90A: 163 – 173CrossRefGoogle Scholar
  45. 45.
    Niculescu CP (1981) Weak compactness in Banach lattices. J Operator Theory 6:217 – 231Google Scholar
  46. 46.
    Niculescu C (1983) Order σ-continuous operators on Banach lattices. In: Banach Space Theory and its Applications, Proceedings, Bucharest 1981, 188 – 201. Springer- Verlag, Berlin Heidelberg New York TokyoCrossRefGoogle Scholar
  47. 47.
    Pelczyński A (1962) Banach spaces on which every unconditionally converging operator is weakly compact. Bull Acad Pol Sci 10:641 – 648Google Scholar
  48. 48.
    Pelczyński A (1965) On strictly singular and strictly cosingular operators. I. Strictly singular and strictly cosingular operators in C(S)-spaces. Bull Acad Pol Sci 13:31 – 41Google Scholar
  49. 49.
    Rosenthal HP (1970) On relatively disjoint families of measures, with some applications to Banach space theory. Stud Math 37:13 – 36, 311 – 313Google Scholar
  50. 50.
    Rosenthal HP (1972/1975) On factors of C[0,1] with non-separable dual. Israel J Math 13:361 – 378, ibid 21:93 – 94CrossRefGoogle Scholar
  51. 51.
    Rosenthal HP (1974) A characterization of Banach spaces containing l 1. Proc Nat Acad Sci (USA) 71:2411 – 2413CrossRefGoogle Scholar
  52. 52.
    Rutovitz D (1965) Some parameters associated with finite-dimensional Banach spaces. J London Math Soc 40:241 – 255CrossRefGoogle Scholar
  53. 53.
    Schachermayer W (1982) On some classical measure-theoretical theorems for non-sigma-complete Boolean algebras. Dissertationes Math 214Google Scholar
  54. 54.
    Schaefer HH (1971) Topological vector spaces, 3rd print. Springer-Verlag, New York Heidelberg BerlinCrossRefGoogle Scholar
  55. 55.
    Schaefer HH (1971) Weak convergence of measures. Math Annalen 193:57 – 64CrossRefGoogle Scholar
  56. 56.
    Schaefer HH (1974) Banach lattices and positive operators. Springer-Verlag, New York Heidelberg BerlinCrossRefGoogle Scholar
  57. 57.
    Seever GL (1968) Measures on F-spaces. Trans Amer Math Soc 133:267 – 280Google Scholar
  58. 58.
    Simons S (1975) On the Dunford-Pettis property and Banach spaces that contain c0. Math Annalen 216:225 – 231CrossRefGoogle Scholar
  59. 59.
    Simons S (1977) Weak compactness in locally convex vector lattices. Arch Math 29:537 – 548CrossRefGoogle Scholar
  60. 60.
    Shoenfield JR (1971) Measurable Cardinals. In Proc of the Summer School and Colloquium in Mathematical Logic, Manchester, 1969, 19–49. North-Holland, Amsterdam LondonGoogle Scholar
  61. 61.
    Sobczyk A (1962) Extension properties of Banach spaces. Bull Amer Math Soc 68:217 – 224CrossRefGoogle Scholar
  62. 62.
    Tokarev EV (1984) Quotient spaces of Banach lattices and Marcinkiewicz spaces. Siberian Math J 25:332 – 338CrossRefGoogle Scholar
  63. 63.
    Veksler AJ, Geiler VA (1972) Order and disjoint completeness of linear partially ordered spaces. Siberian Math J 13:30 – 35CrossRefGoogle Scholar
  64. 64.
    Wilansky A (1978) Modern Methods in Topological Vector Spaces. McGraw-HillGoogle Scholar
  65. 65.
    Grothendieck A (1955) Une caractérisation vectorielle-métrique des espaces L 1 Canad J Math 7:552 – 561CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität TübingenTübingenDeutschland

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