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Tytuł artykułu

Weak sequential completeness and compactness in topological function spaces

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Języki publikacji
EN
Abstrakty
EN
Let (E, r) be a Hausdorff locally convex-solid function space (over a cr-finite measure space) and let E* stand for its topological dual. It is proved that the space (E, r) is weakly sequentially complete if and only if r is a c-Lebesgue and cr-Levy topology. In particular, a characterization of weak sequential completeness of Or-licz spaces L* in terms of Orlicz functions is given. Moreover, it is proved that the Eberlein-Smulian type theorem remains valid for a locally convex space (E, o~(E, E*)). A characterization of conditional and relative weak compactness in (E, r) is obtained.
Twórcy
autor
  • Faculty of Mathematics, Informatics and Econometrics, University of Zielona Góra, ul. Szafrana 4A, 65-516 Zielona Góra, Poland, M.Nowak@wmie.uz.zgora.pl
Bibliografia
  • [A] I. Amemiya, On order topological spaces. In: Proceedings of the International Symposium on Linear Spaces (Jerusalem, 1961), 14-23, Academic Press, Oxford, Pergamon 1961.
  • [AB1] C. D. Aliprantis and O. Burkinshaw, Locally Solid Riesz Spaces, Academic Press, New York, San Francisco, London, 1978.
  • [AB2] C. D. Aliprantis and O. Burkinshaw, Positive Operators, Academic Press, New York, London, 1985.
  • [BD1] O. Burkinshaw and P. Dodds, Weak sequential compactness and completeness in Riesz spaces, Canad. J. Math. 28 (1976), 1332-1339.
  • [BD1] O. Burkinshaw and P. Dodds, Disjoint sequences, compactness and semireflexivity in locally convex Riesz spaces, Illinois J. Math. 21 (1977), 759-775.
  • [DL] L. Drewnowski and I. Labuda, Copies of c0 and l∞ in topological Riesz spaces, Trans. Amer. Math. Soc. 350 (1998), 3555-3570.
  • [G] A. Grothendieck, Critères de compacité dans les espaces functionnels généraux, Amer. Journal of Math. 74 (1952), 168-186.
  • [KA] L. V. Kantorovitch and A. V. Akilov, Functional Analysis, Nauka, Moscow, 1984 (3rd ed) (in Russian).
  • [LZ] W. A. Luxemburg and A. C. Zaanen, Compactness of integral operators in Banach function spaces, Math. Ann. 149 (1963), 150-180.
  • [MW] L. Maligranda and W. Wnuk, Landau type theorem for Orlicz spaces, Math. Z. 208 (1991), 57-64.
  • [MO] S. Mazur and W. Orlicz, On some classes of linear spaces, Studia Math. 17 (1958), 97-119.
  • [MN] P. Meyer-Nieberg, Zur schwachen Kompaktheit in Banach verbänden, Math. Z. 134 (1973), 303-315.
  • [Mu] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Berlin-Heidelberg-New York 1983.
  • [N1] M. Nowak, Singular linear functionals on non-locally convex Orlicz spaces, Indag. Math. N.S. 3 (1992), 336-351.
  • [N2] M. Nowak, Weak sequential completeness of Orlicz spaces, Funct. et Approx. 25 (1997), 91-99.
  • [RR] M. M. Rao, Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker, New York, Basel, Hong Kong, 1991.
  • [Ro] S. Rolewicz, Metric Linear Spaces, Polish Scientific Publ., Warsaw, D. Reidel Publ. Comp., Dordrecht, Boston, Lancaster (2nd ed.), 1984.
  • [S] H. V. Schwartz, Banach Lattices and Operators, Teubner - Texte zur Mathematik, Band 71, B. G. Teubner Verlagsgesellschaft, Leipzig 1984.
  • [V] B. Z. Vulikh, Introduction to the Theory of Partially Ordered Spaces, Moscow, 1961 (in Russian).
  • [W] W. Wnuk, Banach Lattices with Order Continuous Norm, Advances Topics in Mathematics, Polish Scientific Publishers, Warsaw, 1999.
  • [Z] A. C. Zaanen, Riesz Spaces II, North-Holland, Amsterdam, London, 1983.
Uwagi
Dedicated to Professor Julian Musielak on his 75th birthday.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-BUS2-0007-0055
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