Israel Journal of Mathematics

, Volume 16, Issue 2, pp 150–158 | Cite as

On mackey convergence in locally convex spaces

  • Hans Jarchow
  • Johan Swart


After some introductory propositions, we give a dual characterization of those locally convex spaces which satisfy the Mackey convergence condition or the fast convergence condition by means of Schwartz topologies. Making use of the universal Schwartz space (l ,τ(l ,l 1)) we prove some representation theorems for bornological and ultrabornological spaces, that is, every bornological spaceE is a dense subspace of an inductive limit lim indE a, a∈A, ofseparable Banach spacesE a, and every Mackey null sequence inE is a null sequence in someE a. IfE is ultrabornological, thenE can be represented as lim indE a,a∈A, allE a separable Banach spaces, such that every fast null sequence inE is a null sequence in someE a.


Fast Convergence Convergence Condition Topological Vector Space Convex Space Separable Hilbert Space 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. Hogbé-Nlend,Techniques de bornologies en théorie des espaces vectoriels topologiques, Lecture Notes331 (1973), 84–162.CrossRefGoogle Scholar
  2. 2.
    J. Horváth,Topological vector spaces and distributions I, Addison-Wesley, Reading, Massachusetts, 1966.MATHGoogle Scholar
  3. 3.
    H. Jarchow,Duale Charakterisierungen der Schwartz-Räume, Math. Ann.196 (1972), 85–90.CrossRefMathSciNetGoogle Scholar
  4. 4.
    H. Jarehow,Die Universalität des Räumes c 0 für die Klasse der Schwartz-Räume, Math. Ann.203 (1973), 211–214.CrossRefMathSciNetGoogle Scholar
  5. 5.
    G. Köthe,Topologische lineare Räume I, 2. Aufl. Springer, Berlin-Heidelberg-New York, 1966.MATHGoogle Scholar
  6. 6.
    D. Randtke,A simple example of a universal Schwartz space, Proc. Amer. Math. Soc.37, (1973), 185–188.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    T. Terzioĝlu,On Schwartz spaces, Math. Ann.182 (1969), 236–242.CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    M. De Wilde,Réseaux dans les espaces linéaires à semi-normes, Mém. Soc. Roy. Sci. Liège (5)XVIII (1969), 2.Google Scholar
  9. 9.
    M. De Wilde,Ultrabornological spaces and the closed graph theorem, Bull. Soc. Roy. Sci. Liège40 (1971), 116–118.MATHGoogle Scholar

Copyright information

© The Weizmann Science Press of Israel 1973

Authors and Affiliations

  • Hans Jarchow
    • 1
  • Johan Swart
    • 1
  1. 1.Mathematisches InstitutUniversity of ZürichZürichSwitzerland

Personalised recommendations