manuscripta mathematica

, Volume 12, Issue 1, pp 47–65 | Cite as

Über einen Graphensatz für Abbildungen mit normiertem Zielraum

  • Volker Eberhardt


This paper contains a detailed study of those locally convex spaces E-which we call GN-spaces-for which the following closed graph theorem holds: Every closed linear map from E to any normed linear space is continuous. In the first two sections we establish some characterisations and permanence-properties of these spaces. The main result reads as follows: Every separated GN-space is isomorphic to a barrelled subspace of some ωdϕd′, and conversely. Then we determine those GN-spaces which are (DF)-spaces, Schwartz-spaces or nuclear spaces. Finally we show that neither the strong dual nor the tensor product of GN-spaces are GN-spaces.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    BOURBAKI, N.: Espaces vectoriels topologiques, chap. I, II (2. Aufl., 1966), chap. III–V (1964). Paris: Hermann.Google Scholar
  2. 2.
    DIESTEL, J., MORRIS, S.A., SAXON, S.A.: Varieties of linear topological spaces. Trans. Amer. math. Soc.172, 207–230 (1972).Google Scholar
  3. 3.
    EBERHARDT, V.: Durch Graphensätze definierte lokalkonvexe Räume. Diss. München (1972).Google Scholar
  4. 4.
    EBERHARDT, V., ROELCKE, W.: Über einen Graphensatz für Abbildungen mit metrisierbarem Zielraum. In Vorbereitung.Google Scholar
  5. 5.
    FENSKE, C., SCHOCK, E.: Nuclear Spaces of Maximal Diametral Dimension. Erscheint in Composito Math.Google Scholar
  6. 6.
    GROTHENDIECK, A.: Produits tensoriels topologiques et espaces nucléaires. Mem. Amer. math. Soc.16 (1955).Google Scholar
  7. 7.
    HORVATH, J.: Topological Vector Spaces and Distributions, vol. I. Reading, Mass.: Addison-Wesley 1966.Google Scholar
  8. 8.
    IYAHEN, S.O.: Semiconvex Spaces. Glasgow math. J.9, 111–118 (1968).Google Scholar
  9. 9.
    IYAHEN, S.O.: The Domain Space in a Closed Graph Theorem. J. London math. Soc., II. Ser.,2, 19–22 (1970).Google Scholar
  10. 10.
    ILYAHEN, S.O.: On the Closed Graph Theorem. Israel J. Math.10, 96–105 (1971).Google Scholar
  11. 11.
    IYAHEN, S.O.: The Domain Space in a Closed Graph Theorem II. Revue Roumaine Math. pur. appl.17, 39–46 (1972).Google Scholar
  12. 12.
    KALTON, N.J.: Some Forms of the Closed Graph Theorem. Proc. Cambridge philos. Soc.70, 401–408 (1971).Google Scholar
  13. 13.
    KOMURA, Y.: On Linear Topological Spaces. Kumamoto J. Sci., Math.5, 148–157 (1962).Google Scholar
  14. 14.
    KÖTHE, G.: Topological Vector Spaces I. Berlin-Heidelberg-New York: Springer 1969.Google Scholar
  15. 15.
    LURJE, P.: Über topologische Vektorgruppen. Diss. München (1972).Google Scholar
  16. 16.
    PIETSCH, A.: Nuclear Locally Convex Spaces. Berlin-Heidelberg-New York: Springer 1972.Google Scholar
  17. 17.
    ROELCKE, W., LURJE, P., DIEROLF, S., EBERHARDT, V.: Einige Beispiele über Bairesche, normierte und ultrabornologische Räume. Arch. der Math.23, 3–13 (1972).Google Scholar
  18. 18.
    VALDIVIA, M.: Absolutely Convex Sets in Barrelled Spaces. Ann. Inst. Fourier21, 3–13 (1971).Google Scholar
  19. 19.
    VALDIVIA, M.: On Nonbornological Barrelled Spaces. Ann. Inst. Fourier22, 27–30 (1972).Google Scholar

Copyright information

© Springer-Verlag 1974

Authors and Affiliations

  • Volker Eberhardt
    • 1
  1. 1.Mathematisches Institut der Universität MünchenMünchen 2

Personalised recommendations