Results in Mathematics

, Volume 25, Issue 3–4, pp 242–251 | Cite as

Inductive limits of spaces of vector- valued integrable functions

  • Miguel Florencio
  • Fernando Mayoral
  • Pedro J. Paul
Article
First Online:
Received:

Abstract

For an order-continuous Banach function space Λ and a separated inductive limit E:= indn E n, we prove that indn A {En} is a topological subspace of Λ {E}; moreover, both spaces coincide if the inductive limit is hyperstrict. As a consequence, we deduce that if E is an LF-space, then L p {E} is barrelled for 1 ≤ p ≤ ∞.

1991 Mathematics Subject Classification

46A13 46A30 46G10 

En]Keywords

Inductive limits Banach lattices Lusin measurability Bochner integrability 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Bonet, J.; Dierolf, S.: Countable strict inductive limits and the bounded decomposition property. Proc. Roy. Irish Acad. Sect. A, 90, (1990) 63–71.MathSciNetMATHGoogle Scholar
  2. [2]
    Bonet, J.; Dlerolf, S. Y férnandez, C: Inductive limits of vector valued sequence spaces. Publ. Mat., 33 (1989) 363–367.MathSciNetMATHGoogle Scholar
  3. [3]
    Bourbaki, N.: Intégration. Hermann, Paris, 1973.Google Scholar
  4. [4]
    Dieudonné, J.: Sur les espaces de Köthe. J. Analyse Math., 1 (1951) 81–115.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Florencio, M.; Paúl, P.J. Y sáez, C: Köthe echelon spaces à la Dieudonné. Indag. Math. (to appear).Google Scholar
  6. [6]
    Grothendieck, A.: Produits tensoriels topologiques et espaces nucléaires. Mem. Amer. Math. Soc. 16, 1955.Google Scholar
  7. [7]
    Grothendieck, A.: Sur certaines classes de suites dans les espaces de Banach et le thorème de Dvoretzky-Rogers. Bol. Soc. Mat. São Paulo, 8 (1956) 81–110.MathSciNetGoogle Scholar
  8. [8]
    Jarchow, H.: Locally Convex Spaces. B. G. Teubner, Stuttgart, 1981.MATHCrossRefGoogle Scholar
  9. [9]
    Kantorovitch, L.; Akilov, G.: Analyse Fonctionnelle, Tome I. Mir, Moscow, 1981.Google Scholar
  10. [10]
    Köthe, G.: Topological Vector Spaces II. Springer-Verlag, Berlin, Heidelberg and New York, 1979.MATHCrossRefGoogle Scholar
  11. [11]
    Lindenstrauss, J. and Tzafriri, L.: Classical Banach Spaces II, Function spaces. Springer-Verlag, Berlin, Heidelberg and New York, 1979.MATHGoogle Scholar
  12. [12]
    Pérez Carreras, P. and Bonet, J.: Barrelled Locally Convex Spaces. North-Holland, Amsterdam, New York, Oxford and Tokyo, 1987.MATHGoogle Scholar
  13. [13]
    Schmets, J.: Spaces of vector-valued continuous functions. Springer Lecture Notes in Math. 1003, Springer-Verlag Berlin, Heidelberg and New York, 1983.MATHGoogle Scholar
  14. [14]
    Thomas, G.E.F.: The Lebesgue-Nikodym theorem for vector valued Radon measures. Mem. Amer. Math. Soc. 139, 19Google Scholar
  15. [15]
    Thomas, G.E.F.: Totally summable functions with values in locally convex spaces, in Measure Theory (Proc. Oberwolfach, 1975), Springer Lecture Notes in Math. 541, 117–131, Springer-Verlag, Berlin, Heidelberg and New York, 1976.CrossRefGoogle Scholar
  16. [16]
    Zaanen, A.C.: Integration. North-Holland, Amsterdam, 1967.MATHGoogle Scholar

Copyright information

© Birkhäuser Verlag, Basel 1994

Authors and Affiliations

  • Miguel Florencio
    • 1
  • Fernando Mayoral
    • 1
  • Pedro J. Paul
    • 1
  1. 1.E.S. Ingenieros IndustrialesSpain

Personalised recommendations