Positivity

, Volume 15, Issue 1, pp 73–85

On discrete and continuous quotient Riesz spaces

Article
  • 62 Downloads

Abstract

Order properties of quotient Riesz spaces E/N(f) by null ideals N(f) are investigated. We show relationships between properties of a Riesz space E and its order dual E~ and properties of quotients E/N(f) where f runs over some subspaces of E~. A characterization of metrizable locally convex topological Riesz spaces whose all quotients (by proper closed ideals) are discrete is also given.

Keywords

Riesz space Banach lattice Locally solid Riesz space Null ideal Order bounded operator 

Mathematics Subject Classification (2000)

46A40 46B42 46B45 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aliprantis, C., Burkinshaw, O.: Locally Solid Riesz Spaces with Applications to Economics. Mathematical Surveys and Monographs, vol. 105, American Mathematical Society (2003)Google Scholar
  2. 2.
    Aliprantis C., Burkinshaw O.: Positive Operators. Springer, Berlin (2006)MATHGoogle Scholar
  3. 3.
    Alpay S., Altin B., Tonyali C.: On property (b) of vector lattices. Positivity 7, 135–139 (2003)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Jameson, G.: Ordered Linear Spaces. Lecture Notes in Mathematics, vol. 141. Springer, Berlin (1970)Google Scholar
  5. 5.
    Lacey E., Wojtaszczyk P.: Nonatomic Banach lattices can have 1 as a dual space. Proc. Am. Math. Soc. 57, 79–84 (1976)MATHMathSciNetGoogle Scholar
  6. 6.
    Luxemburg W.A.J., Zaanen A.C.: Riesz Spaces I. North-Holland, Amsterdam (1971)MATHGoogle Scholar
  7. 7.
    Meyer-Nieberg P.: Banach Lattices. Springer, Berlin (1991)MATHGoogle Scholar
  8. 8.
    Wiatrowski, B.: On a structure of quotient Riesz spaces, Ph.D.Thesis, A. Mickiewicz University, Poznań (2006) (in Polish)Google Scholar
  9. 9.
    Wnuk W.: On the order-topological properties of the quotient space L/L A. Studia Math. 79, 139–149 (1984)MATHMathSciNetGoogle Scholar
  10. 10.
    Wnuk W.: Banach lattices with order continuous norms, Advanced Topics in Mathematics. Polish Scientific Publishers PWN, Warsaw (1999)Google Scholar
  11. 11.
    Wnuk W., Wiatrowski B.: Order properties of quotient Riesz spaces. Atti Sem. Mat. Fis. Univ. Modena e Regigio Emilia 53, 417–428 (2005)MATHMathSciNetGoogle Scholar
  12. 12.
    Wnuk, W.: Selected topics in quotient Riesz spaces—a short survey. Proceedings of the International Symposium on Banach and Function Spaces II, Kitakyusiu, Japan 2006, Yokohama Publ., 243–276Google Scholar
  13. 13.
    Zaanen A.C.: Riesz Spaces II. North-Holland, Amsterdam (1983)MATHGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceA. Mickiewicz UniversityPoznanPoland

Personalised recommendations