Skip to main content
SpringerLink
Log in

Order Continuous Probabilistic Riesz Norms

  • Published:
Vietnam Journal of Mathematics Aims and scope Submit manuscript

Abstract

The concepts of order continuous norm, σ-order continuous norm, and Fatou norm defined on ordinary normed Riesz spaces are very important in the study of Riesz spaces. In this paper, we introduce the probabilistic analogues of such norms on a topological probabilistic normed Riesz (TPNR) space, and investigate their basic properties. In this context, some well-known theorems of the classical theory of topological Riesz spaces are proved in the setting of TPNR spaces, but now using the tools of probabilistic normed (PN) spaces. However, an interesting and different point here is that, although the classical order continuous Riesz norms are order preserving, the probabilistic Riesz norms considered in this work are order reversing mappings due to the nature of probabilistic distances.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aliprantis, C.D., Burkinshaw, O.: Locally Solid Riesz Spaces with Applications to Economics. American Mathematical Society, Providence, R.I (2003)

  2. Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis. Springer-Verlag, Berlin–Heidelberg (2006)

    MATH  Google Scholar 

  3. Alsina, C., Schweizer, B., Sklar, A.: On the definition of a probabilistic normed space. Aequationes Math. 46, 91–98 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Alsina, C., Schweizer, B., Sklar, A.: Continuity properties of probabilistic norms. J. Math. Anal. Appl. 208, 446–452 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fremlin, D.H.: Measure Theory, vol. 3. Torres Fremlin, Colchester (2002)

    MATH  Google Scholar 

  6. Freudenthal, H.: Teilweise geordnete modulen. Proc. Acad. Amst. 39, 641–651 (1936)

    MATH  Google Scholar 

  7. Guo, T.-X., Ma, R.-P.: Some reviews on various definitions of a random conjugate space together with various kinds of boundedness of a random linear functional. Acta Anal. Funct. Appl. 6, 16–38 (2004)

    MathSciNet  MATH  Google Scholar 

  8. Guo, T.-X., Zeng, X.-L.: Random strict convexity and random uniform convexity in random normed modules. Nonlinear Anal. TMA 73, 1239–1263 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Johnson, W.B., Lindenstrauss, J.: Handbook of the Geometry of Banach Spaces, vol. 1. Elsevier (2001)

  10. Kantorovich, L.V.: Lineare halbgeordnete Raume. Receueil Math. 2, 121–168 (1937)

    MATH  Google Scholar 

  11. Kent, D.C., Richardson, G.D.: Ordered probabilistic metric spaces. J. Aust. Math. Soc., Ser. A 46, 88–99 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lafuerza-Guillén, B., Rodríguez-Lallena, J.A., Sempi, C.: A study of boundedness in probabilistic normed spaces. J. Math. Anal. Appl. 232, 183–196 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lafuerza-Guillén, B.: D-bounded sets in probabilistic normed spaces and in their products. Rend. Mat. 21, 17–28 (2001)

    MathSciNet  MATH  Google Scholar 

  14. Lafuerza-Guillén, B., Rodríguez-Lallena, J.A., Sempi, C.: Normability of probabilistic normed spaces. Note Mat. 29, 99–111 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Menger, K.: Probabilistic geometry. Proc. Natl. Acad. Sci. USA 37, 226–229 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  16. Meyer-Neiberg, P.: Banach Lattices. Springer-Verlag (1991)

  17. Mustari, D.H.: On almost sure convergence in linear spaces of random variables. Theory Probab. Appl. 15, 337–342 (1970)

    Article  MathSciNet  Google Scholar 

  18. Riesz, F.: Sur la décomposition des opérations linéaires. Atti. Congr. Int. Mat. Bologna 3, 143–148 (1930)

    MATH  Google Scholar 

  19. Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. Elsevier/North-Holland, New York, (1983). Reissued with an errata list, notes and supplemental bibliography by Dover Publications, New York (2005)

  20. Sempi, C.: A short and partial history of probabilistic normed spaces. Mediterr. J. Math. 3, 283–300 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Şençi̇men, C., Pehli̇van, S.: Probabilistic normed Riesz spaces. Acta. Math. Sin. Engl. Ser. 28, 1401–1410 (2012)

    Article  MathSciNet  Google Scholar 

  22. Šerstnev, A.N.: On the notion of a random normed space. Dokl. Akad. Nauk. SSSR 149, 280–283 (1963)

    MathSciNet  Google Scholar 

  23. Sherwood, H.: Isomorphically isometric probabilistic normed linear spaces. Stochastica 3, 71–77 (1979)

    MathSciNet  MATH  Google Scholar 

  24. Taylor, R.L.: Convergence of elements in random normed spaces. Bull. Austral. Math. Soc. 12, 156–183 (1975)

    Article  MathSciNet  Google Scholar 

  25. Zaanen, A.C.: Introduction to Operator Theory in Riesz Spaces. Springer-Verlag, Berlin–Heidelberg (1997)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Arts and Sciences, Department of Mathematics, Istiklal Campus, Mehmet Akif Ersoy University, 15030, Burdur, Turkey

    Celaleddin Şençimen

Authors
  1. Celaleddin Şençimen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Celaleddin Şençimen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Şençimen, C. Order Continuous Probabilistic Riesz Norms. Vietnam J. Math. 44, 295–305 (2016). https://doi.org/10.1007/s10013-015-0152-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10013-015-0152-0

Keywords

Mathematics Subject Classification (2010)

Search

Navigation