, Volume 20, Issue 2, pp 413–433 | Cite as

Nonstandard hulls of ordered vector spaces

  • Eduard EmelyanovEmail author
  • Hasan Gül


This paper undertakes the investigation of ordered vector spaces by applying nonstandard analysis. We introduce and study two types of nonstandard hulls of ordered vector spaces. Norm-nonstandard hulls of ordered Banach spaces are also investigated.


Ordered vector space Krein space Nonstandard extension Nonstandard hull 

Mathematics Subject Classification

46A40 46B40 


  1. 1.
    Albeverio, S., Høegh-Krohn, R., Fenstad, J.E., Lindstrøm, T.: Nonstandard methods in stochastic analysis and mathematical physics. In: Pure and Applied Mathematics, vol. 122. Academic Press Inc., Orlando (1986)Google Scholar
  2. 2.
    Aliprantis, C.D., Tourky, R.: Cones and duality. In: Graduate Studies in Mathematics, vol. 84. American Mathematical Society, Providence (2007)Google Scholar
  3. 3.
    Alpay, S., Altin, B., Tonyali, C.: On property \((b)\) of vector lattices. Positivity 7, 135–139 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Davis, M.: Applied nonstandard analysis. In: Pure and Applied Mathematics. New York, London, Sydney (1977)Google Scholar
  5. 5.
    Emel’yanov, E.Yu.: Infinitesimals in ordered vector spaces. Vladikavkaz. Mat. Zh. 15(1), 18–22 (2013)Google Scholar
  6. 6.
    Emel’yanov, E.Yu.: Erratum to “Infinitesimals in ordered vector spaces”. Vladikavkaz. Mat. Zh. 15(2), 82–83 (2013)Google Scholar
  7. 7.
    Emel’yanov, E.Yu.: Infinitesimals in vector lattices. In: Mathematics and its Applications, vol. 525. Kluwer Academic Publishers, Dordrecht, pp 161–230 (2000)Google Scholar
  8. 8.
    Emel’yanov, E.Yu.: Invariant homomorphisms of nonstandard extensions of Boolean algebras and vector lattices. Sibirsk. Mat. Zh. 38, 286–296 (1997) [English translation: Siberian Math. J. 38, 244–252 (1997)]Google Scholar
  9. 9.
    Emel’yanov, E.Yu.: Infinitesimal analysis and vector lattices. Sib. Adv. Math. 6, 19–70 (1996)Google Scholar
  10. 10.
    Emel’yanov, E.Yu.: An infinitesimal approach to the representation of vector lattices by spaces of continuous functions on a compactum. Dokl. Akad. Nauk 344(1), 9–11 (1995)Google Scholar
  11. 11.
    Emel’yanov, E.Yu.: Banach–Kantorovich spaces associated with order-hulls of decomposable lattice-normed spaces. Sibirsk. Mat. Zh. 36, 72–85 (1995) [English translation: Sib. Math. J. 36, 66–77 (1995)]Google Scholar
  12. 12.
    Emel’yanov, E.Yu.: Order hulls of vector lattices. Dokl. Akad. Nauk. 340(3), 303–304 (1995)Google Scholar
  13. 13.
    Emel’yanov, E.Yu.: Ordered and regular hulls of vector lattices. Sibirsk. Mat. Zh. 35, 1243–1252 (1994) [English translation: Sib. Math. J. 35, 1101–1108 (1995)]Google Scholar
  14. 14.
    Emel’yanov, E.Yu.: Nonstandard hulls of vector lattices. Sibirsk. Mat. Zh. 35, 83–95 (1994) [English translation: Sib. Math. J. 35, 77–87 (1994)]Google Scholar
  15. 15.
    Gorokhova, S.G., Emel’yanov, E.Yu.: On the concept of stability of order convergence in vector lattices. Sibirsk. Mat. Zh. 35, 1026–1031 (1994) [English translation: Sib. Math. J. 35, 912–916 (1994)]Google Scholar
  16. 16.
    Henson, C.W., Moore, L.C.: Nonstandard analysis and the theory of Banach spaces. In: Nonstandard Analysis—Recent Developments (Victoria, B.C., 1980). Lecture Notes in Mathematics, vol. 983. Springer, Berlin (1983)Google Scholar
  17. 17.
    Hurd, A., Loeb, P.A.: Introduction to Nonstandard Real Analysis. Academic Press, New York (1985)zbMATHGoogle Scholar
  18. 18.
    Kusraev, A.G., Kutateladze, S.S.: Nonstandard methods of analysis. In: Mathematics and its Applications, vol. 291. Kluwer Academic Publishers Group, Dordrecht (1994)Google Scholar
  19. 19.
    Luxemburg, W.A.J.: A general theory of monads. In: Applications of Model Theory to Algebra, Analysis, and Probability (International Symposium, Pasadena, Calif., 1967). Holt, Rinehart and Winston, New York (1969)Google Scholar
  20. 20.
    Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces I. North-Holland, Amsterdam (1971)zbMATHGoogle Scholar
  21. 21.
    Luxemburg, W.A.J., Stroyan, K.D.: Introduction to the theory of infinitesimals. In: Pure and Applied Mathematics, vol. 72. Academic Press, Boston (1976)Google Scholar
  22. 22.
    Onal, S.: Private communication (2013)Google Scholar
  23. 23.
    Robinson, A.: Nonstandard Analysis. North-Holland, Amsterdam (1966)Google Scholar
  24. 24.
    Schaefer, H.H., Wolff, M.P.: Topological vector spaces. In: Graduate Texts in Mathematics, vol. 3, 2nd edn. Springer, New York (1999)Google Scholar
  25. 25.
    Veksler, A.I.: Archimedean principle in homomorphic images of l-groups and of vector lattices. Izv. Vys. Ucebn. Zaved. Matematika. 54(4), 33–38 (1966)MathSciNetGoogle Scholar
  26. 26.
    Vulikh, B.Z.: Introduction to Theory of Partially Ordered Spaces. Noordhoff, Groningen (1967)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Middle East Technical UniversityAnkaraTurkey

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