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Nonstandard hulls of ordered vector spaces

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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.

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References

  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)

  2. Aliprantis, C.D., Tourky, R.: Cones and duality. In: Graduate Studies in Mathematics, vol. 84. American Mathematical Society, Providence (2007)

  3. Alpay, S., Altin, B., Tonyali, C.: On property \((b)\) of vector lattices. Positivity 7, 135–139 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Davis, M.: Applied nonstandard analysis. In: Pure and Applied Mathematics. New York, London, Sydney (1977)

  5. Emel’yanov, E.Yu.: Infinitesimals in ordered vector spaces. Vladikavkaz. Mat. Zh. 15(1), 18–22 (2013)

  6. Emel’yanov, E.Yu.: Erratum to “Infinitesimals in ordered vector spaces”. Vladikavkaz. Mat. Zh. 15(2), 82–83 (2013)

  7. Emel’yanov, E.Yu.: Infinitesimals in vector lattices. In: Mathematics and its Applications, vol. 525. Kluwer Academic Publishers, Dordrecht, pp 161–230 (2000)

  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)]

  9. Emel’yanov, E.Yu.: Infinitesimal analysis and vector lattices. Sib. Adv. Math. 6, 19–70 (1996)

  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)

  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)]

  12. Emel’yanov, E.Yu.: Order hulls of vector lattices. Dokl. Akad. Nauk. 340(3), 303–304 (1995)

  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)]

  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)]

  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)]

  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)

  17. Hurd, A., Loeb, P.A.: Introduction to Nonstandard Real Analysis. Academic Press, New York (1985)

    MATH  Google Scholar 

  18. Kusraev, A.G., Kutateladze, S.S.: Nonstandard methods of analysis. In: Mathematics and its Applications, vol. 291. Kluwer Academic Publishers Group, Dordrecht (1994)

  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)

  20. Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces I. North-Holland, Amsterdam (1971)

    MATH  Google Scholar 

  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)

  22. Onal, S.: Private communication (2013)

  23. Robinson, A.: Nonstandard Analysis. North-Holland, Amsterdam (1966)

    Google Scholar 

  24. Schaefer, H.H., Wolff, M.P.: Topological vector spaces. In: Graduate Texts in Mathematics, vol. 3, 2nd edn. Springer, New York (1999)

  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)

    MathSciNet  Google Scholar 

  26. Vulikh, B.Z.: Introduction to Theory of Partially Ordered Spaces. Noordhoff, Groningen (1967)

    MATH  Google Scholar 

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Correspondence to Eduard Emelyanov.

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Emelyanov, E., Gül, H. Nonstandard hulls of ordered vector spaces. Positivity 20, 413–433 (2016). https://doi.org/10.1007/s11117-015-0364-2

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  • DOI: https://doi.org/10.1007/s11117-015-0364-2

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