Abstract
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.
Similar content being viewed by others
References
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)
Aliprantis, C.D., Tourky, R.: Cones and duality. In: Graduate Studies in Mathematics, vol. 84. American Mathematical Society, Providence (2007)
Alpay, S., Altin, B., Tonyali, C.: On property \((b)\) of vector lattices. Positivity 7, 135–139 (2003)
Davis, M.: Applied nonstandard analysis. In: Pure and Applied Mathematics. New York, London, Sydney (1977)
Emel’yanov, E.Yu.: Infinitesimals in ordered vector spaces. Vladikavkaz. Mat. Zh. 15(1), 18–22 (2013)
Emel’yanov, E.Yu.: Erratum to “Infinitesimals in ordered vector spaces”. Vladikavkaz. Mat. Zh. 15(2), 82–83 (2013)
Emel’yanov, E.Yu.: Infinitesimals in vector lattices. In: Mathematics and its Applications, vol. 525. Kluwer Academic Publishers, Dordrecht, pp 161–230 (2000)
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)]
Emel’yanov, E.Yu.: Infinitesimal analysis and vector lattices. Sib. Adv. Math. 6, 19–70 (1996)
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)
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)]
Emel’yanov, E.Yu.: Order hulls of vector lattices. Dokl. Akad. Nauk. 340(3), 303–304 (1995)
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)]
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)]
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)]
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)
Hurd, A., Loeb, P.A.: Introduction to Nonstandard Real Analysis. Academic Press, New York (1985)
Kusraev, A.G., Kutateladze, S.S.: Nonstandard methods of analysis. In: Mathematics and its Applications, vol. 291. Kluwer Academic Publishers Group, Dordrecht (1994)
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)
Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces I. North-Holland, Amsterdam (1971)
Luxemburg, W.A.J., Stroyan, K.D.: Introduction to the theory of infinitesimals. In: Pure and Applied Mathematics, vol. 72. Academic Press, Boston (1976)
Onal, S.: Private communication (2013)
Robinson, A.: Nonstandard Analysis. North-Holland, Amsterdam (1966)
Schaefer, H.H., Wolff, M.P.: Topological vector spaces. In: Graduate Texts in Mathematics, vol. 3, 2nd edn. Springer, New York (1999)
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)
Vulikh, B.Z.: Introduction to Theory of Partially Ordered Spaces. Noordhoff, Groningen (1967)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Emelyanov, E., Gül, H. Nonstandard hulls of ordered vector spaces. Positivity 20, 413–433 (2016). https://doi.org/10.1007/s11117-015-0364-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11117-015-0364-2