Nonstandard hulls of ordered vector spaces. (English) Zbl 1348.46075

Emelyanov, Eduard; Gül, Hasan

Positivity 20, No. 2, 413-433 (2016).

The authors extend the technique of nonstandard hulls from vector lattices to ordered vector spaces. They introduce some external subspaces that generalize the subspaces of order-finite and order-infinitesimal elements and apply these concepts to studying the properties of nonstandard hulls, order convergence, Dedekind completion, etc. Some applications are also given to normed ordered vector spaces.

Reviewer: S. S. Kutateladze (Novosibirsk)

MSC:

46S20	Nonstandard functional analysis
46A40	Ordered topological linear spaces, vector lattices

Keywords:

enlargement; order-finite element; order-infinitesimal, o-convergence; nonstandard hull; o-completion; Krein space

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) · Zbl 0605.60005
[2]	Aliprantis, C.D., Tourky, R.: Cones and duality. In: Graduate Studies in Mathematics, vol. 84. American Mathematical Society, Providence (2007) · Zbl 1127.46002
[3]	Alpay, S; Altin, B; Tonyali, C, On property \((b)\) of vector lattices, Positivity, 7, 135-139, (2003) · Zbl 1036.46018
[4]	Davis, M.: Applied nonstandard analysis. In: Pure and Applied Mathematics. New York, London, Sydney (1977) · Zbl 0359.02060
[5]	Emel’yanov, E.Yu.: Infinitesimals in ordered vector spaces. Vladikavkaz. Mat. Zh. 15(1), 18-22 (2013) · Zbl 1326.46062
[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) · Zbl 1036.46018
[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) · Zbl 0511.46070
[17]	Hurd, A., Loeb, P.A.: Introduction to Nonstandard Real Analysis. Academic Press, New York (1985) · Zbl 0583.26006
[18]	Kusraev, A.G., Kutateladze, S.S.: Nonstandard methods of analysis. In: Mathematics and its Applications, vol. 291. Kluwer Academic Publishers Group, Dordrecht (1994) · Zbl 0817.03029
[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) · Zbl 0231.46014
[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) · Zbl 0336.26002
[22]	Onal, S.: Private communication (2013)
[23]	Robinson, A.: Nonstandard Analysis. North-Holland, Amsterdam (1966)
[24]	Schaefer, H.H., Wolff, M.P.: Topological vector spaces. In: Graduate Texts in Mathematics, vol. 3, 2nd edn. Springer, New York (1999) · Zbl 0983.46002
[25]	Veksler, AI, Archimedean principle in homomorphic images of l-groups and of vector lattices, Izv. Vys. Ucebn. Zaved. Matematika., 54, 33-38, (1966)
[26]	Vulikh, B.Z.: Introduction to Theory of Partially Ordered Spaces. Noordhoff, Groningen (1967) · Zbl 0186.44601