Examples of new nonstandard hulls of topological vector spaces

Khalfallah, Adel; Kosarew, Siegmund

doi:10.1090/proc/13930

Examples of new nonstandard hulls of topological vector spaces
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by Adel Khalfallah and Siegmund Kosarew PDF

Proc. Amer. Math. Soc. 146 (2018), 2723-2739 Request permission

Abstract:

In this paper, we construct new nonstandard hulls of topological vector spaces using convex subrings of ${}^*\mathbb {R}$ (or ${}^*\mathbb {C}$) and we show that such spaces are complete. Some examples of locally convex spaces are provided to illustrate our construction. Namely, we show that the new nonstandard hull of the space of polynomials is the algebra of Colombeau’s entire holomorphic generalized functions. The proof is based on the existence of global representatives of entire generalized functions.

References

Jean-François Colombeau, New generalized functions and multiplication of distributions, North-Holland Mathematics Studies, vol. 84, North-Holland Publishing Co., Amsterdam, 1984. Notas de Matemática [Mathematical Notes], 90. MR 738781
Jean-François Colombeau, Elementary introduction to new generalized functions, North-Holland Mathematics Studies, vol. 113, North-Holland Publishing Co., Amsterdam, 1985. Notes on Pure Mathematics, 103. MR 808961
Antoine Delcroix and Dimitris Scarpalezos, Topology on asymptotic algebras of generalized functions and applications, Monatsh. Math. 129 (2000), no. 1, 1–14. MR 1741037, DOI 10.1007/s006050050001
J.-F. Colombeau and J. E. Galé, Holomorphic generalized functions, J. Math. Anal. Appl. 103 (1984), no. 1, 117–133. MR 757628, DOI 10.1016/0022-247X(84)90162-8
Ricardo Estrada and Ram P. Kanwal, Asymptotic analysis, Birkhäuser Boston, Inc., Boston, MA, 1994. A distributional approach. MR 1254657, DOI 10.1007/978-1-4684-0029-8
Robert Goldblatt, Lectures on the hyperreals, Graduate Texts in Mathematics, vol. 188, Springer-Verlag, New York, 1998. An introduction to nonstandard analysis. MR 1643950, DOI 10.1007/978-1-4612-0615-6
C. Ward Henson and L. C. Moore Jr., The nonstandard theory of topological vector spaces, Trans. Amer. Math. Soc. 172 (1972), 405–435. MR 308722, DOI 10.1090/S0002-9947-1972-0308722-5
Lars Hörmander, An introduction to complex analysis in several variables, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0203075
Albert E. Hurd and Peter A. Loeb, An introduction to nonstandard real analysis, Pure and Applied Mathematics, vol. 118, Academic Press, Inc., Orlando, FL, 1985. MR 806135
D. S. Jones, Introduction to asymptotics, World Scientific Publishing Co., Inc., River Edge, NJ, 1997. A treatment using nonstandard analysis. MR 1464943, DOI 10.1142/9789812779373
Adel Khalfallah and Siegmund Kosarew, Complex spaces and nonstandard schemes, J. Log. Anal. 2 (2010), Paper 9, 60. MR 2734197, DOI 10.4115/jla.2010.2.9
Adel Khalfallah, New non-standard topologies, Monatsh. Math. 172 (2013), no. 3-4, 323–344. MR 3127997, DOI 10.1007/s00605-013-0494-1
A. Khalfallah and S. Kosarew, Bounded polynomials and holomorphic mappings between convex subrings of ${}^*\mathbb {C}$, to appear in J. Symbolic Logic
Tom Lindstrøm, An invitation to nonstandard analysis, Nonstandard analysis and its applications (Hull, 1986) London Math. Soc. Stud. Texts, vol. 10, Cambridge Univ. Press, Cambridge, 1988, pp. 1–105. MR 971064
A. H. Lightstone and Abraham Robinson, Nonarchimedean fields and asymptotic expansions, North-Holland Mathematical Library, Vol. 13, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0414354
W. A. J. Luxemburg, A general theory of monads, Applications of Model Theory to Algebra, Analysis, and Probability (Internat. Sympos., Pasadena, Calif., 1967) Holt, Rinehart and Winston, New York, 1969, pp. 18–86. MR 0244931
W. A. J. Luxemburg, On a class of valuation fields introduced by A. Robinson, Israel J. Math. 25 (1976), no. 3-4, 189–201. MR 460102, DOI 10.1007/BF02756999
M. Oberguggenberger, S. Pilipović, and V. Valmorin, Global representatives of Colombeau holomorphic generalized functions, Monatsh. Math. 151 (2007), no. 1, 67–74. MR 2317391, DOI 10.1007/s00605-006-0416-6
Abraham Robinson, Non-standard analysis, North-Holland Publishing Co., Amsterdam, 1966. MR 0205854
K. D. Stroyan and W. A. J. Luxemburg, Introduction to the theory of infinitesimals, Pure and Applied Mathematics, No. 72, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR 0491163
T. D. Todorov, Lecture Notes: Non-Standard Approach to J.F. Colombeau’s Theory of Generalized Functions, arXiv:1010.3482
Imme van den Berg, Nonstandard asymptotic analysis, Lecture Notes in Mathematics, vol. 1249, Springer-Verlag, Berlin, 1987. MR 887738, DOI 10.1007/BFb0077577
Seth Warner, Topological rings, North-Holland Mathematics Studies, vol. 178, North-Holland Publishing Co., Amsterdam, 1993. MR 1240057