Examples of new nonstandard hulls of topological vector spaces
HTML articles powered by AMS MathViewer
- 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
Additional Information
- Adel Khalfallah
- Affiliation: Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
- MR Author ID: 703582
- Email: khelifa@kfupm.edu.sa
- Siegmund Kosarew
- Affiliation: Institut Fourier, Université Grenoble Alpes, 100 rue des maths 38610 Gières, France
- Email: Siegmund.Kosarew@univ-grenoble-alpes.fr
- Received by editor(s): May 15, 2017
- Received by editor(s) in revised form: July 18, 2017, and August 21, 2017
- Published electronically: February 16, 2018
- Communicated by: Heike Mildenberger
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 2723-2739
- MSC (2010): Primary 54J05, 46F30, 26E35, 46S20; Secondary 46S10, 12J25
- DOI: https://doi.org/10.1090/proc/13930
- MathSciNet review: 3778172