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Examples of new nonstandard hulls of topological vector spaces


Authors: Adel Khalfallah and Siegmund Kosarew
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
Published electronically: February 16, 2018
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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.


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Additional Information

Adel Khalfallah
Affiliation: Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
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

DOI: https://doi.org/10.1090/proc/13930
Keywords: Nonstandard analysis, nonstandard hulls, internal polynomials, generalized holomorphicity.
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
Article copyright: © Copyright 2018 American Mathematical Society

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