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Locally convex topological algebras of generalized functions: Compactness and nuclearity in a nonlinear context

Authors: J. Aragona, S. O. Juriaans and J. F. Colombeau
Journal: Trans. Amer. Math. Soc. 367 (2015), 5399-5414
MSC (2010): Primary 46F30, 46F10
Published electronically: March 26, 2015
MathSciNet review: 3347177
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Abstract: In this paper we introduce Hausdorff locally convex algebra topologies on subalgebras of the whole algebra of nonlinear generalized functions. These topologies are strong duals of Fréchet-Schwartz space topologies and even strong duals of nuclear Fréchet space topologies. In particular, any
bounded set is relatively compact and one benefits from all deep properties of nuclearity. These algebras of generalized functions contain most of the classical irregular functions and distributions. They are obtained by replacing the mathematical tool of $ \mathcal {C}^\infty $ functions in the original version of nonlinear generalized functions by the far more evolved tool of holomorphic functions. This paper continues the nonlinear theory of generalized functions in which such locally convex topological properties were strongly lacking up to now.

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Additional Information

J. Aragona
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Sao Paulo-CP 66281, Brazil

S. O. Juriaans
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Sao Paulo-CP 66281, Brazil

J. F. Colombeau
Affiliation: Institut Fourier, Université de Grenoble, 38042 Grenoble Cedex 09, France

Keywords: Functional analysis, nonlinear generalized functions, sharp topology, differential calculus of nonlinear generalized functions, nonlinear operations on distributions, locally convex topological algebras, compact maps, Schwartz locally convex spaces, nuclear maps, nuclear spaces
Received by editor(s): December 25, 2012
Received by editor(s) in revised form: May 27, 2013
Published electronically: March 26, 2015
Additional Notes: The work of the third author was done under the financial support of FAPESP, processo 2011/12532-1, and thanks to the hospitality of the IME-USP. The third author is the corresponding author
Dedicated: Dedicated to the memory of Leopoldo Nachbin
Article copyright: © Copyright 2015 American Mathematical Society

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