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A Convenient Notion of Compact Set for Generalized Functions

Part of: Topological rings and modules Topological linear spaces and related structures Distributions, generalized functions, distribution spaces

Published online by Cambridge University Press:  03 April 2017

Paolo Giordano  and
Michael Kunzinger
Paolo Giordano*
Affiliation:
University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria (paolo.giordano@univie.ac.at; michael.kunzinger@univie.ac.at)
Michael Kunzinger*
Affiliation:
University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria (paolo.giordano@univie.ac.at; michael.kunzinger@univie.ac.at)
*
*Corresponding author.
*Corresponding author.
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Abstract

We introduce the notion of functionally compact sets into the theory of nonlinear generalized functions in the sense of Colombeau. The motivation behind our construction is to transfer, as far as possible, properties enjoyed by standard smooth functions on compact sets into the framework of generalized functions. Based on this concept, we introduce spaces of compactly supported generalized smooth functions that are close analogues to the test function spaces of distribution theory. We then develop the topological and functional–analytic foundations of these spaces.

Keywords

functionally compact setsColombeau generalized functionsgeneralized smooth functionslocally convex modules

MSC classification

Primary: 46F30: Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.) 46A13: Spaces defined by inductive or projective limits (LB, LF, etc.) 13J99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 1 , February 2018 , pp. 57 - 92
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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