The Hahn-Banach theorem in spaces of nonlinear generalized functions

TL;DR

This paper extends the Hahn-Banach theorem to Colombeau spaces of nonlinear generalized functions, enabling better functional analysis tools within this framework and demonstrating the separation of convex sets.

Contribution

It introduces a version of the Hahn-Banach theorem for Colombeau spaces with $oldsymbol{ ext{ extit{ extepsilon}}}$-wise maps, simplifying the extension of linear functionals.

Findings

Established Hahn-Banach theorem in Colombeau spaces

Demonstrated separation of convex sets in these spaces

Simplified extension of linear functionals using $ ext{ extepsilon}$-wise maps

Abstract

In this paper, we establish a suitable version of the Hahn-Banach theorem within the framework of Colombeau spaces, a class of spaces used to model generalized functions. Our approach addresses the case where maps are defined $\varepsilon$-wise, which simplifies the framework and makes the extension of linear functionals more manageable. As an application of our main result, we demonstrate the separation of convex sets in Colombeau spaces.