Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Hahn-Banach theorem in spaces of nonlinear generalized functions

Djamel eddine Kebiche, Paolo Giordano

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.

The Hahn-Banach theorem in spaces of nonlinear generalized functions

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 -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.

Paper Structure

This paper contains 13 sections, 25 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Basic notions of Colombeau spaces
  3. Colombeau spaces and topological $\widetilde{\mathbb C}$-modules
  4. Colombeau spaces
  5. The Robinson-Colombeau ring of generalized numbers
  6. Topological $\widetilde{\mathbb C}$-modules
  7. Internal sets and strongly internal sets
  8. Supremum in $\widetilde{\mathbb{R}}$
  9. $\widetilde{\mathbb C}$-(sub)linear maps and continuity
  10. The analytic form of Hahn-Banach's theorem
  11. The Hahn-Banach theorem in a Colombeau space based on a family of normed spaces
  12. The Hahn-Banach theorem in a Colombeau space based on a family of locally convex topological vector spaces
  13. Geometric forms of the Hahn-Banach theorem: Separation of convex sets

Key Result

Proposition 3

Let $E=(E_{\varepsilon},(p_{\varepsilon n})_{n\in\mathbb{N}})$ be a family of locally convex topological vector spaces, and let $T:\mathcal{G}_{E}\longrightarrow\widetilde{\mathbb C}$ be a $\widetilde{\mathbb C}$-linear map with representatives. If there exists a finite subset $J\subseteq\mathbb{N}$

Theorems & Definitions (61)

  • Definition 1
  • Definition 2
  • Proposition 3
  • proof
  • Theorem 4
  • Proposition 5
  • proof
  • proof : Proof of Thm. \ref{['thm:HBT']}
  • Corollary 6
  • proof
  • ...and 51 more