Topological structures in Colombeau algebras: topological $\widetilde{\C}$-modules and duality theory

TL;DR

This paper explores the topological properties of modules over the ring of complex generalized numbers, focusing on their structure, duality, and continuity within Colombeau algebras of generalized functions.

Contribution

It introduces the concepts of $ ilde{C}$-linear and locally convex topologies for modules over generalized numbers, advancing the understanding of their duality and completeness properties.

Findings

Established topological $ ilde{C}$-modules and their duality theory.

Analyzed continuity of $ ilde{C}$-linear maps.

Applied theory to Colombeau algebras of generalized functions.

Abstract

We study modules over the ring $\widetilde{\C}$ of complex generalized numbers from a topological point of view, introducing the notions of $\widetilde{\C}$-linear topology and locally convex $\widetilde{\C}$-linear topology. In this context particular attention is given to completeness, continuity of $\widetilde{\C}$-linear maps and elements of duality theory for topological $\widetilde{\C}$-modules. As main examples we consider various Colombeau algebras of generalized functions