On the algebraic dual of D(\Omega)

TL;DR

This paper explores the algebraic dual of the space of test functions, comparing its properties and applications to the continuous dual, with a focus on topological aspects and PDE relevance.

Contribution

It provides a detailed analysis of the algebraic dual D*(Ω), highlighting differences from the distribution space D'(Ω) and examining its topological and operational characteristics.

Findings

D*(Ω) has notable differences from D'(Ω) in topological structure.

Certain operations with elements of D*(Ω) are well-defined, while others are not.

Applications to linear PDEs reveal specific advantages and limitations of D*(Ω).

Abstract

This paper is concerned with the algebraic dual D*(\Omega) of the space of test functions D(\Omega). The emphasis is on failures and successes of D*(\Omega) as compared to the continuous dual D'(\Omega), the space of distributions. Topological properties, operations with elements of D*(\Omega) and applications to linear partial differential equations are discussed.