Infinite-Dimensional Linear Algebra and Solvability of Partial Differential Equations

TL;DR

This paper explores the linear algebra of infinite-dimensional spaces using algebraic bases and proves the surjectivity of a broad class of smooth coefficient partial differential operators on the algebraic dual of test functions, ensuring PDE solvability.

Contribution

It introduces a novel approach to PDE solvability by demonstrating surjectivity of regular operators on the algebraic dual space, contrasting with known non-surjectivity on Schwartz distributions.

Findings

Proves surjectivity of regular PDE operators on the algebraic dual of test functions.

Shows PDE solvability within the algebraic dual space.

Contrasts results with non-surjectivity on Schwartz distribution space.

Abstract

We discuss linear algebra of infinite-dimensional vector spaces in terms of algebraic (Hamel) bases. As an application we prove the surjectivity of a large class of linear partial differential operators with smooth ($\mathcal C^\infty$-coefficients) coefficients, called in the article \emph{regular}, acting on the algebraic dual $\mathcal D^*(\Omega)$ of the space of test-functions $\mathcal D(\Omega)$. The surjectivity of the partial differential operators guarantees solvability of the corresponding partial differential equations within $\mathcal D^*(\Omega)$. We discuss our result in contrast to and comparison with similar results about the restrictions of the regular operators on the space of Schwartz distribution $\mathcal D^\prime(\Omega)$, where these operators are often non-surjective.