Infinite-Dimensional Linear Algebra and Solvability of Partial Differential Equations

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 ${\mathcal{D}}^{\ast }(\Omega )$ 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 ${\mathcal{D}}^{\ast }(\Omega )$. 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.

34 pages

References in corpus (1)

• Ordered Fields, the Purge of Infinitesimals from Mathematics and the Rigorousness of Infinitesimal Calculus