Infinite-dimensional linear algebra and solvability of partial differential equations

Todor D. Todorov




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. 


Algebraic basis, Hamel basis, algebraic dual, dual operator, linear partial differential operator, linear partial differential equation, solvability, existence of a solution, generalized solution, Schwartz distribution

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Journal of Logic and Analysis ISSN:  1759-9008