The theory of ordinary linear quasi-differential expressions and operators has been extensively developed in integrable-square Hilbert spaces. There is also an extensive theory of ordinary linear differential expressions and operators in integrable-p Banach spaces.
However, the basic definition of linear quasi-differential expressions involves Lebesgue locally integrable spaces on intervals of the real line. Such spaces are not Banach spaces but can be considered as complete locally convex linear topological spaces where the topology is derived from a countable family of semi-norms. The first conjugate space can also be defined as a complete locally convex linear topological space, but now with the topology derived as a strict inductive limit.
This paper develops the properties of linear quasi-differential operators in a locally integrable space and the first conjugate space. Conjugate and preconjugate operators are defined in, respectively, dense and total domains.