Generalized Ordinary Differential Equations in Abstract Spaces and Applications

and we complete the proof.

Given a function , since , Theorem 1.81holds for instead of . Then, next proposition follows easily (see, also, [70, Corollary after Theorem 5]).

Proposition 1.82: Suppose and . Assume, in addition, that is such that almost everywhere in . Then, and , for every . If, moreover, , then .

In view of Proposition 1.82, we can define equivalence classes of nonabsolute vector integrable functions.

Definition 1.83:Let us assume that . Two functions are equivalent if and only if their indefinite integrals coincide, that is, . By and we mean, respectively, the spaces of equivalence classes of functions of and of , and we endow these spaces with an Alexiewicz-type norm

From Example 1.74, we know that although may belong to the same equivalence class, that is, for all , one cannot conclude that almost everywhere in . It is known, however, that the space of all equivalence classes of real-valued Perron integrable functions , equipped with the usual Alexiewicz norm (see [5]) given by

is noncomplete (see [24], for instance). The same applies to Banach space-valued functions. Let us denote by the space of all equivalence classes of functions , equipped with the Alexiewicz norm The space is noncomplete (see [129]). However, is ultrabornological (see [105]) and, therefore, barrelled. In particular, good functional analytic properties hold, such as the Banach–Steinhaus theorem and the Uniform Boundedness Principle (see, for instance, [142]). The same applies to the space of equivalence classes of functions , endowed with the Alexiewicz norm .