Generalized Ordinary Differential Equations in Abstract Spaces and Applications. Группа авторов

, with , such that .

      The next result follows from Theorem 1.80. A proof of it can be found in [75, Theorem 5].

      Theorem 1.85: Suppose is nonconstant on any nondegenerate subinterval of . Then, the mapping

       is an isometry, that is onto a dense subspace of .

      The next result, known as straddle Lemma, will be useful to prove that the space of regulated functions from to is dense in in the Alexiewicz norm . For a proof of the straddle Lemma, the reader may want to consult [130, 3.4] or [119].

      Lemma 1.86 (Straddle Lemma): Suppose are functions such that is differentiable, with , for all . Then, given , there exists such that

       whenever .

The next result is adapted from [75, Theorem 8].

      Proposition 1.87: Suppose is differentiable and nonconstant on any nondegenerate subinterval of . Then, the Banach space is dense in under the Alexiewicz norm .

      Proof. Assume that and let be given. We need to find a function such that , or equivalently,

(1.13)

      By Corollary 1.50, . Let us denote by the subspace of of functions which are differentiable with continuous derivative. Hence, there is a function such that

(1.14)

      Let be defined by , for all such that , and by . In particular, whenever . Therefore, for almost every , since is differentiable and nonconstant