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

and also exist. We prove (i). The proof of (ii) follows analogously.

      Suppose . Then, property (V2) implies Thus,

      and, hence,

      Conversely, for any given , let . Then for every , there exists such that and , and there exists a division such that

      Then,

      and, hence, which completes the proof.

      Using the fact that , the following corollary follows immediately from Lemma 1.30.

      Corollary 1.31: Let . Then the sets

       are finite for every .

      Thus, we have the next result which can be found in [126, Proposition I.2.10].

      Proposition 1.32: Let . Then the set of points of discontinuity of is countable.

      Let us define

      A proof that equipped with the variation norm, , is complete can be found in [126, Theorem I.2.11]. We reproduce it in the next theorem.

      Theorem 1.33: , equipped with the variation norm, is a Banach space.

Proof. We know that , with the variation norm, is a Banach space. Let be a sequence in converging to in the variation norm. Then, since for every