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

with the space of bounded functions from to , in which case, we write and equip such space with the supremum norm,

      where . Alternatively, we can consider a subspace of formed by all functions such that

      The next result shows that is a Banach space with respect to a special norm. This result, whose proof follows ideas similar to those of [124, 220], will be largely used in. Chapters 5 and 8

      

      Proposition 1.9: The space , equipped with the norm

       is a Banach space.

      Proof. Let be the linear mapping defined by

      for all and .

      Claim. is an isometric isomorphism. Indeed, is an isometry because

      for all . Moreover, if , then defined by

      is such that and , since

      Therefore, is onto and the Claim is proved.

      Once is an isometric isomorphism and is a Banach space, we conclude that is also a Banach space.

      1.1.2