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

may want to check this fact as described, for instance, by the group N. Bourbaki in [32, Corollaire I, p. II.6].

      

      1.1.1 Basic Properties

      Let

be a Banach space with norm

. Here, we describe regulated functions

, where

, with

, is a compact interval of the real line

.

      

      Definition 1.1: A function

is called regulated, if the lateral limits

exist. The space of all regulated functions

will be denoted by

.

      We denote the subspace of all continuous functions

by

and, by

, we mean the subspace of regulated functions

which are left‐continuous on

. Then, the following inclusions clearly hold

      Remark 1.2: Let

. By

, we mean the set of all elements

for which

for every

. Thus, it is clear that

and the range of

belongs to

. Note that, for a given

,

and

do not necessarily belong to

.

      Any finite set

of points in the closed interval

such that

      is called a division of

. We write simply

. Given a division

of

, its elements are usually denoted by

, where, from now on,

denotes the number of intervals in which

is divided through the division

. The set of all divisions

of

is denoted by

.

      Definition 1.3: A function

is called a step function, if there is a division