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

. We write and , where denotes the real line. Next, we present examples, borrowed from [127], of bilinear triples.

      Example 1.21: Let , , and denote Banach spaces. The following are BT:

      1 , , , and ;

      2 , , , and ;

      3 , , , and ;

      4 , and .

      Given a BT , we define, for every , a norm

      and we set . Whenever the space is endowed with the norm , we say that the topological BT is associated with the BT .

      Let be a vector space and be a set of seminorms defined on such that implies Then, defines a topology on , and the sets form a basis of neighborhoods of 0. The sets form a basis of the neighborhood of . Moreover, when endowed with this topology, is called a locally convex space (see [127], p. 3, 4).

      Example 1.22: Every normed or seminormed space is a locally convex space.

      For other examples of locally convex spaces, we refer to [110].

Definition 1.23: Given a BT , and a function , for every division , we define

      Then, is the ‐variation of on . We say that is a function of bounded ‐variation, whenever . In this case, we write .

      The following properties are not difficult to prove. See, e.g. [127, 4.1 and 4.2].

      1 (SB1) is a vector space and the mapping is a seminorm.

      2 (SB2)