alt="upper F"/>. 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.
