alt="f 0 left-parenthesis t right-parenthesis equals limit Underscript k right-arrow infinity Endscripts f Subscript n Sub Subscript k Subscript Baseline left-parenthesis t right-parenthesis period"/> Then, on , by Lemma 1.13. Hence, is the uniform limit of the subsequence in . Finally, any sequence admits a converging subsequence which, in turn, implies that is a relatively compact set, and the proof is finished.
We end this subsection by mentioning an Arzelà–Ascoli‐type theorem for regulated functions taking values in . A slightly different version of it can be found in [96].
Corollary 1.19:The following conditions are equivalent:
1 a set is relatively compact;
2 the set is bounded, and there are an increasing continuous function , with , and a nondecreasing function such that, for every ,for
3 is equiregulated, and for every , the set is bounded.
We point out in [96, Theorem 2.17], item (ii), it is required that is an increasing function. However, it is not difficult to see that if is a nondecreasing function, then taking yields is an increasing function. Therefore, Corollary 1.19 follows as an immediate consequence of [96, Theorem 2.17].
1.2 Functions of Bounded ‐Variation
The concept of a function of bounded ‐variation generalizes the concepts of a function of semivariation and of a function of bounded variation, as we will see in the sequel.
Definition 1.20: A bilinear triple (we write BT) is a set of three vector spaces , , and , where and are normed spaces with a bilinear mapping . For and , we write , and we denote the BT by or simply by . A topological BT is a BT , where is also a normed space and is continuous.
If and are normed spaces, then we denote by the space of all linear continuous functions from to . 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:
alt="f 0 left-parenthesis t right-parenthesis equals limit Underscript k right-arrow infinity Endscripts f Subscript n Sub Subscript k Subscript Baseline left-parenthesis t right-parenthesis period"/> Then, 
