tau Subscript i Baseline right-parenthesis parallel-to 2nd Row 1st Column Blank 2nd Column less-than-or-slanted-equals parallel-to f Subscript n Sub Subscript k Baseline left-parenthesis t right-parenthesis minus f Subscript n Sub Subscript k Baseline left-parenthesis tau Subscript i Baseline right-parenthesis parallel-to plus parallel-to f Subscript n Sub Subscript q Baseline left-parenthesis t right-parenthesis minus f Subscript n Sub Subscript q Baseline left-parenthesis tau Subscript i Baseline right-parenthesis parallel-to plus parallel-to f Subscript n Sub Subscript k Baseline left-parenthesis tau Subscript i Baseline right-parenthesis minus z Subscript i Baseline parallel-to 3rd Row 1st Column Blank 2nd Column plus parallel-to f Subscript n Sub Subscript q Subscript Baseline left-parenthesis tau Subscript i Baseline right-parenthesis minus z Subscript i Baseline parallel-to less-than StartFraction epsilon Over 4 EndFraction plus StartFraction epsilon Over 4 EndFraction plus StartFraction epsilon Over 4 EndFraction plus StartFraction epsilon Over 4 EndFraction equals epsilon period EndLayout"/>
Hence, for every , satisfies the Cauchy condition. Due to the fact that is a complete space and because is a Cauchy sequence, the limit exists.
We conclude by considering 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
