sigma-summation sigma-summation equals equals j 1 vertical-bar vertical-bar d times times of ff left-parenthesis right-parenthesis xi j left-parenthesis right-parenthesis minus minus tjt minus minus j 10 Subscript 2 Baseline equals vertical-bar vertical-bar vertical-bar vertical-bar sigma-summation sigma-summation equals equals j 1 vertical-bar vertical-bar d times times e xi j left-parenthesis right-parenthesis minus minus tjt minus minus j 1 Subscript 2 Baseline 2nd Row 1st Column Blank 2nd Column equals left-bracket sigma-summation Underscript j equals 1 Overscript StartAbsoluteValue d EndAbsoluteValue Endscripts StartAbsoluteValue t Subscript j Baseline minus t Subscript j minus 1 Baseline EndAbsoluteValue squared right-bracket Superscript one half Baseline less-than delta Superscript one half Baseline left-bracket sigma-summation Underscript j equals 1 Overscript StartAbsoluteValue d EndAbsoluteValue Endscripts left-parenthesis t Subscript j Baseline minus t Subscript j minus 1 Baseline right-parenthesis right-bracket Superscript one half Baseline less-than epsilon comma EndLayout"/>
where we applied the Bessel equality. Thus, and , since for every . On the other hand,
for every . Hence, .
1.3.2 Basic Properties
The first result we present in this subsection is known as the Saks–Henstock lemma, and it is useful in many situations. For a proof of it, see [210, Lemma 16], for instance. Similar results hold if we replace by and also by .
Lemma 1.45 (Saks–Henstock Lemma): The following assertions hold.
1 Let and . Given , let be a gauge on such that for every ‐fine ,Then, for every ‐fine , we have
2 Let and Given , let be a gauge on such that for every ‐fine ,Then, for every ‐fine , we have
Now, we define some important sets of functions.
Definition 1.46: Let be the set of all functions which are weakly continuous, that is, for every , the function is continuous, and we denote by the set of all weakly regulated functions , that is, for every , the function is r egulated.
Given and , let us define
By the Banach–Steinhaus theorem, the limits and exist and belong to . Then, by the Uniform Boundedness Principle, is a Banach space when equipped with the usual supremum norm. It is also clear that
