rel="nofollow" href="#fb3_img_img_f1c0d7a7-3191-5c5e-ac1c-72a4fe88a1f7.png" alt="gamma element-of upper K Subscript g Baseline left-parenthesis left-bracket a comma b right-bracket comma upper L left-parenthesis upper W comma upper Y right-parenthesis right-parenthesis"/>. Then, the statement follows from Theorem 1.68.
The next result gives us an integration by parts formula for Perron–Stieltjes integrals. A proof of it can be found in [212, Theorem 13].
Proposition 1.70: Suppose and or and . Then, the Perron–Stieltjes integrals and exist, and the following equality holds:
where , , , and
As an immediate consequence of the previous proposition, we have the following result.
Corollary 1.71: If and is a nondecreasing function, then the integral exists.
We end this subsection by presenting a result, borrowed from [172] and [179, Theorem 5.4.5], which gives us a change of variable formula for Perron–Stieltjes integrals.
Theorem 1.72: Suppose is increasing and maps onto and consider functions . Then, both integrals
exists, whenever one of the integrals exists, in which case, we have
1.3.4 The Fundamental Theorem of Calculus
The first result we present in this section is the Fundamental Theorem of Calculus for the variational Henstock integral. The proof follows standard steps (see [172], p. 43, for instance) adapted to Banach space-valued functions.
Theorem 1.73 (Fundamental Theorem of Calculus): Suppose is a function such that there exists the derivative , for every . Then, and
Next, we give an example, borrowed from [73], of a Banach space-valued function
Example 1.74: Let
Since
