A Riesz Representation Theorem for the Space of Henstock Integrable Vector-Valued Functions

Using a bounded bilinear operator, we define the Henstock-Stieltjes integral for vector-valued functions; we prove some integration by parts theorems for Henstock integral and a Riesz-type theorem which provides an alternative proof of the representation theorem for real functions proved by Alexiewicz.

In [8] we can find some properties of both integrals such as the linearity, integrability over subintervals, and the continuity of the function  : [, ] → , called primitive, given by () = () ∫ (1)  is said to be of strongly bounded variation (BV) on  if the number (, ) fl sup{∑  ‖(  ) − (  )‖  } is finite, where the supremum is taken over all finite sequences {[  ,   ]} of nonoverlapping intervals that have endpoints in .
( where the derivative is in the sense of Frechet. If a function  : [, ] →  is BV or BV * on , then it is bounded on ; that is,  > 0 exists such that ‖()‖  ≤ , for every  ∈ .As an AC() function is BV on  and an AC * () function is BV * on  (immediately from the definitions), then every AC() or AC * () function is also bounded in .It is easy to see that if  is AC() and  0 ⊂ , then  is AC( 0 ), The definition of a function of strongly bounded variation can be extended considering the bilinear operator  :  ×  → .Definition 6.Let  : [, ] →  be a function and  = { 0 ,  1 , . . .,   } a partition of [, ]; we define where the supremum is taken over all possible elections of where the supremum is taken over all partitions of the interval [, ] and (B)var It is straightforward that each function of strongly bounded variation is of strongly bounded B-variation.We recommend the reader interested in this topic to consult the study exposed in [9].

Stieltjes-Type Integrals.
As we mentioned in the introduction, Schwabik in [3] gives the next definition and proves some basic properties such as the Uniform Convergence Theorem.
It is immediate that every Henstock integrable function is Kurzweil integrable and its integrals are the same; we can repeat the proof of this fact for the previous Stieltjes integrals.Similarly, we can prove the properties of linearity and integrability over subintervals for the Henstock-Stieltjes integral directly of the proofs in [8] with slight changes.We omit the formulations and the proofs of such results.
Theorem 9 (see [3,Thm. 11] We will use the following well-known results of the Bochner integral. hence (, ) is Bochner integrable as a consequence of Theorem 10.
Let  > max{ 0 ,   Schwabik in [3] introduces the concept of vector-valued regulated functions; we shall only use the following characterization.
Theorem 23 (see [3,Prop. 2]). : [, ] →  is regulated if and only if it is the uniform limit of step functions.(28) Proof.Let  > 0. Since  is Henstock integrable, with  its primitive, there exists a gauge  1 such that if On the other hand, since  is continuous, it is regulated, and by Theorem 24, () ∫   (, ) exists; then there exists a gauge  2 such that if As we can see, we have two types of integration by parts theorems, one is of the Stieltjes type and the other is non-Stieltjes; it is possible to ask for the conditions so that the integral of the Stieltjes type becomes a non-Stieltjes; for that, we must do the following analysis: (1) The essence in the proof of Theorem 19 is the derivative of the primitive of the function  → ((), ()),that is, the Fundamental Theorem of Calculus.
(2) In Theorem 25 the Fundamental Theorem of Calculus does not apply because  is not necessarily differentiable, and if it is, the primitive of   , in general, is not . (

Representation Theorem
Now, we will establish an important connection between the space of Henstock integrable functions and its dual space: a Riesz representation theorem.
So  is bounded and continuous.