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

1. Introduction

Henstock in  defines a Riemann type integral which is equivalent to Denjoy integral and more general than the Lebesgue integral, called the Henstock integral. Cao in

 extends the Henstock integral for vector-valued functions and provides some basic properties such as the SaksHenstock Lemma.

Schwabik in  considers a bilinear form, defines a Stieltjes type integral, and performs a study about it including ; following his ideas we give integration by parts theorem involving a bilinear operator and, through it, we prove a representation theorem for the space of Henstock vectorvalued functions.

This paper is divided into five sections; in a first step, in Section 2 we present some preliminaries and introduce the Henstock-Stieltjes integral via a bilinear bounded operator and the Bochner integral, together with some basic properties. In Section 3 we provide two useful kinds of integration by parts theorems, one of them in terms of the Bochner integral and the other using Henstock-Stieltjes integral; the representation theorem is proved in Section 4 which, if we consider real-valued functions, provides an alternative proof of the representation theorem proved by Alexiewicz (Theorem 1 in ).

2. Preliminaries

Throughout this paper X, Y, and Z will denote three Banach spaces, [mathematical expression not reproducible], which will denote their respective norms, X* the dual of X, B : X x Y [right arrow] Za bounded bilinear operator fixed, and [a, b] a closed finite interval of the real line with the usual topology and the Lebesgue measure, which we denote by [mu]. For a function f : [a,b] [right arrow] R we denote the Lebesgue integral of f on a measurable E [subset] [a, b], when it exists, by (L) [[integra].sub.E] f.

Definition 1. B : X x Y [right arrow] Z is a bounded bilinear operator if B is linear in each variable and there exists M > 0 such that [mathematical expression not reproducible]; in this case, the norm of the operator B is [mathematical expression not reproducible].

We say that P = {([[t.sub.i]-1,ti], [[xi].sub.i]) : i = 1, ..., n} is a tagged partition of [a,b] if {[[t.sub.i-1], [t.sub.i]] : i = 1, ..., n} is a finite collection of nonoverlapping closed intervals whose union is [a, b] such that [[xi].sub.i] [member of][[t.sub.i-1], [t.sub.i]] for every i. Given a function [delta] from [a,b] to (0, [infinity]), called gauge on [a, b], we say that a tagged partition {([[t.sub.i-1], [t.sub.i]], [[xi].sub.i]) : i = 1, ..., n} is [delta]-fine if

[mathematical expression not reproducible] for every i. (1)

Definition 2. A function f: [a,b] [infinity] X is Kurzweil integrable in [a, b] if there exists w [member of] X such that for every [epsilon] > 0 there exists a gauge [delta] on [a, b] such that if |([t.sub.i-1], [t.sub.i]], [[xi].sub.i]) : i = ..., 1,n] is a [delta]-fine tagged partition of [a, b], then

[mathematical expression not reproducible] (2)

We write w = (K) [[integral].sup.b.sub.a] f.

Definition 3. A function f : [a,b] [right arrow] X is Henstock integrable in [a, b] if there exists F :[a,b] [right arrow] X such that for every [epsilon] > 0 there exists a gauge [delta] on [a,b] such that if {([[t.sub.i-1], [t.sub.i]], [[xi].sub.i]) : i = 1, ..., n} is a [delta]-fine tagged partition of [a, b], then

[mathematical expression not reproducible]. (3)

We write F(b) - F(a) = (H) [[integral].sup.b.sub.a] f.

The Henstock integral is also known as Henstock-Lebesgue integral, briefly HL integral (), or variational Henstock integral ().

In  we can find some properties of both integrals such as the linearity, integrability over subintervals, and the continuity of the function F : [a, b] [right arrow] X, called primitive, given by [mathematical expression not reproducible].

Definition 4. Let F : [a,b] [right arrow] X and let E be a subset of [a,b].

(1) F is said to be of strongly bounded variation (BV) on E if the number [mathematical expression not reproducible] is finite, where the supremum is taken over all finite sequences {[[c.sub.i], [d.sub.i]] of nonoverlapping intervals that have endpoints in E.

(2) F is BV* on E if sup{[[summation].sub.i]; w(F, [[c.sub.i], [d.sub.i]])] is finite, where the supremum is taken over all finite sequences {[[c.sub.i], [d.sub.i]]] of nonoverlapping intervals that have endpoints in E, and [mathematical expression not reproducible] is the oscillation of F on [[c.sub.i], [d.sub.i]].

(3) F is said to be strongly absolutely continuous on E or AC(E) if for every [epsilon] > 0 there exists n > 0 such that, for every finite or infinite sequence of nonoverlapping intervals {[[c.sub.i], [d.sub.i]]}, with [[summation].sub.i] ([d.sub.i] - [c.sub.i]) < [eta], we have [mathematical expression not reproducible] where [c.sub.i], [d.sub.i] [member of] E for all i.

(4) F is AC* (E) if for every [epsilon] > 0 there exists [eta] > 0 such that, for every finite or infinite sequence or nonoverlapping intervals {[[c.sub.i], [d.sub.i]]] satisfying [[summation].sub.i] ([d.sub.i] - [a.sub.i] < [eta], where [c.sub.i], [d.sub.i] [member of] E for all i, we have [[summation].sub.i] w(F; [[c.sub.i], [d.sub.i]]) < [epsilon].

(5) F is ACG* on E if E is the union of a sequence of closed sets {[E.sub.i]} such that, on each [E.sub.i], F is AC* ([E.sub.i]).

The next result gives us a characterization of Henstock integrability.

Theorem 5 (see [8, Thm. 7.4.5, pp. 217]). The function f: [a, b] [right arrow] 0 X is Henstock integrable on [a, b] with the primitive F if and only if F : [a,b] [right arrow] X is continuous and ACG* on [a, b] such that F'(t) = f(t) almost everywhere (a.e.) in [a, b], where the derivative is in the sense of Frechet.

If a function F : [a,b] [right arrow] X is BV or BV* on E, then it is bounded on F; that is, K > 0 exists such that [parallel]F(f)[[parallel].sub.X [less than or equal to] K], for every t [member of] E. As an AC(E) function is BV on E and an AC* (E) function is BV* on E (immediately from the definitions), then every AC(E) or AC* (E) function is also bounded in E. It is easy to see that if F is AC(E) and [E.sub.0] [subset] E, then F is AC([E.sub.0]), similarly if F is AC* (E).

The definition of a function of strongly bounded variation can be extended considering the bilinear operator B : X x Y [right arrow] Z.

Definition 6. Let G : [a,b] [right arrow] Y be a function and D = {[t.sub.0], [t.sub.1], ..., [t.sub.n]] a partition of [a, b]; we define

[mathematical expression not reproducible], (4)

where the supremum is taken over all possible elections of [x.sub.i] [member of] X, i = 1,2, ..., n, with [parallel][x.sub.i][parallel].sub.x] [less than or equal to] 1.

[mathematical expression not reproducible], (5)

where the supremum is taken over all partitions of the interval [a, b] and [mathematical expression not reproducible] is the strong B-variation of G on [a,b]. If we consider

[mathematical expression not reproducible] (6)

in the equality (4), then we define [mathematical expression not reproducible], as the B-variation of G on [a, b].

If [mathematical expression not reproducible] we say that G is of strongly bounded B-variation or G is of bounded B-variation, respectively.

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 .

2.1. Stieltjes-Type Integrals. As we mentioned in the introduction, Schwabik in  gives the next definition and proves some basic properties such as the Uniform Convergence Theorem.

Definition 7. I [member of] Z is the Kurzweil-Stieltjes integral of f : [a, b] [right arrow] X with respect to g : [a,b] [right arrow] Y if for every [member of] >0 there exists a gauge [delta] on [a, b] such that

[mathematical expression not reproducible] (7)

for every [delta]-fine tagged partition {([[t.sub.i-1], [t.sub.1], [[xi].sub.i]) : i = 1, ..., n] of [a, b].

In this case we write I = (KS) [[integral.sup.b.sub.a] B(f, dg).

Now, we introduce the following integral.

Definition 8. A function f: [a,b] [right arrow] X is Henstock-Stieltjes integrable in [a, b] with respect to g : [a,b] [right arrow] Y if there exists H : [a, b] [right arrow] Z such that for every [epsilon] > 0 there exists a gauge [delta] of [a, b] such that if |([[t.sub.i-1], [t.sub.i]][[xi].sub.i]) : i = 1, ..., n] is a [delta]-fine tagged partition of [a, b], then

[mathematical expression not reproducible] (8)

We write H(b) - H(a) = (HS) [[integral].sup.b.sub.a] B(f, dg).

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  with slight changes. We omit the formulations and the proofs of such results.

Theorem 9 (see [3, Thm. 11]). Assume that the functions f, [f.sub.n] : [a,b] [right arrow] X, and g : [a,b] [right arrow] are given. If [mathematical expression not reproducible], the Kurzweil-Stieltjes integrals (KS) [[integral].sup.b.sub.a] B([f.sub.n], dg) exist and the sequence {[f.sub.n]} converges on [a, b] uniformly to f, then the integral (KS) [[integral].sup.b.sub.a] B(f, dg) exists and

[mathematical expression not reproducible]. (9)

2.2. Bochner Integral. Let us recall that a function s : [a,b] [right arrow] X is called simple if there is a finite sequence [mathematical expression not reproducible] of Lebesgue measurable sets such that [E.sub.m] [intersection] [E.sub.l] = [theta] for m [not equal to] l and [mathematical expression not reproducible], and in this case the Bochner integral of s is [mathematical expression not reproducible].

A function f : [a,b] [right arrow] X is strongly measurable if there exists a sequence of simple functions that converges pointwise to f a.e. on [a, b].

A function f: [a,b] [right arrow] X is Bochner integrable if there is a sequence of simple functions fn : [a, b] [right arrow] X, n [member of] N, such that [mathematical expression not reproducible] and

[mathematical expression not reproducible] (10)

the Bochner integral of f: [a,b] [right arrow] X is denoted by [mathematical expression not reproducible] and is defined by

[mathematical expression not reproducible] (11)

We will use the following well-known results of the Bochner integral.

Theorem 10 (see [8, Cor. 1.4.4, pp. 26]). A strongly measurable function f: [a,b] [right arrow] X is Bochner integrable on [a, b] if there exists a function g : [a,b] [right arrow] R, which is Lebesgue integrable such that [mathematical expression not reproducible].

Theorem 11 (see [8, Thm. 7.4.5, pp. 222]). A function f : [a, b] [right arrow] X is Bochner integrable on [a, b] if and only if there exists a function F : [a,b] [right arrow] X, which is AC on [a,b] such that F'(t) = f(t) a.e. on [a,b].

Given F : [a,b] [right arrow] X and g : [a,b] [right arrow] Y we define the function B(F,g) : [a,b] [right arrow] Z, given by B(F,g)(t) = B(F(t), g(t)); we will use this function from now on.

Lemma 12. If F : [a,b] [right arrow] X is a continuous function and g : [a,b] [right arrow] Y is Bochner integrable, then the function B(F, g) is Bochner integrable.

Proof. Since F is continuous, then it is strongly measurable; moreover there exists K > 0 such that [parallel]F(f)[[parallel].sub.x] [less than or equal to] K for all t [member of] [a, b]. Because g is strongly measurable and B is continuous, B(F, g) is strongly measurable.

By Theorem 10 the real function [parallel]g(t)[[parallel].sub.Y] is Lebesgue integrable.

Now

[mathematical expression not reproducible] (12)

hence B(F,g) is Bochner integrable as a consequence of Theorem 10.

It is known that every vector-valued function which is strongly measurable is weakly measurable; that is, [x.sup.*] F : [a, b] [right arrow] R is measurable for each x* [member of] X*; the inverse, in general, is not true (see [10, Example 5, Chapter II, [section]1, and pp. 43]) however under certain conditions is equivalent.

Theorem 13 (see [10, Thm. 2, Chapter II, [section]1, pp. 42] (Pettis)). Let F : [a,b] [right arrow] X be a function. The following conditions are equivalent:

(i) F is strongly measurable.

(ii) F is weakly measurable and there exists a measurable set E [subset] [a, b] with [mu]([a, b] - E) = 0 such that F(E) is separable.

Theorem 14. If F : [a,b] [right arrow] X is continuous a.e., then F is strongly measurable.

Proof. As F is continuous a.e., then F is weakly continuous; that is, x*F : [a, b] [right arrow] R is continuous a.e. and, hence, measurable.

We define E = {t e [a, b] : F(t) is continuous}. E is Lebesgue measurable with [mu]([a, b] - E) = 0, as [a, b] is separable and then E is separable and, furthermore, F is continuous, and then F(E) is separable; hence F is strongly measurable by the Pettis Theorem.

Lemma 15 (see [11, Lemma 6]). If g : [a, b] [right arrow] X is of strongly bounded variation on [a, b], then g is Bochner integrable on [a,b].

As a consequence of Lemmas 12 and 15, we have the following result.

Corollary 16. Let f : [a,b] [right arrow] X be Henstock integrable on [a, b] and F its primitive, g : [a,b] [right arrow] Y of strongly bounded variation, then B(F, g) is Bochner integrable on [a, b].

It is easy to prove that the set of functions AC, AC*, ACG, and ACG* on E [subset] [a, b] form vector spaces with the sum and product by scalars; moreover, these spaces of functions are algebras under the bilinear operator B.

Lemma 17. Let F : [a,b] [right arrow] X and G : [a,b] [right arrow] X be functions and E a subset of [a, b]. If F is ACG* (E) and G is AC(E), then F + G is ACG * (E).

Proof. Since F is ACG* (E) then E = [[universal].sub.n] [E.sub.n], where F is AC*([E.sub.n]).

For every [epsilon] >0 there exists [eta] > 0 such that, for every {[[c.sub.i], [d.sub.i]] : [c.sub.i], [d.sub.i] [member of] [E.sub.n]}, with [[summation].sub.i]([d.sub.i] - [c.sub.i]) < [eta], we have [mathematical expression not reproducible] and

[mathematical expression not reproducible] (13)

and then [mathematical expression not reproducible].

Lemma 18. Let F : [a,b] [right arrow] X and g : [a,b] [right arrow] Y be functions and E a subset of [a, b].IfF is ACG* (E) and g is AC(E), then B(F, g) is ACG* (E).

Proof. The proof is analogous to Lemma 17 changing the inequality 2 by

[mathematical expression not reproducible] (14)

3. Integration by Parts Theorem

3.1. Involving Bochner Integral

Theorem 19. Let f : [a,b] [right arrow] X be a Henstock integrable function with primitive F, g : [a,b] [right arrow] Y of strongly bounded variation, and G the Bochner primitive of g. Then (H) [integral].sup.b.sub.a] B(f, G) exists and

[mathematical expression not reproducible] (15)

Proof. By Corollary 16, B(F, g) is Bochner integrable. Let [phi] : [a, b] [right arrow] Z be a function given by

[mathematical expression not reproducible]. (16)

[phi] is continuous due to the continuity on [a, b] of F, G, and (B) [[integral].sup.t.sub.a] B(F,g).

Theorem 5 implies that F is ACG* on [a,b] and Theorem 11 implies that G is AC on [a,b]; hence the function t [right arrow] B(F(t),G(t)) is ACG* on [a,b] by Lemma 18. Finally, Lemma 17 implies that [phi] is ACG* on [a, b].

In order to prove that [phi] is differentiable on [a, b], using the fact that B(F(t)/(s - t), G(s)) = B(F(t), G(s)/(s - t)) for every s, t 6 [a, b], we calculate

[mathematical expression not reproducible] (17)

which tends to 0 when s [right arrow] t; hence B(F, G) is differentiable on [a,b] and (B(F,G))' = B(F,g) + B(f,G). Then [phi](t) = (B(F, G))' - B(F, g) = B(F, g) + B(f G) - B(F, g) = B(f G).

By Theorem 5, (H) [[integral].sup.b.sub.a] B(f, G) exists and (15) is fulfilled.

3.2. Involving Henstock-Stieltjes Integral

Theorem 20. Let f : [a,b] [right arrow] X and g : [a,b] [right arrow] Y be functions. If the integral (HS) [[integral].sup.b.sub.a] B(f dg) exists and (B)[var.sup.b.sub.a](g) < [infinity], then

[mathematical expression not reproducible] (18)

Proof. Let [member of] >0. There exists a gauge [delta] on [a, b] such that

[mathematical expression not reproducible] (19)

for every [mathematical expression not reproducible], which is [delta]-fine. Then,

[mathematical expression not reproducible] (20)

Now, we shall prove the following Theorem, which is a consequence of Theorem 9.

Theorem 21 (uniform convergence theorem). Let [mathematical expression not reproducible], the integrals (HS) [[integral].sup.b.sub.a] ([f.sub.n],dg) exist for each n, and the sequence [f.sub.n] converges uniformly to f in [a, b], then the integral (HS) [[integral].sup.b.sub.a] B(f, dg) exists and

[mathematical expression not reproducible]. (21)

Proof. Let [epsilon] > 0; because [f.sub.n] converges uniformly to f, there exists [n.sub.0] >0 such that for every n > [n.sub.0] and t [member of] [a, b]

[mathematical expression not reproducible]. (22)

Hence, for every [mathematical expression not reproducible].

Theorem 9 implies the existence of the integral [mathematical expression not reproducible]. Hence, there exists [n.sub.1] [member of] N such that[mathematical expression not reproducible].

Let m > max{[n.sub.0], [n.sub.1]} be fixed; as the integral

[mathematical expression not reproducible] (23)

exists, there exists a gauge [delta] on [a, b] such that, for every [delta]-fine tagged partition [mathematical expression not reproducible],

[mathematical expression not reproducible] (24)

We have

[mathematical expression not reproducible] (25)

Hence, (HS) [[integral].sup.b.sub.a] B(f, dg) exists and

[mathematical expression not reproducible] (26)

Theorem 22. If f : [a, b] [right arrow] X is a step function, then for every g : [a, b] [right arrow] Y, the integral (HS) [[integral].sup.b.sub.a] B(f, dg) exists.

Proof. Analogous to the proof of [12, Lemma 3.2], is enough to prove for functions of the forms [mathematical expression not reproducible], and [X.sub.[b]]x, where [tau] [member of] (a,b) and x [member of] X. Let [tau] [member of] (a,b), x e X, and [mathematical expression not reproducible]. Given [epsilon] >0 we define [delta](t) = [epsilon] if t = [tau] and [delta](t) = (1/2) [absolute value of [tau] - l] if t [right arrow] [tau]; then for any [delta]-fine tagged partition [mathematical expression not reproducible], [tau] is the tag of one subinterval, if [mathematical expression not reproducible] otherwise

[mathematical expression not reproducible] (27)

Hence [mathematical expression not reproducible]. The proofs of the cases [mathematical expression not reproducible] are analogous.

Schwabik in  introduces the concept of vector-valued regulated functions; we shall only use the following characterization.

Theorem 23 (see [3, Prop. 2]). F : [a,b] [right arrow] X is regulated if and only if it is the uniform limit of step functions.

Theorem 24. If F : [a,b] [right arrow] X is regulated and g : [a,b] [right arrow] Y with [mathematical expression not reproducible], then the integral (HS) [[integral.sup.b.sub.a] B(F,dg) exists.

Proof. In as much as F is regulated, there exists a sequence [F.sub.n] : [a,b] [right arrow] X, n = 1,2, ..., of step functions which converges uniformly to F, by Theorem 22; the integrals (HS) [[integral.sup.b.sub.a] B(Fn,dg) exist for each n = 1,2, ... The Uniform Convergence Theorem implies the existence of the integral (HS) [[integral.sup.b.sub.a] B(F,dg).

Theorem 25 (integration by parts theorem). If f: [a,b] [right arrow] X is Henstock integrable, F its primitive, and g : [a,b] [right arrow] Y with [mathematical expression not reproducible] exists and

[mathematical expression not reproducible] (28)

Proof. Let [member of] >0. Since f is Henstock integrable, with F its primitive, there exists a gauge [[delta].sub.1] such that if [mathematical expression not reproducible]; i= 1,2, is a [[delta].sub.1]-fine tagged partition of [a,b],

[mathematical expression not reproducible]. (29)

On the other hand, since F is continuous, it is regulated, and by Theorem 24, (HS) [[integral.sup.b.sub.a] B(F, dg) exists; then there exists a gauge [[delta].sub.2] such that if [P.sub.2] = {([[s.sub.j-1], [s.sub.j]],[[eta].sub.j]); j = 1,2, ..., m} is a [[delta].sub.2]-fine tagged partition of [a, b],

[mathematical expression not reproducible] (30)

We define a gauge [mathematical expression not reproducible] be a [[delta].sup.*]-fine tagged partition of [a, b] and by the left-right process (see [13, Section 1, pp. 6]) we can assume that the tags are the left endpoint of each subinterval; then

[mathematical expression not reproducible] (31)

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 t [right arrow] B(f(t), G(f)),that is, the Fundamental Theorem of Calculus.

(2) In Theorem 25 the Fundamental Theorem of Calculus does not apply because g is not necessarily differentiable, and if it is, the primitive of g, in general, is not g.

(3) The condition of differentiability on a function g : [a, b] [right arrow] Y of strongly bounded variation is equivalent to Y and has the Radon-Nikodym property (see Distel and Uhl [10, Chapter VII [section]6]). Is it fulfilled with functions of strongly bounded B-variation?

Therefore, the condition of g is one that ensures the Fundamental Theorem of Calculus, so we have the following theorems.

Theorem 26. Let F :[a,b] [right arrow] X be continuous function and g : [a,b] [right arrow] Y with (B)[var.sup.b.sub.a](g) < [infinity] such that g exists on [a, b]. Then

[mathematical expression not reproducible] (32)

Proof. (HS) [[integral.sup.b.sub.a] B(F, dg) exists by Theorem 24; then given [epsilon] > 0, we have a gauge [delta] and {([[t.sub.i-1], [t.sub.i]], [[xi].sub.i]) : i = 1, ..., n} a [delta]-fine tagged partition of [a, b]. Since g exists, for every t [member of] [a, b] there exists [[eta].sub.t] > 0 such that if [absolute value of t - s] < [[eta].sub.t] then

[mathematical expression not reproducible] (33)

We define a gauge [[delta].sub.1] : [a,b] [right arrow] (0, [infinity]) by [[delta].sub.1] (t) = min{[[eta].sub.t], [delta](t)}. For all {([[t.sub.j-1], [t.sub.j]],[[xi].sub.j]) : j = l, ..., m} which is [[delta].sub.1]-fine tagged partition of [a, b] and supposing that each tag is the left endpoint of its respective subinterval we have

[mathematical expression not reproducible] (34)

where [??] > 0 is the bound of [??] due to it is continuous.

Obviously, we can change the condition over [??] in Theorem 26 if we ask for strongly bounded variation and impose the Radon-Nikod property on [??]; either with these conditions or with those of Theorem 26 we can write equality (28) as

[mathematical expression not reproducible]. (35)

A function G : [a,b] [right arrow] R satisfies the Lipschitz condition if there exists L > 0 such that [absolute value of G(f) - G(s)] < L[absolute value of t-s], s, t [member of] [a, b], and G is of bounded slope variation (BSV) if

[mathematical expression not reproducible] (36)

is bounded for all divisions a = [t.sub.0] < [t.sub.1] < ... < [t.sub.n] = b.

Lee in [14, p. 75] proves that a function G : [a,b] [right arrow] R is the primitive (in the sense of Henstock) of a function of strongly bounded variation on [a, b] if and only if G satisfies the Lipschitz condition and is of bounded slope variation on [a, b]; this same characterization can be extended for our case (see the proof of [15, Thm. 10]). So we can see that the Fundamental Theorem of Calculus applies and we can restate the above integration by parts formula as follows.

Corollary 27. Let f : [a,b] [right arrow] X be a function and F its primitive. If g : [a,b] [right arrow]Y satisfies the Lipschitz condition and is of bounded slope variation, then (H) [[integral].sup.b.sub.a] B(f, g) exists and

[mathematical expression not reproducible]. (37)

4. Representation Theorem

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

Definition 28. Let H([a,b], X) be the space of all Henstock integrable functions from [a, b] to X. We define a norm on H([a,b], X), called Alexiewicz norm (see e.g.,  or ), by

[mathematical expression not reproducible]. (38)

Theorem 29. Let g : [a,b] [right arrow] Y be a function. If (B) [var.sup.b.sub.a] (g) < [integral], then

T(f) = (H) [[integral].sup.b.sub.a] B(f,g) (39)

defines a continuous linear operator on the space of Henstock integrable functions into Z with

[mathematical expression not reproducible]. (40)

Proof. Let F : [a,b] [right arrow] X the primitive of f and [member of] > 0; by Theorem 24 there exists a gauge [delta] such that if {([[t.sub.i-1], [t.sub.i]], [[xi].sub.i]); i = 1, ..., n} is [delta]-fine,

[mathematical expression not reproducible] (41)

By the integration by parts theorem we have

[mathematical expression not reproducible] (42)

[mathematical expression not reproducible] (43)

It follows that

So T is bounded and continuous.

L(X, Y) denotes the space of continuous linear operators from X to Y and B : Xx L(X, Y) [right arrow] Y is the bilinear bounded operator given by B(x, A) = A(x).

Theorem 30. Let T : H([a,b],X) [right arrow] Y be a linear continuous operator defined on the space of the Henstock integrable functions. There exists a function g : [a, b] [right arrow] L(X, Y) with (B)[var.sup.b.sub.a](g) < [infinity] such that

[mathematical expression not reproducible], (44)

for every f [member of] H([a,b], X).

Proof. For each t [member of] [a,b] and x [member of] X we have [X.sub.[a,t]] x [member of] H([a, b],X). We define a function [[beta].sub.t] : X [right arrow] Y by

[mathematical expression not reproducible] (45)

In this form, [[beta].sub.t] is linear and it is also continuous due to

[mathematical expression not reproducible] (46)

where M = [parallel]T[parallel](b - a).

We define a function g : [a,b] [right arrow] L(X, Y) by

g(t) = [[beta].sub.t]. (47)

Now, (B)[var.sup.b.sub.a] (g) < [infinity], indeed, for every arbitrary partition {[[t.sub.i-1], [t.sub.i]] : i = 1, ..., nj and for every [x.sub.i], i = 1, ..., n, with [mathematical expression not reproducible].

[mathematical expression not reproducible] (48)

Suppose that s : [a,b] [right arrow] X is a step function; by Theorem 22 we have that the integral (HS) [[integral].sup.b.sub.a] (s, dg) exists.

Given f [member of] H([a,b], X), we take its primitive F: [a, b] [right arrow] X. Since F is continuous, there exists ([F.sub.n]) sequence of linear piecewise functions such that [F.sub.n] [right arrow] F uniformly on [a, b]. The integral (HS) [[integral].sup.b.sub.a] ([F.sub.n], dg) exists for every n by Theorem 24.

By the Uniform Convergence Theorem, the integral (HS) [[integral].sup.b.sub.a] B(F, dg) exists; hence

[mathematical expression not reproducible] (49)

Let ([f.sub.n]) be the sequence of derivatives of [F.sub.n]; [f.sub.n] is simple for each n, since [F.sub.n] [right arrow] F uniformly

[mathematical expression not reproducible] (50)

by the continuity of T and the Integration by parts Theorem 25 we have that

[mathematical expression not reproducible] (51)

Finally, we have the following representation theorem.

Corollary 31. T : H([a,b],X) [right arrow] Y is a linear continuous operator if and only if there exists a function g : [a, b] [right arrow] L(X, Y) with (B)[var.sup.b.sub.a](g) < [infinity] such that

T(f) = (H) [[integral].sup.b.sub.a] B(fg), (52)

for every f [member of] H([a,b], X).

Consider X a Banach space and Y = X* in the definition of strongly bounded (B)-variation function; then this is equivalent to the definition of strongly bounded variation function, so we have the next result.

Corollary 32. T is a linear continuous functional on H([a, b], X) if and only if there exists a function g : [a,b] [right arrow] X* and V(g, [a,b]) < [infinity] such that

[mathematical expression not reproducible] (53)

for every f [member of] H([a,b], X).

Hence, the dual space of H([a,b], X) is isometrically isomorphic to the space of functions of strongly bounded variation.

Since the space of Henstock integrable real-valued functions coincides with the Kurzweil integrable functions we will name the integral as the integral of Henstock-Kurzweil.

As a particular case of Corollary 31 (with X = L(X, Y) = Y = R) we have obtained the following representation theorem for the space of Henstock-Kurzweil integrable functions.

Corollary 33. T : HK([a,b], R)[right arrow] R is a linear continuous functional if and only if exists a function g : [a, b] [right arrow] R, V(g[a, b],) < [infinity], such that

T(F) = (HK) [[integral].sup.b.sub.a] [[integral].sup.b.sub.a] fg, (54)

for every f Henstock-Kurzweil integrable function.

Remarks 34. The result above was proved in  by Alexiewicz; later other proofs of the theorem arose, for example, those provided by Sargent and Lee (Theorem 4 in  and Theorem 12.7 in [14, pp. 76], resp.), who use different techniques from those used in this work; for example, HahnBanach-Theoremis not necessary for the proof of Theorem 30.

The Integration by parts Corollary 27 yields a new representation theorem without using Stieltjes integral, which we shall establish next. The proof of the first result below is analogous to Theorem 29; using equality (35), we will only sketch the proof of the second result.

Theorem 35. If g : [a, b] [right arrow] Y is of strongly bounded B-variation and differentiable and g is bounded, then T(f) = (H) [[integral].sup.b.sub.a] B(f g) is a continuous linear operator on the space H([a, b],X) into Z.

If g : [a,b] [right arrow] Y is of bounded slope variation and satisfies the Lipschitz condition then the previous theorem is also true; we shall prove the second part in the new representation theorem.

Theorem 36. Let T : H([a,b], X) [right arrow] Y be a linear continuous operator defined on the space of the Henstock integrable functions. There exists a function g : [a, b] [right arrow] L(X, Y) of bounded slope variation and Lipschitz such that T(f) = (H) [[integral].sup.b.sub.a] B(f, g),for every f [member of] H([a,b],X).

Proof. Let x [member of] X and F be the primitive of f [member of] H([a, b], X). We define the function [[beta].sub.t](x) = T([X.sub.[a,t]] x); this function is in L(X, Y), and we define g : [a, b] [right arrow] L(X, Y) by g(t) = [[beta].sub.t]; g is of bounded slope variation and satisfies the Lipschitz condition (see the proof of [15, Thm. 10]); hence it is differentiable and g is of strongly bounded variation. Following the proof of Theorem 30 using g which is also of strongly bounded variation, by Theorem 26 the integral (HS) [[integral].sup.b.sub.a] B(F, dg) is equal to (H) [[integral].sup.b.sub.a] B(F, g) and by integration by parts Corollary 27, we have T(f) = (H) [[integral].sup.b.sub.a] B(f, g).

The last representation theorem identifies H([a,b], X) with the space of primitives of the functions of strongly bounded variation, unlike the Corollary 31 which identifies it with the space of functions of strongly bounded B-variation.

https://doi.org/10.1155/2018/8169565

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research has been able to see the light thanks to the help, comments, suggestions, and unconditional support of Professor Lee Peng Yee and the work team and the staff of MME/NIE of Nanyang Technological University in Singapore. The authors will be eternally grateful. This research has been supported by Conacyt, VIEP-BUAP, DGRIIA-BUAP, and the Academic Group of Mathematical Modeling and Differential Equations-FCFM-BUAP.

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Tomas Perez Becerra (iD), (1) Juan Alberto Escamilla Reyna, (1) Daniela Rodriguez Tzompantzi, (1) Jose Jacobo Oliveros Oliveros (iD), (1) and Khaing Khaing Aye (2)

(1) Facultad de Ciencias Fisico Matematicas, Benemerita Universidad Autonoma dePuebla, Puebla, PUE, Mexico

(2) Department of Engineering Mathematics, Yangon Technological University, Yangon, Myanmar

Correspondence should be addressed to Tomas Perez Becerra; tompb55@hotmail.com

Received 15 January 2018; Revised 21 March 2018; Accepted 5 April 2018; Published 17 May 2018