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

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A Riesz Representation Theorem for the Space of Henstock Integrable Vector-Valued Functions Hindawi Journal of Function Spaces Volume 2018, Article ID 8169565, 9 pages https://doi.org/10.1155/2018/8169565 Research Article A Riesz Representation Theorem for the Space of Henstock Integrable Vector-Valued Functions Tomás Pérez Becerra ,1 Juan Alberto Escamilla Reyna,1 Daniela Rodríguez Tzompantzi,1 Jose Jacobo Oliveros Oliveros ,1 and Khaing Khaing Aye2 1 Facultad de Ciencias F´ısico Matematicas,´ Benemerita´ Universidad AutonomadePuebla,Puebla,PUE,Mexico´ 2Department of Engineering Mathematics, Yangon Technological University, Yangon, Myanmar Correspondence should be addressed to Tomas´ Perez´ Becerra; [email protected] Received 15 January 2018; Revised 21 March 2018; Accepted 5 April 2018; Published 17 May 2018 Academic Editor: Adrian Petrusel Copyright © 2018 Tomas´ Perez´ Becerra et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Using a bounded bilinear operator, we defne 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. 1. Introduction 2. Preliminaries Henstock in [1] defnes a Riemann type integral which Troughout this paper �, �,and� will denote three Banach is equivalent to Denjoy integral and more general than spaces, ‖⋅‖�, ‖⋅‖�,and‖⋅‖�, which will denote their respective ∗ the Lebesgue integral, called the Henstock integral. Cao in norms, � the dual of �, �:�×�→�a bounded bilinear [2] extends the Henstock integral for vector-valued func- operator fxed, and [�, �] aclosedfniteintervaloftherealline tions and provides some basic properties such as the Saks- with the usual topology and the Lebesgue measure, which we Henstock Lemma. denote by �.Forafunction�:[�,�]→R we denote the Schwabik in [3] considers a bilinear form, defnes a Lebesgue integral of � on a measurable �⊂[�,�],whenit Stieltjes type integral, and performs a study about it including exists, by (�) ∫ �. [4]; following his ideas we give integration by parts theorem � involving a bilinear operator and, through it, we prove a Defnition 1. �:�×�→�is a bounded bilinear operator representation theorem for the space of Henstock vector- if � is linear in each variable and there exists �>0such valued functions. that ‖�(�, �)‖� ≤ �‖�‖�‖�‖�;inthiscase,thenormofthe Tis paper is divided into fve sections; in a frst step, in operator � is ‖�‖ = inf{� > 0 : ‖�(�, �)‖� ≤ �‖�‖�‖�‖�}. Section 2 we present some preliminaries and introduce the Henstock-Stieltjes integral via a bilinear bounded operator We say that �={([��−1,��], ��) : � = 1,...,�}is a tagged and the Bochner integral, together with some basic proper- partition of [�, �] if {[��−1,��]:�=1,...,�}is a fnite collection ties. In Section 3 we provide two useful kinds of integration of nonoverlapping closed intervals whose union is [�, �] such by parts theorems, one of them in terms of the Bochner that �� ∈[��−1,��] for every �.Givenafunction� from [�, �] to integral and the other using Henstock-Stieltjes integral; the (0, ∞),calledgaugeon[�, �], we say that a tagged partition representation theorem is proved in Section 4 which, if {([� ,�], � ):�=1,...,�} � we consider real-valued functions, provides an alternative �−1 � � is -fne if proof of the representation theorem proved by Alexiewicz (Teorem 1 in [5]). [��−1,��] ⊂ (�� −�(��) ,�� +�(��)) for every �. (1) 2 Journal of Function Spaces Defnition 2. Afunction�:[�,�]→�is Kurzweil integrable Teorem 5 (see [8, Tm. 7.4.5, pp. 217]). Te function �: in [�, �] if there exists �∈�such that for every �>0 [�, �] → � is Henstock integrable on [�, �] with the primitive ∗ there exists a gauge � on [�, �] such that if {([��−1,��], ��):�= � if and only if �:[�,�]→�is continuous and ACG on � 1,...,�}is a �-fne tagged partition of [�, �],then [�, �] such that � (�) = �(�) almost everywhere (a.e.) in [�, �], where the derivative is in the sense of Frechet. � � � � � �∑� (� )[� −� ] −�� <�. �:[�,�]→� ∗ � � � � �−1 � (2) If a function is BV or BV on ,thenitis ��=1 � � �� bounded on �;thatis,�>0exists such that ‖�(�)‖� ≤�,for ∗ every �∈�.AsanAC(�) function is BV on � andanAC (�) � ∗ �=(�)∫ � function is BV on � (immediately from the defnitions), then We write � . ∗ every AC(�) or AC (�) function is also bounded in �.Itis �:[�,�]→� easy to see that if � is AC(�) and �0 ⊂�,then� is AC(�0), Defnition 3. Afunction is Henstock inte- ∗ grable in [�, �] if there exists �:[�,�]→�such that for every similarly if � is AC (�). �>0there exists a gauge � on [�, �] such that if {([��−1,��], ��): Te defnition of a function of strongly bounded variation �=1,...,�}is a �-fne tagged partition of [�, �],then can be extended considering the bilinear operator �:�× �→�. � � � ∑ �� (��)[�� −��−1] − [� (��) −�(��−1)]�� <�. (3) Defnition 6. Let �:[�,�]→�be a function and �= �=1 {�0,�1,...,��} a partition of [�, �]; we defne � � �(�) − �(�) = (�) ∫ � � � � We write � . �� (�, �) = sup {∑ ��(��,�(��)−�(��−1))��}, (4) �=1 Te Henstock integral is also known as Henstock- where the supremum is taken over all possible elections of Lebesgue integral, briefy HL integral ([6]), or variational �� ∈�, �=1,2,...,�,with‖��‖� ≤1. Henstock integral ([7]). � � In[8]wecanfndsomepropertiesofbothintegrals (B) var� (�) = sup {�� (�, �)} , (5) such as the linearity, integrability over subintervals, and the continuity of the function �:[�,�]→�,calledprimitive, where the supremum is taken over all partitions of the � � [�, �] (B) �(�) B � given by �(�) = (�) ∫ � or �(�) = (�) ∫ �, �∈[�,�]. interval and var� is the strong -variation of � � on [�, �]. If we consider � � Defnition 4. Let �:[�,�]→�and let � be a subset of [�, �]. � � � ̂� � � �� (�, �) = sup {�∑� (��,�(��) −�(��−1))� } (6) � � (1) � is said to be of strongly bounded variation (BV) on ��=1 �� � if the number �(�, �) fl sup{∑ ‖�(��)−�(��)‖�} � � � intheequality(4),thenwedefne(B)var̂ (�) = sup{�̂ (�, is fnite, where the supremum is taken over all fnite � � �)} as the B-variation of � on [�, �]. sequences {[��,��]} of nonoverlapping intervals that have endpoints in �. � � If (B)var�(�) < ∞ or (B)var̂�(�) < ∞ we say that � is ∗ (2) � is BV on � if sup{∑� �(�, [��,��])} is fnite, where of strongly bounded B-variation or � is of bounded B-varia- the supremum is taken over all fnite sequences tion, respectively. {[��,��]} of nonoverlapping intervals that have end- It is straightforward that each function of strongly points in �,and�(�; [��,��]) = sup{‖�(�) − �(�)‖ : bounded variation is of strongly bounded B-variation. We �, � ∈ [��,��]} is the oscillation of � on [��,��]. recommend the reader interested in this topic to consult the study exposed in [9]. (3) � is said to be strongly absolutely continuous on � or (�) �>0 �>0 AC if for every there exists such that, 2.1. Stieltjes-Type Integrals. As we mentioned in the introduc- for every fnite or infnite sequence of nonoverlapping tion, Schwabik in [3] gives the next defnition and proves {[� ,�]} ∑ (� −�)<� intervals � � ,with � � � ,wehave somebasicpropertiessuchastheUniformConvergence ∑ ‖�(� )−�(�)‖ ≤ � � ,� ∈� � � � � where � � for all . Teorem. � ∗(�) �>0 �> (4) is AC if for every there exists �∈� �: 0 such that, for every fnite or infnite sequence or Defnition 7. is the Kurzweil-Stieltjes integral of [�, �] → � with respect to �:[�,�]→�if for every �>0 nonoverlapping intervals {[��,��]} satisfying ∑ (�� − � there exists a gauge � on [�, �] such that ��)<�,where��,�� ∈�for all �,wehave ∑ �(�; [� ,�]) < �. � � � � � � � � �∑� (� (� ) ,�(� ) −�(� )) −�� <�, ∗ � � � �−1 � (7) (5) � is ACG on � if � is the union of a sequence of ��=1 � ∗ � closed sets {��} such that, on each ��, � is AC (��). for every �-fne tagged partition {([��−1,�1], ��) : � = 1,...,�} of [�, �]. TenextresultgivesusacharacterizationofHenstock � � = (��) ∫ �(�,��) integrability. In this case we write � . Journal of Function Spaces 3 Now, we introduce the following integral. Teorem 10 (see [8, Cor. 1.4.4, pp. 26]). Astronglymeasurable function �:[�,�]→�is Bochner integrable on [�, �] if there Defnition 8. Afunction�:[�,�]→�is Henstock-Stieltjes exists a function �:[�,�]→R, which is Lebesgue integrable [�, �] �:[�,�]→� integrable in with respect to if there such that ‖�(�)‖� ≤ �(�), �∈[�,�]. exists �:[�,�]→�such that for every �>0there exists a gauge � of [�, �] such that if {([��−1,��], ��) : � = 1,...,�}is a Teorem 11 (see [8, Tm. 7.4.5, pp. 222]). Afunction�: �-fne tagged partition of [�, �],then [�, �] → � is Bochner integrable on [�, �] if and only if there exists a function �:[�,�]→�,whichis�� on [�, �] such � ��(�) = �(�) [�, �] � � that a.e. on . ∑ ��(�(��),�(��)−�(��−1)) −[�(��)−�(��−1)]�� �=1 (8) Given �:[�,�]→�and �:[�,�]→�we defne <�. the function �(�, �) : [�, �] → �,givenby�(�, �)(�) = �(�(�), �(�)); we will use this function from now on. � �(�) − �(�) = (��) ∫ �(�,��) We write � . Lemma 12. If �:[�,�]→�is a continuous function and �:[�,�]→�is Bochner integrable, then the function �(�, �) It is immediate that every Henstock integrable function is Bochner integrable. is Kurzweil integrable and its integrals are the same; we canrepeattheproofofthisfactforthepreviousStieltjes Proof. Since � is continuous, then it is strongly measurable; integrals. Similarly, we can prove the properties of linearity moreover there exists �>0such that ‖�(�)‖� ≤�for all �∈ and integrability
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