DOCSLIB.ORG

The Use of an Isometric Isomorphism on the Completion of the Space of Henstock-Kurzweil Integrable Functions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Use of an Isometric Isomorphism on the Completion of the Space of Henstock-Kurzweil Integrable Functions Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 715789, 5 pages http://dx.doi.org/10.1155/2013/715789 Research Article The Use of an Isometric Isomorphism on the Completion of the Space of Henstock-Kurzweil Integrable Functions Luis Ángel Gutiérrez Méndez, Juan Alberto Escamilla Reyna, Francisco Javier Mendoza Torres, and María Guadalupe Morales Macías Facultad de Ciencias F´ısico Matematicas,´ Benemerita´ Universidad, Autonoma´ de Puebla, Avenida San Claudio y 18 Sur, Colonia San Manuel, 72570, Puebla, Mexico Correspondence should be addressed to Luis Angel´ Gutierrez´ Mendez;´ [email protected] Received 19 April 2013; Accepted 5 June 2013 Academic Editor: Nelson Merentes Copyright © 2013 Luis Angel´ Gutierrez´ Mendez´ et al. This 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. Employing an isometrically isomorphic space, we determine new properties for the completion of the space of the Henstock- Kurzweil integrable functions with the Alexiewicz norm. 1. Introduction space of the distributions that are derivatives of the continu- ous functions on [, ], which is an isometrically isomorphic Let [, ] be a compact interval in R.Inthevectorspaceof (HK̂[, ], ‖ ⋅‖ ) Henstock-Kurzweil integrable functions on [, ] with values space to . Making use of this same isometri- in R, the Alexiewicz seminorm is defined as cally isomorphic space, Bongiorno and Panchapagesan in [4] establish characterizations for the relatively weakly compact = ∫ . (HK[, ], ‖ ⋅‖ ) (HK̂[, ], ‖ ⋅‖ ) sup (1) subsets of and . ≤≤ ̂ In this paper, we make an analysis on (HK[, ], ‖ ⋅‖ ) The corresponding normed space is built using the by means of another isometrically isomorphic space to prove ∼ ̂ quotient space determined by the relation if and only if that (HK[, ], ‖ ⋅‖ )has the Dunford-Pettis property, it has = , except in a set of Lebesgue measure zero or, equivalently, a complemented subspace isomorphic to 0, it does not have the if they have the same indefinite integral. This normed space Radon-Riesz property, it is not weakly sequentially complete, (HK[, ], ‖ ⋅‖ ) will be denoted by . and it is not isometrically isomorphic to the dual of any normed It is known that (HK[, ], ‖ ⋅‖ ) is neither complete nor ̂ space; hence, we will also prove that (HK[, ], ‖ ⋅‖ )is of the second category [1]. However, (HK[, ], ‖ ⋅‖ ) is a neither reflexive nor has the Schur property.Then,asanappli- separable space [1] and, consequently, its completion also has ̂ cation of the above results, we prove that (HK[, ], ‖ ⋅‖ ) the same property. In addition, (HK[, ], ‖ ⋅‖ ) has “nice” properties, from the point of view of functional analysis, since is not isomorphic to the dual of any normed space and that (HK[, ], ‖ ⋅‖ ) the space of all bounded, linear, weakly compact operators it is an ultrabornological space [2]. As is ̂ not complete, it is natural to study its completion, which will from (HK[, ], ‖ ⋅‖ ) into itself is not a complemented (HK̂[, ], ‖ ⋅‖ ) subspace in the space of all bounded, linear operators from be denoted by . ̂ Talvila in [3] makes an analysis to determine some pro- (HK[, ], ‖ ⋅‖ ) into itself. ̂ perties of the Henstock-Kurzweil integral on (HK[, ], ‖⋅‖ ) ,suchasintegrationbyparts,Holder¨ inequality, change 2. Preliminaries of variables, convergence theorems, the Banach lattice struc- ture, the Hake theorem, the Taylor theorem, and second mean In this section, we restate the conventions, notations, and value theorem. Talvila makes this analysis by means of the concepts that will be used throughout this paper. 2 Journal of Function Spaces and Applications All the vector spaces are considered over the field of the Definition 4. Let be a subspace of a normed space .Itis real numbers or complex numbers. said that is complemented in if it is closed in and there ∗ Let be a normed space. By , we denote the dual exists a closed subspace in such that =⊕. space of . A topological property that holds with respect totheweaktopologyof is said to be a weak property or Theorem 5 (see [5]). Let be a Banach space with the to hold weakly. On the other hand, if a topological property Dunford-Pettis property. If is a complemented subspace in holds without specifying the topology, the norm topology is ,then has the Dunford-Pettis property. implied. ̂ Let , be two Banach spaces. We denote by (, ) To prove that (HK[, ], ‖ ⋅‖ ) has a complemented sub- ((,, ) resp.) we denote the Banach space of all bounded, space isomorphic to 0 the following result is essential. linear (bounded, linear, weakly compact resp.), operators from into .If=,thenwewrite() (resp. ()) Theorem 6 (see [6]). Let be a compact metric space. If is instead of (, ) (resp. (,). ) an infinite-dimensional complemented subspace of C(),then The symbols 0, 1,and∞ represent, as usual, the contains a complemented subspace isomorphic to 0. vector spaces of all sequences of scalars convergent to 0, all On the other hand, making use again of Theorem 2 we sequences of scalars absolutely convergent, and all bounded ̂ sequences of scalars, respectively, neither one with nor usual will prove that (HK[, ], ‖ ⋅‖ ) is neither weakly sequen- norm. tially complete nor has the Radon-Riesz property. Let be a compact metric space. We denote by C() the { } vector space of all continuous functions of scalar-values on Definition 7. Let be a normed space, and let be a ∈ together with the norm defined by ‖‖∞ = sup{|()| : ∈ sequence in and . } . (i) If {} weakly converges whenever {} is weakly B [, ] By we denote the following collection: Cauchy, then it is said that is weakly sequentially complete. {:[,] → R |is continuous on [,] and () =0} { } { } (2) (ii) If converges to whenever weakly converges to and ‖‖ → ‖‖,thenitissaidthat has the which is a closed subspace of C[, ] and (B[, ], ‖∞ ⋅‖ ) is Radon-Riesz property or the Kadets-Klee property. therefore a Banach space. (iii) If {} converges to whenever {} weakly converges to ,thenitissaidthat has the Schur property. Definition 1. Let be normed spaces and let : → be a lineal operator. We have the following. The following result establishes a characterization of (i) is an isomorphism if it is one-to-one and continuous weakly Cauchy sequences and weakly convergent sequences −1 and its inverse mapping is continuous on the of the space C[, ]. range of .Moreover,if‖()‖ = ,forall‖‖ ∈, Theorem 8 { } it is said that is an isometric isomorphism. (see [7]). Let and be a sequence and an C[, ] ≅ element, respectively, in the space .Thenwehavethe (ii) and are isomorphic, which is denoted by , following. if there exists an isomorphism from onto . (1) The sequence {} is weakly convergent to if and only (iii) and are isometrically isomorphic if there exists an if isometric isomorphism from onto . (i) lim→∞() = (),forall∈[,], The following result is key to our principal results. (ii) there exists >0such that ‖‖∞ ≤,forall ∈N. Theorem 2 (see [4]). The space (B[, ], ‖∞ ⋅‖ ) is isometri- ̂ cally isomorphic to (HK[, ], ‖ ⋅‖ ). (2) The sequence {} isweaklyCauchyifandonlyif ̂ () ∈[,] According to Theorem 2,wewillprovethat(HK[, ], (i) lim→∞ exists, for all , >0 ‖ ‖ ≤ ‖⋅‖) has the Dunford-Pettis property. (ii) there exists such that ∞ ,forall ∈N. Definition 3. Let be a Banach space. It is said that has ̂ We will also prove that (HK[, ], ‖ ⋅‖ ) is not isomet- the Dunford-Pettis property if for every sequence {} in ∗ ∗ ricallyisomorphictothedualofanynormedspaceand,as converging weakly to 0 and every sequence { } in ∗ (HK̂[, ], ‖ ⋅‖ ) converging weakly to 0, the sequence { ()} converges to aconsequence,wewillprovethat is not 0. reflexive, for which we will use again Theorem 2 and the concept of extremal point. If a Banach space has the Dunford-Pettis property, then not necessarily every closed subspace of inherits such Definition 9. Let be a vector space, ⊆,and∈.It property, except when the subspace is complemented in is said that is an extremal point of if for all , ∈ such [5]. that = (1/2)( +) it holds that ==. Journal of Function Spaces and Applications 3 If is a normed space, then its closed unit ball will be However, since is not continuous, it follows that the denoted by and the collection of all extremal points of sequence {} does not converge weakly in (B[, ], ‖∞ ⋅‖ ). as ext(). The following theorem establishes that the extremal points are preserved under isometric isomorphisms. Lemma 15. The space (B[, ], ‖∞ ⋅‖ ) does not have the Radon-Riesz property. Theorem 10 (see [8]). Let be Banach spaces, ⊆ and let :be → an isometric isomorphism. Then is an Proof. Without loss of generality, suppose that |−|.Let ≥1 extremal point of if and only if () is an extremal point of be the function defined by (). () Corollary 11 (see [9]). An infinite-dimensional normed space 1 whoseclosedunitballhasonlyfinitelymanyextremepointsis {( ) (−) , ∈[,− ], { (−) −1 if not isometrically isomorphic to the dual of any normed space. { { 1 1 = {2 (−) −1, if ∈(− ,− ], { 2 3. Principal Results { { 1 2 (−) +1, if ∈(− ,], Lemma 12. The space (B[, ], ‖∞ ⋅‖ ) has the Dunford- { 2 Pettis property. (6) Proof. Let :C[, ] → R be the functional defined by for all ≥2.Thus, () = ().Itisclearthat is bounded and therefore { } ker() is a hyperplane of the space C[, ],thatis, (i) the sequence converges pointwise to ,where () = ( −
Load more

Recommended publications
  • The Closed Graph Theorem and the Space of Henstock-Kurzweil Integrable Functions with the Alexiewicz Norm
    Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 476287, 4 pages http://dx.doi.org/10.1155/2013/476287 Research Article The Closed Graph Theorem and the Space of Henstock-Kurzweil Integrable Functions with the Alexiewicz Norm Luis Ángel Gutiérrez Méndez, Juan Alberto Escamilla Reyna, Maria Guadalupe Raggi Cárdenas, and Juan Francisco Estrada García Facultad de Ciencias F´ısico Matematicas,´ Benemerita´ Universidad Autonoma´ de Puebla, Avenida San Claudio y 18 Sur, ColoniaSanManuel,72570Puebla,PUE,Mexico Correspondence should be addressed to Luis Angel´ Gutierrez´ Mendez;´ [email protected] Received 5 June 2012; Accepted 25 December 2012 Academic Editor: Martin Schechter Copyright © 2013 Luis Angel´ Gutierrez´ Mendez´ et al. This 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. WeprovethatthecardinalityofthespaceHK([a, b]) is equal to the cardinality of real numbers. Based on this fact we show that there exists a norm on HK([a, b]) under which it is a Banach space. Therefore if we equip HK([a, b]) with the Alexiewicz topology then HK([a, b]) is not K-Suslin, neither infra-(u) nor a webbed space. 1. Introduction Afterwards, using the same technique used byHoning,¨ Merino improves Honing’s¨ results in the sense that those [, ] R Let be a compact interval in .Inthevectorspaceof results arise as corollaries of the following theorem. Henstock-Kurzweil integrable functions on [, ] with values in R the Alexiewicz seminorm is defined as Theorem 1 (see [2]). There exists no ultrabornological, infra- (), natural, and complete topology on HK([, ]).
    [Show full text]
  • The Use of an Isometric Isomorphism on the Completion of the Space of Henstock-Kurzweil Integrable Functions
    Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 715789, 5 pages http://dx.doi.org/10.1155/2013/715789 Research Article The Use of an Isometric Isomorphism on the Completion of the Space of Henstock-Kurzweil Integrable Functions Luis Ángel Gutiérrez Méndez, Juan Alberto Escamilla Reyna, Francisco Javier Mendoza Torres, and María Guadalupe Morales Macías Facultad de Ciencias F´ısico Matematicas,´ Benemerita´ Universidad, Autonoma´ de Puebla, Avenida San Claudio y 18 Sur, Colonia San Manuel, 72570, Puebla, Mexico Correspondence should be addressed to Luis Angel´ Gutierrez´ Mendez;´ [email protected] Received 19 April 2013; Accepted 5 June 2013 Academic Editor: Nelson Merentes Copyright © 2013 Luis Angel´ Gutierrez´ Mendez´ et al. This 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. Employing an isometrically isomorphic space, we determine new properties for the completion of the space of the Henstock- Kurzweil integrable functions with the Alexiewicz norm. 1. Introduction space of the distributions that are derivatives of the continu- ous functions on [, ], which is an isometrically isomorphic Let [, ] be a compact interval in R.Inthevectorspaceof (HK̂[, ], ‖ ⋅‖ ) Henstock-Kurzweil integrable functions on [, ] with values space to . Making use of this same isometri- in R, the Alexiewicz seminorm is defined as
    [Show full text]
  • Factorization in the Space of Henstock-Kurzweil Integrable Functions
    Azerbaijan Journal of Mathematics V. 7, No 2, 2017, July ISSN 2218-6816 Factorization in the Space of Henstock-Kurzweil Inte- grable Functions M.G. Morales, J.H. Arredondo∗ Abstract. In this work, we extend the factorization theorem of Rudin and Cohen to HK(R), the space of Henstock-Kurzweil integrable functions. This implies a factorization for the isometric spaces AC and BC . We also study in this context the Banach algebra of functions HK(R)∩BV (R), which is also a dense subspace of L2(R). In some sense this subspace is analogous to L1(R)∩L2(R). However, while L1(R)∩L2(R) factorizes as L1(R)∩L2(R)∗L1(R), via the convolution operation *, it is shown in the paper that HK(R) ∩ BV (R) ∗ L1(R) is a Banach subalgebra of HK(R) ∩ BV (R). Key Words and Phrases: factorization, Banach algebra, Henstock-Kurzweil integral, bounded variation function. 2010 Mathematics Subject Classifications: 42A85, 26A39,46J30 1. Introduction For appropriate functions f, g : R → R, the convolution is defined as f ∗ g(t)= g ∗ f(t) := f(t − y)g(y)dy. (1) ZR Thus, f ∗ g is a superposition of translates of f taken with the weights g(y)dy, and we can expect that f ∗ g inherits properties of f as well as of g, [9]. For example, the space of Lebesgue integrable functions L1(R), is a Banach algebra with the convolution (1). In fact, W. Rudin [18] proved that L1(G) ∗ L1(G)= L1(G), for G = Rn and any locally compact abelian group [19].
    [Show full text]
  • Topologizing the Denjoy Space by Measuring Equiintegrability 1 Introduction
    To appear in Real Analysis Exchange, Volume 31, Number 1 (February 2006) Keywords: barreled, convergence rate, Denjoy space, equiintegrable, Hen- stock integral, inductive limit, KH integral, Kurzweil integral, norm AMS Classi¯cation number(s): 26A39 J. Alan Alewine, [email protected] Division of Science and Mathematics, McKendree College, Lebanon, IL 62254 Eric Schechter, [email protected] Mathematics Department, Vanderbilt University, Nashville, TN 37240 Topologizing the Denjoy Space by Measuring Equiintegrability J. Alan Alewine and Eric Schechter Abstract Basic limit theorems for the KH integral involve equiintegrable sets. We construct a family of Banach spaces X¢ whose bounded sets are precisely the subsets of KH[0; 1] that are equiintegrable andS point- wise bounded. The resulting inductive limit topology on ¢ X¢ = KH[0; 1] is barreled, bornological, and stronger than both pointwise convergence and the topology given by the Alexiewicz seminorm, but it lacks the countability and compatibility conditions that are often associated with inductive limits. 1 Introduction. This paper is concerned with KH[0; 1], the space of all functions f : [0; 1] ! R that are KH integrable (also known as Kurzweil, Henstock, Denjoy-Perron, gauge, nonabsolute, or generalized Riemann integrable). We emphasize that we are considering individual functions, whereas most of the related litera- ture deals with KH[0; 1], the space of equivalence classes of KH integrable 1 functions. Equivalence in this context means agreement outside some set of Lebesgue measure zero. Sections 2 and 3 review basic results about the KH integral, including def- initions of technical terms (gauges, ¿, f(T ), etc.) used in this introduction.
    [Show full text]
  • Banach Spaces for the Schwartz Distributions 3
    BANACH SPACES FOR THE SCHWARTZ DISTRIBUTIONS TEPPER L. GILL Abstract. This paper is a survey of a new family of Banach spaces B that provide the same structure for the Henstock-Kurzweil (HK) integrable func- tions as the Lp spaces provide for the Lebesgue integrable functions. These spaces also contain the wide sense Denjoy integrable functions. They were first use to provide the foundations for the Feynman formulation of quantum mechanics. It has recently been observed that these spaces contain the test functions D as a continuous dense embedding. Thus, by the Hahn-Banach theorem, D′ ⊂ B′. A new family that extend the space of functions of bounded mean oscillation BMO[Rn], to include the HK-integrable functions are also introduced. 1. Introduction Since the work of Henstock [HS1] and Kurzweil [KW], the most important finitely additive measure is the one generated by the Henstock-Kurzweil integral (HK- integral). It generalizes the Lebesgue, Bochner and Pettis integrals. The HK- arXiv:1704.02949v1 [math.FA] 29 Mar 2017 integral is equivalent to the Denjoy and Perron integrals. However, it is much easier to understand (and learn) compared to the these and the Lebesgue integral. It provides useful variants of the same theorems that have made the Lebesgue integral so important. We assume that the reader is acquainted with this integral, 1991 Mathematics Subject Classification. Primary (46) Secondary(47) . Key words and phrases. Henstock-Kurzweil integral, Schwartz distributions, path integral, Navier-Stokes, Markov Processes. 1 2 GILL but more detail can be found in Gill and Zachary [GZ]. (For different perspectives, see Gordon [GO], Henstock [HS], Kurzweil [KW], or Pfeffer [PF].) The most important factor preventing the widespread use of the HK-integral in mathematics, engineering and physics is the lack of a Banach space structure comparable to the Lp spaces for the Lebesgue integral.
    [Show full text]
  • Solving Poisson Equation by Distributional HK-Integral: Prospects and Limitations
    Hindawi International Journal of Mathematics and Mathematical Sciences Volume 2021, Article ID 5511283, 9 pages https://doi.org/10.1155/2021/5511283 Research Article Solving Poisson Equation by Distributional HK-Integral: Prospects and Limitations Amila J. Maldeniya ,1 Naleen C. Ganegoda ,2 Kaushika De Silva,2 and Sanath K. Boralugoda3 1Department of Mathematics, General Sir John Kotelawala Defence University, Kandawala Road, Ratmalana, Sri Lanka 2Department of Mathematics, University of Sri Jayewardenepura, Gangodawila, Nugegoda, Sri Lanka 3Department of Mathematics, University of Colombo, Cumarathunga Munidasa Mawatha, Colombo, Sri Lanka Correspondence should be addressed to Amila J. Maldeniya; [email protected] Received 11 February 2021; Revised 14 May 2021; Accepted 24 June 2021; Published 12 July 2021 Academic Editor: Ra´ul E. Curto Copyright © 2021 Amila J. Maldeniya et al. &is 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. In this paper, we present some properties of integrable distributions which are continuous linear functional on the space of test function D(R2). Here, it uses two-dimensional Henstock–Kurzweil integral. We discuss integrable distributional solution for 3 Poisson’s equation in the upper half space R+ with Dirichlet boundary condition. 1. Introduction improve the accuracy of problems with irregular boundaries, but still abide by approximations as one has to do mini- In this paper, we find an integrable distributional solution mizing functionals with smaller class of functions [1]. Also, for Poisson’s equation in the upper half space. Poisson’s artificial neural network methods can be used for approx- equation is a second-order elliptic partial differential imate nonsmooth solutions.
    [Show full text]
  • Topologies for the Space of BV-Integrable Functions in RN
    Journal of Functional Analysis FU2997 journal of functional analysis 144, 190231 (1997) article no. FU962997 Topologies for the Space of BV-Integrable Functions in RN Thierry De Pauw Departement de Mathematiques, Universite Catholique de Louvain, Chemin du Cyclotron, 2, 1348 Louvain-la-Neuve, Belgium Received December 15, 1995; accepted April 17, 1996 dedicated to my father We introduce two topologies on the space of BV-integrable functions in RN. Among the studied properties are barrelledness and integral representation of linear continuous functionals. In one case, we characterize the dual space. We show the relations with the study of multipliers. Finally, we also introduce and study a topology on the space of bounded BV functions in RN. 1997 Academic Press CONTENTS 1. Introduction. 2. The BV-integral. 3. The Alexiewicz's topology. 4. The natural topology. 5. The 1-dimensional case. 6. ;-bounded :-convergence. 7. The dual space R${(0 ). 8. The space R(0 )[|}|A] is not barrelled. 1. INTRODUCTION As Henri Lebesgue mentioned in his thesis (Integrale, longueur, aire, 1902), his integral doesn't integrate all (partial) derivatives which aren't bounded, and he left open the problem of defining a more general integral which would overcome this drawback. The integral defined independently by Perron, Denjoy, Henstock and Kurzweil solves this problem in dimension 1. Extending the definition (of Kurzweil, for instance) to higher dimensions gives rise to an integral which doesn't either integrate all partial derivatives. It's only in the eighties that several extensions of the Lebesgue integral were found proving the possibility of integrating the divergence of any differentiable vector field (consult for instance [7, 9, 11]).
    [Show full text]
  • Some Results About Completion of the Space of Henstock-Kurzweil Integrable Functions 1 Introduction
    Int. Journal of Math. Analysis, Vol. 8, 2014, no. 7, 307 - 315 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.4116 Some Results about Completion of the Space of Henstock-Kurzweil Integrable Functions Luis Angel´ Guti´errez M´endez Facultad de Ciencias F´isico Matem´aticas Benem´erita Universidad Aut´onoma de Puebla Puebla, Mexico Juan Alberto Escamilla Reyna Facultad de Ciencias F´isico Matem´aticas Benem´erita Universidad Aut´onoma de Puebla Puebla, Mexico Copyright c 2014 Luis Angel´ Guti´errez M´endez and Juan Alberto Escamilla Reyna. This 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. Abstract Continuing with our study begun in [6] and making use of some results of that article, in this paper we establish new properties for the completion of the space of Henstock-Kurzweil integrable functions with Alexiewicz norm. Mathematics Subject Classification: Primary 26A39; Secondary 46B03, 46B04 Keywords: Henstock-Kurzweil integral, Alexiewicz norm, Schauder basis, bounded approximation property, reciprocal Dunford-Pettis property 1 Introduction It is well known that the Henstock-Kurzweil integral is more general than the Lebesgue integral over the real line. However, in the vector space HK[a, b]={f :[a, b] → R | f is Henstock − Kurzweil integrable} 308 Luis Angel´ Guti´errez M´endez and Juan Alberto Escamilla Reyna is impossible to define a similar norm like in L1[a, b], since the Henstock- Kurzweil integral is not an absolute integral. Arguably the “nicest” known seminorm on HK[a, b]istheAlexiewicz semi- norm which is defined as r ||f||A = sup f .
    [Show full text]
  • The Completion of the Henstock-Kurzweil Space: an Alternative Proof
    Int. Journal of Math. Analysis, Vol. 6, 2012, no. 5, 209 - 214 The Completion of the Henstock-Kurzweil Space: An Alternative Proof Juan A. Escamilla Reyna, Francisco J. Mendoza Torres and Nathllely Campos Le´on Facultad de Ciencias F´isico Matem´aticas Benem´erita Universidad Aut´onoma de Puebla, Puebla, M´exico [email protected], [email protected] Abstract Applying results of functional analysis about the extension of bounded linear operators between Banach spaces, we prove that the completion of the space of Henstock-Kurzweil integrable functions, defined on a compact interval I in R under the Alexiewicz norm, is isometrically iso- morphic to the space of the distributions which are the distributional derivative of continuous functions. The proof is different from Bongiorno - Panchapagesan’s. Mathematics Subject Classification: 26A39, 46B04, 54C35 Keywords: Henstock-Kurzweil integral, Alexiewicz norm, bounded linear operator, isometry, completion of a normed space, distribution Introduction Let I =[a, b] be a compact interval in R. In the vector space of Henstock- Kurzweil integrable functions on I, the Alexiewicz seminorm is defined as x ||f||A = sup f(t)dt . (1) x∈I a The respective normed space is built using the quotient space determined by the relation: f ∼ g if and only if f = g a.e, with respect to the Lebesgue measure. We denote this normed space by HK(I). It is known that the space HK(I) is not a Banach space (see [7]). From this arises the important problem of finding “good” characterizations of its completion. One application of this type is studied in ([8]).
    [Show full text]