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

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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 () = ( −
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