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 [𝑎, 𝑏], which is an isometrically isomorphic Let [𝑎, 𝑏] be a compact interval in R.Inthevectorspaceof (HK̂[𝑎, 𝑏], ‖ ⋅‖ ) Henstock-Kurzweil integrable functions on [𝑎, 𝑏] with values space to 𝐴 . 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 󵄩 󵄩 󵄨 𝑟 󵄨 󵄩𝑓󵄩 = 󵄨∫ 𝑓󵄨 . (HK[𝑎, 𝑏], ‖ ⋅‖ ) (HK̂[𝑎, 𝑏], ‖ ⋅‖ ) 󵄩 󵄩𝐴 sup 󵄨 󵄨 (1) subsets of 𝐴 and 𝐴 . 𝑎≤𝑟≤𝑏 󵄨 𝑎 󵄨 ̂ In this paper, we make an analysis on (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) The corresponding normed space is built using the by means of another isometrically isomorphic space to prove 𝑓∼𝑔 ̂ quotient space determined by the relation if and only if that (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ )has the Dunford-Pettis property, it has 𝑓=𝑔 , except in a set of Lebesgue measure zero or, equivalently, a complemented subspace isomorphic to 𝑐0, 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[𝑎, 𝑏], ‖ ⋅‖ ) will be denoted by 𝐴 . and it is not isometrically isomorphic to the dual of any normed It is known that (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) is neither complete nor ̂ space; hence, we will also prove that (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ )is of the second category [1]. However, (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) 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[𝑎, 𝑏], ‖𝐴 ⋅‖ ) the same property. In addition, (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) 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[𝑎, 𝑏], ‖ ⋅‖ ) the space of all bounded, linear, weakly compact operators it is an ultrabornological space [2]. As 𝐴 is ̂ not complete, it is natural to study its completion, which will from (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) into itself is not a complemented (HK̂[𝑎, 𝑏], ‖ ⋅‖ ) subspace in the space of all bounded, linear operators from be denoted by 𝐴 . ̂ Talvila in [3] makes an analysis to determine some pro- (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) into itself. ̂ perties of the Henstock-Kurzweil integral on (HK[𝑎, 𝑏], ‖⋅‖ ) 𝐴 ,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 𝑌 be a subspace of a normed space 𝑋.Itis real numbers or complex numbers. said that 𝑌 is complemented in 𝑋 if it is closed in 𝑋 and there ∗ Let 𝑋 be a normed space. By 𝑋 , we denote the dual exists a closed subspace 𝑊 in 𝑋 such that 𝑋=𝑌⊕𝑊. space of 𝑋. A topological property that holds with respect totheweaktopologyof𝑋 is said to be a weak property or Theorem 5 (see [5]). Let 𝑋 be a Banach space with the to hold weakly. On the other hand, if a topological property Dunford-Pettis property. If 𝑌 is a complemented subspace in holds without specifying the topology, the norm topology is 𝑋,then𝑌 has the Dunford-Pettis property. implied. ̂ Let 𝑋, 𝑌 be two Banach spaces. We denote by 𝐿(𝑋, 𝑌) To prove that (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) has a complemented sub- (𝑊(𝑋,, 𝑌) resp.) we denote the Banach space of all bounded, space isomorphic to 𝑐0 the following result is essential. linear (bounded, linear, weakly compact resp.), operators from 𝑋 into 𝑌.If𝑋=𝑌,thenwewrite𝐿(𝑋) (resp. 𝑊(𝑋)) Theorem 6 (see [6]). Let 𝐾 be a compact metric space. If 𝑋 is instead of 𝐿(𝑋, 𝑋) (resp. 𝑊(𝑋,). 𝑋) an infinite-dimensional complemented subspace of C(𝐾),then 𝑋 𝑐 The symbols 𝑐0, 𝑙1,and𝑙∞ 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[𝑎, 𝑏], ‖𝐴 ⋅‖ ) is neither weakly sequen- norm. tially complete nor has the Radon-Riesz property. Let 𝐾 be a compact metric space. We denote by C(𝐾) the 𝑋 {𝑥 } vector space of all continuous functions of scalar-values on 𝐾 Definition 7. Let be a normed space, and let 𝑛 be a 𝑋 𝑥∈𝑋 together with the norm defined by ‖𝐹‖∞ = sup{|𝐹(𝑥)| : 𝑥∈ sequence in and . 𝐾} . (i) If {𝑥𝑛} weakly converges whenever {𝑥𝑛} is weakly B [𝑎, 𝑏] By 𝑐 we denote the following collection: Cauchy, then it is said that 𝑋 is weakly sequentially complete. {𝐹:[𝑎,] 𝑏 󳨀→ R |𝐹is continuous on [𝑎,] 𝑏 and 𝐹 (𝑎) =0} {𝑥 } 𝑥 {𝑥 } (2) (ii) If 𝑛 converges to whenever 𝑛 weakly converges to 𝑥 and ‖𝑥𝑛‖ → ‖𝑥‖,thenitissaidthat𝑋 has the which is a closed subspace of C[𝑎, 𝑏] and (B𝑐[𝑎, 𝑏], ‖∞ ⋅‖ ) is Radon-Riesz property or the Kadets-Klee property. therefore a Banach space. (iii) If {𝑥𝑛} converges to 𝑥 whenever {𝑥𝑛} weakly converges to 𝑥,thenitissaidthat𝑋 has the Schur property. Definition 1. Let 𝑋𝑎𝑛𝑑𝑌be normed spaces and let 𝑇:𝑋 → 𝑌 be a lineal operator. We have the following. The following result establishes a characterization of (i) 𝑇 is an isomorphism if it is one-to-one and continuous weakly Cauchy sequences and weakly convergent sequences −1 and its inverse mapping 𝑇 is continuous on the of the space C[𝑎, 𝑏]. range of 𝑇.Moreover,if‖𝑇(𝑥)‖ = ,forall‖𝑥‖ 𝑥∈𝑋, Theorem 8 {𝐹 } 𝐹 it is said that 𝑇 is an isometric isomorphism. (see [7]). Let 𝑛 and be a sequence and an C[𝑎, 𝑏] 𝑋 𝑌 𝑋≅𝑌 element, respectively, in the space .Thenwehavethe (ii) and are isomorphic, which is denoted by , following. if there exists an isomorphism from 𝑋 onto 𝑌. (1) The sequence {𝐹𝑛} is weakly convergent to 𝐹 if and only (iii) 𝑋 and 𝑌 are isometrically isomorphic if there exists an if isometric isomorphism from 𝑋 onto 𝑌. (i) lim𝑛→∞𝐹𝑛(𝑥) = 𝐹(𝑥),forall𝑥∈[𝑎,𝑏], The following result is key to our principal results. (ii) there exists 𝑀>0such that ‖𝐹𝑛‖∞ ≤𝑀,forall 𝑛∈N. Theorem 2 (see [4]). The space (B𝑐[𝑎, 𝑏], ‖∞ ⋅‖ ) is isometri- ̂ cally isomorphic to (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ). (2) The sequence {𝐹𝑛} isweaklyCauchyifandonlyif ̂ 𝐹 (𝑥) 𝑥∈[𝑎,𝑏] According to Theorem 2,wewillprovethat(HK[𝑎, 𝑏], (i) lim𝑛→∞ 𝑛 exists, for all , 𝑀>0 ‖𝐹 ‖ ≤𝑀 ‖⋅‖𝐴) has the Dunford-Pettis property. (ii) there exists such that 𝑛 ∞ ,forall 𝑛∈N. Definition 3. Let 𝑋 be a Banach space. It is said that 𝑋 has ̂ We will also prove that (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) is not isomet- the Dunford-Pettis property if for every sequence {𝑥𝑛} in ∗ ∗ ricallyisomorphictothedualofanynormedspaceand,as 𝑋 converging weakly to 0 and every sequence {𝑥𝑛 } in 𝑋 ∗ (HK̂[𝑎, 𝑏], ‖ ⋅‖ ) converging weakly to 0, the sequence {𝑥𝑛 (𝑥𝑛)} converges to aconsequence,wewillprovethat 𝐴 is not 0. reflexive, for which we will use again Theorem 2 and the concept of extremal point. If a Banach space 𝑋 has the Dunford-Pettis property, then not necessarily every closed subspace of 𝑋 inherits such Definition 9. Let 𝑋 be a vector space, 𝐾⊆𝑋,and𝑧∈𝐾.It property, except when the subspace is complemented in 𝑋 is said that 𝑧 is an extremal point of 𝐾 if for all 𝑥, 𝑦 ∈𝐾 such [5]. that 𝑧 = (1/2)(𝑥 +𝑦) it holds that 𝑧=𝑥=𝑦. Journal of Function Spaces and Applications 3

If 𝑋 is a normed space, then its closed unit ball will be However, since 𝐺 is not continuous, it follows that the denoted by 𝐵𝑋 and the collection of all extremal points of 𝐵𝑋 sequence {𝐹𝑛} does not converge weakly in (B𝑐[𝑎, 𝑏], ‖∞ ⋅‖ ). as ext(𝐵𝑋). The following theorem establishes that the extremal points are preserved under isometric isomorphisms. Lemma 15. The space (B𝑐[𝑎, 𝑏], ‖∞ ⋅‖ ) does not have the Radon-Riesz property. Theorem 10 (see [8]). Let 𝑋𝑎𝑛𝑑𝑌be Banach spaces, 𝐾⊆𝑋 and let 𝑇:𝑋be →𝑌 an isometric isomorphism. Then 𝑥 is an Proof. Without loss of generality, suppose that |𝑏−𝑎|.Let ≥1 extremal point of 𝐾 if and only if 𝑇(𝑥) is an extremal point of 𝐹𝑛 be the function defined by 𝑇(𝐾). 𝐹𝑛 (𝑥) Corollary 11 (see [9]). An infinite-dimensional normed space 𝑛 1 whoseclosedunitballhasonlyfinitelymanyextremepointsis {( ) (𝑥−𝑎) , 𝑥∈[𝑎,𝑏− ], { 𝑛 (𝑏−𝑎) −1 if 𝑛 not isometrically isomorphic to the dual of any normed space. { { 1 1 = {2𝑛 (𝑏−𝑥) −1, if 𝑥∈(𝑏− ,𝑏− ], { 𝑛 2𝑛 3. Principal Results { { 1 2𝑛 (𝑥−𝑏) +1, if 𝑥∈(𝑏− ,𝑏], Lemma 12. The space (B𝑐[𝑎, 𝑏], ‖∞ ⋅‖ ) has the Dunford- { 2𝑛 Pettis property. (6)

Proof. Let 𝐻:C[𝑎, 𝑏] → R be the functional defined by for all 𝑛≥2.Thus, 𝐻(𝐹) = 𝐹(𝑎).Itisclearthat𝐻 is bounded and therefore {𝐹 } 𝐹 ker(𝐻) is a hyperplane of the space C[𝑎, 𝑏],thatis, (i) the sequence 𝑛 converges pointwise to ,where 𝐹(𝑥) = (𝑥 − 𝑎)/(𝑏 −𝑎) for all 𝑥∈[𝑎,𝑏], C [𝑎,] 𝑏 = ker (𝐻) ⊕𝑊, (3) (ii) ‖𝐹𝑛‖∞ =1,forall𝑛≥2. where ker(𝐻) = {𝐹 ∈ C[𝑎, 𝑏] : 𝐹(𝑎) =0} B𝑐[𝑎, 𝑏] and 𝑊 is a one-dimension, subspace in C[𝑎, 𝑏].Then,as Therefore, according to Theorem 8 item (1),itfollowsthat C[𝑎, 𝑏] has the Dunford-Pettis property [10] and according the sequence {𝐹𝑛} converges weakly to 𝐹 in (B𝑐[𝑎, 𝑏], ‖∞ ⋅‖ ); to Theorem 5, we obtain the desired conclusion. in addition, as ‖𝐹‖∞ =1,itholdsthat‖𝐹𝑛‖∞ →‖𝐹‖∞. However, the sequence {𝐹𝑛} does not converge to 𝐹 in Lemma 13. The space (B𝑐[𝑎, 𝑏], ‖∞ ⋅‖ ) has a complemented (B𝑐[𝑎, 𝑏], ‖∞ ⋅‖ ). subspace isomorphic to 𝑐0. It is not difficult to prove that if a Banach space has Proof. Using the proof of Lemma 12,wecanseethatB𝑐[𝑎, 𝑏] the Dunford-Pettis property, or if it has a complemented is a complemented subspace in C[𝑎, 𝑏].Then,accordingto subspace isomorphic to 𝑐0,orifitisnotweaklysequentially Theorem 6, we obtain the desired conclusion. complete,orifithastheRadon-Rieszproperty,thenthese properties are preserved under isometric isomorphisms. Lemma 14. (B [𝑎, 𝑏], ‖ ⋅‖ ) The space 𝑐 ∞ is not weakly sequen- Therefore, according to Theorem 2 and Lemmas 12, 13, 14 and tially complete. 15, we obtain the following result. Proof. Without loss of generality, suppose that |𝑏−𝑎|.Let ≥1 ̂ Proposition 16. The space (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) 𝐹𝑛 be the function defined by 1 (1) has the Dunford-Pettis property, {0, if 𝑥∈[𝑎,𝑏− ], 𝐹 (𝑥) = 𝑛 (2) has a complemented subspace isomorphic to 𝑐0, 𝑛 { 1 (4) 𝑛 (𝑥−𝑏) +1, 𝑥∈(𝑏− ,𝑏], { if 𝑛 (3) is not weakly sequentially complete, (4) does not have the Radon-Riesz property. for all 𝑛∈N.Thus,

(i) lim𝑛→∞𝐹𝑛(𝑥) exists, for all 𝑥∈[𝑎,𝑏], Remark 17. According to Definition 7 and Proposition 16 ̂ (ii) ‖𝐹𝑛‖∞ =1,forall𝑛∈N. item (4),itfollowsthat(HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) does not have the Schur property. Therefore, according to Theorem 8 item (2),itfollowsthat {𝐹 } (B [𝑎, 𝑏], ‖ ⋅‖ ) the sequence 𝑛 is weakly Cauchy in 𝑐 ∞ . Lemma 18. The collection of all extremal points of the closed Now, suppose that there exists a function 𝐺∈ unit ball of the space (B𝑐[𝑎, 𝑏], ‖∞ ⋅‖ ) is empty. (B𝑐[𝑎, 𝑏], ‖∞ ⋅‖ ) such that the sequence {𝐹𝑛} converges 𝐺 (1) weakly to . Then according to Theorem 8 item ,itholds Proof. Let 𝐹∈𝐵B [𝑎,𝑏].Since𝐹 is continuous on [𝑎, 𝑏],it 𝐺(𝑥) = 𝐹 (𝑥) 𝑥∈[𝑎,𝑏] 𝑐 that lim𝑛→∞ 𝑛 ,forall ,thatis, holds that for 𝜀=1/2there exits 𝛿>0such that 0, if 𝑥∈[𝑎,) 𝑏 , 1 𝐺 (𝑥) ={ (5) |𝐹 (𝑥)| < ,∀𝑥∈[𝑎, 𝑎) +𝛿 . 1, if 𝑥=𝑏. 2 (7) 4 Journal of Function Spaces and Applications

Now, define the following functions: space. However, we can ask ourselves the following. Is there a ̂ normed space X such that (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) is isomorphic 𝐹 (𝑥) +𝑟(𝑥) , 𝑥∈[𝑎, 𝑎] +𝛿 , 𝐺 (𝑥) ={ if to the dual of 𝑋? To answer this question, we need of the 𝐹 (𝑥) , if 𝑥∈[𝑎+𝛿,𝑏] , following result. (8) 𝐹 (𝑥) −𝑟(𝑥) , 𝑥∈[𝑎, 𝑎] +𝛿 , Lemma 23 𝑋 𝑌 𝐻 (𝑥) ={ if (see [7]). Let be a normed space and let be a 𝐹 (𝑥) , 𝑥∈[𝑎+𝛿,𝑏] , Banach space with the Dunford-Pettis property that does not if ∗ have the Schur property. If 𝑋 contains a copy of 𝑌,then𝑋 where the function 𝑟 can be any continuous function defined contains a copy of 𝑙1. over the interval [𝑎, 𝑎 + 𝛿] such that 𝑟(𝑎) = 0 = 𝑟(𝑎 +𝛿) and ‖𝑟‖ <1/2 ̂ ∞ . Proposition 24. The space (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) is not isomor- Since the functions 𝐺 and 𝐻 are continuous, ‖𝐺‖∞ =1= phic to the dual of any normed space. ‖𝐻‖∞,and𝐹 = (1/2)(𝐺,itholdsthat +𝐻) 𝐹 cannot be an (𝐵 )=0 𝑋 extremal point; therefore, ext B𝑐[𝑎,𝑏] . Proof. Suppose that there exists a normed space such that (𝐵 ) ̂ ∗ It is a known fact that the collection ext C[𝑎,𝑏] is formed (HK [𝑎,] 𝑏 , ‖⋅‖𝐴)≅𝑋 . (9) only by the constant functions ±1. However, since in general there is not a relationship between the extremal points of the Since (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) is separable [1]andconsequently ̂ closed unit ball of a subspace with the extremal points of (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) also is separable, it follows from (9)that ∗ theclosedunitballofallspace,weneedLemma 18 for the 𝑋 is separable. following result. On the other hand, by Proposition 16 item (2) and ∗ according to the isomorphism from (9), it holds that 𝑋 has Proposition 19. (HK̂[𝑎, 𝑏], ‖ ⋅‖ ) The space 𝐴 is not isometri- a complemented subspace isomorphic to 𝑐0.Since𝑐0 has the callyisomorphictothedualofanynormedspace. Dunford-Pettis property [9] and does not have the Schur property [9], it holds that 𝑋 has a copy of 𝑙1,byLemma 23. ∗ Proof. By Lemma 18 and Theorems 2 and 10,itholdsthat As 𝑋 has a copy of 𝑙1,itholdsthat𝑙1 is isometrically ̂ ∗ ⊥ ⊥ ext(𝐵HK̂[𝑎,𝑏])=0. Then, as the space (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) is isomorphic to 𝑋 /𝑙1 ,where𝑙1 denotes the annihilator of 𝑙1. ∗ dimensionality infinite and according to Corollary 11,we Since 𝑙1 is isometrically isomorphic to 𝑙∞ it holds that, in obtain the desired conclusion. particular, ̂ 𝑋∗ Corollary 20. The space (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) is not reflexive. 𝑙 ≅ . ∞ ⊥ (10) 𝑙1 ̂ Proof. Suppose that the space (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) is reflex- ∗ ̂ Therefore, since 𝑋 is separable and according to the ive. Then (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) coincides under the canoni- isomorphism from (10), we obtain that 𝑙∞ is separable, which cal imbedding with its second dual. Therefore, the space ̂ is a contradiction. (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) is isometrically isomorphic to the dual of ̂ ∗ On this way, we can see that Proposition 19 and the space (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) , which is a contradiction by Proposition 19. Corollary 20 are consequences of the above result. We did not do it this way because one of the principal objectives of In general, it is important to know when a Banach space this paper is to show the importance of knowing explicitly a enjoys certain functional analysis properties. However, in closed subspace of C[𝑎, 𝑏] which is isometrically isomorphic ̂ certain contexts, also it is useful to know when a Banach space to (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) anditisaknownfactthatevery does not have certain properties; Propositions 22 and 24 are separable Banach space of infinite dimension is isometrically examples of both facts. isomorphic to a closed subspace of C[𝑎, 𝑏];however,this information is not sufficient to prove, in particular, the results Lemma 21 𝑋, 𝑌 (see [11]). Let be two Banach spaces. Assume that we have shown in this paper. that 𝑋 and 𝑌 contain a complemented copy of 𝑐0.Then𝑊(𝑋, 𝑌) is uncomplemented in 𝐿(𝑋,. 𝑌) References Proposition 22. 𝑊(HK̂[𝑎, 𝑏]) The space is uncomplemented [1] C. Swartz, Introduction to Gauge Integrals, World Scientific, ̂ in the space 𝐿(HK[𝑎, 𝑏]). Singapore, 2001. [2] J. L. Gamez,´ Integracionesdedenjoydefuncionesconvaloresen Proof. According to Proposition 16 item (2),itholdsthat espacios de banach [Ph.D. thesis], Universidad Complutense de ̂ (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) contains a complemented copy of 𝑐0. Madrid, 1997. Therefore, on the basis of Lemma 21, we obtain the desired [3] E. Talvila, “The distributional Denjoy integral,” Real Analysis conclusion. Exchange,vol.33,no.1,pp.51–82,2008. ̂ [4] B. Bongiorno and T. V. Panchapagesan, “On the Alexiewicz By Proposition 19,wecanseethat(HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) is topology of the Denjoy space,” Real Analysis Exchange,vol.21, not isometrically isomorphic to the dual of any normed no. 2, pp. 604–614, 1995-1996. Journal of Function Spaces and Applications 5

[5] J. Diestel, “A survey of results related to the Dunford-Pettis property,” AMS Contemporary Mathematics,vol.2,pp.15–60, 1980. [6] F. Albiac and N. J. Kalton, Topics in Banach Space Theory, Springer,NewYork,NY,USA,2006. [7] J. Diestel, Sequences and Series in Banach Spaces, Springer, New York, NY, USA, 1984. [8] A. Curnock, “An introduction to extreme points and applica- tions in isometric Banach space theory,” in Proceedings of the the Analysis Group, Goldsmiths College, University of London, May 1998, part of early doctoral work. [9]R.E.Megginson,An Introduction to Banach Space Theory, Springer,NewYork,NY,USA,1998. [10] R. G. Bartle, N. Dunford, and J. Schwartz, “Weak compactness and vector measures,” Canadian Journal of Mathematics,vol.7, pp.289–305,1955. [11] G. Emmanuele, “Remarks on the uncomplemented subspace 𝑊(𝐸,,” 𝐹) Journal of Functional Analysis,vol.99,no.1,pp.125– 130, 1991. The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com Volume 2013

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