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

The corresponding normed space is built using the quotient space determined by the relationf ∼ g if and only if f = g, except in a set of Lebesguemeasure zero or, equivalently, if they have the same indefinite integral. This normed space will be denoted by (HK[a, b], ‖ ⋅ ‖ A ). It is known that (HK[a, b], ‖ ⋅ ‖ A ) is neither complete nor of the second category [1]. However, (HK[a, b], ‖ ⋅ ‖ A ) is a separable space [1] and, consequently, its completion also has the same property. In addition, (HK[a, b], ‖ ⋅ ‖ A ) has “nice” properties, from the point of view of functional analysis, since it is an ultrabornological space [2]. As (HK[a, b], ‖ ⋅ ‖ A ) is not complete, it is natural to study its completion, which will be denoted by ( ̂ HK[a, b], ‖ ⋅ ‖ A ). Talvila in [3] makes an analysis to determine some properties of the Henstock-Kurzweil integral on ( ̂ HK[a, b],


Introduction
Let [, ] be a compact interval in R. In the vector space of Henstock-Kurzweil integrable functions on [, ] with values in R, the Alexiewicz seminorm is defined as The corresponding normed space is built using the quotient space determined by the relation  ∼  if and only if  = , except in a set of Lebesgue measure zero or, equivalently, if they have the same indefinite integral.This normed space will be denoted by (HK[, ], ‖ ⋅ ‖  ).
Talvila in [3] makes an analysis to determine some properties of the Henstock-Kurzweil integral on ( ĤK[, ], ‖ ⋅ ‖  ), such as integration by parts, Hölder inequality, change of variables, convergence theorems, the Banach lattice structure, the Hake theorem, the Taylor theorem, and second mean value theorem.Talvila makes this analysis by means of the space of the distributions that are derivatives of the continuous functions on [, ], which is an isometrically isomorphic space to ( ĤK[, ], ‖ ⋅ ‖  ).Making use of this same isometrically isomorphic space, Bongiorno and Panchapagesan in [4] establish characterizations for the relatively weakly compact subsets of (HK[, ], ‖ ⋅ ‖  ) and ( ĤK[, ], ‖ ⋅ ‖  ).
In this paper, we make an analysis on ( ĤK[, ], ‖ ⋅ ‖  ) by means of another isometrically isomorphic space to prove that ( ĤK[, ], ‖ ⋅ ‖  )has the Dunford-Pettis property, it has a complemented subspace isomorphic to  0 , it does not have the Radon-Riesz property, it is not weakly sequentially complete, and it is not isometrically isomorphic to the dual of any normed space; hence, we will also prove that ( ĤK[, ], ‖ ⋅ ‖  )is neither reflexive nor has the Schur property.Then, as an application of the above results, we prove that ( ĤK[, ], ‖ ⋅ ‖  ) is not isomorphic to the dual of any normed space and that the space of all bounded, linear, weakly compact operators from ( ĤK[, ], ‖ ⋅ ‖  ) into itself is not a complemented subspace in the space of all bounded, linear operators from ( ĤK[, ], ‖ ⋅ ‖  ) into itself.
All the vector spaces are considered over the field of the real numbers or complex numbers.
Let  be a normed space.By  * , we denote the dual space of .A topological property that holds with respect to the weak topology of  is said to be a weak property or to hold weakly.On the other hand, if a topological property holds without specifying the topology, the norm topology is implied.
The symbols  0 ,  1 , and  ∞ represent, as usual, the vector spaces of all sequences of scalars convergent to 0, all sequences of scalars absolutely convergent, and all bounded sequences of scalars, respectively, neither one with nor usual norm.
Let  be a compact metric space.We denote by C() the vector space of all continuous functions of scalar-values on  together with the norm defined by ‖‖ ∞ = sup{|()| :  ∈ }.
By B  [, ] we denote the following collection: which is a closed subspace of C[, ] and (B  [, ], ‖ ⋅ ‖ ∞ ) is therefore a Banach space.
Definition 1.Let    be normed spaces and let  :  →  be a lineal operator.We have the following.
(i)  is an isomorphism if it is one-to-one and continuous and its inverse mapping  −1 is continuous on the range of .Moreover, if ‖()‖ = ‖‖, for all  ∈ , it is said that  is an isometric isomorphism.
(ii)  and  are isomorphic, which is denoted by  ≅ , if there exists an isomorphism from  onto .
(iii)  and  are isometrically isomorphic if there exists an isometric isomorphism from  onto .
The following result is key to our principal results.
If a Banach space  has the Dunford-Pettis property, then not necessarily every closed subspace of  inherits such property, except when the subspace is complemented in  [5].Definition 4. Let  be a subspace of a normed space .It is said that  is complemented in  if it is closed in  and there exists a closed subspace  in  such that  =  ⊕ .
Theorem 5 (see [5]).Let  be a Banach space with the Dunford-Pettis property.If  is a complemented subspace in , then  has the Dunford-Pettis property.
To prove that ( ĤK[, ], ‖ ⋅ ‖  ) has a complemented subspace isomorphic to  0 the following result is essential.Theorem 6 (see [6]).Let  be a compact metric space.If  is an infinite-dimensional complemented subspace of C(), then  contains a complemented subspace isomorphic to  0 .
On the other hand, making use again of Theorem 2 we will prove that ( ĤK[, ], ‖ ⋅ ‖  ) is neither weakly sequentially complete nor has the Radon-Riesz property.The following result establishes a characterization of weakly Cauchy sequences and weakly convergent sequences of the space C[, ].
Theorem 8 (see [7]).Let {  } and  be a sequence and an element, respectively, in the space C[, ].Then we have the following.
(1) The sequence {  } is weakly convergent to  if and only if (ii) there exists  > 0 such that ‖  ‖ ∞ ≤ , for all  ∈ N.
(2) The sequence {  } is weakly Cauchy if and only if (ii) there exists  > 0 such that ‖  ‖ ∞ ≤ , for all  ∈ N.
We will also prove that ( ĤK[, ], ‖ ⋅ ‖  ) is not isometrically isomorphic to the dual of any normed space and, as a consequence, we will prove that ( ĤK[, ], ‖ ⋅ ‖  ) is not reflexive, for which we will use again Theorem 2 and the concept of extremal point.Definition 9. Let  be a vector space,  ⊆ , and  ∈ .It is said that  is an extremal point of  if for all ,  ∈  such that  = (1/2)( + ) it holds that  =  = .
If  is a normed space, then its closed unit ball will be denoted by   and the collection of all extremal points of   as ext(  ).
The following theorem establishes that the extremal points are preserved under isometric isomorphisms.
Theorem 10 (see [8]).Let    be Banach spaces,  ⊆  and let  :  →  be an isometric isomorphism.Then  is an extremal point of  if and only if () is an extremal point of ().
Corollary 11 (see [9]).An infinite-dimensional normed space whose closed unit ball has only finitely many extreme points is not isometrically isomorphic to the dual of any normed space.
Proof.Without loss of generality, suppose that | − | ≥ 1.Let   be the function defined by for all  ≥ 2. Thus, (i) the sequence {  } converges pointwise to , where () = ( − )/( − ) for all  ∈ [, ], Therefore, according to Theorem 8 item (1), it follows that the sequence However, the sequence {  } does not converge to  in It is not difficult to prove that if a Banach space has the Dunford-Pettis property, or if it has a complemented subspace isomorphic to  0 , or if it is not weakly sequentially complete, or if it has the Radon-Riesz property, then these properties are preserved under isometric isomorphisms.Therefore, according to Theorem 2 and Lemmas 12, 13, 14 and 15, we obtain the following result.
It is a known fact that the collection ext( C[,] ) is formed only by the constant functions ±1.However, since in general there is not a relationship between the extremal points of the closed unit ball of a subspace with the extremal points of the closed unit ball of all space, we need Lemma 18 for the following result.In general, it is important to know when a Banach space enjoys certain functional analysis properties.However, in certain contexts, also it is useful to know when a Banach space does not have certain properties; Propositions 22 and 24 are examples of both facts.
By Proposition 19, we can see that ( ĤK[, ], ‖ ⋅ ‖  ) is not isometrically isomorphic to the dual of any normed space.However, we can ask ourselves the following.Is there a normed space X such that ( ĤK[, ], ‖ ⋅ ‖  ) is isomorphic to the dual of ?To answer this question, we need of the following result.
Lemma 23 (see [7]).Let  be a normed space and let  be a Banach space with the Dunford-Pettis property that does not have the Schur property.If  * contains a copy of , then  contains a copy of  1 .
On the other hand, by Proposition 16 item (2) and according to the isomorphism from (9), it holds that  * has a complemented subspace isomorphic to  0 .Since  0 has the Dunford-Pettis property [9] and does not have the Schur property [9], it holds that  has a copy of  1 , by Lemma 23.
As  has a copy of  1 , it holds that  * 1 is isometrically isomorphic to  * / ⊥ 1 , where  ⊥ 1 denotes the annihilator of  1 .Since  * 1 is isometrically isomorphic to  ∞ it holds that, in particular, Therefore, since  * is separable and according to the isomorphism from (10), we obtain that  ∞ is separable, which is a contradiction.
On this way, we can see that Proposition 19 and Corollary 20 are consequences of the above result.We did not do it this way because one of the principal objectives of this paper is to show the importance of knowing explicitly a closed subspace of C[, ] which is isometrically isomorphic to ( ĤK[, ], ‖ ⋅ ‖  ) and it is a known fact that every separable Banach space of infinite dimension is isometrically isomorphic to a closed subspace of C[, ]; however, this information is not sufficient to prove, in particular, the results that we have shown in this paper.
Definition 7.Let  be a normed space, and let {  } be a sequence in  and  ∈ .(i)If {  } weakly converges whenever {  } is weakly Cauchy, then it is said that  is weakly sequentially complete.
(ii)If {  } converges to  whenever {  } weakly converges to  and ‖  ‖ → ‖‖, then it is said that  has the Radon-Riesz property or the Kadets-Klee property.(iii) If {  } converges to  whenever {  } weakly converges to , then it is said that  has the Schur property.