The Closed Graph Theorem and the Space of Henstock-Kurzweil Integrable Functions with the Alexiewicz Norm

and Applied Analysis 3 Then, as L[a, b] ⊂ HK([a, b]) it holds that dim(L[a, b]) ≤ dim(HK([a, b])) ≤ card(HK([a, b])). Therefore, by Lemma 8, Corollary 7 and the known fact that c0 = c, we obtain the desired conclusion. Hereafter, the Alexiewicz topology and the topology induced by the norm of Proposition 9 will be denoted as τ A and τ ‖⋅‖ , respectively. Proposition 10. The topology τ ‖⋅‖ on HK([a, b]) is smaller than τ A . Proof. Let ‖ ⋅ ‖ be the norm ensured by Proposition 9. By [3], (HK([a, b]), τ A ) is barrelled and, therefore, the identity function I : (HK ([a, b]) , τA) 󳨀→ (HK ([a, b]) , τ‖⋅‖) , (3) which has a closed graph, is continuous; hence τ ‖⋅‖ ⊆ τ A . In addition, since that τ A is not complete [3], it holds that τ ‖⋅‖ ⊂ τ A . Although (HK([a, b]), τ ‖⋅‖ ) is a Banach space, according to [3] the topology τ ‖⋅‖ is not natural in the sense of the following definition. Definition 11. A vector topology τ on HK([a, b]) is natural if every sequence f n inHK([a, b]) such that f n τ 󳨀 → 0 implies that ∫x a f n → 0, for every x ∈ [a, b]. Now, we shall prove that the space (HK([a, b]), τ A ) is not infra-(u) which follows from the following general result. Proposition 12. Let (X, τ) be a barrelled space that is not complete. If there exists a norm onX under which it is a Banach space, thenX is not an infra-(u) space. Proof. Let ‖ ⋅ ‖ be a norm inX such that (X, ‖ ⋅ ‖) is a Banach space and suppose that (X, τ) is an infra-(u) space. Since (X, τ) is a barrelled space, it holds that the identity function I : (X, τ) 󳨀→ (X, ‖⋅‖) , (4) which has a closed graph, is continuous; hence τ ‖⋅‖ ⊆ τ. Then, as (X, τ) is infra-(u) and (X, ‖ ⋅ ‖) is a Banach space, it follows that I −1 : (X, ‖⋅‖) 󳨀→ (X, τ) (5) is continuous; hence τ ⊆ τ ‖⋅‖ . Therefore τ = τ ‖⋅‖ is a contradiction because τ is not complete. Corollary 13. The space (HK([a, b]), τ A ) is not infra-(u). Proof. On the basis of Proposition 9 there exists a norm on HK([a, b]) under which it is a Banach space. Furthermore, according to [3] the space (HK([a, b]), τ A ) is barrelled but not complete. Therefore, it follows from Proposition 12 that (HK([a, b]), τ A ) is not infra-(u). On the basis of Corollary 13 we see that (HK([a, b]), τ A ) is not a webbed space. Moreover, the same result implies that (HK([a, b]), τ A ) is not an infra-(s) space, a fact that Merino has already proved in [2]. On the other hand, employing the same technique that we have been using entails the following result. Proposition 14. The space (HK([a, b]), τ A ) is not a convexSuslin space. Proof. Suppose that (HK([a, b]), τ A ) is a convexSuslin space. On the basis of Proposition 9 we see that (HK([a, b]), τ ‖⋅‖ ) is a locally convex Baire space; therefore, according toTheorem 6 the identity function I : (HK ([a, b]) , τ‖⋅‖) 󳨀→ (HK ([a, b]) , τA) , (6) which has a closed graph, is continuous, contradicting Proposition 10. Since K-Suslin spaces are a subclass from convex-Suslin spaces, [6], we can see, on the basis of Proposition 14, that (HK([a, b]), τ A ) is not a K-Suslin space. Although apparently the topology τ ‖⋅‖ is not directly related with the Henstock-Kurzweil integral, this topology allowed us to demonstrate some properties that the Alexiewicz topology does not have, which is a topology related to the Henstock-Kurzweil integral since it is a natural topology. Comments (i) There are other types of spaces which enjoy certain versions of Closed Graph Theorem, for example, the spaces: Suslin, quasi-Suslin, [8], and A-complete, [9]; Ptak and infra-Pták [10]; strongly first-countable convergence vector space [11]; and so forth. Then, applying the same technique that we have used in this paper it can be shown that the space (HK([a, b]), τ A ) does not belong to any class of the spaces above. (ii) As another application of the technique which has been used in this paper and reiterating the known fact that L[a, b] ⊂ HK([a, b]) and ‖f‖ A ≤ ‖f‖ 1 , for all f ∈ L[a, b], it can be shown that (L 1 [a, b], ‖ ⋅ ‖ A ) is not a barrelled space and therefore not an ultrabornological space. Acknowledgment This paper is supported by the Committee of Differential Equations and Mathematical Modeling from FCFM-BUAP through Dr. José Jacobo Oliveros Oliveros.