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

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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([, ]). = ∫ . sup (1) ≤≤ On the basis of Corollary 7 we prove the following result: there exists a norm on HK([, ]) under which it is a The corresponding normed space is built using the Banach space; see Proposition 9.Then,basedonthisfactand quotient space determined by the relation ∼if and only if using the same techniques employed by Honing¨ and Merino, =, except in a set of Lebesgue measure zero or, equivalently, we prove that the space (HK([, ]), ‖⋅ ) does not have if they have the same indefinite integral. This normed space certain properties and we give an answer to two problems will be denoted by (HK([, ]), ‖⋅ ) and its elements as posedbyMerinoin[2, page 111]; namely, we will prove, in instead of []. particular, that the space HK([, ]) with the Alexiewicz It is a known fact that the space (HK([, ]), ‖⋅ ) is not topology is neither infra-( )norawebbedspace. complete nor of the second category [1]. However, the space (HK([, ]), ‖⋅ ) has “nice” properties, from the point of 2. Notation and Preliminaries view of functional analysis, since it is an ultrabornological space [2]. For the reader’s convenience, we restate some concepts about Honing¨ [3], using a version of the Closed Graph Theorem, topological vector spaces that will be used later in this paper proved that there is no natural Banach norm on the space which can be found particularly in [4–6]. HK([, ]); even, for functions with values in a Banach space.Also,heobservedthatthesameproofappliesto Definition 2. Let and be topological spaces and let prove that there is no natural Frechet´ topology on the space :be → a function. The graph of is defined by HK([, ]). {(, ()) .| ∈} 2 Abstract and Applied Analysis It is said that has closed graph if its graph is a closed Now, we establish the notations that we will use through- subset of ×with the product topology. out this paper. It is a known fact that if :is → a continuous (i) Let be a vector space. function between topological spaces, where is a Hausdorff space, then hasclosedgraph;however,theconverseisnot (a)Allvectorspacesthatappearthroughoutthis true in general. Precisely, the versions of the Closed Graph paper are considered over the field of the real or Theorem establish under what conditions the converse is complex numbers. fulfilled. Moreover, the importance of these theorems is that (b) The cardinality of will be denoted by card(). in certain contexts it is easier to prove that a function has (c) The cardinality of a Hamel base or algebraic base closed graph than to prove that it is continuous; Propositions is denoted by dim(). It is a known fact that 10, 12,and14 are examples of this fact. every Hamel base in a vector space has the same Definition 3. A topological vector space is locally convex if cardinality. has a local base of convex neighborhoods of 0. ℵ0 (ii) The symbols ℵ0, and will denote the cardinality Definition 4. Let be a locally convex space. Then is of the natural numbers, N,oftherealnumbers,R,and of the set { : N → R |is a function},respec- (i) barrelled if every linear mapping with closed graph tively. from into a Banach space is continuous; (iii) We will denote the set of continuous functions with (ii) infra-() if every linear mapping with closed graph real values whose domain is the compact interval [, ] (C[, ], R) from a Banach space into is continuous; by . 1[, ] (iii) infra-() if every linear mapping with closed graph (iv) will denote the space of equivalence classes of theLebesgueintegrablefunctionswithitsusualnorm. from a barrelled space into is continuous. 1 It is a known fact that [, ] ⊂ HK([, ]) and ‖‖ ≤‖‖ ∈1[, ] The above concepts have other equivalent characteri- 1,forall . zations. However, the characterizations that appear in the preceding definition are the best for the goals of this paper. The following result establishes necessary and sufficient conditions to determine when a vector space has a norm Definition 5. Let be a locally convex space. It is said that under which it is a Banach space. This result is the foundation is a of the goals of this paper. Corollary 7 (i) webbed space or De Wilde space if it admits a C-web; (see [7]). Let be a vector space. Then, is a Banach space under some norm if and only if dim() < ℵ0 or ℵ (ii) convex-Suslin space if it admits a compacting web. dim () 0 = dim(). Note. The definitions of C-web and compacting web are a bit too involved to be here. To see these definitions we refer the 3. Principal Results reader to [5]. Lemma 8. The cardinality of the space HK([, ]) is equal to Although webbed spaces are a subclass of infra-() the cardinality of real numbers. spaces, the webbed spaces have more stability properties than infra-( ) spaces in the sense that every sequentially closed Proof. Since the application :HK([, ]) →(C[, ], R) subspace and every quotient of a webbed space are webbed defined by (),where = is the indefinite integral spaces. Also the inductive and projective limits and the direct associated to ,isinjective,itholdsthatcard(HK([, ])) ≤ sums and direct products of a countable number of webbed ((C[, ], R)) = card . spaces are webbed spaces. Furthermore, if is a webbed On the other hand, as the constant functions are space, then it also will be with a weaker topology. In the other Henstock-Kurzweil integrables, it follows that direction, if isawebbedspace,italsowillbewiththe associated ultrabornological topology. =card ({ : [,] → R |is a constant function}) Some stability properties that enjoy the convex-Suslin (2) spaces are the following: the countable product of convex- ≤ card (HK ([,] )) . Suslin spaces is convex-Suslin and every closed subspace of a convex-Suslin space is a convex-Suslin space. Moreover, these Therefore, card(HK([, ])). = spaces play a version of Closed Graph Theorem as set forth below. Proposition 9. There exists a norm on HK([, ]) under which it is a Banach space. Theorem 6 (see [6]). Let be a locally convex Baire space and 1 let be a convex-Suslin space. If is a linear mapping from Proof. It is a known fact that [, ] is a Banach space of 1 into with closed graph, then is continuous. infinite dimension, which implies that ≤dim( [, ]). Abstract and Applied Analysis 3 1 1 Then, as [, ] ⊂ HK([, ]) it holds that dim( [, ]) ≤ On the basis of Corollary 13 we see that (HK([, ]), ) dim(HK([, ])) ≤ card(HK([, ])). Therefore, by is not a webbed space. Moreover, the same result implies that ℵ Lemma 8, Corollary 7 and the known fact that 0 =,we (HK([, ]), ) is not an infra-()space,afactthatMerino obtain the desired conclusion. has already proved in [2]. On the other hand, employing the same technique that we Hereafter, the Alexiewicz topology and the topology have been using entails the following result. induced by the norm of Proposition 9 will be denoted as Proposition 14. (HK([, ]), ) and ‖⋅‖,respectively. The space is not a convex- Suslin space. Proposition 10. The topology ‖⋅‖ on HK([, ]) is smaller (HK([, ]), ) than . Proof. Suppose that is a convex- Suslin space. On the basis of Proposition 9 we see that Proof. Let ‖⋅‖be the norm ensured by Proposition 9.By (HK([, ]),‖⋅‖ ) is a locally convex Baire space; therefore, [3], (HK([, ]), ) is barrelled and, therefore, the identity according to Theorem 6 the identity function function I : (HK ([,] ) ,‖⋅‖) → (HK ([,] ) ,) , (6) I :(HK ([,] ) ,)→(HK ([,] ) ,‖⋅‖), (3) which has a closed graph, is continuous, contradicting which has a closed graph, is continuous; hence ‖⋅‖ ⊆.In Proposition 10. addition, since that is not complete [3], it holds that ‖⋅‖ ⊂ SinceK-Suslinspacesareasubclassfromconvex-Suslin . spaces, [6], we can see, on the basis of Proposition 14,that (HK([, ]), ) is not a K-Suslin space. Although (HK([, ]),‖⋅‖ ) is a Banach space, according Although apparently the topology ‖⋅‖ is not directly to [3]thetopology‖⋅‖ is not natural in the sense of the following definition. related with the Henstock-Kurzweil integral, this topology allowed us to demonstrate some properties that the Alex- Definition 11. Avectortopology on HK([, ]) is natural iewicz topology does not have, which is a topology related to the Henstock-Kurzweil integral since it is a natural topology. if every sequence in HK([, ]) such that →0 implies that ∫ →0,forevery∈[,].
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