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

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

We prove that the cardinality of the space ℋ𝒦([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 in...

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Main Authors: Luis Ángel Gutiérrez Méndez, Juan Alberto Escamilla Reyna, Maria Guadalupe Raggi Cárdenas, Juan Francisco Estrada García
Format: Article
Language:English
Published: Hindawi Limited 2013-01-01
Series:Abstract and Applied Analysis
Online Access:http://dx.doi.org/10.1155/2013/476287
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spelling doaj-48f69af1c42646d1884da2b36d96e8b82020-11-24T22:54:24ZengHindawi LimitedAbstract and Applied Analysis1085-33751687-04092013-01-01201310.1155/2013/476287476287The Closed Graph Theorem and the Space of Henstock-Kurzweil Integrable Functions with the Alexiewicz NormLuis Ángel Gutiérrez Méndez0Juan Alberto Escamilla Reyna1Maria Guadalupe Raggi Cárdenas2Juan Francisco Estrada García3Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Avenida San Claudio y 18 Sur, Colonia San Manuel, 72570 Puebla, PUE, MexicoFacultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Avenida San Claudio y 18 Sur, Colonia San Manuel, 72570 Puebla, PUE, MexicoFacultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Avenida San Claudio y 18 Sur, Colonia San Manuel, 72570 Puebla, PUE, MexicoFacultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Avenida San Claudio y 18 Sur, Colonia San Manuel, 72570 Puebla, PUE, MexicoWe prove that the cardinality of the space ℋ𝒦([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.http://dx.doi.org/10.1155/2013/476287
collection DOAJ
language English
format Article
sources DOAJ
author Luis Ángel Gutiérrez Méndez
Juan Alberto Escamilla Reyna
Maria Guadalupe Raggi Cárdenas
Juan Francisco Estrada García
spellingShingle Luis Ángel Gutiérrez Méndez
Juan Alberto Escamilla Reyna
Maria Guadalupe Raggi Cárdenas
Juan Francisco Estrada García
The Closed Graph Theorem and the Space of Henstock-Kurzweil Integrable Functions with the Alexiewicz Norm
Abstract and Applied Analysis
author_facet Luis Ángel Gutiérrez Méndez
Juan Alberto Escamilla Reyna
Maria Guadalupe Raggi Cárdenas
Juan Francisco Estrada García
author_sort Luis Ángel Gutiérrez Méndez
title The Closed Graph Theorem and the Space of Henstock-Kurzweil Integrable Functions with the Alexiewicz Norm
title_short The Closed Graph Theorem and the Space of Henstock-Kurzweil Integrable Functions with the Alexiewicz Norm
title_full The Closed Graph Theorem and the Space of Henstock-Kurzweil Integrable Functions with the Alexiewicz Norm
title_fullStr The Closed Graph Theorem and the Space of Henstock-Kurzweil Integrable Functions with the Alexiewicz Norm
title_full_unstemmed The Closed Graph Theorem and the Space of Henstock-Kurzweil Integrable Functions with the Alexiewicz Norm
title_sort closed graph theorem and the space of henstock-kurzweil integrable functions with the alexiewicz norm
publisher Hindawi Limited
series Abstract and Applied Analysis
issn 1085-3375
1687-0409
publishDate 2013-01-01
description We prove that the cardinality of the space ℋ𝒦([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.
url http://dx.doi.org/10.1155/2013/476287
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