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...
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2013-01-01
|
| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2013/476287 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832545570727657472 |
|---|---|
| author | Luis Ángel Gutiérrez Méndez Juan Alberto Escamilla Reyna Maria Guadalupe Raggi Cárdenas Juan Francisco Estrada García |
| author_facet | Luis Ángel Gutiérrez Méndez Juan Alberto Escamilla Reyna Maria Guadalupe Raggi Cárdenas Juan Francisco Estrada García |
| author_sort | Luis Ángel Gutiérrez Méndez |
| collection | DOAJ |
| 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. |
| format | Article |
| id | doaj-art-48f69af1c42646d1884da2b36d96e8b8 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-48f69af1c42646d1884da2b36d96e8b82025-02-03T07:25:22ZengWileyAbstract 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 |
| 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 |
| title | 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_short | 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 |
| url | http://dx.doi.org/10.1155/2013/476287 |
| work_keys_str_mv | AT luisangelgutierrezmendez theclosedgraphtheoremandthespaceofhenstockkurzweilintegrablefunctionswiththealexiewicznorm AT juanalbertoescamillareyna theclosedgraphtheoremandthespaceofhenstockkurzweilintegrablefunctionswiththealexiewicznorm AT mariaguadaluperaggicardenas theclosedgraphtheoremandthespaceofhenstockkurzweilintegrablefunctionswiththealexiewicznorm AT juanfranciscoestradagarcia theclosedgraphtheoremandthespaceofhenstockkurzweilintegrablefunctionswiththealexiewicznorm AT luisangelgutierrezmendez closedgraphtheoremandthespaceofhenstockkurzweilintegrablefunctionswiththealexiewicznorm AT juanalbertoescamillareyna closedgraphtheoremandthespaceofhenstockkurzweilintegrablefunctionswiththealexiewicznorm AT mariaguadaluperaggicardenas closedgraphtheoremandthespaceofhenstockkurzweilintegrablefunctionswiththealexiewicznorm AT juanfranciscoestradagarcia closedgraphtheoremandthespaceofhenstockkurzweilintegrablefunctionswiththealexiewicznorm |