A Version of the Hahn-Banach Theorem for R-Vector Spaces
Recently, $R$-metric spaces have been introduced to generalized metric spaces. This extension is based on the construction of a new universe with interesting properties. In this paper, some topological properties of $R$-metric spaces are studied and compared to the classical metric spaces via severa...
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| Format: | Article |
| Language: | English |
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University of Maragheh
2024-10-01
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| Series: | Sahand Communications in Mathematical Analysis |
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| Online Access: | https://scma.maragheh.ac.ir/article_713438_dcfae09d3cc9154ad27593c9326e34fa.pdf |
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| author | Azadeh Alijani Zohreh Heydarpour Maryam Naderi Parizi |
| author_facet | Azadeh Alijani Zohreh Heydarpour Maryam Naderi Parizi |
| author_sort | Azadeh Alijani |
| collection | DOAJ |
| description | Recently, $R$-metric spaces have been introduced to generalized metric spaces. This extension is based on the construction of a new universe with interesting properties. In this paper, some topological properties of $R$-metric spaces are studied and compared to the classical metric spaces via several examples. Also, some properties of a metric space with different relations are considered. Then, the elementary tools needed for the study of two important theorems of functional analysis are presented. For example, $R$-sequentially bounded sets, $R$-bounded sets, $R$-sequentially bounded functions and $R$-bounded functions are introduced in $R$-metric spaces. Moreover, a condition is given under which an $R$-continuous function is $R$-sequentially bounded. Finally, variants of the Heine--Borel theorem and the Hahn--Banach theorem are proved for $R$-metric spaces and $R$-vector spaces, respectively. |
| format | Article |
| id | doaj-art-ae358c84fba34bcfa8afe97f0308c8af |
| institution | Kabale University |
| issn | 2322-5807 2423-3900 |
| language | English |
| publishDate | 2024-10-01 |
| publisher | University of Maragheh |
| record_format | Article |
| series | Sahand Communications in Mathematical Analysis |
| spelling | doaj-art-ae358c84fba34bcfa8afe97f0308c8af2025-02-11T05:27:48ZengUniversity of MaraghehSahand Communications in Mathematical Analysis2322-58072423-39002024-10-01214274210.22130/scma.2024.2009923.1442713438A Version of the Hahn-Banach Theorem for R-Vector SpacesAzadeh Alijani0Zohreh Heydarpour1Maryam Naderi Parizi2Vali-e-Asr University of Rafsanjan, Department of Mathematics, Rafsanjan, Iran. ORDID: 0000-0001-7851-8684.Department of mathematics, Payame Noor University (PNU), P. O. Box 19395-4697, Tehran, Iran.Department of mathematics, Payame Noor University (PNU), P. O. Box 19395-4697, Tehran, Iran.Recently, $R$-metric spaces have been introduced to generalized metric spaces. This extension is based on the construction of a new universe with interesting properties. In this paper, some topological properties of $R$-metric spaces are studied and compared to the classical metric spaces via several examples. Also, some properties of a metric space with different relations are considered. Then, the elementary tools needed for the study of two important theorems of functional analysis are presented. For example, $R$-sequentially bounded sets, $R$-bounded sets, $R$-sequentially bounded functions and $R$-bounded functions are introduced in $R$-metric spaces. Moreover, a condition is given under which an $R$-continuous function is $R$-sequentially bounded. Finally, variants of the Heine--Borel theorem and the Hahn--Banach theorem are proved for $R$-metric spaces and $R$-vector spaces, respectively.https://scma.maragheh.ac.ir/article_713438_dcfae09d3cc9154ad27593c9326e34fa.pdfhahn banach theoremr-connected setr-compact setr-continuous functionr-metric space |
| spellingShingle | Azadeh Alijani Zohreh Heydarpour Maryam Naderi Parizi A Version of the Hahn-Banach Theorem for R-Vector Spaces Sahand Communications in Mathematical Analysis hahn banach theorem r-connected set r-compact set r-continuous function r-metric space |
| title | A Version of the Hahn-Banach Theorem for R-Vector Spaces |
| title_full | A Version of the Hahn-Banach Theorem for R-Vector Spaces |
| title_fullStr | A Version of the Hahn-Banach Theorem for R-Vector Spaces |
| title_full_unstemmed | A Version of the Hahn-Banach Theorem for R-Vector Spaces |
| title_short | A Version of the Hahn-Banach Theorem for R-Vector Spaces |
| title_sort | version of the hahn banach theorem for r vector spaces |
| topic | hahn banach theorem r-connected set r-compact set r-continuous function r-metric space |
| url | https://scma.maragheh.ac.ir/article_713438_dcfae09d3cc9154ad27593c9326e34fa.pdf |
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