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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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Maragheh
2024-10-01
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| Series: | Sahand Communications in Mathematical Analysis |
| Subjects: | |
| Online Access: | https://scma.maragheh.ac.ir/article_713438_dcfae09d3cc9154ad27593c9326e34fa.pdf |
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| Summary: | 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. |
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| ISSN: | 2322-5807 2423-3900 |