A Version of the Hahn-Banach Theorem for R-Vector Spaces
Year: 2024
Type: article
Abstract: 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.
Institutions: Vali Asr University of Rafsanjan, Payame Noor University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6963895646
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