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 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.