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.

کلیدواژه ها:

Hahn Banach theorem ، R-connected set ، R-compact set ، R-continuous function ، R-metric space