Hahn Wijsman Sequence Space

Resumo

In this paper, we introduce and examine the Hahn-Wijsman sequence space $ h(W^u )$ , a generalized sequence space defined for a metric space $(X,d)$ and a positive sequence $\{{u_k }\}$, where a sequence $\{{x_k }\}\in h(W^u )\; if \;\sum_{k=1}|x_k | u_k< \infty$. We investigate its structural characteristics by establishing that it forms a vector space and construct an appropriate norm under which the space $h(W^u )$ becomes a normed linear space. We further demonstrate that it is complete and hence a Banach space. The dual space is characterized by bounded linear functionals, and we explore isomorphic relationships between the Hahn-Wijsman space and classical sequence spaces such as$ c_0 (p), c(p)\;and \;\ell(p)$. These results not only provide a deeper understanding of the topological and geometric aspects of the Hahn Wijsman space but also position it as a potential tool for further analysis in functional spaces and contribute to the ongoing development of sequence space theory.