On the Double Sequence Space ${\mathcal{H}}_{\vartheta }$ as an Extension of Hahn Space $h$

Abstract

Double sequence spaces have become a significant area of research within functional analysis due to their applications in various branches of mathematics and mathematical physics. In this study, we investigate Hahn double sequence space denoted as ${\mathcal{H}}_{\vartheta }$, where $\vartheta \in \{p,bp,r\}$, as an extension of the Hahn sequence space $h$. Our investigation begins with an analysis of several topological properties of ${\mathcal{H}}_{\vartheta }$, apart from a comprehensive analysis of the relationship between Hahn double sequences and some other classical double sequence spaces. The $\alpha -$dual, algebraic dual and $\beta (bp)-$dual, and $\gamma -$dual of the space ${\mathcal{H}}_{\vartheta }$ are detrmined. Furthermore, we define the determining set of ${\mathcal{H}}_{\vartheta }$ and we state the conditions concerning the characterization of four-dimensional (4D) matrix classes $({\mathcal{H}}_{\vartheta },\lambda )$, where $\lambda =\{{\mathcal{H}}_{\vartheta },\mathcal{B}\mathcal{V},{\mathcal{B}\mathcal{V}}_{\vartheta 0},{\mathcal{CS}}_{\vartheta },{\mathcal{CS}}_{\vartheta 0},\mathcal{BS}\}$ and $(\mu ,{\mathcal{H}}_{\vartheta })$, where $\mu =\{{\mathcal{L}}_u,{\mathcal{C}}_{\vartheta 0},{\mathcal{C}}_{\vartheta },{\mathcal{M}}_u\}$. In conclusion, this research contributes non-standard investigation and various significant results into the space ${\mathcal{H}}_{\vartheta }$. The conducted results are deepen the understanding of the space ${\mathcal{H}}_{\vartheta }$ and open up new avenues for further research and applications in sequence space theory.