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 $\lambda={\mathcal{H}{\vartheta},\mathcal{BV}, \mathcal{BV}{\vartheta 0}, \mathcal{CS}{\vartheta},\mathcal{CS}{\vartheta 0},\mathcal{BS}}$ 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.