  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.
