The generalised double Hahn sequence space $H_\vartheta^{d}$

Abstract

Suppose that $\vartheta\in\{bp,r\}$. In this study, we introduce the generalised double Hahn sequence space $ H_\vartheta^{d} $ as an extension of generalised Hahn sequence space $h_d$ defined by Goes [J. Math. Anal. Appl. \textbf{39} (1972), 477--494]. We give some topological properties of this space. Then, we characterize the classes $ (H_\vartheta^{d} : W) $ of four dimensional infinite matrices, where $ W \in \{\mathcal{C_\vartheta, \mathcal{M}_u, \mathcal{L}_u, H_\vartheta^{d}\} $. Finally, we determine the $ \alpha $-dual of the space $ H_\vartheta^{d} $ and $ \beta(bp) $- and $ \gamma$-duals of the space H_r^{d} $.