In this study, we define the spaces \documentclass{aastex}
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$$\tilde M_u ,\,\tilde C_p ,\,\tilde C_{0p} ,\,\tilde C_{bp} ,\,\tilde C_r \,{\text{and}}\,\tilde L_q$$
\end{document} of double sequences whose Cesàro transforms are bounded, convergent in the Pringsheim’s sense, null in the Pringsheim’s sense, both convergent in the Pringsheim’s sense and bounded, regularly convergent and absolutely q-summable, respectively, and also examine some properties of those sequence spaces. Furthermore, we show that these sequence spaces are Banach spaces. We determine the alpha-dual of the space \documentclass{aastex}
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$$\tilde M_u$$
\end{document} and the β(bp)-dual of the space \documentclass{aastex}
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$$\tilde C_r$$
\end{document}, and β(ϑ)-dual of the space \documentclass{aastex}
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$$\tilde C_\eta$$
\end{document} of double sequences, where ϑ, η ∈ {p, bp, r}. Finally, we characterize the classes (\documentclass{aastex}
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$$\tilde C_{bp}$$
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$$\tilde C_\vartheta$$
\end{document}) for ϑ ∈ {p, bp, r} of four dimensional matrix transformations, where μ is any given space of double sequences.
On the paranormed space $\mathcal{M}_{u}(t)$ of double sequences
