The spaces BS, BS(t), CSp, CSbp, CSr and BV of double sequences have recently been studied by Altay and Başar [J. Math. Anal. Appl. 309(1)(2005), 70–90]. In this work, following Altay and Başar [1], we introduce the Banach space Lq of double sequences corresponding to the well-known space ℓq of single sequences and examine some properties of the space Lq. Furthermore, we determine the β(υ)-dual of the space and establish that the α- and γ-duals of the space Lq coincide with the β(υ)-dual; where 1 ≤ q < ∞ and υ ∈ {p, bp, r}.