Abstract In this study, the concept of infinite quasi-Sobolev spaces ℓ ∞ m , where m ∈ ℝ is considered. These spaces have been proved as quasi-Banach spaces, as well as, Banach spaces, while they neither Hilbert spaces nor quasi-Hilbert spaces. Some kinds of linear operators such as continuous, bounded, closed and completely continuous for operators which map ℓ ∞ m or ℓ 1 m into ℓ ∞ m are discussed.