On Topological Structure of Total Paranormed Double Sequence Space (ℓ 2 ((X, ||.||), γ, ¯ w¯), G)

The aim of this paper is to introduce and study a new class ℓ2 ((X, ||.||), γ̅, ω̅ ) of double sequences with their terms in a normed space X as a generalization of the familiar sequence space ℓ. Besides the investigation of the condition pertaining to the containment relations of the class ℓ2 ((X, ||.||), γ̅, ω̅ ) of same kind in terms of γ̅ and ω̅, our primary interest is to explore some of the preliminary results that characterize the linear topological structures of ℓ2 ((X, ||.||), γ̅, ω̅ ) when topologized it with suitable natural paranorm.