Global Journal of Science Frontier Research, F: Mathematics and Decision Science, Volume 21 Issue 4

1 Global Journal of Science Frontier Research Volume XXI Issue IV Year 2021 29 ( F ) © 2021 Global Journals Version I Difference Sequence Spaces of Second order Defined by a Sequence of Moduli and showed that these are Banach spaces with the norm || x || 4 = m X i =1 | x i | + ||4 m x || ∞ . Let U be the set of all sequences u = ( u k ) such that u k 6 = 0 ( k = 1 , 2 , 3 .... ) . Malkowsky[13] defined the following sequence spaces ` ∞ ( u, 4 ) = { x = ( x k ) ∈ ω : ( u k 4 x k ) ∈ ` ∞ } c ( u, 4 ) = { x = ( x k ) ∈ ω : ( u k 4 x k ) ∈ c } c 0 ( u, 4 ) = { x = ( x k ) ∈ ω : ( u k 4 x k ) ∈ c 0 } where u ∈ U. The concept of paranorm (see[12]) is closely related to linear metric spaces. It is a generalization of that of absolute value. Let X be a linear space. A function g : X −→ R is called paranorm, if for all x, y ∈ X, (PI) g ( x ) = 0 if x = 0 , (P2) g ( − x ) = g ( x ) , (P3) g ( x + y ) ≤ g ( x ) + g ( y ) , (P4) If ( λ n ) is a sequence of scalars with λ n → λ ( n → ∞ ) and x n , a ∈ X with x n → a ( n → ∞ ) , in the sense that g ( x n − a ) → 0 ( n → ∞ ) , in the sense that g ( λ n x n − λa ) → 0 ( n → ∞ ) . A paranorm g for which g ( x ) = 0 implies x = 0 is called a total paranorm on X , and the pair ( X, g ) is called a totally paranormed space. The idea of modulus was structured by Nakano[16]. A function f : [0, ∞ ) −→ [0, ∞ ) is called a modulus if (P1) f (t) = 0 if and only if t = 0, (P2) f (t+u) ≤ f (t)+ f (u) for all t,u ≥ 0, (P3) f is increasing, and (P4) f is continuous from the right at zero. Ruckle [17-19] used the idea of a modulus function f to construct the sequence space X ( f ) = { x = ( x k ) : ∞ X k =1 f ( | x k | ) < ∞} 12. Maddox, I.J. Some properties of paranormed sequence spaces, J. London. Math. Soc.1 (1969), 316-322. R ef