Some multiplier difference sequence spaces defined by a sequence ...

180 Hemen Duttak ∈ N.Now we have for all i, j ≥ n 0 ,Then we havesup((f k (|∆ n (m)λ k x i k − ∆ n (m)λ k x j k |))p k) 1 Hk≥1< εlim [sup ((f k (|∆ n (m)λ k x i k − ∆ n (m)λ k x jj→∞k |))p k) 1 H ] < ε, for all i ≥ n0k≥1This implies thatsup((f k (|∆ n (m)λ k x i k − ∆ n (m)λ k x k |)) p k) 1 Hk≥1< εIt follows that(x i − x) ∈ l ∞ (F , ∆ n (m), Λ, p).Since (x i ) ∈ l ∞ (F , ∆ n (m) , Λ, p) and l ∞(F , ∆ n (m), Λ, p) is a linear space, so wehave x = x i − (x i − x) ∈ l ∞ (F , ∆ n (m), Λ, p).This completes the proof of the Theorem.Theorem 3 If 0 < p k ≤ q k < ∞ for each k, then Z(F, ∆ n (m), Λ, p) ⊆Z(F, ∆ n (m) , Λ, q), for Z = c 0 and c.Proof. We prove the result for the case Z = c 0 and for the other case itwill follow on applying similar arguments.Let (x k ) ∈ c 0 (F , ∆ n (m), Λ, p). Then we haveThis implies thatlim (f k(|∆ n (m)λ k x k |)) p k= 0.k→∞f k (|∆ n (m)λ k x k |) < ε(0 < ε ≤ 1) for sufficiently large k.