A Note on Some Strongly Sequence Spaces

and Applied Analysis 3 P:4 p x y ≤ p x p y , for all x, y ∈ X triangle inequality , P:5 if λn is a sequence of scalars, with λn → λ n → ∞ , and xn is a sequence of vectors with p xn −x → 0 n → ∞ , then p λnxn −λx → 0 n → ∞ continuity of multiplication by scalars . A complete linear metric space is said to be a Fréchet space. A Fréchet sequence space X is said to be an FK space, if its metric is stronger than the metric of w on X, that is, convergence in the sequence space X implies coordinatewise convergence the letters F and K stand for Fréchet and Koordinate, the German word for coordinate . Note that, by Ruckle in 5 , a modulus function f is a function from 0,∞ to 0,∞ such that i f x 0, if and only if, x 0, ii f x y ≤ f x f y , for all x, y ≥ 0, iii f increasing, iv f is continuous from the right at zero. Since |f x − f y | ≤ f |x − y| , it follows from condition iv that f is continuous on 0,∞ . Furthermore, from condition ii , we have f nx ≤ nf x for all n ∈ N, and thus f x f ( nx 1 n ) ≤ nf ( x n ) , 1.8 hence 1 n f x ≤ f ( x n ) , ∀n ∈ N. 1.9 In 5 , Ruckle used the idea of a modulus function f in order to construct a class of FK spaces L ( f ) {