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

© 2021 Global Journals 1 Global Journal of Science Frontier Research Volume XXI Issue IV Year 2021 30 ( F ) Version I Difference Sequence Spaces of Second order Defined by a Sequence of Moduli This space is an FK space. Ruckle[17-19] proved that the intersection of all such X ( f ) spaces is φ , the space of all finite sequences. The space X ( f ) is closely related to the space l 1 which is an X ( f ) space with f ( x ) = x for all real x ≥ 0. Thus Ruckle[17-19] proved that, for any modulus f . X ( f ) ⊂ l 1 and X ( f ) α = ` ∞ The space X ( f ) is a Banach space with respect to the norm || x || = ∞ X k =1 f ( | x k | ) < ∞ . Spaces of the type X ( f ) are a special case of the spaces structured by Gramsch in[4]. From the point of view of local convexity, spaces of the type X ( f ) are quite patho- logical. Symmetric sequence spaces, which are locally convex have been frequently studied by Garling[2-3], K¨othe[9] and Ruckle[17-19]. Kolk [7-8] gave an extension of X ( f ) by considering a sequence of moduli F = ( f k ) and defined the sequence space X ( F ) = { x = ( x k ) : ( f k ( | x k | )) ∈ X } Khan and Lohani [5] defined the following sequence spaces ` ∞ ( u, 4 , F ) = { x = ( x k ) ∈ ω : sup k ≥ 0 f k ( | u k 4 x k | ) < ∞} c ( u, 4 , F ) = { x = ( x k ) ∈ ω : lim k →∞ f k ( | u k 4 x k − l | ) = 0 , l ∈ C } c 0 ( u, 4 , F ) = { x = ( x k ) ∈ ω : lim k →∞ f k ( | u k 4 x k | ) = 0 } where u ∈ U. If we take x k instead of 4 x ,then we have the following sequence spaces ` ∞ ( u, F ) = { x = ( x k ) ∈ ω : sup k ≥ 0 f k ( | u k x k | ) < ∞} c ( u, F ) = { x = ( x k ) ∈ ω : lim k →∞ f k ( | u k x k − l | ) = 0 , l ∈ C } c 0 ( u, F ) = { x = ( x k ) ∈ ω : lim k →∞ f k ( | u k x k | ) = 0 } 9. K Ö the, G. Topological Vector spaces. 1.(Springer,Berlin,1970.) R ef