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

R eferences R éférences R eferencias 1 Global Journal of Science Frontier Research Volume XXI Issue IV Year 2021 35 ( F ) © 2021 Global Journals Version I Difference Sequence Spaces of Second order Defined by a Sequence of Moduli Hence x ∈ c 0 ( u, 4 2 , F ) . (b) Let p k ≥ 1 for each k and sup k p k < ∞ . Suppose that x ∈ c 0 ( u, 4 2 , F ) . Then for each > 0 there exists a positive integer N such that f k ( | u k ( 4 2 x k ) | ) ≤ for all k ≥ N Since 1 ≤ p k ≤ sup k p k < ∞ , we have lim k →∞ ( f k ( | u k ( 4 2 x k ) | )) p k ≤ lim k →∞ ( f k ( | u k ( 4 2 x k ) | )) ≤ < 1 Therefore x ∈ c 0 ( u, 4 2 , F, p ) . 1. Asma, C., Colak, R. On the K Ö the-Toeplitz duals of generalised sets of difference sequences. Demonstratio Math; 33(4), (2000)797-803. 2. Garling, D.J.H. On Symmetric Sequence Spaces, Proc. London. Math. Soc.16(1966), 85-106. 3. Garling, D.J.H. Symmetric bases of locally convex spaces, Studia Math. Soc. 30(1968), 163-181. 4. Gramsch, B. Die Klasse metrisher linearer Raume L( ), Math.Ann.171(1967), 61-78. 5. Khan.V.A., Lohani, Q.M.D. Difference sequence spaces defined by a sequence of moduli. Southeast Asian Bulletin of Mathematics. 30:, (2006), 1061-1067. 6. Kizmaz,H. On Certain sequence spaces, Canad.Math.Bull. 24(1981) 169-176. 7. Kolk, E. On strong boundedness and summability with respect to a sequence of modulii , Acta Comment.Univ.Tartu, 960(1993), 41-50. 8. Kolk, E. Inclusion theorems for some sequence spaces defined by a sequence of modulii, Acta Comment.Univ.Tartu, 970(1994), 65-72. 9. K Ö the, G. Topological Vector spaces. 1.(Springer,Berlin,1970.) 10. Maddox, I.J. Elements of Functional Analysis, Cambridge University Press.(1970) 11. Maddox, I.J. Sequence spaces defined by a modulus, Math. Camb. Phil. Soc. 100(1986), 161-166. 12. Maddox, I.J. Some properties of paranormed sequence spaces, J. London. Math. Soc.1 (1969), 316-322. 13. Malkowsky, E.A. Note on the K Ö the-Toeplitz duals of generalized sets of bounded and convergent difference sequences, J.Analysis. 4,(1996)81-91. 14. Mikail. On some difference sequence spaces, Dogra-Tr:J:Math., 17,(1993)18- 24. φ 15. Mikail and Colak, R. On some generalized difference sequence spaces. Soochow: J: of Math: 21(4)(1995)377-386. 16. Nakano, H. Concave modulars. J. Math Soc. Japan, 5(1953)29-49. 17. Ruckle, W.H. On perfect Symmetric BK-spaces., Math.Ann.175(1968)121- 126. 18. Ruckle, W.H. Symmetric coordinate spaces and symmetric bases, Canad.J.Math.19 (1967)828-838. 19. Ruckle, W.H. FK-spaces in which the sequence of coordinate vectors is bounded, Canad.J.Math.25(5)(1973)973-975. N otes