Paranormed Norlund Nᵗ- difference sequence spaces and their α-, β- and γ-duals
DOI:
https://doi.org/10.22199/issn.0717-6279-5289Keywords:
paranormed sequence space, Nᵗ-Difference sequence space, Norlund matrix, Schauder basis, α-, β- and γ-dualsAbstract
Kizmaz [4] defined some difference spaces viz., ℓ∞ (∆), c(∆) and c0(∆) and studied by Et and Colak [1] thoroughly. In this paper, Norlund Nt- difference sequence spaces Nt(c0, p, ∆), Nt(c, p, ∆) and Nt(ℓ∞, p, ∆) contain the sequences whose Nt∆-transforms in c0, c and ℓ∞ are defined and the paranormed linear structures are developed on these spaces. It has been shown that the spaces Nt(c0, p, ∆), Nt(c, p, ∆) & Nt(ℓ∞, p, ∆) are linearly isomorphic and are of non-absolute type. Further, it is verified that Nt(c, p, ∆), Nt(c0, p, ∆)and Nt(l∞, p, ∆) of non-absolute form are isomorphic to Nt(c0, p), Nt(c, p) and Nt(ℓ∞, p), respectively. Topological properties such as the completeness and the isomorphism are also discussed. Some inclusion relations among these spaces are also verified. Finally, the α-, β- and γ- dual of these spaces are determined and constructed the Schauder-basis of Nt(c0, p, ∆) and Nt(c, p, ∆).
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References
M. Et and R. Colak R., “On some generalized sequence spaces”, Soochow Journal of Mathematics, vol. 21, pp. 377-386, 1995.
K. G. Grosse-Erdmann, “Matrix transformation between the sequence spaces of Maddox”, Journal of Mathematical Analysis and Applications, vol. 180, no. 1, pp. 223-238, 1993. https://doi.org/10.1006/JMAA.1953.1398
A. M. Jarrah and E. Makowsky, “Ordinary, absolute and strong summability and matrix transformations”, Filomat, vol. 17, pp. 59-78, 2003. https://doi.org/10.2298/FIL0317059J
H. Kizmaz, “On certain sequence spaces”, Canadian Mathematical Bulletin, vol. 24, no. 2, pp. 169-176, 1981. https://doi.org/10.4153/CMB-1981-027-5
G. C. Lascarides and I.J. Maddox, “Matrix Transformations between some classes of sequences”, Proceedings Cambridge Philosophical Society, vol. 68, no. 1, pp. 99-104, 1970. https://doi.org/10.1017/S0305004100001109
I. J. Maddox, Elements of Functional Analysis, 2nd ed. Cambridge: The University Press, 1988.
I. J. Maddox, “Paranormed sequence spaces generated by infinite matrices”, Proceedings Cambridge Philosophical Society, vol. 64, pp. 335-340, 1968. https://doi.org/10.1017/S0305004100042894
I. J. Maddox, “Spaces of strongly summable sequences”, Quarterly Journal of Mathematics, vol. 18, no. 2, pp. 345-355, 1967. https://doi.org/10.1093/qmath/18.1.345
E. Malkowsky and E. Savas, “Matrix transformation between sequence spaces of generalized weighted means”, Applied Mathematics and Computation, vol. 147, no. 2, pp. 333-345, 2004. https://doi.org/10.1016/S0096-3003(02)00670-7
M. F. Mears, “The inverse Norlund means”, Annals of Mathematics, vol. 44, no. 3, pp. 401-409, 1943. https://doi.org/10.2307/1968971
H. Nakano, “Modulared sequence spaces”, Proceedings of the Japan Academy, vol. 27, no. 2, pp. 508-512, 1951. https://doi.org/10.3792/pja/1.195571225
A. Peyerimhoff, Lectures on Summability. Lectures Notes in Mathematics. New York: Springer, 1969.
S. Simons., “The sequence spaces l(pv) and m(pv)”, Proceedings of the London Mathematical Society, vol. 15, no. 3, pp. 422-436, 1965. https://doi.org/10.1112/plms/53-15.1.422.
M. Yesilkayagil and F. Basar, “On the paranormed Norlund sequence space of nonabsolute type”, Abstract and Applied Analysis, 2014, Article ID 858704. https://doi.org/10.1155/2014/858704
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