## Abstract and Applied Analysis

### On the Paranormed Nörlund Sequence Space of Nonabsolute Type

#### Abstract

Maddox defined the space $\ell (p)$ of the sequences $x=({x}_{k})$ such that ${\sum }_{k=\mathrm{0}}^{\mathrm{\infty }}\mathrm{‍}|{x}_{k}{|}^{{p}_{k}}<\mathrm{\infty }$, in Maddox, 1967. In the present paper, the Nörlund sequence space ${N}^{t}(p)$ of nonabsolute type is introduced and proved that the spaces ${N}^{t}(p)$ and $\ell (p)$ are linearly isomorphic. Besides this, the alpha-, beta-, and gamma-duals of the space ${N}^{t}(p)$ are computed and the basis of the space ${N}^{t}(p)$ is constructed. The classes $({N}^{t}(p):\mu )$ and $(\mu :{N}^{t}(p))$ of infinite matrices are characterized. Finally, some geometric properties of the space ${N}^{t}(p)$ are investigated.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 858704, 9 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412606524

Digital Object Identifier
doi:10.1155/2014/858704

Mathematical Reviews number (MathSciNet)
MR3191072

Zentralblatt MATH identifier
07023210

#### Citation

Yeşilkayagil, Medine; Başar, Feyzi. On the Paranormed Nörlund Sequence Space of Nonabsolute Type. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 858704, 9 pages. doi:10.1155/2014/858704. https://projecteuclid.org/euclid.aaa/1412606524

#### References

• I. J. Maddox, “Spaces of strongly summable sequences,” Quarterly Journal of Mathematics, vol. 18, no. 1, pp. 345–355, 1967.
• H. Nakano, “Modulared sequence spaces,” Proceedings of the Japan Academy, vol. 27, no. 2, pp. 508–512, 1951.
• S. Simons, “The sequence spaces $\ell ({p}_{\upsilon })$ and $m({p}_{\upsilon })$,” Proceedings of the London Mathematical Society, vol. 15, no. 3, pp. 422–436, 1965.
• A. Peyerimhoff, Lectures on Summability, Lecture Notes in Mathematics, Springer, New York, NY, USA, 1969.
• F. M. Mears, “The inverse Nörlund mean,” Annals of Mathematics, vol. 44, no. 3, pp. 401–409, 1943.
• B. Choudhary and S. K. Mishra, “On Köthe-Toeplitz duals of certain sequence spaces and thair matrix transformations,” Indian Journal of Pure and Applied Mathematics, vol. 24, no. 59, pp. 291–301, 1993.
• F. Başar and B. Altay, “Matrix mappings on the space bs(p) and its $\alpha$-, $\beta$- and $\gamma$-duals,” The Aligarh Bulletin of Mathematics, vol. 21, no. 1, pp. 79–91, 2002.
• F. Başar, “Infinite matrices and almost boundedness,” Bollettino della Unione Matematica Italiana A, vol. 6, no. 7, pp. 395–402, 1992.
• B. Altay and F. Başar, “On the paranormed Riesz sequence space of non-absolute type,” Southeast Asian Bulletin of Mathematics, vol. 26, pp. 701–715, 2002.
• C. S. Wang, “On Nörlund sequence space,” Tamkang Journal of Mathematics, vol. 9, pp. 269–274, 1978.
• K. G. Grosse-Erdmann, “Matrix transformations between the sequence spaces of Maddox,” Journal of Mathematical Analysis and Applications, vol. 180, no. 1, pp. 223–238, 1993.
• C. G. Lascarides and I. J. Maddox, “Matrix transformations between some classes of sequences,” Proceedings of the Cambridge Philosophical Society, vol. 68, pp. 99–104, 1970.
• S. Chen, “Geometry of Orlicz spaces,” Dissertationes Mathematicae, vol. 356, pp. 1–224, 1996.
• J. Diestel, Geomety of Banach Spaces-Selected Topics, Springer, Berlin, Germany, 1984.
• L. Maligranda, Orlicz Spaces and Interpolation, Institute of Mathematics Polish Academy of Sciences, Poznan, Poland, 1985.
• H. Nergiz and F. Başar, “Some geometric properties of the domain of the double sequential band matrix $B(\widetilde{r},\widetilde{s})$ in the sequence space $\ell (p)$,” Abstract and Applied Analysis, vol. 2013, Article ID 421031, 7 pages, 2013. \endinput