Journal of Operators

Volume 2013, Article ID 904838, 7 pages

http://dx.doi.org/10.1155/2013/904838

Research Article

## Some Properties of the Sequence Space

School of Mathematics, Shri Mata Vaishno Devi University, Katra, Jammu and Kashmir 182 320, India

Received 29 March 2013; Revised 6 June 2013; Accepted 6 June 2013

Academic Editor: Palle E. Jorgensen

Copyright © 2013 Kuldip Raj and Sunil K. Sharma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- S. Gähler, “Lineare
*2*-normierte Räume,”*Mathematische Nachrichten*, vol. 28, pp. 1–43, 1965. View at Google Scholar · View at MathSciNet - A. Misiak, “
*n*-inner product spaces,”*Mathematische Nachrichten*, vol. 140, pp. 299–319, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Gunawan, “On
*n*-inner products,*n*-norms, and the Cauchy-Schwarz inequality,”*Scientiae Mathematicae Japonicae*, vol. 5, pp. 47–54, 2001. View at Google Scholar · View at MathSciNet - H. Gunawan, “The space of
*p*-summable sequences and its natural*n*-norm,”*Bulletin of the Australian Mathematical Society*, vol. 64, no. 1, pp. 137–147, 2001. View at Publisher · View at Google Scholar · View at MathSciNet - H. Gunawan and M. Mashadi, “On
*n*-normed spaces,”*International Journal of Mathematics and Mathematical Sciences*, vol. 27, no. 10, pp. 631–639, 2001. View at Publisher · View at Google Scholar · View at MathSciNet - J. Lindenstrauss and L. Tzafriri, “On Orlicz sequence spaces,”
*Israel Journal of Mathematics*, vol. 10, pp. 379–390, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Maligranda,
*Orlicz Spaces and Interpolation*, vol. 5 of*Seminários de Matemática*, Polish Academy of Science, Warszawa, Poland, 1989. View at MathSciNet - J. Musielak,
*Orlicz Spaces and Modular Spaces*, vol. 1034 of*Lecture Notes in Mathematics*, Springer, Berlin, Germany, 1983. View at MathSciNet - A. Wilansky,
*Summability through Functional Analysis*, vol. 85 of*North-Holland Mathematics Studies*, North-Holland, Amsterdam, The Netherland, 1984. View at MathSciNet - F. Basar,
*Summability Theory and Its Applications*, Monographs, Bentham Science Publishers, E-Books, Istanbul, Turkey, 2012. - F. Başar, B. Altay, and M. Mursaleen, “Some generalizations of the space
*bv(p)*of*p*-bounded variation sequences,”*Nonlinear Analysis: Theory, Methods and Applications A*, vol. 68, no. 2, pp. 273–287, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - C. Belen and S. A. Mohiuddine, “Generalized weighted statistical convergence and application,”
*Applied Mathematics and Computation*, vol. 219, no. 18, pp. 9821–9826, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - T. Bilgin, “Some new difference sequences spaces defined by an Orlicz function,”
*Filomat*, vol. 17, pp. 1–8, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. L. Braha and M. Et, “The sequence space ${E}_{n}^{q}\left(M,p,s\right)$ and ${N}_{k}$-lacunary statistical convergence,”
*Banach Journal of Mathematical Analysis*, vol. 7, no. 1, pp. 88–96, 2013. View at Google Scholar · View at MathSciNet - R. Çolak, B. C. Tripathy, and M. Et, “Lacunary strongly summable sequences and
*q*-lacunary almost statistical convergence,”*Vietnam Journal of Mathematics*, vol. 34, no. 2, pp. 129–138, 2006. View at Google Scholar · View at MathSciNet - A. M. Jarrah and E. Malkowsky, “The space
*bv(p)*, its*β*-dual and matrix transformations,”*Collectanea Mathematica*, vol. 55, no. 2, pp. 151–162, 2004. View at Google Scholar · View at MathSciNet - I. J. Maddox, “Statistical convergence in a locally convex space,”
*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 104, no. 1, pp. 141–145, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Mursaleen, “Generalized spaces of difference sequences,”
*Journal of Mathematical Analysis and Applications*, vol. 203, no. 3, pp. 738–745, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Mursaleen, “Matrix transformations between some new sequence spaces,”
*Houston Journal of Mathematics*, vol. 9, no. 4, pp. 505–509, 1983. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Mursaleen, “On some new invariant matrix methods of summability,”
*The Quarterly Journal of Mathematics*, vol. 34, no. 133, pp. 77–86, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Raj and S. K. Sharma, “Some sequence spaces in
*2*-normed spaces defined by Musielak-Orlicz function,”*Acta Universitatis Sapientiae. Mathematica*, vol. 3, no. 1, pp. 97–109, 2011. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Raj and S. K. Sharma, “Some generalized difference double sequence spaces defined by a sequence of Orlicz-functions,”
*Cubo*, vol. 14, no. 3, pp. 167–189, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - K. Raj and S. K. Sharma, “Some multiplier sequence spaces defined by a Musielak-Orlicz function in
*n*-normed spaces,”*New Zealand Journal of Mathematics*, vol. 42, pp. 45–56, 2012. View at Google Scholar · View at MathSciNet - H. Fast, “Sur la convergence statistique,”
*Colloquium Mathematicum*, vol. 2, pp. 241–244, 1951. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. J. Schoenberg, “The integrability of certain functions and related summability methods,”
*The American Mathematical Monthly*, vol. 66, pp. 361–375, 1959. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. A. Fridy, “On statistical convergence,”
*Analysis*, vol. 5, no. 4, pp. 301–303, 1985. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Connor, “A topological and functional analytic approach to statistical convergence,” in
*Analysis of Divergence*, Applied and Numerical Harmonic Analysis, pp. 403–413, Birkhäuser, Boston, Mass, USA, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Šalát, “On statistically convergent sequences of real numbers,”
*Mathematica Slovaca*, vol. 30, no. 2, pp. 139–150, 1980. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Mursaleen, “
*λ*-statistical convergence,”*Mathematica Slovaca*, vol. 50, no. 1, pp. 111–115, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Işik, “On statistical convergence of generalized difference sequences,”
*Soochow Journal of Mathematics*, vol. 30, no. 2, pp. 197–205, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Savaş, “Strong almost convergence and almost
*λ*-statistical convergence,”*Hokkaido Mathematical Journal*, vol. 29, no. 3, pp. 531–566, 2000. View at Google Scholar · View at MathSciNet - E. Malkowsky and E. Savas, “Some
*λ*-sequence spaces defined by a modulus,”*Archivum Mathematicum*, vol. 36, no. 3, pp. 219–228, 2000. View at Google Scholar · View at MathSciNet - E. Kolk, “The statistical convergence in Banach spaces,”
*Acta et Commentationes Universitatis Tartuensis*, vol. 928, pp. 41–52, 1991. View at Google Scholar · View at MathSciNet - I. J. Maddox, “A new type of convergence,”
*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 83, no. 1, pp. 61–64, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. A. Mohiuddine and M. Aiyub, “Lacunary statistical convergence in random
*2*-normed spaces,”*Applied Mathematics & Information Sciences*, vol. 6, no. 3, pp. 581–585, 2012. View at Google Scholar · View at MathSciNet - A. R. Freedman, J. J. Sember, and M. Raphael, “Some Cesàro-type summability spaces,”
*Proceedings of the London Mathematical Society*, vol. 37, no. 3, pp. 508–520, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet