`Abstract and Applied AnalysisVolume 2012 (2012), Article ID 719729, 9 pageshttp://dx.doi.org/10.1155/2012/719729`
Research Article

## Statistical Convergence of Double Sequences in Locally Solid Riesz Spaces

1Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

Received 26 May 2012; Accepted 13 August 2012

Copyright © 2012 S. A. Mohiuddine et al. 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.

1. H. Fast, “Sur la convergence statistique,” Colloquium Mathematicum, vol. 2, pp. 241–244, 1951.
2. H. Steinhaus, “Sur la convergence ordinaire et la convergence asymptotique,” Colloquium Mathematicum, vol. 2, pp. 73–34, 1951.
3. A. Zygmund, Trigonometric Series, Cambridge University Press, New York, NY, USA, 1st edition, 1959.
4. G. G. Lorentz, “A contribution to the theory of divergent sequences,” Acta Mathematica, vol. 80, pp. 167–190, 1948.
5. J. A. Fridy, “On statistical convergence,” Analysis, vol. 5, no. 4, pp. 301–313, 1985.
6. R. Çolak and Ç. A. Bektaş, “$\lambda$-statistical convergence of order $\alpha$,” Acta Mathematica Scientia Series B, vol. 31, no. 3, pp. 953–959, 2011.
7. V. Kumar and M. Mursaleen, “On $\left(\lambda ,\mu \right)$-statistical convergence of double sequences on intuitionistic fuzzy normed spaces,” Filomat, vol. 25, no. 2, pp. 109–120, 2011.
8. S. A. Mohiuddine and Q. M. D. Lohani, “On generalized statistical convergence in intuitionistic fuzzy normed space,” Chaos, Solitons & Fractals, vol. 42, no. 3, pp. 1731–1737, 2009.
9. 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.
10. M. Mursaleen and A. Alotaibi, “On $I$-convergence in random 2-normed spaces,” Mathematica Slovaca, vol. 61, no. 6, pp. 933–940, 2011.
11. S. A. Mohuiddine, A. Alotaibi, and S. M. Alsulami, “Ideal convergence of double sequences in 2 random 2-normed spaces,” Advances in Difference Equations, vol. 2012, article 149, 2012.
12. M. Mursaleen and S. A. Mohiuddine, “On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space,” Journal of Computational and Applied Mathematics, vol. 233, no. 2, pp. 142–149, 2009.
13. M. Mursaleen and S. A. Mohiuddine, “On ideal convergence of double sequences in probabilistic normed spaces,” Mathematical Reports, vol. 12, no. 4, pp. 359–371, 2010.
14. M. Mursaleen and S. A. Mohiuddine, “On ideal convergence in probabilistic normed spaces,” Mathematica Slovaca, vol. 62, no. 1, pp. 49–62, 2012.
15. M. Mursaleen, S. A. Mohiuddine, and O. H. H. Edely, “On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces,” Computers & Mathematics with Applications, vol. 59, no. 2, pp. 603–611, 2010.
16. E. Savaş and S. A. Mohiuddine, “$\overline{\lambda }$-statistically convergent double sequences in probabilistic normed spaces,” Mathematica Slovaca, vol. 62, no. 1, pp. 99–108, 2012.
17. 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.
18. H. Cakalli, “On statistical convergence in topological groups,” Pure and Applied Mathematika Sciences, vol. 43, no. 1-2, pp. 27–31, 1996.
19. H. Çakalli and E. Savaş, “Statistical convergence of double sequences in topological groups,” Journal of Computational Analysis and Applications, vol. 12, no. 2, pp. 421–426, 2010.
20. S. A. Mohiuddine and E. Savaş, “Lacunary statistically convergent double sequences in probabilistic normed spaces,” Annali dell'Universita di Ferrara. In press.
21. M. Mursaleen and S. A. Mohiuddine, “Statistical convergence of double sequences in intuitionistic fuzzy normed spaces,” Chaos, Solitons & Fractals, vol. 41, no. 5, pp. 2414–2421, 2009.
22. S. A. Mohiuddine, H. Şevli, and M. Cancan, “Statistical convergence in fuzzy 2-normed space,” Journal of Computational Analysis and Applications, vol. 12, no. 4, pp. 787–798, 2010.
23. M. Mursaleen, “On statistical convergence in random 2-normed spaces,” Acta Scientiarum Mathematicarum, vol. 76, no. 1-2, pp. 101–109, 2010.
24. H. Albayrak and S. Pehlivan, “Statistical convergence and statistical continuity on locally solid Riesz spaces,” Topology and its Applications, vol. 159, no. 7, pp. 1887–1893, 2012.
25. A. C. Zaanen, Introduction to Operator Theory in Riesz Spaces, Springer, Berlin, Germany, 1997.
26. G. T. Roberts, “Topologies in vector lattices,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 48, pp. 533–546, 1952.
27. A. Pringsheim, “Zur Theorie der zweifach unendlichen Zahlenfolgen,” Mathematische Annalen, vol. 53, no. 3, pp. 289–321, 1900.
28. M. Mursaleen, C. Çakan, S. A. Mohiuddine, and E. Savaş, “Generalized statistical convergence and statistical core of double sequences,” Acta Mathematica Sinica, vol. 26, no. 11, pp. 2131–2144, 2010.
29. M. Mursaleen and O. H. H. Edely, “Generalized statistical convergence,” Information Sciences, vol. 162, no. 3-4, pp. 287–294, 2004.
30. Mursaleen and O. H. H. Edely, “Statistical convergence of double sequences,” Journal of Mathematical Analysis and Applications, vol. 288, no. 1, pp. 223–231, 2003.