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**On Some I-Convergent Double Sequence Spaces Defined by a Modulus Function** ()

In 2000, Kostyrko, Salat, and Wilczynski introduced and studied the concept of

*I-*convergence of sequences in metric spaces where*I is an ideal. The concept of**I-*convergence has a wide application in the field of Number Theory, trigonometric series, summability theory, probability theory, optimization and approximation theory. In this article we introduce the double sequence spaces and ,for a modulus function*f*and study some of the properties of these spaces.

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V. Khan and N. Khan, "On Some

*I*-Convergent Double Sequence Spaces Defined by a Modulus Function,"*Engineering*, Vol. 5 No. 5A, 2013, pp. 35-40. doi: 10.4236/eng.2013.55A006.Conflicts of Interest

The authors declare no conflicts of interest.

[1] | H. Fast, “Sur la Convergence Statistique,” Colloqium Mathematicum, Vol. 2, No. 1, 1951, pp. 241-244. |

[2] | J. A. Fridy, “On Statistical Convergence,” Analysis, Vol. 5, 1985, pp. 301-313. |

[3] | J. A. Fridy, “Statistical Limit Points,” Proceedings of American Mathematical Society, Vol. 11, 1993, pp. 11871192. doi:10.1090/S0002-9939-1993-1181163-6 |

[4] | P. Kostyrko, T. Salat and W. Wilczynski, “I-Convergence,” Real Analysis Exchange, Vol. 26, No. 2, 1999, pp. 193-200. |

[5] | T. Salat, B. C. Tripathy and M. Ziman, “On Some Properties of I-Convergence,” Tatra Mountain Mathematical Publications, 2000, pp. 669-686. |

[6] | K. Demirci, “I-Limit Superior and Limit Inferior,” Mathematical Communications, Vol. 6, 2001, pp. 165-172. |

[7] | T. J. I. Bromwich, “An Introduction to the Theory of Infinite Series,” MacMillan Co. Ltd., New York, 1965. |

[8] | M. Basarir and O. Solancan, “On Some Double Sequence Spaces,” Journal of the Indian Academy of Mathematics, Vol. 21, No. 2, 1999, pp. 193-200. |

[9] | H. Nakano, “Concave Modulars,” Journal of Mathematical Society, Japan, Vol. 5, No. 1, 1953, pp. 29-49. doi:10.2969/jmsj/00510029 |

[10] | W. H. Ruckle, “On Perfect Symmetric BK-Spaces,” Mathematische Annalen, Vol. 175, No. 2, 1968, pp. 121-126. doi:10.1007/BF01418767 |

[11] | W. H. Ruckle, “FK-Spaces in Which the Sequence of Coordinate Vectors is Bounded,” Canadian Journal of Mathematics, Vol. 25, No. 5, 1973, pp. 973-975. doi:10.4153/CJM-1973-102-9 |

[12] | B. Gramsch, “Die Klasse Metrisher Linearer Raume L(φ),” Mathematische Annalen, Vol. 171, 1967, pp. 6178. doi:10.1007/BF01433094 |

[13] | D. J. H. Garling, “On Symmetric Sequence Spaces,” Proceedings of London Mathematical Society, Vol. 16, 1966, pp. 85-106. doi:10.1112/plms/s3-16.1.85 |

[14] | D. J. H. Garling, “Symmetric Bases of Locally Convex Spaces,” Studia Mathematica, Vol. 30, No. 2, 1968, pp. 163-181. |

[15] | G. Kothe, “Topological Vector Spaces,” Springer, Berlin, 1970. |

[16] | W. H. Ruckle, “Symmetric Coordinate Spaces and Symmetric Bases,” Canadian Journal of Mathematics, Vol. 19, 1967, pp. 828-838. doi:10.4153/CJM-1967-077-9 |

[17] | V. A. Khan and S. Tabassum, “On Some New Double Sequence Spaces of Invariant Means Defined by Orlicz Function,” Communications, Faculty of Sciences, University of Ankara, Vol. 60, 2011, pp. 11-21. |

[18] | J. Singer, “Bases in Banach Spaces. 1,” Springer, Berlin, 1970. |

[19] | M. Sen and S. Roy, “Some I-Convergent Double Classes of Sequences of Fuzzy Numbers Defined by Orlicz Functions,” Thai Journal of Mathematics, Vol. 10, No. 4, 2013, pp. 1-10. |

[20] | I. J. Maddox, “Some Properties of Paranormed Sequence Spaces,” Journal of the London Mathematical Society, Vol. 1, 1969, pp. 316-322. |

[21] | J. Connor and J. Kline, “On Statistical Limit Points and the Consistency of Statistical Convergence,” Journal of Mathematical Analysis and Applications, Vol. 197, No. 2, 1996, pp. 392-399. doi:10.1006/jmaa.1996.0027 |

[22] | K. Dems, “On I-Cauchy Sequences,” Real Analysis Exchange, Vol. 30, No. 1, 2005, pp. 123-128. |

[23] | M. Gurdal, “Some Types Of Convergence,” Doctoral Dissertation, Sleyman Demirel University, Isparta, 2004. |

[24] | O. T. Jones and J. R. Retherford, “On Similar Bases in Barrelled Spaces,” Proceedings of American Mathematical Society, Vol. 18, 1967, pp. 677-680. doi:10.1090/S0002-9939-1967-0217552-8 |

[25] | P. K. Kamthan and M. Gupta, “Sequence Spaces and Series,” Marcel Dekker Inc., New York, 1981. |

[26] | I. J. Maddox, “Elements of Functional Analysis,” Cambridge University Press, Cambridge, 1970. |

[27] | I. J. Maddox, “Sequence Spaces Defined by a Modulus,” Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 100, 1986, pp. 161-166. doi:10.1017/S0305004100065968 |

[28] | T. Salat, “On Statistically Convergent Sequences of Real Numbers,” Mathematica Slovaca, Vol. 30, 1980, pp. 139150. |

[29] | A. K. Vakeel and K. Ebadullah, “On Some I-Convergent Sequence Spaces Defined by a Modulus Function,” Theory and Applications of Mathematics and Computer Science, Vol. 1, No. 2, 2011, pp. 22-30. |

[30] | A. Wilansky, “Functional Analysis,” Blaisdell, New York, 1964. |

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