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

Vakeel. A. Khan, Nazneen Khan

Department of Mathematics, Aligarh Muslim University, Aligarh, India.

DOI: 10.4236/eng.2013.55A006                

Abstract

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.

Keywords

Ideal; Filter; Modulus Function; Lipschitz Function; I-Convergence Field; I-Convergent; Monotone and Solid Double Sequence Spaces

Conflicts of Interest

The authors declare no conflicts of interest.

References

[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.