THE v-INVARIANT χ2 SEQUENCE SPACES
Year 2011,
Volume: 01 Issue: 2, 173 - 184, 01.12.2011
N. Subramanian
U. K. Misra
Abstract
In this paper we define v− invariatness of a double sequence space of χ and examine the v− invariatness of the double sequence space of χ. Furthermore, we give duals of double sequence space of χ.
References
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[1] Aposto, T., (1978), Mathematical Analysis, Addison-Wesley, London.
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[2] Basarir, M. and Solancan, O., (1999), On some double sequence spaces, J. Indian Acad. Math., 21(2), 193-200.
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[3] Bektas, C. and Altin, Y., (2003), The sequence space `M (p, q, s) on seminormed spaces, Indian J. Pure Appl. Math., 34(4), 529-534.
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[4] Bromwich, T.J.I’A., (1965), An introduction to the theory of infinite series, Macmillan and Co.Ltd., New York.
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[6] Krasnoselskii, M.A. and Rutickii, Y.B., (1961), Convex functions and Orlicz spaces, Gorningen, Netherlands.
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[7] Lindenstrauss, J. and Tzafriri, L., (1971), On Orlicz sequence spaces, Israel J. Math., 10, 379-390.
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[8] Maddox, I.J., (1986), Sequence spaces defined by a modulus, Math. Proc. Cambridge Philos. Soc, 100(1), 161-166.
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[9] Moricz, F., (1991), Extentions of the spaces c and c0 from single to double sequences, Acta. Math. Hung., 57(1-2), 129-136.
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[10] Moricz, F. and Rhoades, B.E., (1988), Almost convergence of double sequences and strong regularity of summability matrices, Math. Proc. Camb. Phil. Soc., 104, 283-294.
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[11] Mursaleen, M., Khan, M.A. and Qamaruddin, (1999), Difference sequence spaces defined by Orlicz functions, Demonstratio Math., Vol. XXXII, 145-150.
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[12] Nakano, H., (1953), Concave modulars, J. Math. Soc. Japan, 5, 29-49.
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[13] Orlicz, W., (1936), U¨ber Raume ` L M´ Bull. Int. Acad. Polon. Sci. A, 93-107.
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[14] Parashar, S.D. and Choudhary, B., (1994), Sequence spaces defined by Orlicz functions, Indian J. Pure Appl. Math., 25(4), 419-428.
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[15] Chandrasekhara Rao, K. and Subramanian, N., (2004), The Orlicz space of entire sequences, Int. J. Math. Math. Sci., 68, 3755-3764.
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[16] Ruckle, W.H., (1973), FK spaces in which the sequence of coordinate vectors is bounded, Canad. J. Math., 25, 973-978.
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[17] Tripathy, B.C., (2003), On statistically convergent double sequences, Tamkang J. Math., 34(3), 231- 237.
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[18] Tripathy, B.C., Et, M. and Altin, Y., (2003), Generalized difference sequence spaces defined by Orlicz function in a locally convex space, J. Anal. Appl., 1(3), 175-192.
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[19] Turkmenoglu, A., (1999), Matrix transformation between some classes of double sequences, J. Inst. Math. Comp. Sci. Math. Ser., 12(1), 23-31.
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[20] Kamthan, P.K. and Gupta, M., (1981), Sequence spaces and series, Lecture notes, Pure and Applied Mathematics, 65 Marcel Dekker, In c., New York.
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[21] G¨okhan, A. and C¸ olak, R., (2004), The double sequence spaces c P 2 (p) and c P B 2 (p), Appl. Math. Comput., 157(2), 491-501.
-
[22] G¨okhan, A. and C¸ olak, R., (2005), Double sequence spaces ` ∞2 , ibid., 160(1), 147-153.
-
[23] Zeltser, M., (2001), Investigation of Double Sequence Spaces by Soft and Hard Analitical Methods, Dissertationes Mathematicae Universitatis Tartuensis 25, Tartu University Press, Univ. of Tartu, Faculty of Mathematics and Computer Science, Tartu.
-
[24] Mursaleen, M. and Edely, O.H.H., (2003), Statistical convergence of double sequences, J. Math. Anal. Appl., 288(1), 223-231.
-
[25] Mursaleen, M., (2004), Almost strongly regular matrices and a core theorem for double sequences, J. Math. Anal. Appl., 293(2), 523-531.
-
[26] Mursaleen, M. and Edely, O.H.H., (2004), Almost convergence and a core theorem for double sequences, J. Math. Anal. Appl., 293(2), 532-540.
-
[27] Altay, B. and Basar, F., (2005), Some new spaces of double sequences, J. Math. Anal. Appl., 309(1), 70- 90.
-
[28] Basar, F. and Y.Sever, Y., (2009), The space Lp of double sequences, Math. J. Okayama Univ, 51, 149-157.
-
[29] Subramanian, N. and Misra, U.K., (2010), The semi normed space defined by a double gai sequence of modulus function, Fasciculi Math., 46.
-
[30] Kizmaz, H., (1981), On certain sequence spaces, Cand. Math. Bull., 24(2), 169-176.
-
[31] Kuttner, B., (1946), Note on strong summability, J. London Math. Soc., 21, 118-122.
-
[32] Maddox, I.J., (1979), On strong almost convergence, Math. Proc. Cambridge Philos. Soc., 85(2), 345- 350.
-
[33] Cannor, J., (1989), On strong matrix summability with respect to a modulus and statistical convergence, Canad. Math. Bull., 32(2), 194-198.
-
[34] Pringsheim, A., (1900), Zurtheorie derzweifach unendlichen zahlenfolgen, Math. Ann., 53, 289-321.
-
[35] Hamilton, H.J., (1936), Transformations of multiple sequences, Duke Math. J., 2, 29-60.
-
[36] ———-, (1938), A Generalization of multiple sequences transformation, Duke Math. J., 4, 343-358.
-
[37] ———-, (1938), Change of Dimension in sequence transformation , Duke Math. J., 4, 341-342.
-
[38] ———-, (1939), Preservation of partial Limits in Multiple sequence transformations, Duke Math. J., 4, 293-297.
-
[39] , Robison, G.M., (1926), Divergent double sequences and series, Amer. Math. Soc. Trans., 28, 50-73.
-
[40] Silverman, L.L., On the definition of the sum of a divergent series, unpublished thesis, University of Missouri studies, Mathematics series.
-
[41] Toeplitz, O., (1911), U¨ber allgenmeine linear mittel bridungen, Prace Matemalyczno Fizyczne (warsaw), 22.
-
[42] Basar, F. and Altay, B., (2003), On the space of sequences of p− bounded variation and related matrix mappings, Ukrainian Math. J., 55(1), 136-147.
-
[43] Altay, B. and Basar, F., (2007), The fine spectrum and the matrix domain of the difference operator ∆ on the sequence space `p,(0 < p < 1), Commun. Math. Anal., 2(2), 1-11.
-
[44] C¸ olak, R., Et, M. and Malkowsky, E., (2004), Some Topics of Sequence Spaces, Lecture Notes in Mathematics, Firat Univ. Elazig, Turkey, 1-63, Firat Univ. Press, ISBN: 975-394-0386-6.
Year 2011,
Volume: 01 Issue: 2, 173 - 184, 01.12.2011
N. Subramanian
U. K. Misra
References
-
[1] Aposto, T., (1978), Mathematical Analysis, Addison-Wesley, London.
-
[2] Basarir, M. and Solancan, O., (1999), On some double sequence spaces, J. Indian Acad. Math., 21(2), 193-200.
-
[3] Bektas, C. and Altin, Y., (2003), The sequence space `M (p, q, s) on seminormed spaces, Indian J. Pure Appl. Math., 34(4), 529-534.
-
[4] Bromwich, T.J.I’A., (1965), An introduction to the theory of infinite series, Macmillan and Co.Ltd., New York.
-
[5] Hardy, G.H., (1917), On the convergence of certain multiple series, Proc. Camb. Phil. Soc., 19, 86-95.
-
[6] Krasnoselskii, M.A. and Rutickii, Y.B., (1961), Convex functions and Orlicz spaces, Gorningen, Netherlands.
-
[7] Lindenstrauss, J. and Tzafriri, L., (1971), On Orlicz sequence spaces, Israel J. Math., 10, 379-390.
-
[8] Maddox, I.J., (1986), Sequence spaces defined by a modulus, Math. Proc. Cambridge Philos. Soc, 100(1), 161-166.
-
[9] Moricz, F., (1991), Extentions of the spaces c and c0 from single to double sequences, Acta. Math. Hung., 57(1-2), 129-136.
-
[10] Moricz, F. and Rhoades, B.E., (1988), Almost convergence of double sequences and strong regularity of summability matrices, Math. Proc. Camb. Phil. Soc., 104, 283-294.
-
[11] Mursaleen, M., Khan, M.A. and Qamaruddin, (1999), Difference sequence spaces defined by Orlicz functions, Demonstratio Math., Vol. XXXII, 145-150.
-
[12] Nakano, H., (1953), Concave modulars, J. Math. Soc. Japan, 5, 29-49.
-
[13] Orlicz, W., (1936), U¨ber Raume ` L M´ Bull. Int. Acad. Polon. Sci. A, 93-107.
-
[14] Parashar, S.D. and Choudhary, B., (1994), Sequence spaces defined by Orlicz functions, Indian J. Pure Appl. Math., 25(4), 419-428.
-
[15] Chandrasekhara Rao, K. and Subramanian, N., (2004), The Orlicz space of entire sequences, Int. J. Math. Math. Sci., 68, 3755-3764.
-
[16] Ruckle, W.H., (1973), FK spaces in which the sequence of coordinate vectors is bounded, Canad. J. Math., 25, 973-978.
-
[17] Tripathy, B.C., (2003), On statistically convergent double sequences, Tamkang J. Math., 34(3), 231- 237.
-
[18] Tripathy, B.C., Et, M. and Altin, Y., (2003), Generalized difference sequence spaces defined by Orlicz function in a locally convex space, J. Anal. Appl., 1(3), 175-192.
-
[19] Turkmenoglu, A., (1999), Matrix transformation between some classes of double sequences, J. Inst. Math. Comp. Sci. Math. Ser., 12(1), 23-31.
-
[20] Kamthan, P.K. and Gupta, M., (1981), Sequence spaces and series, Lecture notes, Pure and Applied Mathematics, 65 Marcel Dekker, In c., New York.
-
[21] G¨okhan, A. and C¸ olak, R., (2004), The double sequence spaces c P 2 (p) and c P B 2 (p), Appl. Math. Comput., 157(2), 491-501.
-
[22] G¨okhan, A. and C¸ olak, R., (2005), Double sequence spaces ` ∞2 , ibid., 160(1), 147-153.
-
[23] Zeltser, M., (2001), Investigation of Double Sequence Spaces by Soft and Hard Analitical Methods, Dissertationes Mathematicae Universitatis Tartuensis 25, Tartu University Press, Univ. of Tartu, Faculty of Mathematics and Computer Science, Tartu.
-
[24] Mursaleen, M. and Edely, O.H.H., (2003), Statistical convergence of double sequences, J. Math. Anal. Appl., 288(1), 223-231.
-
[25] Mursaleen, M., (2004), Almost strongly regular matrices and a core theorem for double sequences, J. Math. Anal. Appl., 293(2), 523-531.
-
[26] Mursaleen, M. and Edely, O.H.H., (2004), Almost convergence and a core theorem for double sequences, J. Math. Anal. Appl., 293(2), 532-540.
-
[27] Altay, B. and Basar, F., (2005), Some new spaces of double sequences, J. Math. Anal. Appl., 309(1), 70- 90.
-
[28] Basar, F. and Y.Sever, Y., (2009), The space Lp of double sequences, Math. J. Okayama Univ, 51, 149-157.
-
[29] Subramanian, N. and Misra, U.K., (2010), The semi normed space defined by a double gai sequence of modulus function, Fasciculi Math., 46.
-
[30] Kizmaz, H., (1981), On certain sequence spaces, Cand. Math. Bull., 24(2), 169-176.
-
[31] Kuttner, B., (1946), Note on strong summability, J. London Math. Soc., 21, 118-122.
-
[32] Maddox, I.J., (1979), On strong almost convergence, Math. Proc. Cambridge Philos. Soc., 85(2), 345- 350.
-
[33] Cannor, J., (1989), On strong matrix summability with respect to a modulus and statistical convergence, Canad. Math. Bull., 32(2), 194-198.
-
[34] Pringsheim, A., (1900), Zurtheorie derzweifach unendlichen zahlenfolgen, Math. Ann., 53, 289-321.
-
[35] Hamilton, H.J., (1936), Transformations of multiple sequences, Duke Math. J., 2, 29-60.
-
[36] ———-, (1938), A Generalization of multiple sequences transformation, Duke Math. J., 4, 343-358.
-
[37] ———-, (1938), Change of Dimension in sequence transformation , Duke Math. J., 4, 341-342.
-
[38] ———-, (1939), Preservation of partial Limits in Multiple sequence transformations, Duke Math. J., 4, 293-297.
-
[39] , Robison, G.M., (1926), Divergent double sequences and series, Amer. Math. Soc. Trans., 28, 50-73.
-
[40] Silverman, L.L., On the definition of the sum of a divergent series, unpublished thesis, University of Missouri studies, Mathematics series.
-
[41] Toeplitz, O., (1911), U¨ber allgenmeine linear mittel bridungen, Prace Matemalyczno Fizyczne (warsaw), 22.
-
[42] Basar, F. and Altay, B., (2003), On the space of sequences of p− bounded variation and related matrix mappings, Ukrainian Math. J., 55(1), 136-147.
-
[43] Altay, B. and Basar, F., (2007), The fine spectrum and the matrix domain of the difference operator ∆ on the sequence space `p,(0 < p < 1), Commun. Math. Anal., 2(2), 1-11.
-
[44] C¸ olak, R., Et, M. and Malkowsky, E., (2004), Some Topics of Sequence Spaces, Lecture Notes in Mathematics, Firat Univ. Elazig, Turkey, 1-63, Firat Univ. Press, ISBN: 975-394-0386-6.