### THE v-INVARIANT χ2 SEQUENCE SPACES

Year 2011, Volume 01, Issue 2, 173 - 184, 01.12.2011

### 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

• [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.
Year 2011, Volume 01, Issue 2, 173 - 184, 01.12.2011

### 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.

### Details

Primary Language English Research Article N. SUBRAMANİAN This is me Department of Mathematics, SASTRA University, Thanjavur-613 401, India U. K. MİSRA This is me Department of Mathematics and Statistics, Berhampur University, Berhampur-760 007,Odissa, India December 1, 2011 Year 2011, Volume 01, Issue 2

### Cite

 Bibtex @ { twmsjaem761793, journal = {TWMS Journal of Applied and Engineering Mathematics}, issn = {2146-1147}, eissn = {2587-1013}, address = {Işık University ŞİLE KAMPÜSÜ Meşrutiyet Mahallesi, Üniversite Sokak No:2 Şile / İstanbul}, publisher = {Turkic World Mathematical Society}, year = {2011}, volume = {01}, number = {2}, pages = {173 - 184}, title = {THE v-INVARIANT χ2 SEQUENCE SPACES}, key = {cite}, author = {Subramanian, N. and Misra, U. K.} }