Double Sequence Spaces Definedby a Sequence of Modulus Functions over $n$-normed Spaces

Sharma, Sunil K.   ; Esi, Ayhan

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 53 (2014) no. 1, pp. 117-134

Abstract (VO)

In the present paper we introduce some double sequence spaces defined by a sequence of modulus function $ F = (f_{k,l})$ over $n$-normed spaces. We also make an effort to study some topological properties and inclusion relations between these spaces.

Détail

Classification : 40C05, 42B15
Keywords: double sequences; $P$-convergent; modulus function; paranorm space

Bibliographie

[1] Altay, B., Başar, F.: Some new spaces of double sequencs. J. Math. Anal. Appl. 309 (2005), 70–90. | DOI | MR

[2] Altin, Y.: Properties of some sets of sequences defined by a modulus function. Acta Math. Sci. Ser. B Engl. Ed. 29 (2009), 427–434. | DOI | MR | Zbl

[3] Altin, Y., Et, M.: Generalized difference sequence spaces defined by a modulus function in a locally convex space. Soochow J. Math. 31 (2005), 233–243. | MR | Zbl

[4] Altin, Y., Işik, M., Çolak, R.: A new sequence space defined by a modulus. Stud. Univ. Babes–Bolyai Math. 53 (2008), 3–13. | MR | Zbl

[5] Altinok, H., Altin, Y., Işik, M.: The sequence space $Bv_\sigma (M,P,Q,S)$ on seminormed spaces. Indian J. Pure Appl. Math. 39 (2008), 49–58. | MR | Zbl

[6] Başarir, M., Sonalcan, O.: On some double sequence spaces. J. Indian Acad. Math. 21 (1999), 193–200. | MR | Zbl

[7] Başar, F., Sever, Y.: The space $\mathcal {L}_p$ of double sequences. Math. J. Okayama Univ. 51 (2009), 149–157. | MR

[8] Bromwich, T. J.: An Introduction to the Theory of Infinite Series. Macmillan and co., Ltd., New York, 1965.

[9] Connor, J.: On strong matrix summability with respect to a modulus and statistical convergence. Canad. Math. Bull. 30 (1989), 194–198. | MR | Zbl

[10] Esi, A.: Some new sequence spaces defined by a sequence of moduli. Turk. J. Math. 21 (1997), 61–68. | MR | Zbl

[11] Esi, A.: Strongly $[V_2, \lambda _2, M, p]$-summable double sequence spaces defined by Orlicz function. Int. J. Nonlinear Anal. Appl. 2 (2011), 110–115. | Zbl

[12] Et, M., Çolak, R.: On generalized difference sequence spaces. Soochow J. Math. 21, 4 (1995), 377–386.

[13] Gähler, S.: Linear 2-normietre RumeStatistical convergence in 2-normed spaces. Math. Nachr. 28 (1965), 1–43. Statistical convergence in 2-normed spaces Southeast Asian Bull. Math. 33, 2 (2009), 257–264.

[14] Gunawan, H.: On $n$-inner product, $n$-norms, and the Cauchy-Schwartz inequality. Sci. Math. Jap. 5 (2001), 47–54. | MR

[15] Gunawan, H.: The space of $p$-summable sequence and its natural $n$-norm. Bull. Aust. Math. Soc. 64 (2001), 137–147. | DOI | MR

[16] Gunawan, H., Mashadi, M.: On n-normed spaces. Int. J. Math. Math. Sci. 27 (2001), 631–639. | DOI | MR | Zbl

[17] Hamilton, H. J.: Transformations of multiple sequences. Duke Math. J. 2 (1936), 29–60. | DOI | MR | Zbl

[18] Hardy, G. H.: On the convergence of certain multiple series. Proc. Camb. Phil., Soc. 19 (1917), 86–95.

[19] Hardy, G. H.: Divergent Series. Oxford at the Clarendon Press, 1949. | MR | Zbl

[20] Kizmaz, H.: On certain sequences spaces. Canad. Math. Bull. 24, 2 (1981), 169–176. | DOI | MR

[21] Maddox, I. J.: Sequence spaces defined by a modulus. Math. Proc. Cambridge Philos. Soc. 100 (1986), 161–166. | DOI | MR | Zbl

[22] Malkowsky, E., Savaş, E.: Some $ \lambda $-sequence spaces defined by a modulus. Archivum Math. 36 (2000), 219–228. | MR | Zbl

[23] Misiak, A.: $n$-inner product spaces. Math. Nachr. 140 (1989), 299–319. | DOI | MR | Zbl

[24] Moricz, F.: Extension of the spaces $c $ and $c_0$ from single to double sequences. Acta Math. Hungarica 57 (1991), 129–136. | DOI | MR

[25] Moricz, F., Rhoades, B. E.: Almost convergence of double sequences and strong reqularity of summability matrices. Math. Proc. Camb. Phil. Soc. 104 (1988), 283–294. | DOI | MR

[26] Mursaleen, M.: Almost strongly regular matrices and a core theorem for double sequences. J. Math. Anal. Appl. 293, 2 (2004), 523–531. | DOI | MR | Zbl

[27] Mursaleen, M., Edely, O. H. H.: Statistical convergence of double sequences. J. Math. Anal. Appl. 288, 1 (2003), 223–231. | DOI | MR | Zbl

[28] Mursaleen, M., Edely, O. H. H.: Almost convergence and a core theorem for double sequences. J. Math. Anal. Appl. 293, 2 (2004), 532–540. | DOI | MR | Zbl

[29] Pringsheim, A.: Zur Theori der zweifach unendlichen Zahlenfolgen. Math. Ann. 53 (1900), 289–321. | DOI | MR

[30] Raj, K., Sharma, S. K.: Difference sequence spaces defined by sequence of modulus function. Proyecciones J. Math. 30 (2011), 189–199. | MR

[31] Raj, K., Sharma, S. K.: Some difference sequence spaces defined by sequence of modulus function. Int. J. Math. Archive 2 (2011), 236–240.

[32] Robinson, G. M.: Divergent double sequences and series. Trans. Amer. Math. Soc. 28 (1926), 50–73. | DOI | MR

[33] Savaş, E.: On some generalized sequence spaces defined by a modulus. Indian J. Pure Appl. Math. 30 (1999), 459–464. | MR | Zbl

[34] Savaş, E., Patterson, R. F.: Double sequence spaces defined by a modulus. Math. Slovaca 61 (2011), 245–256. | DOI | MR | Zbl

[35] Tripathy, B. C.: Generalized difference paranormed statistically convergent sequences defined by Orlicz function in a locally convex spaces. Soochow J. Math. 30 (2004), 431–446. | MR

[36] Tripathy, B. C.: Statistically convergent double sequences. Tamkang J. Math. 34 (2003), 231–237. | MR | Zbl

[37] Wilansky, A.: Summability through Functional Analysis. 85 North–Holland Math. Stud. 1984. | MR | Zbl

[38] Zeltser, M.: Investigation of Double Sequence Spaces by Soft and Hard Analytical Methods. Dissertationes Mathematicae Universitatis Tartuensis 25, Tartu University Press, Univ. of Tartu, Faculty of Mathematics and Computer Science, Tartu, 2001. | MR | Zbl