Double sequence spaces over $n$-normed spaces

Raj, Kuldip   ; Sharma, Sunil K.

Archivum mathematicum, Tome 50 (2014) no. 2, pp. 65-76

Abstract (VO)

In this paper, we define some classes of double sequences over $n$-normed spaces by means of an Orlicz function. We study some relevant algebraic and topological properties. Further some inclusion relations among the classes are also examined.

Détail

DOI : 10.5817/AM2014-2-65

Classification : 40A05, 40C05, 40D05, 46A70
Keywords: paranorm space; Orlicz function; solid; monotone; double sequences; $n$-normed space

Bibliographie

[1] Başarir, M., Sonalcan, O.: On some double sequence spaces. J. Indian Acad. Math. 21 (1999), 193–220. | MR | Zbl

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

[3] Dutta, H.: An application of lacunary summability method to $n$-norm. Int. J. Appl. Math. Stat. 15 (09) (2009), 89–97. | MR

[4] Dutta, H.: Characterization of certain matrix classes involving generalized difference summability spaces. Appl. Sci. 11 (2009), 60–67. | MR | Zbl

[5] Dutta, H.: On n-normed linear space valued strongly (C,1)-summable difference sequences. Asian-Eur. J. Math. 3 (4) (2010), 565–575. | DOI | MR | Zbl

[6] Dutta, H.: On sequence spaces with elements in a sequence of real linear n-normed spaces. Appl. Math. Lett. 23 (9) (2010), 1109–1113. | DOI | MR | Zbl

[7] Dutta, H.: On some $n$-normed linear space valued difference sequences. J. Franklin Inst. B 348 (2011), 2876–2883. | DOI | MR | Zbl

[8] Dutta, H.: An Orlicz extension of difference sequences on real linear n-normed spaces. J. Inequal. Appl. 2013:232 (2013), 13pp. | DOI | MR | Zbl

[9] Dutta, H., Başar, F.: A generalization of Orlicz sequence spaces by Cesàro mean of order one. Acta Math. Univ. Comenian. (N.S.) 80 (2) (2011), 185–200. | MR | Zbl

[10] Dutta, H., Bilgin, T.: Strongly $(V^\lambda , A, \Delta _{vm}^n, p)$-summable sequence spaces defined by an Orlicz function. Appl. Math. Lett. 24 (7) (2011), 1057–1062. | DOI | MR

[11] Dutta, H., Reddy, B.S.: On non-standard $n$-norm on some sequence spaces. Int. J. Pure Appl. Math. 68 (1) (2011), 1–11. | MR | Zbl

[12] Dutta, H., Reddy, B.S., Cheng, S.S.: Strongly summable sequences defined over real $n$-normed spaces. Appl. Math. E-Notes 10 (2010), 199–209. | MR | Zbl

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

[14] Gähler, S.: Linear 2-normietre Rume. Math. Nachr. 28 (1965), 1–43. | DOI

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

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

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

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

[19] Karakaya, V., Dutta, H.: On some vector valued generalized difference modular sequence spaces. Filomat 25 (3) (2011), 15–27. | DOI | MR | Zbl

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

[21] Lindenstrauss, J., Tzafriri, L.: On Orlicz sequence spaces. Israel J. Math. 10 (1971), 379–390. | DOI | MR | Zbl

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

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

[24] 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

[25] Nakano, H.: Modular sequence spaces. Proc. Japan Acad. Ser. A Math. Sci. 27 (1951), 508–512. | MR

[26] Raj, K., Sharma, A.K., Sharma, S.K.: A sequence space defined by Musielak-Orlicz functions. Int. J. Pure Appl. Math. 67 (2011), 472–484. | MR

[27] Raj, K., Sharma, S.K., Sharma, A.K.: Some difference sequence spaces in $n$-normed spaces defined by Musielak-Orlicz function. Armen. J. math. 3 (2010), 127–141. | MR

[28] Raj, K., Sharma, S.K., Sharma, A.K.: Some new sequence spaces defined by a sequence of modulus functions in $n$-normed spaces. Int. J. Math. Sci. Engg. Appl. 5 (2011), 395–403. | MR

[29] Savaş, E.: On some new sequence spaces in 2-normed spaces using ideal convergence and an Orlicz function. J. Inequal. Appl. (2010), 8 pages, Article ID 482392. | MR | Zbl

[30] Simons, S.: The sequence spaces $l(p_v)$ and $m(p_v)$. Proc. Japan Acad. 27 (1951), 508–512.

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

[32] 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

[33] Tripathy, B.C., Altin, Y., Et, M.: Generalized difference sequence spaces on seminormed space defined by Orlicz function. Math. Slovaca 58 (2008), 315–324. | DOI | MR

[34] Tripathy, B.C., Dutta, H.: On some lacunary difference sequence spaces defined by a sequence of Orlicz functions and q-lacunary $\Delta _m^ n$ statistical convergence. An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat. 20 (1) (2012), 417–430. | MR

[35] Tripathy, B.C., Esi, A., Tripathy, B.K.: On a new type of generalized difference Cesàro sequence spaces. Soochow J. Math. 31 (3) (2005), 333–340. | MR | Zbl

[36] Wilansky, A.: Summability through functional analysis. North-Holland Mathematics Studies, vol. 85, Amsterdam - New York - Oxford: North-Holland, 1984. | MR | Zbl