Topological properties of some sequences defined over n-normed spaces




Orlicz function, n−norm, Paranormed spaces, Completeness, Solidity


Some classes of real number sequences over n? normed spaces defined by means of Orlicz functions, a bounded sequence of strictly positive real numbers, a multiplier and a normal paranormed sequence space are investigated. Relevant properties of such classes have been investigated. Moreover, relationships among different such classes of sequences have also been studied under various parameters and conditions. Finally, the spaces are investigated for some other useful properties.

Author Biography

Tanweer Jalal, National Institute of Technology.

 Dept. of Mathematics.


Z. U. Ahmad and A. H. A. Bataineh, “Some new sequences defined by Orlicz function”, The Aligarh bulletin mathematics, vol. 20, no. 2, pp. 39-51, 2001.

Y. Altin, “Properties of some sets of sequences defined by a modulus function,” Acta Mathematica Scientia, vol. 29, no. 2, pp. 427–434, Mar. 2009, doi: 10.1016/S0252-9602(09)60042-4

C. A. Bekta? and Y. Altin, “The sequence space on ????M (p, q, s) seminormed spaces”, Indian journal pure and applications mathematics, vol. 34, no. 4, pp. 529-534, Apr. 2003. [On line]. Available:

S. Debnath and S. Saha, “On some I -convergent generalized difference sequence spaces associated with multiplier sequence defined by a sequence of modulli”, Proyecciones (Antofagasta), vol. 34, no. 2, pp. 137–146, Jun. 2015, doi: 10.4067/S0716-09172015000200003

H. Dutta, “On some n-normed linear space valued difference sequences”, Journal of the Franklin Institute, vol. 348, no. 10, pp. 2876–2883, Dec. 2011, doi: 10.1016/j.jfranklin.2011.09.010

H. Dutta and F. Ba?ar, “A generalization of Orlicz sequence spaces by Cesaro mean of order one”, Acta mathematica universitatis comenianae, vol. 80, no. 2, pp. 185-200, 2011. [On line]. Available:

H. Dutta and I. H. Jebril, “An extension of modular sequence spaces”, Abstract and applied analysis, Art ID. 371806, 2013, doi: 10.1155/2013/371806

A. Esi, M. Isik and A. Esi, “On some new sequence spaces defined by Orlicz functions”, Indian journal pure and applications mathematics, vol. 35, no. 1, pp. 31-36, Jan. 2004. [On line]. Available:

S. Gähler, “Lineare 2-normierte Räume”, Mathematische nachrichten, vol. 28, no. 1-2, pp. 1–43, 1964, doi: 10.1002/mana.19640280102

D. Ghosh and P. D. Srivastava, “On some vector valued sequence spaces defined using a modulus function”, Indian journal pure and applications mathematics, vol. 30, no. 8, pp. 819-826, Aug. 1999. [On line]. Available:

H. Gunawan, “On n-inner product, n-norms, and the Cauchy-Schwarz inequality”, Scientiae mathematicae japonicae online, vol. 5, pp. 47-54, 2001. [On line]. Available:

H. Gunawan, “The space of p- summable sequences and its natural n-norm”, Bulletin of the Australian Mathematical Society, vol. 64, no. 1, pp. 137–147, Aug. 2001, doi: 10.1017/S0004972700019754

H. Gunawan and M. Mashadi, “On n-normed spaces”, International journal of mathematics and mathematical sciences, vol. 27, no. 10, pp. 631-639, 2001. [On line]. Available:

H. Gurdal and M. Sahiner, “Ideal convergence in n-normed spaces and some new sequence spaces via n-norm”, Malaysian journal of fundamental and applied sciences vol. 4, no. 1, pp. 233-244, Jun. 2008, doi: 10.11113/mjfas.v4n1.32

M. I?ik, “Some classes of almost convergent paranormed sequence spaces defined by Orlicz functions”, Demonstratio mathematica, vol. 45, no. 3, 2012, doi: 10.1515/dema-2013-0403

T. Jalal and R. Ahmad, “A new generalized vector-valued paranormed sequence space using modulus function”, Malaya journal of matematik, vol. 3, no. 1, pp. 110-118, 2015. [On line]. Available:

T. Jalal, “Some new lacunary sequence spaces of Invariant means defined by Musielak-Orlicz functions on n-normed space”, International journal of pure and applied mathematics, vol. 119, no. 7, pp. 1-11, 2018. [On line]. Available:

T. Jalal, “Some new I-lacunary generalized difference sequence spaces in n-normed space,” in Modern mathematical methods and high performance computing in science and technology, vol. 171, S. Singh, H. Srivastava, E. Venturino, M. Resch, and V. Gupta, Eds. Singapore: Springer, 2016, pp. 249–258, doi:

T. Jalal, “New A-generalized sequence spaces defined by ideal convergence and a sequence of modulus functions on multiple normed spaces”, International journal of open problems in computer science and mathematics, vol. 8, no. 1, pp. 87- 98, Mar. 2015, doi: 10.12816/0010708

T. Jalal, “Some new convergent sequence spaces defined by using a sequence of modulus functions in n-normed spaces”, International journal of mathematical archive, vol. 5, no. 9, pp. 202-209, Sep. 2014. [On line]. Available:

P. K. Kamthan and M. Gupta, Sequence spaces and series. New York, NY: Marcel Dekker, 1981.

V. Karakaya and H. Dutta, “On some vector valued generalized difference modular sequence spaces”, Filomat, vol. 25, no. 3, pp. 15-27, 2011, doi: 10.2298/FIL1103015K

M. A. Krasnoselskii and Y. B. Rutisky, Convex functions and Orlicz spaces, Groningen: P. Noordhoff, 1961.

J. Lindenstrauss and L. Tzafriri, “On Orlicz sequence spaces”, Israel journal of mathematics, vol. 10, pp. 379-390, Sep. 1971, doi: 10.1007/BF02771656

I. J. Maddox, Elements of functional analysis, Cambridge: Cambridge University Press, Cambridge, 1970.

I. J. Maddox, “Sequence spaces defined by a modulus”, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 100, no. 1, pp. 161-166, 1986, doi: 10.1017/S0305004100065968

A. Misiak, “n-inner product spaces”, Mathematische nachrichten, vol. 140, no. 1, pp. 299-319, 1989, doi: 10.1002/mana.19891400121

M. Mursaleen, Q. A. Khan, and T. A. Chishti, “Some new convergent sequences spaces defined by Orlicz functions and statistical convergence”, Italian journal of pure and applied mathematics, no. 9, pp. 25-32, 2001. [On line]. Available:

F. Nuray and A. Gülcü, “Some new sequence spaces defined by Orlicz functions”, Indian journal of pure and applied mathematics, vol. 26, no. 12, pp. 1169-1176, 1995. Available. [On line].

M. M. Rao and Z. D. Ren, Theory on Orlicz spaces, New York: Marcel Dekker, 1991.

W. H. Ruckle, “FK spaces in which the sequence of coordinate vectors in bounded”, Canadian journal of mathematics, vol. 25, no. 5, pp. 973-978, Oct. 1973, doi: 10.4153/CJM-1973-102-9

E. Sava?, “?m- strongly summable sequences spaces in 2- normed spaces defined by ideal convergence and an Orlicz function”, Applied mathematics and computation, vol. 217, no. 1, pp. 271-276, Sep. 2010, doi: 10.1016/j.amc.2010.05.057

E. Sava?, “Some new double sequences spaces defined by Orlicz functions in n-normed spaces”, Journal of inequalities and applications, Art. ID. 592840, 2011, doi: 10.1155/2011/592840



How to Cite

T. Jalal, “Topological properties of some sequences defined over n-normed spaces”, Proyecciones (Antofagasta, On line), vol. 39, no. 5, pp. 1137-1155, Oct. 2020.