Some Difference Paranormed Sequence Spaces over n-normed Spaces Defined by a Musielak-Orlicz Function

  • Raj, Kuldip ;
  • Sharma, Sunil K. ;
  • Gupta, Amit
  • Received : 2011.08.18
  • Accepted : 2012.08.22
  • Published : 2014.03.23


In the present paper we introduce difference paranormed sequence spaces $c_0(\mathcal{M},{\Delta}^n_m,p,u,{\parallel}{\cdot},{\cdots},{\cdot}{\parallel})$, $c(\mathcal{M},{\Delta}^n_m,p,u,{\parallel}{\cdot},{\cdots},{\cdot}{\parallel})$ and $l_{\infty}(\mathcal{M},{\Delta}^n_m,p,u,{\parallel}{\cdot},{\cdots},{\cdot}{\parallel})$ defined by a Musielak-Orlicz function $\mathcal{M}$ = $(M_k)$ over n-normed spaces. We also study some topological properties and some inclusion relations between these spaces.


Paranorm space;difference sequence spaces;Orlicz function;Musielak-Orlicz function;solid;monotone etc


  1. M. Et and R. Colak, On generalized difference sequence spaces, Soochow J. Math., 21(1995), 377-386.
  2. Y. Altin, Properties of some sets of sequences defined by a modulus function, Acta Math. Sci. Ser. B Engl. Ed., 29(2009), 427-434.
  3. M. Et, H. Altinok and Y. Altin, On generalized sequence spaces, Appl. Math. Comput., 154(2004), 167-173.
  4. M. Et, Y. Altin, B. Choudhary and B. C. Tripathy, On some classes of sequences defined by sequences of Orlicz functions, Math. Inequal. Appl., 9(2006), 335-342.
  5. S. Gahler, Linear 2-normietre Rume, Math. Nachr., 28(1965), 1-43.
  6. H. Gunawan, On n-inner product, n-norms, and the Cauchy-Schwartz inequality, Sci. Math. Jap., 5(2001), 47-54.
  7. H. Gunawan, The space of p-summable sequence and its natural n-norm, Bull. Aust. Math. Soc., 64(2001), 137-147.
  8. H. Gunawan and M. Mashadi, On n-normed spaces, Int. J. Math. Math. Sci., 27(2001), 631-639.
  9. H. Kizmaz, On certain sequence spaces, Canad. Math-Bull., 24(1981), 169-176.
  10. J. Lindenstrauss and L. Tzafriri,On Orlicz sequence spaces, Israel J. Math., 10(1971), 379-390.
  11. L. Maligranda, Orlicz spaces and interpolation, Seminars in Mathematics 5, Polish Academy of Science (1989).
  12. A. Misiak, n-inner product spaces, Math. Nachr., 140(1989), 299-319.
  13. J. Musielak, Orlicz spaces and modular spaces, Lecture Notes in Matthematics, 1034(1983).
  14. K. Raj, A. K. Sharma and S. K. Sharma, A Sequence space defined by a Musielak-Orlicz function, Int. J. Pure Appl. Math., 67(2011), 475-484.
  15. K. Raj, S. K. Sharma and A. K. Sharma, Some difference sequence spaces in n-normed spaces defined by a Musielak-Orlicz function, Armen. J. Math., 3(2010), 127-141.
  16. K. Raj and S. K. Sharma, Some sequence spaces in 2-normed spaces defined by a Musielak-Orlicz function, Acta Univ. Sapientiae Math., 3(2011), 97-109.
  17. B. C. Tripathy and H. Dutta, Some difference paranormed sequence spaces defined by Orlicz functions, Fasciculi Math., Nr, 42(2009), 121-131.
  18. A. Wilansky, Summability through Functional Analysis, North-Holland Math. Stud., 85(1984).

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