We introduce the sequence space defined by a Musielak-Orlicz function . We also study some topological properties and prove some inclusion relations involving this space. 1. Introduction and Preliminaries The concept of 2-normed spaces was initially developed by G？hler [1] in the mid-1960s, while one can see that of -normed spaces in Misiak [2]. Since then, many others have studied this concept and obtained various results; see Gunawan [3, 4] and Gunawan and Mashadi [5]. Let and be a linear space over the field , where is the field of real or complex numbers of dimension , where . A real valued function on satisfying the following four conditions: (1) if and only if are linearly dependent in ; (2) is invariant under permutation; (3) for any ; and (4) is called an -norm on , and the pair is called an -normed space over the field . For example, we may take being equipped with the -norm = the volume of the -dimensional parallelopiped spanned by the vectors which may be given explicitly by the formula where for each . Let be an -normed space of dimension and be linearly independent set in . Then the following function on defined by defines an -norm on with respect to . A sequence in an -normed space is said to converge to some if A sequence in an -normed space is said to be Cauchy if If every Cauchy sequence in converges to some , then is said to be complete with respect to the -norm. Any complete -normed space is said to be -Banach space. An Orlicz function is a function which is continuous, nondecreasing, and convex with , for and as . Lindenstrauss and Tzafriri [6] used the idea of Orlicz function to define the following sequence space. Let be the space of all real or complex sequences ; then which is called an Orlicz sequence space. The space is a Banach space with the norm It is shown in [6] that every Orlicz sequence space contains a subspace isomorphic to . The -condition is equivalent to for all values of and for . A sequence of Orlicz function is called a Musielak-Orlicz function; see [7, 8]. A sequence defined by is called the complementary function of a Musielak-Orlicz function . For a given Musielak-Orlicz function , the Musielak-Orlicz sequence space and its subspace are defined as follows: where is a convex modular defined by We consider equipped with the Luxemburg norm or equipped with the Orlicz norm Let be a linear metric space. A function : is called paranorm if (1) for all , (2) for all , (3) for all , (4)if is a sequence of scalars with as , and is a sequence of vectors with as ; then as . A paranorm for which implies is called total
References
[1]
S. G？hler, “Lineare 2-normierte R？ume,” Mathematische Nachrichten, vol. 28, pp. 1–43, 1965.
[2]
A. Misiak, “n-inner product spaces,” Mathematische Nachrichten, vol. 140, pp. 299–319, 1989.
[3]
H. Gunawan, “On n-inner products, n-norms, and the Cauchy-Schwarz inequality,” Scientiae Mathematicae Japonicae, vol. 5, pp. 47–54, 2001.
[4]
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, 2001.
[5]
H. Gunawan and M. Mashadi, “On n-normed spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 27, no. 10, pp. 631–639, 2001.
[6]
J. Lindenstrauss and L. Tzafriri, “On Orlicz sequence spaces,” Israel Journal of Mathematics, vol. 10, pp. 379–390, 1971.
[7]
L. Maligranda, Orlicz Spaces and Interpolation, vol. 5 of Seminários de Matemática, Polish Academy of Science, Warszawa, Poland, 1989.
[8]
J. Musielak, Orlicz Spaces and Modular Spaces, vol. 1034 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1983.
[9]
A. Wilansky, Summability through Functional Analysis, vol. 85 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherland, 1984.
[10]
F. Basar, Summability Theory and Its Applications, Monographs, Bentham Science Publishers, E-Books, Istanbul, Turkey, 2012.
[11]
F. Ba？ar, B. Altay, and M. Mursaleen, “Some generalizations of the space bv(p) of p-bounded variation sequences,” Nonlinear Analysis: Theory, Methods and Applications A, vol. 68, no. 2, pp. 273–287, 2008.
[12]
C. Belen and S. A. Mohiuddine, “Generalized weighted statistical convergence and application,” Applied Mathematics and Computation, vol. 219, no. 18, pp. 9821–9826, 2013.
[13]
T. Bilgin, “Some new difference sequences spaces defined by an Orlicz function,” Filomat, vol. 17, pp. 1–8, 2003.
[14]
N. L. Braha and M. Et, “The sequence space and -lacunary statistical convergence,” Banach Journal of Mathematical Analysis, vol. 7, no. 1, pp. 88–96, 2013.
[15]
R. ？olak, B. C. Tripathy, and M. Et, “Lacunary strongly summable sequences and q-lacunary almost statistical convergence,” Vietnam Journal of Mathematics, vol. 34, no. 2, pp. 129–138, 2006.
[16]
A. M. Jarrah and E. Malkowsky, “The space bv(p), its β-dual and matrix transformations,” Collectanea Mathematica, vol. 55, no. 2, pp. 151–162, 2004.
[17]
I. J. Maddox, “Statistical convergence in a locally convex space,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 104, no. 1, pp. 141–145, 1988.
[18]
M. Mursaleen, “Generalized spaces of difference sequences,” Journal of Mathematical Analysis and Applications, vol. 203, no. 3, pp. 738–745, 1996.
[19]
M. Mursaleen, “Matrix transformations between some new sequence spaces,” Houston Journal of Mathematics, vol. 9, no. 4, pp. 505–509, 1983.
[20]
M. Mursaleen, “On some new invariant matrix methods of summability,” The Quarterly Journal of Mathematics, vol. 34, no. 133, pp. 77–86, 1983.
[21]
K. Raj and S. K. Sharma, “Some sequence spaces in 2-normed spaces defined by Musielak-Orlicz function,” Acta Universitatis Sapientiae. Mathematica, vol. 3, no. 1, pp. 97–109, 2011.
[22]
K. Raj and S. K. Sharma, “Some generalized difference double sequence spaces defined by a sequence of Orlicz-functions,” Cubo, vol. 14, no. 3, pp. 167–189, 2012.
[23]
K. Raj and S. K. Sharma, “Some multiplier sequence spaces defined by a Musielak-Orlicz function in n-normed spaces,” New Zealand Journal of Mathematics, vol. 42, pp. 45–56, 2012.
[24]
H. Fast, “Sur la convergence statistique,” Colloquium Mathematicum, vol. 2, pp. 241–244, 1951.
[25]
I. J. Schoenberg, “The integrability of certain functions and related summability methods,” The American Mathematical Monthly, vol. 66, pp. 361–375, 1959.
[26]
J. A. Fridy, “On statistical convergence,” Analysis, vol. 5, no. 4, pp. 301–303, 1985.
[27]
J. Connor, “A topological and functional analytic approach to statistical convergence,” in Analysis of Divergence, Applied and Numerical Harmonic Analysis, pp. 403–413, Birkh？user, Boston, Mass, USA, 1999.
[28]
T. ？alát, “On statistically convergent sequences of real numbers,” Mathematica Slovaca, vol. 30, no. 2, pp. 139–150, 1980.
[29]
M. Mursaleen, “λ-statistical convergence,” Mathematica Slovaca, vol. 50, no. 1, pp. 111–115, 2000.
[30]
M. I？ik, “On statistical convergence of generalized difference sequences,” Soochow Journal of Mathematics, vol. 30, no. 2, pp. 197–205, 2004.
[31]
E. Sava？, “Strong almost convergence and almost λ-statistical convergence,” Hokkaido Mathematical Journal, vol. 29, no. 3, pp. 531–566, 2000.
[32]
E. Malkowsky and E. Savas, “Some λ-sequence spaces defined by a modulus,” Archivum Mathematicum, vol. 36, no. 3, pp. 219–228, 2000.
[33]
E. Kolk, “The statistical convergence in Banach spaces,” Acta et Commentationes Universitatis Tartuensis, vol. 928, pp. 41–52, 1991.
[34]
I. J. Maddox, “A new type of convergence,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 83, no. 1, pp. 61–64, 1978.
[35]
S. A. Mohiuddine and M. Aiyub, “Lacunary statistical convergence in random 2-normed spaces,” Applied Mathematics & Information Sciences, vol. 6, no. 3, pp. 581–585, 2012.
[36]
A. R. Freedman, J. J. Sember, and M. Raphael, “Some Cesàro-type summability spaces,” Proceedings of the London Mathematical Society, vol. 37, no. 3, pp. 508–520, 1978.
