Double sequence spaces over n-normed spaces defined by a sequence of Orlicz functions
© Alotaibi et al.; licensee Springer. 2014
Received: 21 February 2014
Accepted: 9 May 2014
Published: 29 May 2014
In the present paper we introduce double sequence space defined by a sequence of Orlicz functions over n-normed space. We examine some of its topological properties and establish some inclusion relations.
Keywordsdouble sequence spaces paranormed space Orlicz function n-normed space
1 Introduction and preliminaries
The initial works on double sequences is found in Bromwich . Later on, it was studied by Hardy , Moricz , Moricz and Rhoades , Başarır and Sonalcan  and many others. Hardy  introduced the notion of regular convergence for double sequences. Quite recently, Zeltser  in her PhD thesis has essentially studied both the theory of topological double sequence spaces and the theory of summability of double sequences. Mursaleen and Edely  have recently introduced the statistical convergence which was further studied in locally solid Riesz spaces . Nextly, Mursaleen  and Mursaleen and Savas  have defined the almost regularity and almost strong regularity of matrices for double sequences and applied these matrices to establish core theorems and introduced the M-core for double sequences and determined those four dimensional matrices transforming every bounded double sequences into one whose core is a subset of the M-core of x. More recently, Altay and Başar  have defined the spaces , , , , and of double sequences consisting of all double series whose sequence of partial sums are in the spaces , , , , and , respectively and also examined some properties of these sequence spaces and determined the α-duals of the spaces , , and the -duals of the spaces and of double series. Recently Başar and Sever  have introduced the Banach space of double sequences corresponding to the well known space of single sequences and examined some properties of the space . Now, recently Raj and Sharma  have introduced entire double sequence spaces. By the convergence of a double sequence we mean the convergence in the Pringsheim sense i.e. a double sequence has Pringsheim limit L (denoted by ) provided that given there exists such that whenever , see . The double sequence is bounded if there exists a positive number M such that for all k and l.
Throughout this paper, ℕ and ℂ denote the set of positive integers and complex numbers, respectively. A complex double sequence is a function x from into ℂ and briefly denoted by . If for all , there is such that where and , then a double sequence is said to be convergent to . A real double sequence is non-decreasing, if for . A double series is infinite sum and its convergence implies the convergence of partial sums sequence , where (see ). For recent development on double sequences, we refer to [16–20] and [21–23].
A double sequence space E is said to be solid if for all double sequences of scalars such that for all whenever .
An Orlicz function is a continuous, non-decreasing, and convex function such that , for and as .
where η, known as the kernel of M, is right differentiable for , , , η is non-decreasing and as .
for all ,
for all ,
for all ,
if is a sequence of scalars with as and is a sequence of vectors with as , then as .
A paranorm p for which implies is called a total paranorm and the pair is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see , Theorem 10.4.2, p.183).
if and only if are linearly dependent in X;
is invariant under permutation;
for any , and
defines an -norm on X with respect to .
If every Cauchy sequence in X converges to some , then X is said to be complete with respect to the n-norm. Any complete n-normed space is said to be n-Banach space.
where if converges for each .
where and .
We examine some topological properties of and establish some inclusion relations.
2 Main results
Theorem 2.1 Let be a sequence of Orlicz functions and be a bounded sequence of positive real numbers, then the space is linear space over the field of complex number ℂ.
This proves that . Hence is a linear space. This completes the proof of the theorem. □
Hence where and as . This proves that is a paranormed space with the paranorm defined by g. This completes the proof of the theorem. □
Theorem 2.3 Let ϕ and ψ be two double sequences then if and only if .
This is a contradiction as . This completes the proof of the theorem. □
Corollary 2.4 Let ϕ and ψ be two double sequences then if and only if and .
Proof It is easy to prove so we omit the details. □
- (ii)Let . Then there exists a such that
This completes the proof of the theorem. □
Theorem 2.6 The sequence space is solid.
This implies that . This proves that the space is a solid. □
Corollary 2.7 The sequence space is monotone.
Proof It is trivial so we omit the details. □
- Bromwich TJ: An Introduction to the Theory of Infinite Series. Macmillan Co., New York; 1965.MATHGoogle Scholar
- Hardy GH: On the convergence of certain multiple series. Proc. Camb. Philos. Soc. 1917, 19: 86–95.Google Scholar
- Moricz F: Extension of the spaces c and from single to double sequences. Acta Math. Hung. 1991, 57: 129–136. 10.1007/BF01903811MathSciNetView ArticleMATHGoogle Scholar
- Moricz F, Rhoades BE: Almost convergence of double sequences and strong regularity of summability matrices. Math. Proc. Camb. Philos. Soc. 1988, 104: 283–294. 10.1017/S0305004100065464MathSciNetView ArticleMATHGoogle Scholar
- Başarır M, Sonalcan O: On some double sequence spaces. J. Indian Acad. Math. 1999, 21: 193–200.MathSciNetMATHGoogle Scholar
- Zeltser M Diss. Math. Univ. Tartu 25. In Investigation of Double Sequence Spaces by Soft and Hard Analytical Methods. Tartu University Press, Tartu; 2001.Google Scholar
- Mursaleen M, Edely OHH: Statistical convergence of double sequences. J. Math. Anal. Appl. 2003,288(1):223–231. 10.1016/j.jmaa.2003.08.004MathSciNetView ArticleMATHGoogle Scholar
- Mohiuddine SA, Alotaibi A, Mursaleen M: Statistical convergence of double sequences in locally solid Riesz spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 719729 10.1155/2012/719729Google Scholar
- Mursaleen M: Almost strongly regular matrices and a core theorem for double sequences. J. Math. Anal. Appl. 2004,293(2):523–531. 10.1016/j.jmaa.2004.01.014MathSciNetView ArticleMATHGoogle Scholar
- Mursaleen M, Savas E: Almost regular matrices for double sequences. Studia Sci. Math. Hung. 2003, 40: 205–212.MathSciNetMATHGoogle Scholar
- Altay B, Başar F: Some new spaces of double sequences. J. Math. Anal. Appl. 2005, 309: 70–90. 10.1016/j.jmaa.2004.12.020MathSciNetView ArticleMATHGoogle Scholar
- Başar F, Sever Y: The space of double sequences. Math. J. Okayama Univ. 2009, 51: 149–157.MathSciNetMATHGoogle Scholar
- Raj K, Sharma SK: Some multiplier double sequence spaces. Acta Math. Vietnam. 2012, 37: 391–406.MathSciNetMATHGoogle Scholar
- Pringsheim A: Zur Theorie der zweifach unendlichen Zahlenfolgen. Math. Ann. 1900, 53: 289–321. 10.1007/BF01448977MathSciNetView ArticleGoogle Scholar
- Limaye BV, Zeltser M: On the Pringsheim convergence of double series. Proc. Est. Acad. Sci. 2009, 58: 108–121. 10.3176/proc.2009.2.03MathSciNetView ArticleMATHGoogle Scholar
- Cakan C, Altay B, Mursaleen M: The σ -convergence and σ -core of double sequences. Appl. Math. Lett. 2006, 19: 1122–1128. 10.1016/j.aml.2005.12.003MathSciNetView ArticleMATHGoogle Scholar
- Mursaleen M, Mohiuddine SA: Double σ -multiplicative matrices. J. Math. Anal. Appl. 2007, 327: 991–996. 10.1016/j.jmaa.2006.04.081MathSciNetView ArticleMATHGoogle Scholar
- Mursaleen M, Mohiuddine SA: Regularly σ -conservative and σ -coercive four dimensional matrices. Comput. Math. Appl. 2008, 56: 1580–1586. 10.1016/j.camwa.2008.03.007MathSciNetView ArticleMATHGoogle Scholar
- Mursaleen M, Mohiuddine SA: On σ -conservative and boundedly σ -conservative four dimensional matrices. Comput. Math. Appl. 2010, 59: 880–885. 10.1016/j.camwa.2009.10.006MathSciNetView ArticleMATHGoogle Scholar
- Mursaleen M, Mohiuddine SA: Convergence Methods for Double Sequences and Applications. Springer, Berlin; 2014.View ArticleMATHGoogle Scholar
- Mohiuddine SA, Alotaibi A: Some spaces of double sequences obtained through invariant mean and related concepts. Abstr. Appl. Anal. 2013., 2013: Article ID 507950Google Scholar
- Mohiuddine SA, Raj K, Alotaibi A: Some paranormed double difference sequence spaces for Orlicz functions and bounded-regular matrices. Abstr. Appl. Anal. 2014., 2014: Article ID 419064Google Scholar
- Sharma SK, Raj K, Sharma AK: Some new double sequence spaces over n -normed space. Int. J. Appl. Math. 2012, 25: 255–269.MathSciNetMATHGoogle Scholar
- Lindenstrauss J, Tzafriri L: On Orlicz sequence spaces. Isr. J. Math. 1971, 10: 345–355.MathSciNetView ArticleMATHGoogle Scholar
- Musielak J Lecture Notes in Mathematics 1034. Orlicz Spaces and Modular Spaces 1983.Google Scholar
- Maligranda L Seminars in Mathematics 5. In Orlicz Spaces and Interpolation. Polish Academy of Science, Warsaw; 1989.Google Scholar
- Raj K, Sharma SK: Some multiplier sequence spaces defined by a Musielak-Orlicz function in n -normed spaces. N.Z. J. Math. 2012, 42: 45–56.MathSciNetMATHGoogle Scholar
- Raj K, Sharma SK: Some double sequence spaces defined by a sequence of Orlicz function. J. Math. Anal. 2012, 3: 12–20.MathSciNetMATHGoogle Scholar
- Raj K, Sharma SK: Some generalized difference double sequence spaces defined by a sequence of Orlicz-function. CUBO 2012, 14: 167–189. 10.4067/S0719-06462012000300011MathSciNetView ArticleMATHGoogle Scholar
- Wilansky A North-Holland Math. Stud. 85. Summability Through Functional Analysis 1984.Google Scholar
- Gähler S: Lineare 2-normierte Räume. Math. Nachr. 1965, 28: 1–43.View ArticleMATHGoogle Scholar
- Misiak A: n -Inner product spaces. Math. Nachr. 1989, 140: 299–319. 10.1002/mana.19891400121MathSciNetView ArticleMATHGoogle Scholar
- Gunawan H: On n -inner product, n -norms, and the Cauchy-Schwartz inequality. Sci. Math. Jpn. 2001, 5: 47–54.Google Scholar
- Gunawan H: The space of p -summable sequence and its natural n -norm. Bull. Aust. Math. Soc. 2001, 64: 137–147. 10.1017/S0004972700019754MathSciNetView ArticleMATHGoogle Scholar
- Gunawan H, Mashadi M: On n -normed spaces. Int. J. Math. Math. Sci. 2001, 27: 631–639. 10.1155/S0161171201010675MathSciNetView ArticleMATHGoogle Scholar
- Sargent WLC: Some sequence spaces related to the spaces. J. Lond. Math. Soc. 1960, 35: 161–171.MathSciNetView ArticleMATHGoogle Scholar
- Malkowsky E, Mursaleen M: Matrix transformations between FK-spaces and the sequence spaces and . J. Math. Anal. Appl. 1995, 196: 659–665. 10.1006/jmaa.1995.1432MathSciNetView ArticleMATHGoogle Scholar
- Tripathy BC, Sen M: On a new class of sequences related to the space . Tamkang J. Math. 2002, 33: 167–171.MathSciNetMATHGoogle Scholar
- Mursaleen M: On some geometric properties of a sequence space related to . Bull. Aust. Math. Soc. 2003, 67: 343–347. 10.1017/S0004972700033803MathSciNetView ArticleMATHGoogle Scholar
- Duyar C, Oǧur O: On a new space of double sequences. J. Funct. Spaces Appl. 2013., 2013: Article ID 509613Google Scholar
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