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Double sequence spaces over n-normed spaces defined by a sequence of Orlicz functions
Journal of Inequalities and Applications volume 2014, Article number: 216 (2014)
Abstract
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.
MSC:40A05, 46A45.
1 Introduction and preliminaries
The initial works on double sequences is found in Bromwich [1]. Later on, it was studied by Hardy [2], Moricz [3], Moricz and Rhoades [4], Başarır and Sonalcan [5] and many others. Hardy [2] introduced the notion of regular convergence for double sequences. Quite recently, Zeltser [6] 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 [7] have recently introduced the statistical convergence which was further studied in locally solid Riesz spaces [8]. Nextly, Mursaleen [9] and Mursaleen and Savas [10] 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 [11] 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 [12] 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 [13] 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 [14]. 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 [15]). 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 .
Let be a double sequence. A set is defined by
If for all , then E is said to be symmetric. Now let be a family of subsets σ having at most elements s in ℕ. Also denotes the class of subsets in such that the element numbers of and are at most s and t, respectively. Besides is taken as a non-decreasing double sequence of the positive real numbers such that
An Orlicz function is a continuous, non-decreasing, and convex function such that , for and as .
Lindenstrauss and Tzafriri [24] used the idea of Orlicz function to define the following sequence space:
which is called an Orlicz sequence space. Also is a Banach space with the norm
Also, it was shown that every Orlicz sequence space contains a subspace isomorphic to (). The -condition is equivalent to , for all L with . An Orlicz function M can always be represented in the following integral form:
where η, known as the kernel of M, is right differentiable for , , , η is non-decreasing and as .
For further reading on Orlicz spaces, we refer to [25–29].
Let X be a linear metric space. A function is called a 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 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 [30], Theorem 10.4.2, p.183).
The concept of 2-normed spaces was initially developed by Gähler [31] in the mid-1960s, while that of n-normed spaces one can see in Misiak [32]. Since then, many others have studied this concept and obtained various results; see Gunawan [33, 34] and Gunawan and Mashadi [35] and references therein. Let and X be a linear space over the field , where is the field of real or complex numbers of dimension d, where . A real valued function on satisfying the following four conditions:
-
(1)
if and only if are linearly dependent in X;
-
(2)
is invariant under permutation;
-
(3)
for any , and
-
(4)
is called a n-norm on X, and the pair is called a n-normed space over the field . For example, we may take being equipped with the Euclidean n-norm , the volume of the n-dimensional parallelepiped spanned by the vectors which may be given explicitly by the formula
where for each . Let be a n-normed space of dimension and be linearly independent set in X. Then the function on defined by
defines an -norm on X with respect to .
A sequence in a n-normed space is said to converge to some if
A sequence in a n-normed space is said to be Cauchy if
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.
The space was introduced by Sargent [36]:
which was further studied in [37, 38] and [39]. Recently, Duyar and Oǧur [40] introduced the sequence space and studied some of its properties.
Let be an infinite double matrix of complex numbers, be a sequence of Orlicz functions, and be a bounded double sequence of positive real numbers. In the present paper we define the following sequence space:
where if converges for each .
If , we have
The following inequality will be used throughout the paper:
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.
Proof Let and . Then there exist positive numbers and such that
and
Let . Since ℳ is a non-decreasing and convex function, we have
Thus, we have
This proves that . Hence is a linear space. This completes the proof of the theorem. □
Theorem 2.2 be a sequence of Orlicz functions and be a bounded sequence of positive real numbers, then the space is a paranormed space with the paranorm defined by
Proof It is clear that and if . Then there exist positive numbers and such that
and
Then, by using Minkowski’s inequality, we have
where . This shows that . Using this triangle inequality we can write
Thus we have
Thus
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 .
Proof Let . Then for all . If , then
Thus
and hence . This shows that
Conversely, let and for all , and suppose . Then there exists a subsequence of such that . If , then we have
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. □
Theorem 2.5 Let , and be sequences of Orlicz functions satisfying -condition. Then
-
(i)
,
-
(ii)
.
Proof (i) Let . Then there exists such that
By the continuity of ℳ, we can take a number δ with such that , whenever , for arbitrary . Now let
Thus we have
By the properties of the Orlicz function we have
Again, we have
for . If ℳ satisfies the -condition, then we have
and so
Hence, we have
Thus, we have and hence .
-
(ii)
Let . Then there exists a such that
and
By the inequality, we have
Hence
This completes the proof of the theorem. □
Theorem 2.6 The sequence space is solid.
Proof Let be a double sequence of scalars such that and . Then we have
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. □
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Alotaibi, A., Mursaleen, M. & Sharma, S.K. Double sequence spaces over n-normed spaces defined by a sequence of Orlicz functions. J Inequal Appl 2014, 216 (2014). https://doi.org/10.1186/1029-242X-2014-216
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DOI: https://doi.org/10.1186/1029-242X-2014-216