- Open Access
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.
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. □
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