Some generalized difference statistically convergent sequence spaces in 2-normed space
© Başarır et al.; licensee Springer 2013
Received: 12 December 2012
Accepted: 2 April 2013
Published: 16 April 2013
In this paper, we define a new generalized difference matrix and introduce some -difference statistically convergent sequence spaces in a real linear 2-normed space. We also investigate some topological properties of these spaces.
MSC:40A05, 46A45, 46E30.
Keywordsstatistical convergence generalized difference sequence space 2-norm paranorm completeness solidity
The concept of 2-normed spaces has been initially introduced by Gähler in the 1960s , as an interesting non-linear generalization of a normed linear space, which has been subsequently studied by many authors [27–29]. Since then, a lot of activities have been started to study summability, sequence spaces and related topics on 2-normed spaces [30–33]. Recently, some difference sequence spaces have been introduced in 2-normed spaces by several authors [30, 31, 34].
where and .
where the vertical bar denotes the cardinality of the enclosed set.
2 Definitions and preliminaries
A sequence space E is said to be solid (or normal) if implies for all sequences of scalars with for all .
A linear topological space X over the real field R is said to be a paranormed space if there is a sub-additive function such that , , and scalar multiplication is continuous, i.e. and imply that for all λ’s in ℝ and all x’s in X, where θ is the zero vector in the linear space X.
The following inequality will be used throughout the paper:
and for .
if and only if , are linearly dependent,
for any ,
where stands for the inner product on X .
Now we will give the following known example for 2-normed spaces.
where and . Then is a 2-norm on Z.
If every Cauchy sequence in X converges to some , then X is said to be complete with respect to the 2-norm. Any complete 2-normed space is said to be a 2-Banach space .
for each nonzero z in X. For , we say this is statistically null .
Firstly, we give the following lemma, which we need to establish our main results.
Lemma 2.2 
Every closed linear subspace F of an arbitrary linear normed space E, different from E, is a nowhere dense set in E.
Throughout the paper , , , , , , and denote the spaces of all, convergent, null, statistically convergent, statistically null, bounded, bounded statistically convergent and bounded statistically null X valued sequence spaces, where is a real 2-normed space. By , we mean the zero element of X.
3 Main results
In this section, we define the generalized difference matrix and introduce difference sequence spaces , , , , , , , , which are defined on a real linear 2-normed space. We investigate some topological properties of the spaces , , and including linearity, existence of paranorm and solidity. Further, we show that the sequence spaces and are complete paranormed spaces when the base space is a 2-Banach space. Moreover, we give some inclusion relations.
If we take then the above sequence spaces are reduced to , , , , , , and , respectively.
If we take , , then the sequence spaces , , , , , are reduced to , , , , and , respectively, which are studied in .
By taking for all , then these sequence spaces are denoted by , , , , , , and , respectively.
If we replace the base space X, which is a real linear 2-normed space by ℂ, complete normed linear space, and take and take , , then the above sequence spaces are denoted by , , , , , , and , respectively.
If we take , , for all , then these sequence spaces are denoted by , , , , , , and , respectively.
If we replace the base space X, which is a real linear 2-normed space by ℂ, we obtain the spaces , , , , , , and , respectively.
Moreover, if we take , and for all , we get the spaces , , , c, , W, m and , respectively.
Theorem 3.1 Let be a sequence of strictly positive real numbers. Then the sequence spaces are linear spaces where .
Proof The proof of the theorem can be obtained by similar techniques in . □
Theorem 3.2 For any two sequences and of positive real numbers and for any two 2-norms and on X we have , where .
Proof The proof follows from the fact that the zero element belongs to each of the sequence spaces involved in the intersection. □
where and , .
Proof We will prove the theorem for the sequence space . It can be proved for the space similarly.
This implies that .
which tends to zero as . Hence, g is a paranorm on the sequence space .
for all . It follows that . Since and is a linear space, so we have . This completes the proof. □
If , then and the inclusion is strict, where .
If , then and the inclusion is strict, where .
Proof The parts of proof and are easy. To show the inclusions are strict, choose , , and consider the 2-norm as defined in (2.1), let for all , , , , , then but . If we choose , and for all , , , , , then but . These complete the proofs of parts (1) and (2) of the theorem, respectively. □
and the inclusion is strict.
and the inclusion is strict.
and overlap but neither one contains the other.
- (1)It is clear that . To show that the inclusion is strict, choose the sequence such that,(3.2)
It is easy to see that . To show that the inclusion is strict, let us take and consider the 2-norm as defined in (2.1), , , , , then but .
Since the sequence belongs to each of the sequence spaces, the overlapping part of the proof is obvious. For the other part of the proof, consider the sequence defined by (3.2) and the 2-norm as defined in (2.1). Then , but . Conversely if we choose where for all , then but . That is, but .
Theorem 3.6 The space is not solid in general, where .
Proof To show that the space is not solid in general, consider the following examples. □
Example 3.7 Let , , , and consider the 2-normed space as defined in Example 2.1. Let for all . Consider the sequence , where is defined by for each fixed . Then for . Let , then for . Thus for is not solid in general.
Example 3.8 Let , , , and consider the 2-normed space as defined in Example 2.1. Let for all odd k and for all even k. Consider the sequence , where is defined by for each fixed . Then for . Let , then for . Thus for is not solid in general.
Theorem 3.9 The spaces and are nowhere dense subsets of .
are strict, the spaces and are nowhere dense subsets of by Lemma 2.2. □
If we take the limit for , it follows that from the inequality above. Since , we have the result. □
Dedicated to Professor Hari M Srivastava.
The authors would like to thank the anonymous reviewers for their comments and suggestions to improve the quality of the paper.
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