On Paranormed Ideal Convergent Generalized Difference Strongly Summable Sequence Spaces Defined over n-Normed Spaces
Bipan Hazarika1
Academic Editor: G. Γlafsson
Received25 Jan 2011
Accepted03 Apr 2011
Published16 Jun 2011
Abstract
We introduce a new class of ideal convergent (shortly I-convergent) sequence spaces using an Orlicz function and difference operator of order defined over the n-normed spaces. We investigate these spaces for some linear topological structures. These investigations will enhance the acceptability of the notion of n-norm by giving a way to construct different sequence spaces with elements in n-normed spaces. We also give some relations related to these sequence spaces.
1. Introduction
Throughout the paper w, , c, , and ββdenote the classes of all, bounded, convergent, null, and p-absolutely summable sequences of complex numbers.
The notion of statistical convergence is a very useful functional tool for studying the convergence problems of numerical sequences/matrices (double sequences) through the concept of density. It was first introduced by Fast [1], and Schoenberg [2], independently for the real sequences. Later on, it was further investigated from sequence point of view and linked with the summability theory by Fridy [3] and many others. The idea is based on the notion of natural density of subsets of , the set of positive integers, which is defined as follows: the natural density of a subset E of is denoted by Ξ΄(E) and is defined by
where the vertical bar denotes the cardinality of the respective set.
The notion of I-convergence (I denotes the ideal of subsets of ), which is a generalization of statistical convergence, was introduced by Kastyrko et al. [4] and further studied by many other authors.
The notion of difference sequence space was introduced by Kizmaz [5], who studied the difference sequence spaces , , and . The notion was further generalized by Et and Γolak [6] by introducing the sequence spaces , , and . Another type of generalization of the difference sequence spaces is due to Tripathy and Esi [7], who studied the spaces , , and . Tripathy et al. [8] generalized the above notion and unified these as follows.
For nonnegative integers m, s, the generalized difference sequence spaces are defined as follows: for a given sequence space Z, we have
where = , for all , the difference operator is equivalent to the following binomial representation:
Taking , we get the spaces , , and , introduced and studied by Tripathy and Et [7]. Taking , we get the spaces , , and studied by Et and Γolak [6]. Taking , we get the spaces , , and introduced and studied by Kizmaz [5].
Let m, s be nonnegative integers, then for a given sequence space Z, we introduced
where ,, for allββ, the difference operator is equivalent to the following binomial representation:
where , for .
The concept of 2-normed space was initially introduced by GΓ€hler [9], in the mid of 1960s, while that of -normed spaces can be found in Misiak [10]. Since then, many others authors have used this concept and obtained various results, see, for instance, Gunawan [11] and Gunawan and Mashadi [12, 13]. Recently, a lot of activities have started to study summability, sequence spaces, and related topics in these spaces (see [14, 15]).
The notion of ideal-convergence in 2-normed spaces was initially introduced by GΓΌrdal [16]. Later on, it was extended to n-normed spaces by GΓΌrdal and Sahnier [17].
An Orlicz function is a function , which is continuous, nondecreasing and convex with , , as and , as .
An Orlicz function can always be represented in the following integral form:
where p is the known kernel of M, right differentiable for , , , for and , as .
If convexity of Orlicz function M is replaced by , then this function is called Modulus function, which was presented and discussed by Maddox [18].
Lindenstrauss and Tzafriri [19] used the idea of Orlicz function to construct the sequence space
The space with the norm
becomes a Banach space which is called an Orlicz sequence space.
The space ββis closely related to the space which is an Orlicz sequence space with , for .
Subsequently, Orlicz function was used to define sequence spaces by Parashar and Choudhary [20] and others (see [10, 15, 21β23]).
Remark 1.1. It is well known if M is a convex function and M(0) = 0, then
In this paper, we introduce some new ideal convergent sequences over n-normed spaces by using Orlicz functions and generalized difference sequence spaces.
2. Definitions and Preliminaries
Let n be a nonnegative integer and X a real vector space of dimension (d may be infinite). A real-valued function on satisfying the following conditions:(1) if and only if are linearly dependent,(2) is invariant under permutation,(3), for any Ξ±β ,(4), is called an n-norm on and the pair () is called an -normed space.
A trivial example of an n-normed space is , equipped with the Euclidean n-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 n-normed space of dimension and a linearly independent set in . Then, the function is defined by
defines as -norm on with respect to , and this is known as the derived -norm (see [12] for details).
The standard n-norm on , a real inner product space of dimension , is as follows:
where denotes the inner product on . If we take , then this n-norm is exactly the same as the Euclidean n-norm ββmentioned earlier.
For , this n-norm is the usual norm .
Definition 2.1. A sequence () in an n-normed space () is said to converge to some with respect the n-norm if for each , there exists an positive integer such that
Definition 2.2. A sequence () in an n-normed space is said to be Cauchy with respect the n-norm if for each , there exists an positive integer such that
If every Cauchy sequence in converges to some , then is said to be complete with respect to the n-norm. Any complete n-normed space is said to be n-Banach space.
Definition 2.3. Let S be a nonempty set. A nonempty family of sets (power set of S) is called an ideal in S if (i) for each , we have ; (ii) for each and , we have .
Definition 2.4. Let S be a nonempty set. A family (power set of S) is called a filter on S if (i) ; (ii) for each A, , we have ; (iii) for each and , we have .
An ideal is called nontrivial if and . It is clear that is a nontrivial ideal if and only if the class is a filter on S. The filter is called the filter associated with the ideal .
A nontrivial ideal is called an admissible ideal in if and only if it contains all singletons, that is, if it contains .
Definition 2.5. A sequence () in a normed space is said to be statistically convergent to if for each , the set has natural density zero.
Definition 2.6. A sequence () in a normed space is said to be I-convergent to if for each , the set
Definition 2.7. A sequence () in a normed space is said to be I-bounded if there exists such that the set belongs to .
Definition 2.8. A sequence () in a normed space () is said to be I-Cauchy if for each , there exists an positive integer such that the set
Definition 2.9. A sequence () in an n-normed space () is said to be I-convergent to with respect the n-norm, if for each , the set
Definition 2.10. A sequence () in an n-normed space () is said to be I-Cauchy with respect to the n-norm if for each , there exists a positive integer such that the set
Definition 2.11. A sequence space is said to be normal (or solid) if , whenever and for all sequence () of scalars with , for all .
Let and a sequence space.A K-step space of is a sequence space .
A canonical preimage of a sequence is a sequence defined as
A canonical preimage of a step space is a set of canonical preimages of all elements in; that is, y is in canonical preimage of if and only if y is canonical preimage of some .
Definition 2.12. A sequence space is said to be monotone if it contains the canonical preimages of all its step spaces.
Example 2.13. If we take . Then, is a nontrivial admissible ideal ofββββand the corresponding convergence coincide with usual convergence of sequences in n-normed spaces.
Example 2.14. If we take , where denote the asymptotic density of the set A then is a nontrivial admissible ideal of and the corresponding convergence coincide with statistical convergence in n-normed spaces.
Now, we state the following lemmas (see [12] for details).
Lemma 2.15. Every n-normed space is an -normed space for all . In particular, every n-normed space is a normed space.
Lemma 2.16. A standard n-normed space is complete if and only if it is complete with respect to the usual norm .
Lemma 2.17. On a standard n-normed space , the derived -norm defined with respect to the orthogonal set is equivalent to the standard -norm . To be precise, one has
for all , where.
Lemma 2.18 (Kamthan and Gupta [24]). Every normal space is monotone.
3. Main Results
In this section, we define some new ideal convergent sequence spaces and investigate their linear topological structures. Also, we find out some relations related to these sequence spaces.
Let be an admissible ideal of , an Orlicz function, and () an n-normed space. Further, let be any bounded sequence of positive real numbers, and let , denote the spaces of all X-valued sequences. For each , we define the following sequence spaces:
The following well-known inequality will be used throughout the article. Let be any sequence of positive real numbers with , thenββ, for all and . Also, ,ββfor all .
Theorem 3.1. The spaces , ,ββand , β βare linear.
Proof. We will prove the result for the space only and the others can be proved similarly. Let and be any two elements of the space . Then, there exist and such that the sets
belong to . Let be two scalars. Since is an n-norm, the operator is linear and the Orlicz function M is continuous, the following inequality holds:
where . From the above relation, we get
Since both the sets on the right hand side of the relation (3.7) belong to , this completes the proof.
Note 1. It is easy to verify that the space is a linear space.
Theorem 3.2. The space , , , , ββand , are paranormed spaces (not totally paranormed ) with respect to the paranorm defined by
where .
Proof. Clearly, and . Let and be any two elements of the space ), where . Then,for , we set
Let and . If , then we have
Thus, we have
Let , where , and let as . We have to show that as . We set
If andββ, we observe by the continuity of the Orlicz function M that
From the above inequality, it follows that
and consequently
Note that , for all . Hence, by our assumption, the right hand side (3.17) tends to 0 as , and the result follows. This completes the proof of the theorem.
Theorem 3.3. Let , and ββbe Orlicz functions. Then, the following hold: (a), provided be such that ,(b).
Proof. (a) Let be given. Choose such that . Using the continuity of the Orlicz function M, choose such that implies that . Let be any element in . Put
Then, by definition of ideal, we have the set . If , then we have
Using the continuity of the Orlicz function M, then from the relation (3.19), we have
Consequently, we get
This shows that
This proves the result. (b) Let . Then, by the following inequality, the result follows:
Theorem 3.4. The inclusions β βare strict for . In general, , for ,ββand the inclusion is strict, where .
Proof. We will give the proof for only. The others can be proved by similar arguments. Let be any element in the space . Let be given. Then, there exists Ο > 0 such that the set
Since M is nondecreasing and convex, it follows that
where . Thus, we have
Since both the sets in the right side of the relation (3.26) belongs to I, we get the set
Let , for all , , for all . Consider a sequence . Then, x belongs to but does not belong to . This shows that the inclusion is strict.
Theorem 3.5. Let ββand ββbe bounded, then
Proof. Let . We put
For all . Let be such that , for all . Define the sequences and as follows:ββfor , let and ; for , let and . Then, clearly, for all , we have and . Therefore, we have
Hence, ).
Theorem 3.6. For any two sequences and ββof positive real numbers and for any two n-norms ββand ββββon X, the following holds:
where .
Proof. Proof of the theorem is obvious, because the zero element belongs to each of the sequence spaces involved in the intersection.
Theorem 3.7. The sequence spaces ) are normal as well as monotone, where .
Proof. We will give the proof for only. Let be any element in , and let ββbe a sequence of scalars such that , for all . Then, we have
where . Hence, , for all sequence of scalars with , for all , whenever . By Lemma 2.18, we have the space ) is monotone.
Remark 3.8. If we replace the difference operator by , then for each , we get the following sequence spaces:
Note 2. For , we write the above spaces as ), where , , ,ββand . It is clear from definitions that
Corollary 3.9. The sequence spaces ) are paranormed spaces (not totally paranormed ) with respect to the paranormed ββdefined by
where and , , , and .
Remark 3.10. It is obvious that ) if and only if ), for , , ββand . Also it is clear that the paranorm and are equivalent.
We state the following Theorem in view of the Lemma 2.17.
Theorem 3.11. Let X be a standard n-normed space and an orthogonal set in X. Then, the following hold: (a),
(b),
(c),
(d),β βwhere is the derived -norm defined with respect to the set and is the standard -norm on X.
Remark 3.12. Theorem 3.11 also holds if we replace the difference operator ββby the difference operator .
Theorem 3.13. The spaces ) and ) are equivalent as topological spaces, where , and .
Proof. Consider the mapping defined by
Then, clearly, T is a linear homeomorphism and the proof follows.
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