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Abstract and Applied Analysis
Volume 2014, Article ID 790950, 8 pages
Research Article

The Ideal Convergence of Strongly of   in Metric Spaces Defined by Modulus

1Department of Mathematics, SASTRA University, Thanjavur 613 401, India
2Department of Mathematics, Srinivasa Ramanujan Centre, SASTRA University, Kumbakonam 612 001, India

Received 9 January 2014; Accepted 25 April 2014; Published 20 May 2014

Academic Editor: Feyzi Başar

Copyright © 2014 N. Subramanian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The aim of this paper is to introduce and study a new concept of the space via ideal convergence defined by modulus and also some topological properties of the resulting sequence spaces were examined.

1. Introduction

Let be a double sequence of real or complex numbers. Then the series is called a double series. The double series is said to be convergent if and only if the double sequence is convergent, where We denote as the class of all complex double sequences . A sequence is said to be double analytic if The vector space of all prime sense double analytic sequences is usually denoted by . A sequence is called double entire sequence if The vector space of all prime sense double entire sequences is usually denoted by . The space is a metric space with the metric The space is a metric space with the metric for all and in .

Consider a double sequence . The section of the sequence is defined by for all , with 1 in the position and zero otherwise. An FK-space (or a metric space) is said to have AK property if is a Schauder basis for . Or equivalently . We need the following inequality in the sequel of the paper.

Lemma 1. For and , one has

Some initial work on double sequence spaces is found in Bromwich. Later on it was investigated by Moricz [1], Moricz and Rhoades [2], Basarir and Solancan [3], Tripathy [4], Turkmenoglu [5], Subramanian and Misra [6, 7], and many others. Tripathy and Dutta [8] introduced and investigated different types of fuzzy real valued double sequence spaces. Generalizing the concept of ordinary convergence for real sequences Kostyrko et al. introduced the concept of ideal convergence which is a generalization of statistical convergence, by using the ideal of the subsets of the set of natural numbers.

The notion of different sequence spaces (for single sequences) was introduced by Kizmaz [9] as follows: for and , where for all . Here , , , and denote the classes of all, convergent, null, and bounded scalar valued single sequences, respectively. The above spaces are Banach spaces normed by Later on the notion was further investigated by many others. We now introduce the following difference double sequence spaces defined by where and , respectively. for all . We further generalized this notion and introduced the following notion. For , An Orlicz function is a function which is continuous, nondecreasing, and convex with , , for and as . If convexity of Orlicz function is replaced by , then this function is called modulus function. A modulus function is said to satisfy -condition for all values , if there exists such that , .

Remark 2. A modulus function satisfies the inequality for all with .

Lemma 3. Let be a modulus function which satisfies -condition and let . Then for each , one has for some constant .

Spaces of strongly summable sequences were discussed by Kuttner, Maddox, and others. The class of sequences which are strongly Cesàro summable with respect to a modulus was introduced by Maddox as an extension of the definition of strongly Cesàro summable sequences. Connor further extended this definition to a definition of strong -summability with respect to a modulus where is a nonnegative regular matrix and established some connections between strong -summability, strong -summability with respect to a modulus, and -statistical convergence. The notion of convergence of double sequences was presented by A. Pringsheim. Also, the four-dimensional matrix transformation was studied extensively by Robison and Hamilton.

2. Definitions and Preliminaries

Let be a nonempty set. A nonvoid class (power set, of ) is called an ideal if is additive (i.e., ) and hereditary (i.e., and ). A nonempty family of sets is said to be a filter on if ; and , . For each ideal there is a filter given by . A nontrivial ideal is called admissible if and only if .

A double sequence space is said to be solid or normal if , whenever and for all double sequences of scalars with , for all .

Let and let be a real vector space of dimension , where . A real valued function on satisfies the following four conditions:(i) if and and only if are linearly dependent,(ii) is invariant under permutation,(iii), ,(iv) for , or(v), for , , is called the product metric of the Cartesian product of metric spaces which is the norm of the -vector of the norms of the subspaces.

A trivial example of product metric of metric spaces is the norm space equipped with the following Euclidean metric in the product space which is the norm: where for each .

3. Main Results

In this section we introduce the notion of different types of -convergent double sequences. This generalizes and unifies different notions of convergence for . We will denote the ideal of by .

Let be an ideal of , a modulus function, a double analytic sequence of strictly positive real numbers, and a -product of metric spaces which is the norm of the -vector of the norms of the subspaces. Further denotes -valued sequence space. Now, we define the following sequence spaces: If for all we obtain The following well-known inequality will be used in this study: , ; then for all and . Also for all .

Theorem 4. The sets and are linear spaces over the complex field

Proof. Now only prove and the others can be proved similarly. Let and . Then Since is a -product of metric spaces which is the norm of the -vector of the norms of the subspaces and is a modulus function, the following inequality holds: From the above inequality we get This completes the proof.

Theorem 5. paranormed space with respect to the paranorm is defined by

Proof. and are easy to prove, so we omit them. Let us take . Let Then we have Thus and .
Now, let , where and as . We have to prove that as . Let We observe that From this inequality, it follows that and consequently Hence by our assumption the right-hand side tends to zero as , and . This completes the proof.

Theorem 6. (i) If , then .
(ii) If , then .
(iii) If and is analytic, then .

Proof. The proof is easy. Therefore omit it.

Lemma 7. If a sequence is solid, then it is monotone. (See [10, page 53].)

Theorem 8. is solid and also monotone.

Proof. Let and be scalars such that for . Then we have where . Hence with for all whenever . Also by Lemma 7, it follows that is monotone. This completes the proof.

Theorem 9. Let , , and be modulus functions. Then one has (i),(ii).

Proof. (i) Let . For given , we first choose such that . Now using the continuity of , choose such that implies . Let .
We observe that Thus if then Hence from above inequality and using continuity of , we must have Hence we have
(ii) Let . Then This completes the proof.

Theorem 10. Let the double sequence be analytic. Then
and the inclusion are strict.

Theorem 11. The class of sequence is sequence algebras.

Proof. Let and . Then the result follows from the following inclusion relation: Similarly we can prove the result for other cases.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


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