- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2013 (2013), Article ID 507950, 11 pages

http://dx.doi.org/10.1155/2013/507950

## Some Spaces of Double Sequences Obtained through Invariant Mean and Related Concepts

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 29 November 2012; Accepted 11 March 2013

Academic Editor: Elena Braverman

Copyright © 2013 S. A. Mohiuddine and Abdullah Alotaibi. 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.

#### Abstract

We introduce some double sequences spaces involving the notions of invariant mean (or -mean) and -convergence for double sequences while the idea of -convergence for double sequences was introduced by Çakan et al. 2006, by using the notion of invariant mean. We determine here some inclusion relations and topological results for these new double sequence spaces.

#### 1. Preliminaries, Background, and Notation

In 1900, Pringsheim [1] presented the following notion of convergence for double sequences. A double sequence is said to *converge to the limit * *in Pringsheim’s sense* (shortly, -*convergent to* ) if for every there exists an integer such that whenever . In this case, is called the -*limit* of .

A double sequence of real or complex numbers is said to be *bounded* if . The space of all bounded double sequences is denoted by .

If and is -convergent to , then is said to be *boundedly *-*convergent to * (shortly, *-convergent to *). In this case, is called the -*limit* of . The assumption of -convergent was made because a double sequence on which -convergent is not necessarily bounded.

In general, for any notion of convergence , the space of all -convergent double sequences will be denoted by , the space of all -convergent to double sequences by and the limit of a -convergent double sequence by -, where .

Let denote the vector space of all double sequences with the vector space operations defined coordinatewise. Vector subspaces of are called *double sequence spaces*. In addition to the above-mentioned double sequence spaces, we consider the double sequence space as
of all absolutely summable double sequences.

All considered double sequence spaces are supposed to contain where

We denote the pointwise sums , , and by , , and , respectively.

Let be a one-to-one mapping from the set of natural numbers into itself. A continuous linear functional on the space of bounded single sequences is said to be an *invariant mean* or a -*mean* if and only if (i) when the sequence has for all , (ii) , where , and (iii) for all .

Throughout this paper we consider the mapping which has no finite orbits, that is, for all integer and , where denotes the th iterate of at . Note that a -mean extends the limit functional on the space of convergent single sequences in the sense that for all , (see [2]). Consequently, the set of bounded sequences all of whose -means are equal. We say that a sequence is -*convergent* if and only if . Using this concept, Schaefer [3] defined and characterized -conservative, -regular, and -coercive matrices for single sequences. If is translation then is reduced to the set of almost convergent sequences [4]. Recently, Mohiuddine [5] has obtained an application of almost convergence for single sequences in approximation theorems and proved some related results.

In 2006, Çakan et al. [6] presented the following definition of -convergence for double sequences and further studied by Mursaleen and Mohiuddine [7–9]. A double sequence of real numbers is said to be -*convergent* to a number if and only if , where
while here the limit means -limit. Let us denote by the space of -convergent double sequences . For , the set is reduced to the set of almost convergent double sequences [10]. Note that .

Maddox [11] has defined the concepts of strong almost convergence and -convergence for single sequences and established inclusion relation between strong almost convergence, -convergence, and almost convergence for single sequence. Başarir [12] extended the notion of strong almost convergence from single sequences to double sequences and proved some interesting results involving this idea and the notion of almost convergence for double sequences. In the recent past, Mursaleen and Mohiuddine [13] presented the notions of absolute and strong -convergence for double sequences. A bounded double sequence is said to be *strongly *-*convergent* if there exists a number such that
while here the limit means -limit. In this case, we write -. Let us denote by the set of all strongly -convergent sequences . If is translation then is reduced to the set of strong almost convergence double sequences due to Başarir [12].

For more details of spaces for single and double sequences and related concepts, we refer to [14–31] and references therein.

In this paper, we define and study some new spaces involving the idea of invariant mean and -convergence for double sequences and establish a relation between these spaces. Further, we extend above spaces to more general spaces by considering the double sequences such that for all , and and prove some topological results.

#### 2. The Double Sequence Spaces

We construct the following spaces involving the idea of invariant mean and -convergence for double sequences: where with for all ;

*Remark 1. *If -, that is,
as , uniformly in , ; then

We remark that by using Abel’s transformation for single series We get Abel’s transformation for double series where

In the recent past, Altay and Başar [32] also presented another form of Abel’s transformation for double series.

#### 3. Inclusion Relations

In the following theorem, we establish a relationship between spaces defined in Section 2. Before proceeding further, first we prove the following lemmas which will be used to prove our inclusion relations.

Lemma 2. *Consider that - if and only if *(L1)*-; *(L2)* as (uniformly in , ); *(L3)* as (uniformly in , ); *(L4)* as (uniformly in , ),**
where
*

*Proof. *Suppose that -. Thus, we have -, that is, (L1) holds. We see that conditions (L2) and (L3) follows from the Remark 1. Write
By our assumption, that is, -, as uniformly in , . The condition (L1) implies that tends to zero as tending to uniformly in ; therefore as uniformly in and as uniformly in by the conclusion (L2) and (L3), respectively. Thus, (14) tends to zero as uniformly in , that is, (L4) holds.

Conversely, let (L1)–(L4) hold. Then,
() uniformly in .

Lemma 3. *One has
*

*Proof. *Since
First, we solve the expression in the first bracket
Now, the expression in the second bracket
Substituting (18) and (19) in (17), we get
We know that
From (21), we have

Thus, (20) becomes
Also (22) can be written as
Similarly, we can write
Using (24) and (25) in (23), we get
This implies that

Theorem 4. *One has the following inclusions and the limit is preserved in each case:*(i)*, *(ii)* if the conditions (L2) and (L3) of Lemma 2 hold, *(iii)*. *

*Proof. *(i) Let with -, say. Then,
This implies that
Also, we have
Hence, and
(ii) We have to show that . If , then we have
as , uniformly in , ; and
that is, -.

In order to prove that , it is enough to show that condition (L4) of Lemma 2 holds. Now,
Replacing and by and , respectively, we have
By Lemma 3, we have
So that we have
By using Abel’s transformation for double series in the right hand side of above equation, we have
0 as , uniformly in (by (32)). Hence, by Lemma 2, .

(iii) Let , and we have to show that
where is an absolute constant. Since , there exist integers such that
Hence, it is left to show that for fixed
From (40), we have
for every fixed , and for all , . Since
Accordingly,
This implies that
Using (42) and (45), we have
for every fixed , and for all , , where is a constant depending upon .

Now, for any given infinite double series denoted as “”, let us write
and be monotonically increasing. For simplicity in notation, we denote
Again from the definition of , it is easy to obtain
for all and with , , , . Further, we calculate

Thus, we have
Hence, it follows from (46) that for each fixed , ,
Hence, it follows from (52) that
where is independent of . By (49), we have
Also from (43) and (54), we have

#### 4. Topological Results

Here, we extend the spaces , , to more general spaces, respectively, denoted by , , , where is a double sequence of positive real numbers for all , and . First, we recall the notion of paranorm as follows.

A *paranorm* is a function defined on a linear space such that for all (P1) if (P2)(P3)(P4)If is a sequence of scalars with and with in the sense that , then , in the sense that .

A paranorm for which implies is called a *total paranorm *on , and the pair is called a *total paranormed space*. Note that each seminorm (norm) on is a paranorm (total) but converse needs not be true.

A paranormed space is a topological linear space with the topology given by the paranorm .

Now, we define the following spaces:

We remark that if is a constant sequence, then we write in place of .

Theorem 5. *Let be a bounded double sequence of strictly positive real numbers. Then, is a complete linear topological space paranormed by
**
where . If then is a Banach space and is a -normed space if .*

*Proof. *Let and be two double sequences. Then,
where and . Since
therefore
where and . From (60), we have that if , then . Thus, is a linear space. Without loss of generality, we can take

Clearly, , . From (58) and Minkowski’s inequality, we have
Hence,
Since is bounded away from zero, there exists a constant such that for all , . Now for , and so
that is, the scaler multiplication is continuous. Hence, is a paranorm on .

Let be a Cauchy sequence in , that is,
Since
it follows that
In particular,
Hence, is a Cauchy sequence in . Since is complete, there exists such that coordinatewise as . It follows from (66) that given , there exists such that
for . Now taking and in (69), we have for . This proves that and . Hence is complete. If is a constant then it is easy to prove the rest of the theorem.

Theorem 6. *One has the following:** is a complete paranormed space, paranormed by
**
which is defined on . If then is a Banach space and if , is -normed space.*

*Proof. *In order for the paranorm in (70) be defined, we require that
which is proved in the next theorem (i.e., Theorem 7). Using the standard technique as in the previous theorem, we can prove that is subadditive.

Now, we have to prove the continuity of scalar multiplication. Suppose that . Then, for there exist integers such that
If , then by (72) we have
Since for fixed
as , it follows from (73) and (74) that for fixed , as . Also, since implies that
It follows that for fixed , as . This proves the continuity of scalar multiplication. Hence, is a paranorm. The proof of the completeness of