- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 507950, 11 pages
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.
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  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 ). 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  defined and characterized -conservative, -regular, and -coercive matrices for single sequences. If is translation then is reduced to the set of almost convergent sequences . Recently, Mohiuddine  has obtained an application of almost convergence for single sequences in approximation theorems and proved some related results.
In 2006, Çakan et al.  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 . Note that .
Maddox  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  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  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 .
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  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 , ),
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
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
(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
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 can be achieved by using the same technique as in Theorem 5.
Theorem 7. Suppose that is bounded double sequence of strictly positive real numbers. Then,(i) is a complete linear space paranormed by the function defined in (70). In particular, if is a constant, is a Banach space for