Research Article | Open Access

Ali Karaisa, Ümıt Karabıyık, "Almost Sequence Spaces Derived by the Domain of the Matrix ", *Abstract and Applied Analysis*, vol. 2013, Article ID 783731, 8 pages, 2013. https://doi.org/10.1155/2013/783731

# Almost Sequence Spaces Derived by the Domain of the Matrix

**Academic Editor:**Feyzi Başar

#### Abstract

By using , we introduce the sequence spaces , , and of normed space and -space and prove that , and are linearly isomorphic to the sequence spaces , , and , respectively. Further, we give some inclusion relations concerning the spaces , , and the nonexistence of Schauder basis of the spaces and is shown. Finally, we determine the - and -duals of the spaces and . Furthermore, the characterization of certain matrix classes on new almost convergent sequence and series spaces has exhaustively been examined.

#### 1. Preliminaries, Background and Notation

By , we will denote the space of all real or complex valued sequences. Any vector subspace of is called sequence space. We will write , , , and for the spaces of all bounded, null, convergent, and absolutely -summable sequences, respectively, which are -space with the usual sup-norm defined by and , for , where, here and in what follows, the summation without limits runs from to . Further, we will write , for the spaces of all sequences associated with bounded and convergent series, respectively, which are -spaces with their natural norm [1].

Let and be two sequence spaces and an infinite matrix of real or complex numbers , where . Then, we say that defines a matrix mapping from into and we denote it by writing that and if for every sequence the sequence , the -transform of is in , where

The notation denotes the class of all matrices such that . Thus, if and only if the series on the right hand side of (1) converges for each and every and we have for all . The matrix domain of an infinite matrix in a sequence space is defined by

The approach constructing a new sequence space by means of the matrix domain of a particular triangle has recently been employed by several authors in many research papers. For example, they introduced the sequence spaces in [2], and in [3], in [4], and in [5], and and in [6]. Recently, matrix domains of the generalized difference matrix and triple band matrix in the sets of almost null and almost convergent sequences have been investigated by Başar and Kirişçi [7] and Sönmez [8], respectively. Later, Kayaduman and Şengönül introduced some almost convergent spaces which are the matrix domains of the Riesz matrix and Cesàro matrix of order in the sets of almost null and almost convergent sequences (see [9, 10]).

We now focus on the sets of almost convergent sequences. A continuous linear functional on is called a Banach limit if (i) for and for every , (ii) , where is shift operator which is defined on by , and (iii) , where . A sequence is said to be almost convergent to the generalized limit if all Banach limits of are [11] and denoted by . In other words, uniformly in if and only if The characterization given above was proved by Lorentz in [11]. We denote the sets of all almost convergent sequences and series by where We know that the inclusions strictly hold. Because of these inclusions, norms and of the spaces and are equivalent. So the sets and are BK-spaces with the norm .

The rest of this paper is organized, as follows. We give foreknowledge on the main argument of this study and notations in this section. In Section 2, we introduce the almost convergent sequence and series spaces and which are the matrix domains of the matrix in the almost convergent sequence and series spaces and , respectively. In addition, we give some inclusion relations concerning the spaces , , and the non-existence of Schauder basis of the spaces and is shown to give certain theorems related to behavior of some sequences. In Section 3, we determine the beta- and gamma-duals of the spaces and and characterize the classes , , and , where , , , and , where , , and denote the space of Maddox convergent, null and bounded sequence spaces defined by Maddox [12].

Lemma 1 (see [13]). *The set has no Schauder basis.*

#### 2. The Sequence Spaces , , and Derived by the Domain of the Matrix

In the present section, we introduce the sequence spaces , , and as the set of all sequences such that -transforms of them are in the spaces , , and , respectively. Further, this section is devoted to examination of the basic topological properties of the sets , , and . Recently, Aydın and Başar [14] studied the sequence spaces and : where denotes the matrix defined by

Now we introduce the sequence spaces , , and as the sets of all sequences such that their -transforms are in the spaces , , and , respectively; that is, We can redefine the spaces , , and by the notation of (2): It is known by Başar [15] that the method is regular for . We assume unless stated otherwise that .

Define the sequence , which will be frequently used, as the -transform of a sequence ; that is,

Theorem 2. *The spaces and have no Schauder basis.*

* Proof. *Since it is known that the matrix domain of a normed sequence space has a basis if and only if has a basis whenever is a triangle [16, Remark 2.4] and the space has no Schauder basis by [7, Corollary 3.3], we have that has no Schauder basis. Since the set has no basis in Lemma 1, has no Schauder basis.

Theorem 3. *The following statements hold.*(i)* The sets and are linear spaces with the coordinatewise addition and scalar multiplication which are -spaces with the norm*(ii)

*The set**is a linear space with the coordinatewise addition and scalar multiplication which is a**-space with the norm**Proof. *Since the second part can be similarly proved, we only focus on the first part. Since the sequence spaces and endowed with the norm are -spaces (see [1, Example (b)]) and the matrix is normal, Theorem of Wilansky [17, p.61] gives the fact that the spaces and are -spaces with the norm in (11).

Now, we may give the following theorem concerning the isomorphism between our spaces and the sets , , and .

Theorem 4. *The sequence spaces , , and are linearly isomorphic to the sequence spaces , , and , respectively; that is, , , and .*

* Proof. *To prove the fact that , we should show the existence of a linear bijection between the spaces and . Consider the transformation defined with the notation of (2) from to by . The linearity of is clear. Further, it is clear that whenever , and hence, is injective.

Let , and define the sequence by
whence
which implies that . As a result, is surjective. Hence, is a linear bijection which implies that the spaces and are linearly isomorphic, as desired. Similarly, the isomorphisms and can be proved.

Theorem 5. *The inclusion strictly holds.*

* Proof. *Let . Since , . Because is regular for , . Therefore, since , we see that . So we have that the inclusion holds. Further, consider the sequence defined by . Then, since , . One can easily see that . Thus, , and this completes the proof.

Theorem 6. *The sequence spaces and overlap, but neither of them contains the other.*

* Proof. *Let us consider the sequence defined by for all . Then, since , . It is clear that . This means that the sequence spaces and are not disjoint. Now, we show that the sequence space and do not include each other. Let us consider the sequence defined as in proof of Theorem 5 above and where the blocks of 0’s are increasing by factors of and the blocks of ’s are increasing by factors of . Then, since , , but . Therefore, . Also, the sequence since where the blocks of 0’s are increasing by factors of and the blocks of 1’s are increasing by factors of , but is bounded. This means that . Hence, the sequence spaces and overlap, but neither of them contains the other. This completes the proof.

Theorem 7. *Let the spaces , , and be given. Then, *(i)* strictly hold;*(ii)

*strictly hold.** Proof. *(i) Let which means that . Since , . This implies that . Thus, we have .

Now, we show that this inclusion is strict. Let us consider the sequence defined as in proof of Theorem 6 for all . Consider the following:
which means that ; that is to say, the inclusion is strict.(ii) Let which means that . Since , . This implies that . Thus, we have . Furthermore, let us consider the sequence defined as in proof of Theorem 5 for all . Then, since , . This completes the proof.

#### 3. Certain Matrix Mappings on the Sets , and Some Duals

In this section, we will characterize some matrix transformations between the spaces of almost convergent sequence and almost convergent series in addition to paranormed and classical sequence spaces after giving - and -duals of the spaces and . We start with the definition of the beta- and gamma-duals.

If and are sequences and and are subsets of , then we write , and for the multiplier space of and . One can easily observe for a sequence space with and that inclusions and hold, respectively. The --, and -duals of a sequence space, which are, respectively, denoted by , and , are defined by It is obvious that . Also, it can easily be seen that the inclusions , , and hold whenever .

Lemma 8 (see [18]). * if and only if
*

Lemma 9 (see [18]). * if and only if (18) holds and there are such that
*

Theorem 10. *Define the sets and by
**
where for all . Then .*

* Proof. *Take any sequence , and consider the following equality:
where is
for all . Thus, we deduce from (23) that whenever if and only if whenever where is defined in (24). Therefore, with the help of Lemma 8, .

Theorem 11. *The -dual of the space is the intersection of the sets
**
where for all . Then, .*

* Proof. *Let us take any sequence . By (23), whenever if and only if whenever . It is obvious that the columns of that matrix in where defined in (24), we derive the consequence by Lemma 9 that .

Theorem 12. *The -dual of the space is the intersection of the sets
**
In other words, we have .*

* Proof. *We obtain from (23) that whenever if and only if whenever , where is defined in (24). Therefore, by Lemma 19(viii), .

Theorem 13. *Define the set by
**
Then, .*

* Proof. *This may be obtained in the same way as mentioned in the proof of Theorem 12 with Lemma 19(viii) instead of Lemma 19(vii). So we omit details.

For the sake of brevity, the following notations will be used: for all . Assume that the infinite matrices and map the sequences and which are connected with relation (10) to the sequences and , respectively; that is, One can easily conclude here that the method is directly applied to the terms of the sequence , while the method is applied to the -transform of the sequence . So the methods and are essentially different.

Now, suppose that the matrix product exists which is a much weaker assumption than the conditions on the matrix belonging to any matrix class, in general. It is not difficult to see that the sequence in (30) reduces to the sequence in (29) under the application of formal summation by parts. This leads us to the fact that exists and is equal to and formally holds if one side exists. This statement is equivalent to the following relation between the entries of the matrices and which are connected with the relation

Note that the methods and are not necessarily equivalent since the order of summation may not be reversed. We now give the following fundamental theorem connected with the matrix mappings on/into the almost convergent spaces and .

Theorem 14. *Suppose that the entries of the infinite matrices and are connected with relation (31) for all , and let be any given sequence space. Then, if and only if
*

*Proof. *Suppose that and are connected with the relation (31), and let be any given sequence space, and keep in mind that the spaces and are norm isomorphic.

Let , and take any sequence , and keep in mind that . Then, ; that is, (32) holds for all and exists which implies that for each . Thus, exists for all , and thus, we have in the equality
for all , and we have (31) which means that . On the other hand, assume that (32) holds and . Then, we have for all which gives together with for each that exists. Then, we obtain from the equality
for all , as , that , and this shows that .

Theorem 15. *Suppose that the entries of the infinite matrices and are connected with the relation
**
for all and is any given sequence space. Then, if and only if .*

* Proof. *Let , and consider the following equality:
for all , which yields as that whenever if and only if whenever . This step completes the proof.

Theorem 16. *Let be any given sequence space, and the matrices and are connected with the relation (31). Then, if and only if and for all .*

* Proof. *The proof is based on the proof of Theorem 14.

Theorem 17. *Let be any given sequence space, and the elements of the infinite matrices and are connected with relation (35). Then, if and only if .*

* Proof. *The proof is based on the proof of Theorem 15.

By Theorems 14, 15, 16, and 17, we have quite a few outcomes depending on the choice of the space to characterize certain matrix mappings. Hence, by the help of these theorems, the necessary and sufficient conditions for the classes , , and may be derived by replacing the entries of and by those of, and , respectively, where the necessary and sufficient conditions on the matrices and are read from the concerning results in the existing literature

Lemma 18. *Let be an infinite matrix. Then, the following statements hold:*(i)* if and only if*(ii)

*(iii)*

*if and only if (37) and*

*if and only if (37) and*Lemma 19. *Let be an infinite matrix. Then, the following statements hold:*(i)* (Duran, [19]) if and only if (18) holds and*(ii)

*(iii)*

*(King, [20])**if and only if (18), (40) hold and**(iv)*

*(Başar and Çolak, [21])**if and only if (40) holds and**(v)*

*(Başar and Çolak, [21])**if and only if (40), (43) hold and**(vi)*

*(Duran, [19])**if and only if (18), (40), and (42) hold and**(vii)*

*(Başar, [22])**if and only if (40), (44), (46), and (45) hold;**(viii)*

*(Öztürk, [23])**if and only if (19), (43), and (44) hold and**(ix)*

*if and only if (43) and (44) hold;**(x)*

*(Başar and Solak, [24])**if and only if (44), (45) hold and**(xi)*

*(Başar, [22])**if and only if (45), (48) hold and**(xii)*

*(Başar and Çolak, [21])**if and only if (48) holds;*

*(Başar, [25])**if and only if*Now we give our main results which are related to matrix mappings on/into the spaces of almost convergent series and sequences .

Corollary 20. *Let be an infinite matrix. Then, the following statements hold.*(i)* if and only if for all and (40), (44) hold with instead of , (46) holds with instead of , and (45) holds with instead of .*(ii)

*(iii)*

*if and only if**for all**and (19), (43), (44), and (47) hold with**instead of**.**(iv)*

*if and only if**for all**and (43) and (44) hold with**instead of**.**(v)*

*if and only if**for all**and (45), (48), and (49) hold with**instead of**.**(vi)*

*if and only if (48) holds with**instead of**.**(vii)*

*if and only if (44) holds with**instead of**and (45), (48) hold with**instead of**.*

*if and only if (45), (48), and (49) hold with**instead of**.*Corollary 21. *Let be an infinite matrix. Then, the following statements hold.*(i)* if and only if (37)and (38) hold with instead of .*(ii)

*(iii)*

*if and only if (37) holds with**instead of**.*

*if and only if (37) and (39) hold with**instead of**.*Corollary 22. *Let be an infinite matrix. Then, the following statements hold.*(i)* if and only if for all and (18) holds with instead of .*(ii)

*(iii)*

*if and only if**for all**and (18), (19), (20), and (21) hold with**instead of**.*

*if and only if**for all**and (50),(53) hold with**instead of**and (51),(52) hold with**instead of**.*Corollary 23. *Let be an infinite matrix. Then, the following statements hold.*(i)* if and only if (18), (40) hold with instead of and (41) holds with instead of .*(ii)

*(iii)*

*if and only if (18), (40), and (46) hold with**instead of**and (42) holds with**instead of**.**(iv)*

*if and only if (18), (40), and (42) hold with**instead of**.**(v)*

*if and only if (40), (43), and (44) hold with**instead of**and (45) holds with**instead of**.**(vi)*

*if and only if (40), (44) hold with**instead of**, (46) holds with**instead of**, and (45) holds with**instead of**.*

*if and only if (40) and (43) hold with**instead of**.**Remark 24. *Characterization of the classes , , , and is redundant since the spaces of almost bounded sequences and are equal.

#### Acknowledgment

The authors thank the referees for their careful reading of the original paper and for the valuable comments.

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Copyright © 2013 Ali Karaisa and Ümıt Karabıyık. 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.