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Kaya, Meltem; Furkan, Hasan

doi:https://doi.org/10.1155/2015/458324

Journal of Function Spaces

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Volume 2015 | Article ID 458324 | https://doi.org/10.1155/2015/458324

Some Topological and Geometric Properties of Some New Spaces of -Convergent and Bounded Series

Meltem Kaya¹and Hasan Furkan¹

Academic Editor: Alberto Fiorenza

Received16 Mar 2015

Accepted14 Apr 2015

Published01 Jun 2015

Abstract

The main purpose of this study is to introduce the spaces , , and which are -spaces of nonabsolute type. We prove that these spaces are linearly isomorphic to the spaces , , and , respectively, and derive some inclusion relations. Additionally, Schauder bases of the spaces and have been constructed and the -, -, and -duals of these spaces have been computed. Besides, we characterize some matrix classes from the spaces , , and to the spaces , , and , where . Finally, we examine some geometric properties of these spaces as Gurarǐ’s modulus of convexity, property , property , property WORTH, nonstrict Opial property, and weak fixed point property.

1. Introduction

By a sequence space, we understand a linear subspace of the space , where and denotes the complex field. A sequence space with a linear topology is called a -space provided each of the maps defined which is continuous for all . A -space is called an -space provided is a complete linear metric space. An -space whose topology is normable is called a -space (see [1, pages 272-273]) which contains , the set of all finitely nonzero sequences. We write , , and for the spaces of all bounded, convergent, and null sequences, respectively. Also by , we denote the space of all -absolutely summable sequences, where . Moreover, we write , , and for the spaces of all bounded, convergent, and null series, respectively.

Let and be two sequence spaces, and let be an infinite matrix of complex numbers , where . Then we say that defines a matrix transformation from into , and we denote it by writing if for every sequence , the sequence , the -transform of , is in , whereand by denotes the subspace of consisting of for which the sum exists as a finite sum. For simplicity in notation, here and in what follows, the summation without limits runs from to and we will use the convention that any term with a negative subscript is equal to naught; for example, and .

By , we denote the class of all matrices such that . Thus if and only if the series on the right side of (1) converges for each and each and we have for all . For an arbitrary sequence space , the matrix domain of an infinite matrix in is defined bywhich is a sequence space. If is triangle, then one can easily observe that the normed sequence spaces and are norm isomorphic; that is, . If is a sequence space, then the continuous dual of the space is defined by We denote the collection of all finite subsets of by . Also, we will write for the sequence whose only nonzero term is in the th place for each . Throughout this paper, let be a strictly increasing sequence of positive real numbers tending to infinity; that is, We define the matrix of weighted mean relative to the sequence by for all ; . With a direct calculation we derive the equalityIt is easy to show that the matrix Λ is regular and is reduced, in the special case for all to the matrix of Cesàro means of order one. Introducing the concept of -strong convergence, several results on -strong convergence of numerical sequences and Fourier series were given by Móricz [2]. Since we have in the special case for all , the matrix is also reduced to the Riesz means with respect to the sequence .

We summarize the knowledge in the existing literature concerning domain of the matrix over some sequence spaces. Mursaleen and Noman [3–6] introduced the spaces , , , and of lambda-bounded, lambda-convergent, lambda-null, and lambda-absolutely -summable sequences and gave the inclusion relations between these spaces and the classical sequence spaces , , and . Later, Mursaleen and Noman [7] investigated the difference spaces and obtained from the spaces and . Recently, paranormed -sequence spaces of nonabsolute type have been studied by Karakaya et al. [8]. More recently, Sönmez and Başar [9] introduce the difference sequence spaces and , which are the generalization of the spaces and . Quite recently, some new sequence spaces of nonabsolute type and matrix transformations have been studied by Ganie and Sheikh [10]. The same authors have studied the spaces of -convergent sequences and almost convergence [11]. Also, the fine spectrum of the operator defined by lambda matrix over the spaces of null and convergent sequences has been studied by Yeşilkayagil and Başar [12].

In this work, our purpose is to construct the sequence spaces , , and by the domain of the matrix in the spaces , , and , respectively, of the series whose sequence of partial sums are in the spaces , , and [3].

We define the sequence by the -transform of a sequence ; that is, , and so we haveAlso, we say that a sequence is -convergent if . In particular, we say that is -null or -bounded if or , respectively.

2. The Sequence Spaces , , and

In the present section, we introduce the sequence spaces , , and as the sets of all sequences whose -transforms are in the spaces , , and , respectively; that is,

With the notation of (2), we can redefine the spaces , , and as the matrix domains of the triangle in the spaces , , and by

Then, it is immediate by (12) that the sets , , and are linear spaces with coordinatewise addition and scalar multiplication; that is, , , and are the sequence spaces consisting of all sequences which are -convergent, -null, and -bounded series of type , respectively.

Now, we may begin with the following theorem which is essential in the text.

Theorem 1. The sequence spaces , , and are -spaces with the same norm ; that is,

Proof. Since (12) holds and , , and are -spaces with the respect to their natural norms and the matrix is a triangle, Theorem 4.3.12 of Wilansky [13, page 63] gives the fact that , , and are -spaces with the given norms. This completes the proof.

Remark 2. One can easily check that the absolute property does not hold on the spaces , , and ; that is, , , and for at least one sequence in the spaces , , and , and this shows that , , and are the sequence spaces of nonabsolute type, where .

Now, we give the final theorem of this section.

Theorem 3. The sequence spaces , , and of nonabsolute type are isometrically isomorphic to the spaces , , and , respectively; that is, , , and .

Proof. To prove this, we should show the existence of an isometric isomorphism between the spaces and . Consider the transformation defined, with the notation of (8), from to by . Then, for every and the linearity of is clear. Also, it is trivial that whenever and hence is injective. Furthermore, let be given and define the sequence byThen, by using (8) and (14), we have for every thatThis shows that and since , we obtain that . Thus, we deduce that and . Hence is surjective. Moreover, one can easily see for every that which means that is norm preserving. Therefore is isometry. Consequently is an isometric isomorphism which shows that the spaces and are isometrically isomorphic.
It is clear that if the spaces and are replaced by the respective one of the spaces and or and , then we obtain the fact that and . This completes the proof.

3. The Inclusion Relations

In the present section, we establish some inclusion relations concerning the spaces , , and . We may begin with the following lemma.

Lemma 4 (see [3]). For any sequence , the equalityholds, where is the sequence defined by

Theorem 5. The inclusions strictly hold.

Proof. It is obvious that the inclusions hold. Let us consider the sequence defined by In the present case, we obtain for every that which shows that . Thus, the sequence is in but not in . Hence is a strict inclusion.
To show the strictness of the inclusion , we define the sequence by Then, we have for every that This shows . Thus, the sequence is in but not in and hence is a strict inclusion. This concludes the proof.

Lemma 6 (see [3, Theorem 4.1.]). The inclusions strictly hold.

Theorem 7. The inclusions and strictly hold.

Proof. It is clear that the inclusion holds, since implies and hence which means that . Consider the sequence defined by Then, and hence , since the inclusion holds. On the other hand, we have for every that which shows that and hence . Thus, the sequence is in but not in . Therefore, the inclusion is strict.
Similarly, it is also trivial that the inclusion holds. To show that this inclusion is strict, we define the sequence by . In the present case, we have for every that which shows that . Thus, the sequence is in but not in and hence is a strict inclusion. This completes the proof.

Theorem 8. The inclusion holds if and only if for every sequence .

Proof. Suppose that the inclusion holds, and take any . Then and by the hypothesis. Thus, we deduce from (17) thatHence, we obtain from (26) by letting thatAs and , the right-hand side of equality (27) is convergent as . Thereby, the series converges and so .
Conversely, let be given. Then, we have by the hypothesis that . Again, it follows by (26) that which shows that while and . Hence, the inclusion holds and this concludes the proof.

Theorem 9. The inclusion holds if and only if for every sequence .

Proof. One can see by analogy to Theorem 8 that the inclusion also holds if and only if for every sequence . This completes the proof.

Theorem 10. The inclusion holds if and only if for every sequence .

Proof. Suppose that the inclusion holds, and take any . Then, by the hypothesis. Thus, we obtain from equality (17)which yields that .
Conversely, assume that for every . Again, we obtain from equality (17) This shows that . Hence, the inclusion holds. This completes the proof.

4. The Basis for the Spaces , , and

In the present section, we give a sequence of the points of the spaces and which forms a basis for these spaces. If a normed sequence space contains a sequence with the property that for every there is a unique sequence of scalars such that then is called a Schauder basis (or briefly basis) for . The series which has the sum is then called the expansion of with respect to and is written as .

Now, since the transformation defined from to in the proof of Theorem 3 is an isomorphism, we have the following theorem.

Theorem 11. Define the sequence for every fixed by for all .
Then, one has the following:(a)The sequence is a Schauder basis for the spaces and and every or has a unique representation of the form(b) has no Schauder basis.

Proof. (a) It is clear that is a basis for since is a basis for and [14, Corollary 2.3]. Let be given. Then, and for a unique sequence of scalars. Therefore, we obtain that where is the inverse of the matrix . Since , we get that Consequently, Thus, we deduce that , which shows that is represented as in (33).
Finally, let us show the uniqueness of the representation (33) of . Suppose on the contrary that there exists another representation . Since the linear transformation defined from to , in the proof of Theorem 3, is continuous, we have Therefore, the representation (33) of is unique. It can be proved similarly for . This completes the proof.
(b) As a direct consequence of Remark 2.2. of Malkowsky and Rakocevic [14], since has no Schauder basis also has no Schauder basis.

As a result, it easily follows from Theorem 1 that and are the Banach spaces with their natural norms. Then, by Theorem 11 we obtain the following corollary.

Corollary 12. The sequence spaces and of nonabsolute type are separable.

5. The -, -, and -Duals of the Spaces , , and

In this section, we state and prove the theorems determining the -, -, and -duals of the sequence spaces , , and of nonabsolute type. For arbitrary sequence spaces and , the set defined byis called the multiplier space of and . One can easily observe for a sequence space with and that the inclusions and hold, respectively. With the notation of (39), the -, -, and -duals of a sequence space , which are, respectively, denoted by , , and , are defined by It is clear that . Also, it can be obviously seen that the inclusions , , and hold whenever .

The following known results [15] are fundamental for this section.

Lemma 13. Consider if and only if

Lemma 14. Consider if and only if

Lemma 15. Consider if and only if (42) holds and

Lemma 16. Consider if and only if

Lemma 17. Consider if and only if (44) holds:

Lemma 18. Consider if and only if (43) and (45) hold and

Lemma 19. Consider if and only if

Lemma 20. Consider if and only if (44) holds.

Lemma 21. Consider if and only if (43) and (44) hold.

Now, we prove the following result.

Theorem 22. Define the sets and as follows:where the matrix is defined via the sequence by for all , . Then and .

Proof. Let . Then, by bearing in mind relations (8) and (14), it is immediate that the equalityholds for all . We therefore observe by (51) that whenever if and only if whenever . This means that the sequence if and only if . Hence, we obtain by Lemma 13 with instead of that if and only if which yields the result that .
Similarly, we deduce from Lemma 15 with (51) that if and only if . Then, it is clear that the columns of the matrix are in the space , since for all . Therefore, we derive from (42) thatThis shows that . This completes the proof.

Theorem 23. Define the sets , , and as follows: wherefor all . Then and .

Proof. Because the proof may also be obtained for the space in a similar way, we omit it. Take any and consider the equationwhere the matrix is defined by for all . Thus, we deduce by (57) that where if and only if whenever . This means that if and only if . Therefore, by using Lemma 16, we derive from (44) and (45) that respectively. Thereby, we conclude that .

Theorem 24. The -dual of the spaces , , and is the set .

Proof. The proof of this result follows the same lines as those in the proof of Theorem 23 using Lemmas 19, 20, and 21 instead of Lemma 16.

6. Certain Matrix Mappings on the Spaces , , and

In this present section, we characterize the matrix classes , , , , , , , , and , where .

For an infinite matrix , we write for brevity that

The following lemmas [15] will be needed in proving our results.

Lemma 25. Consider if and only if (44) holds and

Lemma 26. Consider if and only if (44) holds and

Lemma 27. Consider if and only if (43) holds and

Lemma 28. Consider if and only if

Lemma 29. Consider if and only if

Lemma 30. Consider if and only if (43) and (65) hold.

Now, we give the following results on the matrix transformations.

Theorem 31. (i) Consider if and only if(ii) Consider if and only if (68) holds and(iii) Consider if and only if (69) holds and

Proof. Suppose that conditions (67) and (68) hold and take any . Then, we have by Theorem 23 that for all and this implies the existence of the -transform of ; that is, exists. Further, it is clear that the associated sequence is in and hence .
Let us now consider the following equality derived by using relation (8) from the th partial sum of the series :Therefore, by using (67) and (68), from (71) as , we obtain thatFurther, since the matrix is in the class by Lemma 19 and (67), we have . Therefore, we deduce from (1) and (72) that and hence .
Conversely, suppose that . Then for all and this, with Theorem 23, implies both (68) and which together imply that relation (72) holds for all sequences and . Further, since by the hypothesis, we obtain by (72) that which shows that , where . Hence, the necessity of (67) is immediate by (48). This concludes the proof of part .
Since parts (ii) and (iii) can be proved similarly, we omit their proofs.

Corollary 32. (i) Consider if and only if (68) and (69) hold and(ii) Consider if and only if (68) and (69) hold and(iii) Consider if and only if (70) and (74) hold and

Corollary 33. (i) Consider if and only if (68) and (69) hold and(ii) Consider if and only if (68) and (69) hold and(iii) Consider if and only if (70) hold and

Corollary 34. (i) Consider if and only if (68) holds and(ii) Consider if and only if (68) and (80) hold and(iii) Consider if and only if (70), (80), and (82) hold.

Corollary 35. (i) Consider if and only if (68) and (80) hold and(ii) Consider if and only if (68) and (80) hold and(iii) Consider if and only if (70), (80), and (84) hold.

Since Corollaries 32, 33, 34, and 35 can be proved similarly with Theorem 31, we omit their proofs.

7. Some Geometric Properties of the Spaces , , and

In this section, we investigate some geometric properties for the sequence spaces and .

Let be a normed linear space, and let and be the unit sphere and unit ball of (for the brevity ), respectively. Consider Clarkson’s modulus of convexity (Clarkson [16] and Day [17]) defined by where . The inequality for all characterizes the uniformly convex spaces.

In [18], Gurarǐ’s modulus of convexity is defined by where . It is easily shown that for any . Also if , then is uniformly convex, and if , then is strictly convex.

Opial property [19] states that

If the strict inequality becomes , this condition becomes a nonstrict Opial property.

The coefficient , introduced by García-Falset [20], is defined asSo and it is not hard to see that, in the definition of , “” can be replaced by “.” Some values of are and , .

A Banach space has property if whenever , then is a function of only. Property which is introduced by Kalton [21] is equivalent to

Sims [22] introduced a property called weak orthogonality (WORTH) for Banach spaces. A Banach space is said to have property WORTH if

It remains unknown if property WORTH implies fixed point property. In many situations, the fixed point property can be easily obtained when we assume, in addition, that the spaces are considered to have the property WORTH.

The following result will be used in our results.

Proposition 36 (see [23, Proposition 2.1]). For the following conditions on a Banach space , one has (i) (ii) (iii) (iv).(i) has property .(ii) has property WORTH.(iii)If , then for each we have that is an increasing function of on .(iv) satisfies the nonstrict Opial property.

In [24, 25] it has been shown that implies has property .

It has been shown that Banach space has property (resp., ) [26] if for all , whenever ,Clearly the above properties imply property and property implies Opial property.

Theorem 37 (see [27]). A Banach space has property if and only if .

Remark 38 (see [28]). A Banach space with has the weak fixed point property.

Now, let us give our first theorem in this section.

Theorem 39. Gurarǐ’s modulus of convexity for the normed spaces and iswhere .

Proof. Assume . Then we haveLet and consider the following sequences:where is the inverse of the matrix . Since and , we haveBy using sequences given above, we obtain the following equalities: To complete the conditions of or for Gurarǐ’s modulus of convexity, it remains to evaluate the infimum of for . We have Consequently we get This is the desired result. Thereby the proof is completed.

Corollary 40. Since and for , and hence and are strictly convex.

Theorem 41. Consider if .

Proof. In (or ), we have and , where is the standard basis. Since , we havethus .

Now, by Proposition 36 and Theorem 41, we obtain the following results.

Corollary 42. The spaces and have property and so the spaces and have property .

Corollary 43. The spaces and have property WORTH.

Corollary 44. If , then for each or we have which is an increasing function of on .

Corollary 45. The spaces and satisfy the nonstrict Opial property.

Hence, by Remark 38 and Theorem 41, we have the following result.

Corollary 46. The spaces and have the weak fixed point property.

Conflict of Interests

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

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Copyright © 2015 Meltem Kaya and Hasan Furkan. 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.

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