Some Functional Sections in Topological Sequence Spaces
Summability is a particularly fertile field for functional analysis application. Summability through functional analysis has become one of the most fascinating disciplines since it contains both interesting and challenging issues. In this paper, we aim to introduce four new sectional properties for topological sequence spaces: sectional weakly absolute convergence (WAC), sectional weak boundedness (WB), sectional weakly -absolute convergence , and sectional weakly bounded variation (WBV). We have, also, investigate some of their relations and identities.
1. Introduction and Preliminaries
Functional analysis studies, particularly in the early twentieth century, underwent significant methodological modifications as a result of the influence of contemporary thinking styles. Banach’s contributions to the theory of linear operators, in particular, are known as fundamental in terms of methodology. With its tight connection to other disciplines of mathematics, topology, which was initially employed exclusively for certain geometry issues, has resulted in technique changes in every subject throughout time and has given solid foundations for the areas in which it is applied. Frechet-Coordinate space ( space) theory has been important in the creation of certain areas, such as topological sequence spaces, as well as in resolving the overwhelming majority of issues involving summability, particularly matrix fields. Topological vector space is a linear space with a topology that enables continuous vector space operations. If this vector space has a full metric space structure, it is called Frechet space, and furthermore, if it has a topology with continuous coordinate functions, it is referred to as Frechet-Coordinate space. The theory of spaces has acquired more prominence in recent years and has found applications in a variety of fields, thanks to the efforts of many researchers. and spaces are now widely used in the research of sequence spaces and matrix transformations. By accumulating from spaces, one may build new sequence spaces that include or encompass previously existing sequence spaces. Between these spaces, matrix characterizations may be provided, and certain duals and their characteristics can be disclosed. We will begin with the following definitions, concepts, and properties that will be necessary for our major findings. By , we denote the vector space containing all real- or complex-valued sequences that are topologized through coordinatewise convergence. Any vector subspace of is said to be a sequence space. A sequence space with a locally convex topology is referred to as a -space if the inclusion mapping is continuous, when has the topology of coordinatewise convergence. Additionally, if is complete and metrizable, is referred to be an -space. A -space is an -space with a normable topology. Among important -spaces are as follows:
: the space of all bounded sequences
: the space of all convergent sequences
: the space of all null sequences
: the space of all absolutely -summable sequences, that is,
Also, is called sup norm. Here and throughout, will denote the set of positive integers. As is conventional, is replaced by , indicates the norm , and indicates the norm on , , and .
Definition 1. (i)Let and be any sequence spaces . Then, the multiplier space is defined bywhere is the termwise product, namely, for and . (ii)Likewise, for any spaces in ,(iii)If for some spaces , in , the sequence space is referred to as -space(iv)Recall the following spaces in :For a space in , , , and are called -, -, and -duals of , respectively.
Remark 2. For spaces in , the following statements are known: (i)These are Banach spaces with their natural norms(ii)(iii)If , then , for any space , in particular, being one of the -, -, and -duals(iv), , where is one of the -, -, and -duals
Definition 3. (i)Suppose that represent the sequence of ones and symbolize the sequences with the one in the -th position, namely, , where denotes the Kronecker delta. And also, suppose that signify the linear span of the ’s, in other words, the space of all finitely nonzero sequences(ii)Here and throughout, will denote the topological dual of the space . Then, for some ,is referred to as the -dual of a space . Here, is also a space with (iii)Thereby, if , then . Also, if holds, an space containing is called a semiconservative space .
Definition 4 (see [1–4]). For an space , we denote the -th section of a sequence by Then, the following is valid for the sequence : (i)Section boundedness (abschnitt beschränktheit) denoted by (ii)Section convergence (abschnitt konvergenz) denoted by if (iii)Coordinatewise boundedness (koordinatenweise beschränkt) denoted by if (iv)Functional section convergence denoted by if converges for every (equivalently, if is Cauchy in the weak topology )(v)Weak section convergence denoted by if converges to in the weak topology (vi)Section density (abschnitt dichte) denoted by if (closure of in ).A space is AD space if , where is closure of in . Via Hahn-Banach theorem, it is clear that .
Definition 5 (see [2, 3, 5]). Let be the collection of all finite subsets of . (i)A series in a topological vector space is unconditionally convergent to , if the net converges to directed by set inclusion(ii)The series is unconditionally Cauchy, if the net is a Cauchy net
Definition 6 (see [2, 3]). Assume that denotes the set of all sequences in consisting of ’s and ’s, such that,
If , we have , where denotes the coordinatewise product . If is a -space, then for any sequence . An unrestricted section of a sequence is any sequence of the form , where .
Let be a -space. Then, the following is valid for a sequence : (i)Unconditional section boundedness in , if is a bounded subset of (ii)Unconditional section convergence , if the net converges to in (iii)Unconditional weak section convergence , if the net converges to in (iv)Unconditional functional section convergence , if the net is Cauchy in
Recently, some researchers have attempted to assign properties to , , , , , , , , , and in spaces and topological properties of sequence spaces (see, e.g., [6, 7]). Additional information and techniques on summability and related concepts can be found in [4, 8–12].
-space theory, which has an important role in the characterization of matrix transformations between sequence spaces and contributes to the proof of the results in the summability theory in an easier and shorter way, is a concept that has been extensively studied by researchers. Sequence spaces are also widely used concepts in many areas of functional analysis and mathematical analysis. In this context, as in the studies on sequence spaces, the main motivation in this study is to reveal new sequence spaces and to determine the relationships between them by looking at the topological structures and properties of these sequence spaces. In addition, investigations have been made on the duals of these new sequence spaces and their relations.
In this article, we aim to introduce four new sectional properties for spaces, (sectional weakly absolute convergence), (sectional weak boundedness), (sectional weakly -absolute convergence), and (sectional weakly bounded variation), and explore some of their relations and identities. Also, we examine certain relationships between those subspaces of spaces which are made from various existing sequence properties.
2. Relationships between the Subspaces of Spaces
This section explores further relationships between the subspaces of spaces. We will start with some well-known properties on the sections and duals of a sequence space, as stated in the following lemmas.
Lemma 7 (see [1, 2]). Assume that is a space which contains the space . Then (1) has -property if and only if the inclusion(2) has -property if and only if the inclusion(3) has -property if and only if the inclusionHere, denotes the set of all sequences that have property , that is, So are the others.
Lemma 9 (see ). Let has the property . Then, for any spaces , if and only if .
Lemma 10 (see ). Let be an space, a sequence, and where . Then, is an space and if and only if , , .
Remark 12. For the properties in space such as , , , , , , , , , , and are equivalent to . By Definition 11, obviously, we can write .
Theorem 13. Assume that and are spaces containing . If , then , where is one of the and duals.
Proof. Suppose that and . Then, for all . Since, , for all . This implies that . Therefore, .
Theorem 14. Assume that and are spaces containing . If , then and .
Proof. Suppose that . Taking -dual of the inclusion gives . Again, taking -dual of the resultant inclusion yields . It follows from Definition 11 that . Likewise, the other inclusion may be established.
Theorem 15. Let and be spaces. Also, let contain and has the property . Then, the following two statements hold: (i)Assume that and are -spaces. Then(ii)Assume that and are -spaces. Then
Proof. For (i), let . By Theorem 14, the inclusion follows. Conversely, let . It can be concluded from Definition 11 that . By the assumption, since and are -spaces, we have . Since, has the property , by Lemma 9, we find . This completes the proof of the statement (i).
Likewise, the assertion (ii) may be established. We omit the details.
Theorem 16. Let be an space containing If is a space, then .
Proof. Since is space, holds. Then, taking -dual of the inclusion gives . From Definition 11 and , we have .
The following corollary summarizes some inclusion relations between spaces.
Corollary 17. Let be an space containing Then (i) or (ii) or (iii) or (iv) or (v) or (vi) or
Proof. We will only prove the statement (i). Let be . Obviously, . Conversely, assume that . Taking -dual on the assumed inclusion, we have . Using , we can find . Since, has the property , we obtain . Similarly, the statement can be proved.
Likewise, the other statements can be shown. The details are omitted.
3. Sectional Weakly Properties
We define and investigate four new sectional properties: sectional weakly absolute convergence , sectional weak boundedness , sectional weakly -absolute convergence (), and sectional weakly bounded variation of a sequence in space, as given in the following definition.
Definition 18. Let be an space containing . Then, a sequence in has the following properties: (a)Sectional weakly absolute convergence (denoted by ), if(b)Sectional weak boundedness (denoted by ), if(c)Sectional weakly -absolute convergence (denoted by ), where , if(d)Sectional weakly bounded variation (denoted by ), if
The following notations in connection to Definition 18 are introduced:
Also, in association with the Definition 18 and the above notations, assume that has one of the properties , , , and . Then,
and an space has the property , if , and so .
The following theorems will demonstrate some features connected with the definitions and notations presented in this and previous sections.
Theorem 19. Let be an space containing . Then, (i)(ii)(iii)(iv)
Proof. For the identity (i), we find from the property that if and only if for each if and only if .
Likewise, the other identities (ii), (iii), and (iv) can be verified. We omit the details.
Theorem 20. Let be an space containing . Then, (i) has if and only if , i.e., (ii) has if and only if , i.e.,
Proof. For (i), let be an space containing . By part (i) of Theorem 19, . Therefore, . Also, by Lemma 8, we have . Hence, . The opposite implication can be proved by a similar way.
The proof of the statement in (ii) would run in parallel with the proof of the statement (i). We omit the details.
Theorem 21. Let and be spaces containing which have one (say ) of the properties , , , and . If , then .
Proof. By Definition 18 and the subsequent notations, the result is obvious.
Theorem 22. Let be an space and have the property . (i)If and have property WAC, then(ii)If and have property WBV, then
Proof. We will only prove the statement (i). Suppose that . By Theorem 21, . Conversely, let . By Theorem 19, . Then, by part (iii) in Remark 2, we have . By part (i) in Theorem 20, . By part (iv) in Remark 2, we find . By part (i) in Theorem 20 again, we obtain . Since has the property , we obtain from Lemma 9.
The proof of the statement (ii) would flow in parallel with the proof of statement (i). The details are omitted.
Theorem 23. Let be an space . Then,
Proof. By applying the inclusion to Theorem 19, this yields the result.
Theorem 24. Let be an space and . Then, (i) implies (ii) implies (iii) implies (iv) implies
Proof. We will prove the statement (i). Assume that . Then, we find from Lemma 10 that
for all and . Hence, if and only if .
Similarly, the other statements (ii), (iii), and (iv) can be proved. We omit the details.
Theorem 25. Let be an space including For , if and only if .
Proof. The assertion follows from the definitions of and .
Corollary 26. Let be an space including and has the property . Then, the following statements are equivalent: (i) has the property (ii)(iii)
Proof. The assertion is a consequence of Theorem 25.
Theorem 27. Let be an space including . Then, for , if and only if .
Proof. The assertion follows from the definitions of and .
From Theorem 27, we conclude the following result.
Corollary 28. Let be an space including and has the property . Then, the following statements hold true: (i) has the property (ii)(iii)(iv)
Theorem 29. The property is weaker than the property .
Proof. Let has the property and . Since has property, there exists such that for all . It follows that for all . This implies that the sequence is bounded. The Banach-Mackey theorem states that is bounded if and only if it is weakly bounded (see ). Therefore, is weakly bounded, namely, This means that has the property .
Corollary 30. Property is weaker than the property .
Proof. Let has the property . Then, for all and , we have . Since , for those and . This implies that has the property .
Definition 31. Let and be spaces including . Also, assume that . A sequence has the property , if for all . Further, consider the following set Clearly, . Equivalently, an space including has the property , if ; that is, .
Theorem 32. Let be an space including and . If is an space, then if and only if .
Proof. It is obvious from Definition 31 that .
Using above definition, it is obvious that .
Sufficiency: let . For each , ; and By the equality , , for , we get and and so Thus, we obtain Since is an space, then .
Necessity: let For , there exists a such that for all . Hence, we get Since , we have Let Then, there exists an such that for all . Then, we get for and So, Thus, and hold.
4. Concluding Remarks
It is known that the foundations of the theory of spaces were set in the first half of the twentieth century by mathematicians such as Mazur and Orlicz, and then, Zeller began his work in 1951. One of the primary benefits of space theory is that it is the most powerful and widely used instrument for proving various classical results in summability theory and for describing matrix transformations between sequence spaces in an easy and concise manner. Along with the theory of summability, sequence spaces are important in other fields where functional analysis and summability are employed. Numerous duals of these spaces play a significant role in summability theory and topological sequence spaces. Due to the discoveries acquired from and spaces in the work on sequence spaces and matrix transformations, new sequence spaces may be constructed that are contained or encompassed by an array of previously existing spaces.
Motivated by this idea, in this article, we introduced four new sectional properties for topological sequence spaces, sectional weakly absolute convergence , sectional weak boundedness , sectional weakly -absolute convergence , and sectional weakly bounded variation , and investigated some of their properties and identities. Further studies will focus on advanced features and inclusions of these spaces.
Researchers interested in the subject can define new spaces that can be functional in terms of summability theory and sequence spaces.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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