# Some Functional Sections in Topological Sequence Spaces

Authors: Ahmet Ocak Akdemir; Merve Temizer Ersoy; Hasan Furkan; Maria Alessandra Ragusa
Journal: Journal of Function Spaces
 (2022)

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## Abstract

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 weaklyp-absolute convergence WpAC, and sectional weakly bounded variation (WBV). We have, also, investigate some of their relations and identities.


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## Body

## 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 (FK 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 FK spaces has acquired more prominence in recent years and has found applications in a variety of fields, thanks to the efforts of many researchers. K and FK spaces are now widely used in the research of sequence spaces and matrix transformations. By accumulating from FK 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 ω, we denote the vector space containing all real- or complex-valued sequences that are topologized through coordinatewise convergence. Any vector subspace of ω is said to be a sequence space. A sequence space X with a locally convex topology τ is referred to as a K-space if the inclusion mapping X,τ⟶ω is continuous, when ω has the topology of coordinatewise convergence. Additionally, if τ is complete and metrizable, X,τ is referred to be an FK-space. A BK-space is an FK-space with a normable topology. Among important BK-spaces are as follows:m: the space of all bounded sequencesc: the space of all convergent sequencesc0: the space of all null sequencesℓp1≤p<∞: the space of all absolutely p-summable sequences, that is,
(1)ℓp=x=xkk∈ℕ∈ω:xp≔∑k∈ℕxkp1/p<∞.Also,x∞≔supxk:k∈ℕ is called sup norm. Here and throughout, ℕ will denote the set of positive integers. As is conventional, ℓ1 is replaced by ℓ, ·1 indicates the norm ℓ, and ·∞ indicates the norm on m, c, and c0.Definition 1.
(i)
Letλ and μ be any sequence spaces ⊂ω. Then, the multiplier space λμ=λ⟶μ is defined by(2)λμ≔y∈ω∣xy∈μforallx∈λ,
wherexy is the termwise product, namely, xy=xkykk∈ℕ for x=xkk∈ℕ∈ω and y=ykk∈ℕ∈ω.
(ii)
Likewise, for any spacesλ,μ,ν in ω,(3)λμν=λμν≔y∈ω∣xy∈νforallx∈λμ(iii)
Ifλ=λμμ for some spaces λ, μ in ω, the sequence space λ is referred to as μ-space(iv)
Recall the following spaces inω:(4)ℓ=x=xk∈ω:x1≔∑k∈ℕxk<∞,cs≔xk∈ω:∑k=1∞xkisconvergent,bs≔x=xk∈ω:xbs=supn∑k=1nxk<∞
For a spaceλ in ω, λℓ, λcs, and λbs are called α-, β-, and γ-duals of λ, respectively.Remark 2.
For spacesλ,μ in ω, the following statements are known:
(i)
These are Banach spaces with their natural norms(ii)
ϕ⊂λα⊂λβ⊂λγ(iii)
Ifλ⊂μ, then μζ⊂λζ, for any space ζ⊂ω, in particular, ζ being one of the α-, β-, and γ-duals(iv)
λζ=λζζζ, λ⊂λζζ, where ζ is one of the α-, β-, and γ-dualsDefinition 3.
(i)
Suppose thate represent the sequence of ones 1,1,⋯ and ekk∈ℕ symbolize the sequences 0,⋯,0,1,0,⋯ with the one in the k-th position, namely, ek=δkjj∈ℕ, where δkj denotes the Kronecker delta. And also, suppose that ϕ signify the linear span of the ek’s, in other words, the space of all finitely nonzero sequences(ii)
Here and throughout,X′ will denote the topological dual of the space X. Then, for some f∈λ′,(5)λf=fekk∈ℕ,is referred to as the f-dual of a BK space λ⊃ϕ. Here, λf is also a BK space with fλ′=fekk∈ℕλf(iii)
Thereby, ifλ⊂μ, then μf⊂λf. Also, if λf⊂cs holds, an FK space λ containing ϕ is called a semiconservative space sc.Definition 4 (see [1–4]).
For anFK space λ⊃ϕ, we denote the n-th section of a sequence x=xk∈λ by
(6)xn=∑k=1nxkekn∈ℕ.
Then, the following is valid for the sequencex:
(i)
Section boundedness (abschnitt beschränktheit) denoted byABifsupn∈ℕxnλ<∞(ii)
Section convergence (abschnitt konvergenz) denoted byAK if limn⟶∞x−xnλ=0(iii)
Coordinatewise boundedness (koordinatenweise beschränkt) denoted byKB if supn∈ℕxnenλ<∞(iv)
Functional section convergence denoted byFAK if fxn converges for every f∈λ′ (equivalently, if xn is Cauchy in the weak topology σλ,λ′)(v)
Weak section convergence denoted bySAK if xn converges to x in the weak topology σλ,λ′(vi)
Section density (abschnitt dichte) denoted byAD if xn∈ϕ¯ (closure of ϕ in λ).
AK space λ is AD space if λ=ϕ¯, where ϕ¯ is closure of ϕ in λ. Via Hahn-Banach theorem, it is clear that λf=ϕ¯f.Definition 5 (see [2, 3, 5]).
LetΩ be the collection of all finite subsets of ℕ.
(i)
A series∑xk in a topological vector space X is unconditionally convergent to x∈X, if the net ∑k∈FxkF∈Ω converges to x directed by set inclusion(ii)
The series∑xk is unconditionally Cauchy, if the net ∑k∈FxkF∈Ω is a Cauchy netDefinition 6 (see [2, 3]).
Assume thatH denotes the set of all sequences in ϕ consisting of 0’s and 1’s, such that,
(7)H≔h=hk∈ϕ:hk=0orhk=1,k∈ℕ.
Ifx∈ω, we have Hx=hx:h∈H, where hx denotes the coordinatewise product hx=hkxk. If X is a K-space, then Hx⊂X for any sequence x. An unrestricted section of a sequence x=xk is any sequence of the form ∑k∈Fxkek, where F∈Ω.
Letx be a K-space. Then, the following is valid for a sequence x∈ω:
(i)
Unconditional section boundednessUAB in X, if Hx is a bounded subset of E(ii)
Unconditional section convergenceUAK, if the net Hx converges to x in X(iii)
Unconditional weak section convergenceUSAK, if the net Hx converges to x in σX,X′(iv)
Unconditional functional section convergenceUFAK, if the net Hx is Cauchy in σX,X′Recently, some researchers have attempted to assign properties toAK, AB, KB, SAK, FAK, AD, UAB, UAK, USAK, and UFAK in FK 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].FK-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 forFK spaces, WAC (sectional weakly absolute convergence), WB (sectional weak boundedness), WpAC (sectional weakly p-absolute convergence), and WBV (sectional weakly bounded variation), and explore some of their relations and identities. Also, we examine certain relationships between those subspaces of FK spaces which are made from various existing sequence properties.


## 2. Relationships between the Subspaces ofFK Spaces

This section explores further relationships between the subspaces ofFK 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 thatX is a BK space which contains the space ϕ. Then
(1)
X has AB-property XAB⊃X if and only if the inclusion(8)bv0X⊂Xholds(2)
X has KB-property XKB⊃X if and only if the inclusion(9)ℓX⊂Xholds(3)
X has UAB-property XUAB⊃X if and only if the inclusion(10)cX⊂Xholds
Here,XAB denotes the set of all sequences x∈X that have property AB, that is,
(11)XAB=x∈X:xhasAB
So are the others.Lemma 8 (see [4, 9]).
Assume thatX is an FK space which contains ϕ. Then
(1)
Xβ⊂Xγ⊂Xf(2)
IfX has property AK, then Xβ=Xf(3)
IfX has property AD, then Xβ=XγLemma 9 (see [4]).
LetX has the property AD. Then, for any FK spaces Y, Y⊃X if and only if Yf⊂Xf.Lemma 10 (see [4]).
LetX be an FK space, z a sequence, and
(12)Y=z−1∗X≔x∈ω:xz∈X,
wherexz=xnzn. Then, Y is an FK space and f∈Y′ if and only if fx=αx+gzx, α∈ϕ, g∈X′.Definition 11 (see [1, 3, 4]).
LetX be an FK space including ϕ. Then,
(i)
XAB≔B+X=x∈w:xhasABinX=Xfγ(ii)
XFAK≔F+X=x∈w:xhasFAKinX=Xfβ(iii)
XUAB=x∈w:xhasUABinX(iv)
XKB=x∈w:xhasKBinXRemark 12.
For the properties inFK space E such as AK, AB, KB, SAK, FAK, AD, UAB, UAK, USAK, UFAK, and XE are equivalent to EX. By Definition 11, obviously, we can write F+X⊂B+X.Theorem 13.
Assume thatX and Y are FK spaces containing ϕ. If X⊂Y⊂ω, then Xμ⊃Yμ, where μ is one of the α,β,γ,bv, and ℓ∞ duals.Proof.
Suppose thatX⊂Y and z∈Yμ. Then, yz∈μ for all y∈Y. Since, X⊂Y, xz∈μ for all x∈X. This implies that z∈Xμ. Therefore, Yμ⊂Xμ.Theorem 14.
Assume thatX and Y are FK spaces containing ϕ. If X⊂Y, then F+X⊂F+Y and B+X⊂B+Y.Proof.
Suppose thatX⊂Y. Taking f-dual of the inclusion X⊂Y gives Yf⊂Xf. Again, taking β-dual of the resultant inclusion Yf⊂Xf yields Xfβ⊂Yfβ. It follows from Definition 11 that F+X⊂F+Y. Likewise, the other inclusion may be established.Theorem 15.
LetX and Y be FK spaces. Also, let X contain ϕ and has the property AD. Then, the following two statements hold:
(i)
Assume thatYf and Xf are β-spaces. Then(13)X⊂Y⇔F+X⊂F+Y(ii)
Assume thatYf and Xf are γ-spaces. Then(14)X⊂Y⇔B+X⊂B+YProof.
For (i), letX⊂Y. By Theorem 14, the inclusion F+X⊂F+Y follows. Conversely, let F+X⊂F+Y. It can be concluded from Definition 11 that Xfββ⊃Yfββ. By the assumption, since Yf and Xf are β-spaces, we have Xf⊃Yf. Since, X has the property AD, by Lemma 9, we find X⊂Y. This completes the proof of the statement (i).
Likewise, the assertion (ii) may be established. We omit the details.Theorem 16.
LetX be an FK space containing ϕ. If X is a sc space, then B+⊃bv.Proof.
SinceX is sc space, Xf⊂cs holds. Then, taking γ-dual of the inclusion Xf⊂cs gives Xfγ⊃csγ. From Definition 11 and csγ=bv, we have B+⊃bv.The following corollary summarizes some inclusion relations between spaces.Corollary 17.
LetX be an FK space containing ϕ. Then
(i)
X⊃ℓ⇔F+⊃ℓ or B+⊃ℓ(ii)
X⊃bv⇒F+⊃bv0 or B+⊃bv(iii)
X⊃bs⇒F+⊃cs or B+⊃bs(iv)
X⊃bv0⇒F+⊃bv0 or B+⊃bv(v)
X⊃c0⇔F+⊃ℓ∞ or B+⊃ℓ∞(vi)
X⊃cs⇔F+⊃cs or B+⊃bsProof.
We will only prove the statement (i). Let beX⊃ℓ. Obviously, F+⊃ℓ. Conversely, assume that F+⊃ℓ. Taking β-dual on the assumed inclusion, we have ℓβ⊃F+β. Using ℓβ=ℓ∞, we can find Xf⊂Xfββ⊂ℓβ=ℓ∞. Since, ℓ has the property AD, we obtain Xf⊂ℓ∞⇔X⊃ℓ. Similarly, the statement X⊃ℓ⇔B+⊃ℓ 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 convergenceWAC, sectional weak boundedness WB, sectional weakly p-absolute convergence (WpAC), and sectional weakly bounded variation WBV of a sequence in FK space, as given in the following definition.Definition 18.
LetX be an FK space containing ϕ. Then, a sequence x=xn in X has the following properties:
(a)
Sectional weakly absolute convergence (denoted byWAC), if(15)∑nfxnen<∞forallf∈X′(b)
Sectional weak boundedness (denoted byWB), if(16)supnfxnenforallf∈X′(c)
Sectional weaklyp-absolute convergence (denoted by WpAC), where 1≤p<∞, if(17)∑nfxnenp<∞forallf∈X′(d)
Sectional weakly bounded variation (denoted byWBV), if(18)∑nfxnen−fxn−1en−1<∞forallf∈X′The following notations in connection to Definition18 are introduced:
(19)WAC+X≔x=xn∈w∣xnfen∈ℓforallf∈X′,WB+X≔x∈w∣xnfen∈ℓ∞forallf∈X′,WpAC+X≔x∈w∣xnfen∈ℓpforallf∈X′,WBV+X≔x∈w∣xnfen∈bvforallf∈X′.Also, in association with the Definition18 and the above notations, assume that E has one of the properties WAC, WB, WpAC, and WBV. Then,
(20)EX≔E+X∩X,and anFK space X has the property E, if X⊂E+X, and so EX=X.The following theorems will demonstrate some features connected with the definitions and notations presented in this and previous sections.Theorem 19.
LetX be an FK space containing ϕ. Then,
(i)
WAC+X=Xfℓ(ii)
WB+X=Xfℓ∞(iii)
WqAC+X=Xfℓq(iv)
WBV+X=XfbvProof.
For the identity (i), we find from the propertyWAC that x∈WAC+X if and only if xu∈ℓ for each u∈Xf if and only if x∈Xfℓ.
Likewise, the other identities (ii), (iii), and (iv) can be verified. We omit the details.Theorem 20.
LetX be an FK space containing ϕ. Then,
(i)
X has WAC if and only if Xf⊂Xℓ, i.e., Xf=Xℓ(ii)
X has WBV if and only if Xf⊂Xbv, i.e., Xf=XbvProof.
For (i), letX be an FK space containing ϕ. By part (i) of Theorem 19, X⊂WAC+X=Xfℓ. Therefore, Xℓ⊃Xfℓℓ⊃Xf. Also, by Lemma 8, we have Xℓ⊂Xf. Hence, Xf=Xℓ. 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.
LetX and Y be FK spaces containing ϕ which have one (say E) of the properties WAC, WB, WqAC, and WBV. If X⊂Y, then E+X⊂E+Y.Proof.
By Definition18 and the subsequent notations, the result is obvious.Theorem 22.
LetY be an FK space ⊃ϕ and X have the property AD.
(i)
IfX and Y have property WAC, then(21)X⊂Y⇔WAC+X⊂WAC+Y(ii)
IfX and Y have property WBV, then(22)X⊂Y⇔WBV+X⊂WBV+YProof.
We will only prove the statement (i). Suppose thatX⊂Y. By Theorem 21, WAC+X⊂WAC+Y. Conversely, let WAC+X⊂WAC+Y. By Theorem 19, Xfℓ⊂Yfℓ. Then, by part (iii) in Remark 2, we have Xfℓℓ⊃Yfℓℓ. By part (i) in Theorem 20, Xℓℓℓ⊃Yℓℓℓ. By part (iv) in Remark 2, we find Xℓ⊃Yℓ. By part (i) in Theorem 20 again, we obtain Xf⊃Yf. Since X has the property AD, we obtain X⊂Y 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.
LetX be an FK space ⊃ϕ. Then,
(23)WAC+X⊂WBV+X⊂WB+X.Proof.
By applying the inclusionℓ⊂bv⊂ℓ∞ to Theorem 19, this yields the result.Theorem 24.
LetX be an FK space ⊃ϕ and z∈ω. Then,
(i)
z−1∗X⊃ℓ implies z∈WAC+X(ii)
z−1∗X⊃ℓ∞ implies z∈WB+X(iii)
z−1∗X⊃ℓp implies z∈WqAC+X(iv)
z−1∗X⊃bs implies z∈WBV+XProof.
We will prove the statement (i). Assume thatf∈z−1∗X′. Then, we find from Lemma 10 that
(24)fen=αn+gzen=αn+gznen=αn+zngen,
for allα∈ϕ and g∈X′. Hence, fen∈ℓ if and only if z∈WAC+X.
Similarly, the other statements (ii), (iii), and (iv) can be proved. We omit the details.Theorem 25.
LetX be an FK space including ϕ. For x∈X, x∈UABX if and only if x∈WAC+X.Proof.
The assertion follows from the definitions ofWAC+X and UABX.Corollary 26.
LetX be an FK space including ϕ and X has the property WAC+. Then, the following statements are equivalent:
(i)
X has the property UAB(ii)
cX=X(iii)
cX⊂WAC+XProof.
The assertion is a consequence of Theorem25.Theorem 27.
LetX be an FK space including ϕ. Then, for x∈X, x∈KBX if and only if x∈WB+X.Proof.
The assertion follows from the definitions ofWB+X and KBX.From Theorem27, we conclude the following result.Corollary 28.
LetX be an FK space including ϕ and X has the property WB+. Then, the following statements hold true:
(i)
X has the property KB(ii)
Xf⊂Xm(iii)
ℓX⊂X(iv)
X⊂XfmTheorem 29.
The propertyWB+ is weaker than the property AB.Proof.
LetX has the property AB and x=xk∈X. Since X has AB property, there exists M>0 such that xk≤M for all k∈ℕ. It follows that
(25)xkek=xk−xk−1≤xk+xk−1≤2M,
for allk∈ℕ\1. This implies that the sequence xkekk∈ℕ is bounded. The Banach-Mackey theorem states that X is bounded if and only if it is weakly bounded (see [13]). Therefore, xkek is weakly bounded, namely,
(26)supk∈ℕxkfek<∞forallf∈X′.
This means thatX has the property WB+.Corollary 30.
PropertyFAK is weaker than the property WAC+.Proof.
LetX has the property WAC+. Then, for all x∈X and f∈X′, we have xnfen∈ℓ. Since ℓ⊂cs, xnfen∈cs for those x∈X and f∈X′. This implies that X has the property FAK.Definition 31.
LetX and λ be FK spaces including ϕ. Also, assume that f∈X′. A sequence z=zn∈X has the property Wλf, if zngen∈λf for all g∈X′. Further, consider the following set
(27)Wλf+X≔z∈ω:zngen∈λfforallg∈X′.
Clearly,WλfX=Wλf+X∩X. Equivalently, an FK space X including ϕ has the property Wλf, if Wλf+X⊃X; that is, WλfX=X.Theorem 32.
Letλ be an FK space including ϕ and z∈ω. If λ is an AD space, then z−1∗X⊃λ if and only if z∈Wλf+X.Proof.
It is obvious from Definition31 that Wλf+X=Xfλf.
Using above definition, it is obvious thatWλf+X=Xfλf.
Sufficiency: letz∈Wλf+X. For each u∈Xf, zu∈λf;
(28)zu=zkuk=zkgek=gzkek,g∈X′,
andgzkek∈λf. By the equality fx=αx+gzx, α∈ϕ, for fek∈z−1∗Xf, we get fek=αk+gzkek and gzkek∈λf and so fek∈λf. Thus, we obtain z−1∗Xf⊂λf. Since λ is an AD space, then z−1∗X⊃λ.
Necessity: letz−1∗X⊃λ. For u=uk∈Xf, there exists a g∈X′ such that uk=gek for all k∈ℕ. Hence, we get
(29)zu=zkuk=zkgek=gzkek.
Sincez−1∗X⊃λ, we have z−1∗Xf⊂λf. Let v∈z−1∗Xf. Then, there exists an f∈z−1∗X′ such that vk=fek for all k∈ℕ. Then, we get fek=αk+gzkek for α∈ϕ and g∈X′. So, gzkek∈λf. Thus, gzkek∈λf and z∈Wλf+X hold.


## 4. Concluding Remarks

It is known that the foundations of the theory ofFK 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 FK 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 K and FK 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 convergenceWAC, sectional weak boundedness WB, sectional weakly p-absolute convergence WpAC, and sectional weakly bounded variation WBV, 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.



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