# Some Topological and Geometrical Properties of a New Difference Sequence Space

Authors: Serkan Demiriz; Celal Çakan
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We introduce the new difference sequence spaceapr(Δ) . Further, it is
proved that the space  apr(Δ) is the BK-space including the space bvp, which is the space of sequences of pbounded
variation. We also show that the spaces  apr(Δ), and ℓp are linearly isomorphic for 1≤p<∞.
Furthermore, the basis and the α-, β- and
γ-duals of the space apr(Δ) are determined. We devote
the final section of the paper to examine some geometric properties of the space apr(Δ).


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## Body

## 1. Preliminaries, Background, and Notation

Byω, we will denote the space of all real valued sequences. Any vector subspace of ω is called as a sequence space. We will write ℓ∞,c, and c0 for the spaces of all bounded, convergent and null sequences, respectively. Also, by bs,cs,ℓ1 and ℓp; we denote the spaces of all bounded, convergent, absolutely, and p-absolutely convergent series, respectively, where 1<p<∞.A sequence spaceλ with a linear topology is called a K-space, provided each of the maps pi:λ→ℂ defined by pi(x)=xi is continuous for all i∈ℕ, where ℂ denotes the complex field and ℕ={0,1,2,…}. A K-space λ is called an FK-space, provided λ is a complete linear metric space. An FK-space whose topology is normable is called a BK-space (see [1, pages 272-273]).Letλ,μ be two sequence spaces and A=(ank) an infinite matrix of real or complex numbers ank, where n,k∈ℕ. Then, we say that A defines a matrix mapping from λ into μ, and we denote it by writing A:λ→μ; if for every sequence x=(xk)∈λ, the sequence Ax={(Ax)n}, the A-transform of x, is in μ, where
(1.1)(Ax)n=∑kankxk(n∈N).
For simplicity in notation, here and in what follows, the summation without limits runs from 0 to ∞. By (λ:μ) we denote the class of all matrices A such that A:λ→μ. Thus, A∈(λ:μ) if and only if the series on the right side of (1.1) converges for each n∈ℕ and every x∈λ, and we have Ax={(Ax)n}n∈ℕ∈μ for all x∈λ. A sequence x is said to be A-summable to α if Ax converges to α which is called as the A-limit of x.If a normed sequence spaceλ contains a sequence (bn) with the property that for every x∈λ, there is a unique sequence of scalars (αn) such that (1.2)lim⁡n⟶∞⁡‖x-(α0b0+α1b1+⋯+αnbn)‖=0,
then (bn) is called a Schauder basis (or briefly basis) for λ. The series ∑αkbk which has the sum x is then called the expansion of x with respect to (bn) and written as x=∑αkbk.For a sequence spaceλ, the matrix domainλA of an infinite matrix A is defined by(1.3)λA={x=(xk)∈ω:Ax∈λ},
which is a sequence space. The new sequence space λA generated by the limitation matrix A from the space λ either includes the space λ or is included by the space λ, in general; that is, the space λA is the expansion or the contraction of the original space λ.We will defineBr=(bnkr) by(1.4)bnkr={1-rn+1rk,0≤k≤n-1,rn+1n+1,k=n,0,k>n,(0<r<1),
for all n,k∈ℕ and denote the collection of all finite subsets of ℕ by ℱ. We will also use the convention that any term with negative subscript is equal to naught.The approach constructing a new sequence space by means of the matrix domain of a particular limitation method has been recently employed by Wang [2], Ng and Lee [3], Malkowsky [4], and Altay et al. [5]. They introduced the sequence spaces (ℓp)Nq in [2], (ℓp)C1=Xp in [3], (ℓ∞)Rt=r∞t,cRt=rct and (c0)Rt=r0t in [4] and (ℓp)Er=epr in [5]; where Nq,C1,Rt and Er denote the Nörlund, arithmetic, Riesz and Euler means, respectively, and 1≤p≤∞.Recently, there has been a lot of interest in investigating geometric properties of sequence spaces besides topological and some other usual properties. In literature, there are many papers concerning the geometric properties of different sequence spaces. For example, in [6], Mursaleen et al. studied some geometric properties of normed Euler sequence space. Şimşek and Karakaya [7] investigated the geometric properties of sequence space ℓρ(u,v,p) equipped with Luxemburg norm. Further information on geometric properties of sequence space can be found in [8, 9].The main purpose of the present paper is to introduce the difference sequence spaceapr(Δ) together with matrix domain and is to derive some inclusion relations concerning with apr(Δ). Also, we investigate some topological properties of this new space and furthermore characterize geometric properties concerning Banach-Saks type p.


## 2.apr(Δ) Difference Sequence Space

In the present section, we introduce the difference sequence spaceapr(Δ) and emphasize its some properties. Although the difference sequence space λ(Δ) corresponding to the space λ was defined by Kızmaz [10] as follows: (2.1)λ(Δ)={x=(xk)∈ω:(xk-xk+1)∈λ},
the difference sequence space corresponding to the space ℓp was not examined, where λ denotes the anyone of the spaces c0,c or ℓ∞. So, Başar and Altay have recently studied the sequence space bvp, the space of p-bounded variation, in [11] defined by (2.2)bvp={x=(xk)∈ω:(xk-xk-1)∈lp},1≤p<∞,
which fills up the gap in the existing literature. Recently, Aydín and Başar [12] studied the sequence spaces a0r and acr, defined by (2.3)a0r={x=(xk)∈ω:lim⁡n⟶∞⁡1n+1∑k=0n(1+rk)xk=0},acr={x=(xk)∈ω:lim⁡n⟶∞⁡1n+1∑k=0n(1+rk)xkexists}.
Aydín and Başar [13] introduced the difference sequence spaces a0r(Δ) and acr(Δ), defined by (2.4)a0r(Δ)={x=(xk)∈ω:lim⁡n⟶∞⁡1n+1∑k=0n(1+rk)(xk-xk-1)=0},acr(Δ)={x=(xk)∈ω:lim⁡n⟶∞⁡1n+1∑k=0n(1+rk)(xk-xk-1)exists}.
Aydín [14] introduced apr sequence space, defined by (2.5)apr={x=(xk)∈ω:∑n|1n+1∑k=0n(1+rk)xk|p<∞};1≤p<∞.
Define the matrix Δ=(δnk) by (2.6)δnk={(-1)n-k,n-1≤k≤n,0,0≤k<n-1ork>n.
As was made by Başar and Altay in [11], we treat slightly more different than Kızmaz and the other authors following him and employ the technique obtaining a new sequence space by the matrix domain of a triangle limitation method. We will introduce the sequence space apr(Δ) which is a natural continuation of Aydín and Başar [13], as follows: (2.7)apr(Δ)={x=(xk)∈ω:∑n|1n+1∑k=0n(1+rk)(xk-xk-1)|p<∞};1≤p<∞.
With the notation of (1.3), we may redefine the space apr(Δ) by(2.8)apr(Δ)=(apr)Δ.Define the sequencey={yn(r)} which will be frequently used as the Br-transform of a sequence x=(xk), that is,(2.9)yn(r)=∑k=0n-1(1-r)rk1+nxk+1+rn1+nxn,n∈N.
Now, we may begin with the following theorem which is essential in the text.Theorem 2.1.
The setapr(Δ) becomes the linear space with the coordinatewise addition and scalar multiplication which is the BK-space with the norm
(2.10)‖x‖apr(Δ)=‖y‖lp,
where 1≤p<∞.Proof.
Since the proof is routine, we omit the details of the proof.Theorem 2.2.
The spaceapr(Δ) is linearly isomorphic to the space ℓp; that is, apr(Δ)≅ℓp, where 1≤p<∞.Proof.
It is enough to show the existence of a linear bijection between the spacesapr(Δ) and ℓp for 1≤p<∞. Consider the transformation T defined, with the notation of (2.9), from apr(Δ) to ℓp by x↦y=Tx. The linearity of T is clear. Furthermore, it is trivial that x=θ whenever Tx=θ, and hence, T is injective.
We assume thaty∈ℓp for 1≤p<∞ and define the sequence x=(xk) by
(2.11)xn(r)=∑k=0n∑j=k-1k(-1)k-j1+j1+rkyj;k∈N.
Then, since
(2.12)(Δx)n=∑j=n-1n(-1)n-j1+j1+rnyj;n∈N,
we get that
(2.13)∑n|11+n∑k=0n(1+rk)∑j=k-1k(-1)k-j1+j1+rkyj|p=∑n|yn|p<∞.
Thus, we have that x∈apr(Δ). In addition, one can derive that
(2.14)‖x‖apr(Δ)=(∑n|11+n∑k=0n(1+rk)(xk-xk-1)|p)1/p=(∑n|11+n∑k=0n(1+rk)∑j=k-1k(-1)k-j1+j1+rkyj|p)1/p=‖y‖lp
which means that T is surjective and is norm preserving. Hence, T is a linear bijection.We wish to exhibit some inclusion relations concerning with the spaceapr(Δ).Theorem 2.3.
The inclusionbvp⊂apr(Δ) strictly holds for 1<p<∞.Proof.
To prove the validity of the inclusionbvp⊂apr(Δ) for 1<p<∞, it suffices to show the existence of a number K>0 such that ∥x∥apr(Δ)≤K∥x∥bvp for every x∈bvp.
Letx∈bvp and 1<p<∞. Then, we obtain
(2.15)∑n|11+n∑k=0n(1+rk)(xk-xk-1)|p≤∑n[2∑k=0n|xk-xk-1|1+n]p<2p(pp-1)p∑n|xn-xn-1|p,
as expected,
(2.16)‖x‖apr(Δ)≤(2pp-1)‖x‖bvp,
for 1<p<∞.
Furthermore, let us consider the sequencex={xn(r)} defined by
(2.17)xn(r)=∑k=0n(-1)n1+rk,n∈N.
Then, the sequence x is in apr(Δ)-bvp, as asserted.Lemma 2.4 (see [11, Theorem  2.4]).
The inclusionℓp⊂bvp strictly holds for 1<p<∞.Combining Lemma2.4 and Theorem 2.3, we get the following corollary.Corollary 2.5.
The inclusionℓp⊂apr(Δ) strictly holds for 1<p<∞.


## 3. The Basis for the Spaceapr(Δ)

In the present section, we will give a sequence of the points of the spaceapr(Δ) which forms a basis for the space apr(Δ), where 1≤p<∞.Theorem 3.1.
Define the matrixBr={bn(k)(r)}n∈ℕ of elements of the space apr(Δ) for every fixed k∈ℕ by
(3.1)bn(k)(r)={1+k1+rk-1+k1+rk+1,0≤k≤n-1,1+n1+rn,k=n,0,k>n,
for every fixed k∈ℕ. Then, the sequence {b(k)(r)}k∈ℕ is a basis for the space apr(Δ), and any x∈apr(Δ) has a unique representation of the form
(3.2)x=∑kλk(r)b(k)(r),
where λk(r)=(Brx)k for all k∈ℕ and 1≤p<∞.Proof.
It is clear that{b(k)(r)}⊂apr(Δ), since
(3.3)Brb(k)(r)=e(k)∈lp,k=0,1,2,…,
for 1≤p<∞; here, e(k) is the sequence whose only nonzero term is 1 in the kth place for each k∈ℕ. Let x∈apr(Δ) be given. For every nonnegative integer m, we set
(3.4)x[m]=∑k=0mλk(r)b(k)(r).
Then, by applying Br to (3.4), we obtain with (3.3) that
(3.5)Brx[m]=∑k=0mλk(r)Brb(k)(r)=∑k=0m(Brx)ke(k),{Br(x-x[m])}i={0,0≤i≤m,(Brx)i,i>m,
where i,m∈ℕ. For a given ε>0, there is an integer m0 such that
(3.6)[∑i=m∞|(Brx)i|p]1/p<ε2,
for all m≥m0. Hence,
(3.7)‖x-x[m]‖apr(Δ)=[∑i=m∞|(Brx)i|p]1/p≤[∑i=m0∞|(Brx)i|p]1/p<ε2<ε,
for all m≥m0, which proves that x∈apr(Δ) is represented as in (3.2).
Let us show the uniqueness of representation forx∈apr(Δ) given by (3.2). Assume, on the contrary, that there exists a representation x=∑kμk(r)b(k)(r). Since the linear transformation T, from apr(Δ) to ℓp, used in Theorem 2.2 is continuous, at this stage, we have
(3.8)(Brx)n=∑kμk(r){Brb(k)(r)}n=∑kμk(r)en(k)=μn(r);n∈N,
which contradicts the fact that (Brx)n=λn(r) for all n∈ℕ. Hence, the representation (3.2) of x∈apr(Δ) is unique. This step concludes the proof.


## 4. Theα-, β-, and γ-Duals of the Space apr(Δ)

In this section, we state and prove theorems determining theα-, β-, and γ-duals of the space apr(Δ). Since the case p=1 can be proved by the same analogy and can be found in the literature, we omit the proof of that case and consider only the case 1<p<∞ in the proof of Theorems 4.4–4.6.For the sequence spacesλ and μ, define the set S(λ,μ) by(4.1)S(λ,μ)={z=(zk)∈ω:xz=(xkzk)∈μ,∀x∈λ}.
With the notation of (4.1), α-, β- and γ-duals of a sequence space λ, which are, respectively, denoted by λα,λβ and λγ, are defined by (4.2)λα=S(λ,l1),λβ=S(λ,cs),λγ=S(λ,bs).
It is well-known for the sequence spaces λ and μ that λα⊆λβ⊆λγ and λη⊃μη whenever λ⊂μ, where η∈{α,β,γ}.We begin with to quoting the lemmas due to Stieglitz and Tietz [15], which are needed in the proof of the following theorems.Lemma 4.1.
A∈(ℓp:ℓ1) if and only if
(4.3)sup⁡K∈F∑k|∑n∈Kank|q<∞(1<p≤∞).Lemma 4.2.
A∈(ℓp:c) if and only if
(4.4)lim⁡n⟶∞⁡ankexistsfor eachk∈N,(4.5)sup⁡n∈N∑k|ank|q<∞1<p<∞.Lemma 4.3.
A∈(ℓp:ℓ∞) if and only if (4.5) holds.Theorem 4.4.
Define the setaqr by
(4.6)aqr={a=(ak)∈ω:sup⁡K∈F∑k|∑n∈Kcnkr|q<∞},
where Cr=(cnkr) is defined via the sequence a=(an) by
(4.7)cnkr={(1+k1+rk-1+k1+rk+1)an,0≤k≤n-1,1+n1+rnan,k=n,0,k>n.
for all n,k∈ℕ. Then, {apr(Δ)}α=aqr, where 1<p<∞.Proof.
Bearing in mind the relation (2.9), we immediately derive that
(4.8)anxn=∑k=0n∑j=k-1k(-1)k-j1+j1+rkanyj=(Cry)n,n∈N.
It follows from (4.8) that ax=(anxn)∈ℓ1 whenever x∈apr(Δ) if and only if Cry∈ℓ1 whenever y∈ℓp. This means that a=(an)∈{apr(Δ)}α if and only if Cr∈(ℓp:ℓ1). Then, we derive by Lemma 4.1 with Cr instead of A that
(4.9)sup⁡K∈F∑k|∑n∈Kcnkr|q<∞.
This yields the desired consequence that {apr(Δ)}α=aqr.Theorem 4.5.
Define the setsa1r,a2r, and a3r by
(4.10)a1r={a=(ak)∈ω:sup⁡n∈N∑k|enkr|q<∞},a2r={a=(ak)∈ω:∑j=k∞ajexists for each fixedk∈N},a3r={a=(ak)∈ω:{1+n1+rnan}∈cs},
where E=(enkr) is defined by
(4.11)enkr={(k+1)[ak1+rk+(11+rk-11+rk+1)∑j=k+1naj],0≤k≤n-1,1+n1+rnan,k=n,0,k>n,
for all n,k∈ℕ. Then, {apr(Δ)}β=a1r∩a2r∩a3r, where 1<p<∞.Proof.
Consider the equation(4.12)∑k=0nakxk=∑k=0n{∑j=0k[∑i=j-1j(-1)i-j1+i1+rjyj]}ak=∑k=0n-1(k+1)[ak1+rk+(11+rk-11+rk+1)∑j=k+1naj]yk+1+n1+rnanyn=(Ey)nn∈N.
Thus, we deduce from Lemma 4.2 with (4.12) that ax=(akxk)∈cs whenever x=(xk)∈apr(Δ) if and only if Ey∈c whenever y=(yk)∈ℓp. That is to say that a=(ak)∈{apr(Δ)}β if and only if E∈(ℓp:c). Therefore, we derive from (4.4) and (4.5) that {apr(Δ)}β=a1r∩a2r∩a3r.Theorem 4.6.
{apr(Δ)}γ=a1r, where 1<p<∞.Proof.
It is natural that the present theorem may be proved by the same technique used in the proof of Theorems4.4 and 4.5, above. But, we prefer here the following classical way.
Leta=(ak)∈a1r and x=(xk)∈apr(Δ). Then, we obtain by applying Hölder’s inequality that
(4.13)|∑k=0nakxk|=|∑k=0n{∑j=0k[∑i=j-1j(-1)i-j1+i1+rjyj]}ak|=|∑k=0nenkryk|≤(∑k=0n|enkr|q)1/q(∑k=0n|yk|p)1/p,
which gives us by taking supremum over n∈ℕ that
(4.14)sup⁡n∈N|∑k=0nakxk|≤sup⁡n∈N[(∑k=0n|enkr|q)1/q(∑k=0n|yk|p)1/p]≤‖y‖lp⋅(sup⁡n∈N∑k=0n|enkr|q)1/q<∞.
This means that a=(ak)∈{apr(Δ)}γ. Hence,
(4.15)a1r⊂{apr(Δ)}γ.
Conversely, let a=(ak)∈{apr(Δ)}γ and x=(xk)∈apr(Δ). Then, one can easily see that (∑k=0nenkryk)n∈ℕ∈ℓ∞ whenever (akxk)∈bs. This shows that the triangle matrix E=(enkr), defined by (4.11), is in the class (ℓp:ℓ∞). Hence, the condition (4.5) holds with enkr instead of ank which yields that a=(ak)∈a1r. That is to say that
(4.16){apr(Δ)}γ⊂a1r.
Therefore, by combining the inclusions (4.15) and (4.16), we deduce that the γ-dual of the space apr(Δ) is the set a1r, and this step completes the proof.


## 5. Some Geometric Properties of the Spaceapr(Δ)

In this section, we study some geometric properties of the spaceapr(Δ).A Banach spaceX is said to have the Banach-Saks property if every bounded sequence (xn) in X admits a subsequence (zn) such that the sequence {tk(z)} is convergent in the norm in X [16], where (5.1)tk(z)=1k+1(z0+z1+⋯+zk)(k∈N).
A Banach space X is said to have the weak  Banach-Saks  property whenever given any weakly null sequence (xn)⊂X and there exists a subsequence (zn) of (xn) such that the sequence {tk(z)} strongly convergent to zero.In [17], García-Falset introduce the following coefficient: (5.2)R(X)=sup⁡{liminf⁡n⟶∞‖xn-x‖:(xn)⊂B(X),xn→ω0,x∈B(X)},
where B(X) denotes the unit ball of X.Remark 5.1.
A Banach spaceX with R(X)<2 has the weak fixed point property, [18].Let1<p<∞. A Banach space is said to have the Banach-Saks typep or property (BS)p, if every weakly null sequence (xk) has a subsequence (xkl) such that for some C>0, (5.3)‖∑l=0nxkl‖<C(n+1)1/p,
for all n∈ℕ (see [19]).Now, we may give the following results related to the some geometric properties, mentioned above, of the spaceapr(Δ).Theorem 5.2.
The spaceapr(Δ) has the Banach-Saks type p.Proof.
Let(ɛn) be a sequence of positive numbers for which ∑εn≤1/2, and also let (xn) be a weakly null sequence in B(apr(Δ)). Set b0=x0=0 and b1=xn1=x1. Then, there exists m1∈ℕ such that
(5.4)‖∑i=m1+1∞b1(i)e(i)‖apr(Δ)<ε1.
Since (xn) is a weakly null sequence implies xn→0 coordinatewise, there is an n2∈ℕ such that
(5.5)‖∑i=0m1xn(i)e(i)‖apr(Δ)<ε1,
where n≥n2. Set b2=xn2. Then, there exists an m2>m1 such that
(5.6)‖∑i=m2+1∞b2(i)e(i)‖apr(Δ)<ε2.
By using the fact that xn→0 coordinatewise, there exists an n3>n2 such that
(5.7)‖∑i=0m2xn(i)e(i)‖apr(Δ)<ε2,
where n≥n3.
If we continue this process, we can find two increasing subsequences(mi) and (ni) such that
(5.8)‖∑i=0mjxn(i)e(i)‖apr(Δ)<εj,
for each n≥nj+1 and
(5.9)‖∑i=mj+1∞bj(i)e(i)‖apr(Δ)<εj,
where bj=xnj. Hence,
(5.10)‖∑j=0nbj‖apr(Δ)=‖∑j=0n(∑i=0mj-1bj(i)e(i)+∑i=mj-1+1mjbj(i)e(i)+∑i=mj+1∞bj(i)e(i))‖apr(Δ)≤‖∑j=0n(∑i=0mj-1bj(i)e(i))‖apr(Δ)+‖∑j=0n(∑i=mj-1+1mjbj(i)e(i))‖apr(Δ)+‖∑j=0n(∑i=mj+1∞bj(i)e(i))‖apr(Δ)≤‖∑j=0n(∑i=mj-1+1mjbj(i)e(i))‖apr(Δ)+2∑j=0nεj.
On the other hand, it can be seen that ∥xn∥apr(Δ)<1. Therefore, ∥xn∥apr(Δ)p<1. We have
(5.11)‖∑j=0n(∑i=mj-1+1mjbj(i)e(i))‖apr(Δ)p=∑j=0n∑i=mj-1+1mj|∑k=0i-1(1-r)rk1+ixj(k)+1+ri1+ixi(k)|p≤∑j=0n∑i=0∞|∑k=0i-1(1-r)rk1+ixj(k)+1+ri1+ixi(k)|p≤(n+1).
Hence, we obtain
(5.12)‖∑j=0n(∑i=mj-1+1mjbj(i)e(i))‖apr(Δ)≤(n+1)1/p.
By using the fact 1≤(n+1)1/p for all n∈ℕ, we have
(5.13)‖∑j=0nbj‖apr(Δ)≤(n+1)1/p+1≤2(n+1)1/p.
Hence, apr(Δ) has the Banach-Saks type p. This completes the proof of the theorem.Remark 5.3.
Note thatR(apr(Δ))=R(ℓp)=21/p, since apr(Δ) is linearly isomorphic to ℓp.Hence, by the Remarks5.1 and 5.3, we have the following.Theorem 5.4.
The spaceapr(Δ) has the weak fixed point property, where 1<p<∞.



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