# On Paranorm ZweierI-Convergent Sequence Spaces

Authors: Vakeel A. Khan; Khalid Ebadullah; Ayhan Esi; Nazneen Khan; Mohd Shafiq
Journal: Journal of Mathematics
 (2013)

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## Abstract

In this paper, we introduce the paranorm ZweierI-convergent sequence spaces 𝒵I(q), 𝒵I0(q), and 𝒵∞I(q)𝒵∞I(q)forq=qk, a sequence of positive real numbers. We study some topological properties, prove the decomposition theorem, and study some inclusion relations on these spaces.


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## Body

## 1. Introduction

Letℕ,ℝ, and ℂ be the sets of all natural, real, and complex numbers, respectively. We write
(1)ω={x=(xk):xk∈ℝorℂ},
the space of all real or complex sequences.Letl∞, c, and c0 denote the Banach spaces of bounded, convergent, and null sequences, respectively, normed by ||x||∞=supk|xk|.The following subspaces ofω were first introduced and discussed by Maddox [1]:l(p):={x∈ω:∑k|xk|pk<∞},l∞(p):={x∈ω:supk|xk|pk<∞},c(p):={x∈ω:limk|xk-l|pk=0,forsomel∈ℂ},c0(p):={x∈ω:limk|xk|pk=0},where p=(pk) is a sequence of strictly positive real numbers.After that Lascarides [2, 3] defined the following sequence spaces:
(2)l∞{p}={supk|xkr|pktkx∈ω:thereexistsr>0suchthatsupk|xkr|pktk<∞},c0{p}={limk|xkr|pktkx∈ω:thereexistsr>0suchthatlimk|xkr|pktk=0},l{p}={{∑k=1∞|xkr|pktk<∞}x∈ω:thereexistsr>0suchthat∑k=1∞|xkr|pktk<∞{∑k=1∞|xkr|pktk<∞}},
where tk=pk-1, for all k∈ℕ.Each linear subspace ofω, for example, λ,μ⊂ω, is called a sequence space.A sequence spaceλ with linear topology is called a K-space provided each map pi→ℂ defined by pi(x)=xi is continuous for all i∈ℕ.AK-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.Letλ and 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 to μ, and we denote it by writing A:λ→μ.If for every sequencex=(xk)∈λ the sequence Ax={(Ax)n}, the A transform of x is in μ, where
(3)(Ax)n=∑k‍ankxk,(n∈ℕ).
By (λ:μ), we denote the class of matrices A such that A:λ→μ.Thus,A∈(λ:μ) if and only if series on the right side of (3) converges for each n∈ℕ and every x∈λ.The approach of constructing the new sequence spaces by means of the matrix domain of a particular limitation method has been recently employed by Altay et al. [4], Başar and Altay [5], Malkowsky [6], Ng and Lee [7], and Wang [8].Şengönül[9] defined the sequence y=(yi) which is frequently used as the Zp transform of the sequence x=(xi), that is,
(4)yi=pxi+(1-p)xi-1,
where x-1=0,p≠1, 1<p<∞ and Zp denotes the matrix Zp=(zik) defined by
(5)𝒵ik={p,(i=k),1-p,(i-1=k);(i,k∈ℕ),0,otherwise.
Following Başar and Altay [5], Şengönül[9] introduced the Zweier sequence spaces 𝒵 and 𝒵0 as follows:
(6)𝒵={x=(xk)∈ω:Zpx∈c},𝒵0={x=(xk)∈ω:Zpx∈c0}.
Here we quote below some of the results due to Şengönül [9] which we will need in order to establish the results of this paper.Theorem 1 (see [9, Theorem  2.1]).
The sets𝒵 and 𝒵0 are the linear spaces with the coordinate wise addition and scalar multiplication which are the BK-spaces with the norm
(7)∥x∥𝒵=∥x∥𝒵0=∥Zpx∥c.Theorem 2 (see [9, Theorem  2.2]).
The sequence spaces𝒵 and 𝒵0 are linearly isomorphic to the spaces c and c0, respectively, that is, 𝒵≅c and 𝒵0≅c0.Theorem 3 (see [9, Theorem  2.3]).
The inclusions𝒵0⊂𝒵 strictly hold for p≠1.Theorem 4 (see [9, Theorem  2.6]).
𝒵
0 is solid.Theorem 5 (see [9, Theorem  3.6]).
𝒵 is not a solid sequence space.The concept of statistical convergence was first introduced by Fast [10] and also independently by Buck [11] and Schoenberg [12] for real and complex sequences. Further this concept was studied by Connor [13, 14], Connor et al. [15], and many others. Statistical convergence is a generalization of the usual notion of convergence that parallels the usual theory of convergence. A sequence x=(xk) is said to be statistically convergent to L if for a given ϵ>0 as
(8)limk1k|{i:|xi-L|≥ϵ,i≤k}|=0.
The notion of I-convergence is a generalization of the statistical convergence. At the initial stage, it was studied by Kostyrko et al.  [16]. Later on, it was studied by Šalát et al. [17, 18], Demirci [19], Tripathy and Hazarika [20, 21], and Khan et al. [22–24].Here we give some preliminaries about the notion ofI-convergence.Let X be a nonempty set. Then a family of setsI⊆2X (denoting the power set of X) is said to be an ideal if I is additive, that is, A,B∈I⇒A∪B∈I, and hereditary, that is,  A∈I, B⊆A⇒B∈I.A nonempty family of sets£⊆2X is said to be a filter on X if and only if ϕ∉£, for A,B∈£ we have A∩B∈£ and for each A∈£ and A⊆B implies B∈£.An idealI⊆2X is called nontrivial if I≠2X.A non-trivial idealI⊆2X is called admissible if {{x}:x∈X}⊆I.A non-trivial idealI is maximal if there cannot exist any non-trivial ideal J≠I containing I as a subset.For each idealI, there is a filter £(I) corresponding to I. that is, £(I) = {K⊆N:Kc∈I}, where Kc=N-K.Definition 6.
A sequence(xk)∈ω is said to be I-convergent to a number L if {k∈N:|xk-L|≥ϵ}∈I for every ε>0. In this case we write I-limxk=L.
The spacecI of all I-convergent sequences converging to L is given by
(9)cI={(xk)∈ω:{k∈ℕ:|xk-L|≥ϵ}∈I,forsomeL∈ℂ{(xk)∈ω:{k∈ℕ:|xk-L|≥ε}}}.Definition 7.
A sequence(xk)∈ω is said to be I-null if L=0. In this case we write I-limxk=0.Definition 8.
A sequence(xk)∈ω is said to be I-Cauchy if for every ϵ>0 there exists a number k0∈ℕ such that {k∈N:|xk-xm|≥ϵ}∈I for all k,m≥k0.Definition 9.
A sequence(xk)∈ω is said to be I-bounded if there exists M>0 such that {k∈N:|xk|>M}∈I.Definition 10.
Let(xk),(yk) be two sequences. We say that (xk)=(yk) for almost all k relative toI (a.a.k.r.I), if {k∈ℕ:xk≠yk}∈I.The following lemma will be used for establishing some results of this paper.Lemma 11.
IfI⊂2N and M⊆N. If M∉I, then M∩N∉I (see [20, 21]) cf. ([17, 18, 20–24]).Recently Khan and Ebadullah [25] introduced the following classes of sequence spaces:
(10)𝒵I={k∈ℕ:{x=(xk)∈ω:I-limZpx=LforsomeL{x=(xk)∈ω:I-lim}}{x=(xk)∈ω:I-limZp}}∈I,𝒵0I={k∈ℕ:{x=(xk)∈ω:I-limZpx=0}}∈I,𝒵∞I={k∈ℕ:{x=(xk)∈ω:supk|Zpx|<∞}}∈I,
We also denote by
(11)m𝒵I=𝒵∞I∩𝒵I,m𝒵0I=𝒵∞I∩𝒵0I.In this paper we introduce the following classes of sequence spaces:(12)𝒵I(q)={x=(xk)∈ω:{k∈ℕ:|Zpx-L|qk≥ϵ}∈I,forsomeL∈ℂ{{k∈ℕ:|Zpx-L|qk≥ϵ}}};𝒵0I(q)={x=(xk)∈ω:{k∈ℕ:|Zpx|qk≥ϵ}∈I};𝒵∞I(q)={x=(xk)∈ω:supk|Zpx|qk<∞}.
We also denote by
(13)m𝒵I(q)=𝒵∞I(q)∩𝒵I(q),m𝒵0I(q)=𝒵∞I(q)∩𝒵0I(q),
where q=(qk), is a sequence of positive real numbers.Throughout the paper, for the sake of convenience now we will denote byZpx=x/,Zpy=y/,Zpz=z/ for all x,y,z∈ω.


## 2. Main Results

Theorem 12.
The classes of sequences𝒵I(q),𝒵0I(q),m𝒵I(q) and m𝒵0I(q) are linear spaces.Proof.
We shall prove the result for the space𝒵I(q).
The proof for the other spaces will follow similarly.
Let(xk),(yk)∈𝒵I(q), and let α,β be scalars. Then for a given ϵ>0: we have
(14){k∈ℕ:|xk/-L1|qk≥ϵ2M1,forsomeL1∈ℂ}∈I{k∈ℕ:|yk/-L2|qk≥ϵ2M2,forsomeL2∈ℂ}∈I,
where
(15)M1=D·max{1,supk|α|qk},M2=D·max{1,supk|β|qk},D=max{1,2H-1}whereH=supkqk≥0.
Let
(16)A1={k∈ℕ:|xk/-L1|qk<ϵ2M1,forsomeL1∈ℂ}∈IA2={k∈ℕ:|yk/-L2|qk<ϵ2M2,forsomeL2∈ℂ}∈I
be such that A1c,A2c∈I.
Then(17)A3={k∈ℕ:|(αxk/+βyk/)-(αL1+βL2)|qk<ϵ}⊇{k∈ℕ:|α|qk|xk/-L1|qk<ϵ2M1|α|qk·D}∩{k∈ℕ:|β|qk|yk/-L2|qk<ϵ2M2|β|qk·D}.
Thus A3c=A1c∩A2c∈I. Hence (αxk+βyk)∈𝒵I(q). Therefore 𝒵I(q) is a linear space. The rest of the result follows similarly.Theorem 13.
Let(qk)∈l∞. Then m𝒵I(q) and m𝒵0I(q) are paranormed spaces, paranormed by g(x)=supk|xk|qk/M where M=max{1,supkqk}.Proof.
Letx=(xk),y=(yk)∈m𝒵I(q).(1)
Clearly,g(x)=0 if and only if x=0.(2)
g(x)=g(-x) is obvious.(3)
Sinceqk/M≤1 and M>1, using Minkowski’s inequality, we have
(18)supk|xk+yk|qk/M≤supk|xk|qk/M+supk|yk|qk/M(4)
Now for any complexλ, we have (λk) such that λk→λ, (k→∞).
Letxk∈m𝒵I(q) such that |xk-L|qk≥ϵ.
Therefore,g(x-Le)=supk|xk-L|qk/M≤supk|xk|qk/M+supk|L|qk/M,  where e=(1,1,1,…).
Henceg((λnxk-λL))≤g((λnxk))+g(λL)=λng(x)+λg(L) as (k→∞).
Hencem𝒵I(q) is a paranormed space.
The rest of the result follows similarly.Theorem 14.
m
𝒵
I
(
q
) is a closed subspace of l∞(q).Proof.
Let(xk(n)) be a Cauchy sequence in m𝒵I(q) such that x(n)→x.
We show thatx∈m𝒵I(q).
Since(xk(n))∈m𝒵I(q), then there exists an such that
(19){k∈ℕ:|x(n)-an|≥ϵ}∈I.
We need to show that (1)
(an) converges to a,(2)
ifU={k∈ℕ:|xk-a|<ϵ}, then Uc∈I.
(1) Since(xk(n)) is a Cauchy sequence in m𝒵I(q) then for a given ϵ>0, there exists k0∈ℕsuch that
(20)supk|xk(n)-xk(i)|<ϵ3,∀n,i≥k0.
For a given ε>0, we have
(21)Bni={k∈ℕ:|xk(n)-xk(i)|<ϵ3},Bi={k∈ℕ:|xk(i)-ai|<ϵ3},Bn={k∈ℕ:|xk(n)-an|<ϵ3}.
Then Bnic,Bic,Bnc∈I.
LetBc=Bnic∩Bic∩Bnc, where B={k∈ℕ:|ai-an|<xϵ}.
ThenBc∈I.
We choosek0∈Bc, then for each n,i≥k0, we have
(22){k∈ℕ:|ai-an|<ϵ}⊇{k∈ℕ:|xk(i)-ai|<ϵ3}∩{k∈ℕ:|xk(n)-xk(i)|<ϵ3}∩{k∈ℕ:|xk(n)-an|<ϵ3}.
Then (an) is a Cauchy sequence of scalars in ℂ, so there exists a scalar a∈ℂ such that an→a, as n→∞.
(2) Let0<δ<1 be given. Then we show that if U={k∈ℕ:|xk-a|qk<δ}, then Uc∈I.
Sincex(n)→x, then there exists q0∈ℕsuch that
(23)P={k∈ℕ:|x(q0)-x|<(δ3D)M}
which implies that Pc∈I.
The numberq0 can be so chosen that together with (23), we have
(24)Q={k∈ℕ:|aq0-a|qk<(δ3D)M}
such that Qc∈I.
Since{k∈ℕ:|x(q0)-aq0|qk≥δ}∈I. Then we have a subset S of ℕsuch that Sc∈I, where
(25)S={k∈ℕ:|x(q0)-aq0|qk<(δ3D)M}.
Let Uc=Pc∩Qc∩Sc, where U={k∈ℕ:|xk-a|qk<δ}.
Therefore for eachk∈Uc, we have
(26){k∈ℕ:|xk-a|qk<δ}⊇{k∈ℕ:|x(q0)-x|qk<(δ3D)M}∩{k∈ℕ:|x(q0)-aq0|qk<(δ3D)M}∩{k∈ℕ:|aq0-a|qk<(δ3D)M}.
Then the result follows.Since the inclusionsm𝒵I(q)⊂l∞(q) and m𝒵0I(q)⊂l∞(q) are strict, so in view of Theorem 14 we have the following result.Theorem 15.
The spacesm𝒵I(q) and m𝒵0I(q) are nowhere dense subsets of l∞(q).Theorem 16.
The spacesm𝒵I(q) and m𝒵0I(q) are not separable.Proof.
We shall prove the result for the spacem𝒵I(q).
The proof for the other spaces will follow similarly.
LetM be an infinite subset of ℕ of increasing natural numbers such that M∈I.
Let(27)qk={1,ifk∈M,2,otherwise.
Let P0={(xk):xk=0or1,fork∈Mandxk=0,otherwise}.
ClearlyP0 is uncountable.
Consider the class of open ballsB1={B(z,1/2):z∈P0}.
LetC1 be an open cover of m𝒵I(q) containing B1.
SinceB1 is uncountable, so C1 cannot be reduced to a countable subcover for m𝒵I(q).
Thusm𝒵I(q) is not separable.Theorem 17.
LetG=supkqk<∞ and I an admissible ideal. Then the following is equivalent.(a)
(xk)∈𝒵I(q);(b)
there exists(yk)∈𝒵(q) such that xk=yk, for a.a.k.r.I;(c)
there exists(yk)∈𝒵(q) and (xk)∈𝒵0I(q) such that xk=yk+zk for all k∈ℕand {k∈ℕ:|yk-L|qk)≥ϵ}∈I;(d)
there exists a subsetK={k1<k2<⋯} of ℕsuch that K∈£(I) and limn→∞|xkn-L|qkn=0.Proof.
(a) implies (b).
Let(xk)∈𝒵I(q). Then there exists L∈ℂsuch that
(28){k∈ℕ:|xk/-L|qk≥ϵ}∈I.
Let (mt) be an increasing sequence with mt∈ℕ such that
(29){k≤mt:|xk/-L|qk≥t-1}∈I.
Define a sequence (yk) as
(30)yk=xk,∀k≤m1.
For mt<k≤mt+1,t∈ℕ. 							(31)yk={xk,if|xk/-L|qk<t-1,L,otherwise.
Then (yk)∈𝒵(q) and form the following inclusion:
(32){k≤mt:xk≠yk}⊆{k≤mt:|xk/-L|qk≥ϵ}∈I,
we get xk=yk, for a.a.k.r.I.
(b) implies (c).
For(xk)∈𝒵I(q). Then there exists (yk)∈𝒵(q) such that xk=yk, for a.a.k.r.I.
LetK={k∈ℕ:xk≠yk}, then k∈I.
Define a sequence(zk) as
(33)zk={xk-yk,ifk∈K,0,otherwise.
Then zk∈𝒵0I(q) and yk∈𝒵(q).
(c) implies (d).
Suppose (c) holds.
Letϵ>0 be given.
LetP1={k∈ℕ:|zk/|qk≥ϵ}∈I and
(34)K=P1c={k1<k2<k3<⋯}∈£(I).
Then we have limn→∞|xkn/-L|qkn=0.
(d) implies (a).
LetK={k1<k2<k3<⋯}∈£(I) and limn→∞|xkn/-L|qkn=0.
Then for anyϵ>0, and Lemma 11, we have
(35){k∈ℕ:|xk/-L|qk≥ϵ}⊆Kc∪{k∈K:|xk/-L|qk≥ϵ}.
Thus (xk)∈𝒵I(q).Theorem 18.
Leth=infkqk and G=supkqk. Then the following results are equivalent.(a)
G<∞ and h>0.(b)
𝒵0I(q)=𝒵0I.Proof.
Suppose thatG<∞ and h>0, then the inequalities min{1,sh}≤sqk≤max{1,sG} hold for any s>0 and for all k∈ℕ.
Therefore the equivalence of (a) and (b) is obvious.Theorem 19.
Let(qk) and (rk) be two sequences of positive real numbers. Then m𝒵0I(q)⊇m𝒵0I(r) if and only if limk∈Kinf(qk/rk)>0, where Kc⊆ℕsuch that K∈I.Proof.
Letlimk∈Kinf(qk/rk)>0 and (xk)∈m𝒵0I(r). Then there exists β>0 such that qk>βrk, for all sufficiently large k∈K.
Since(xk)∈m𝒵0I(r) for a given ϵ>0, we have
(36)B0={k∈ℕ:|xk|rk≥ϵ}∈I.
Let G0=Kc∪B0 Then G0∈I.
Then for all sufficiently largek∈G0,
(37){k∈ℕ:|xk|qk≥ϵ}⊆{k∈ℕ:|xk|βrk≥ϵ}∈I.
Therefore (xk)∈m𝒵0I(q).
The converse part of the result follows obviously.Theorem 20.
Let(qk) and (rk) be two sequences of positive real numbers. Then m𝒵0I(r)⊇m𝒵0I(q) if and only if limk∈Kinf(rk/qk)>0, where Kc⊆ℕ such that K∈I.Proof.
The proof follows similarly as the proof of Theorem19.Theorem 21.
Let(qk) and (rk) be two sequences of positive real numbers. Then m0I(r)=m0I(q) if and only if limk∈Kinf(qk/rk)>0, and limk∈Kinf(rk/qk)>0, where K⊆ℕ such that Kc∈I.Proof.
By combining Theorems19 and 20, we get the required result.



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