# Some Spaces of Double Sequences Obtained through Invariant Mean and Related Concepts

Authors: S. A. Mohiuddine; Abdullah Alotaibi
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

We introduce some double sequences spaces involving the notions of invariant mean (orσ-mean) and σ-convergence for double sequences while the idea of σ-convergence for double sequences was introduced by Çakan et al. 2006, by using the notion of invariant mean. We determine here some inclusion relations and topological results for these new double sequence spaces.


---

## Body

## 1. Preliminaries, Background, and Notation

In 1900, Pringsheim [1] presented the following notion of convergence for double sequences. A double sequence x=(xjk) is said to converge to the limitL  in Pringsheim’s  sense (shortly, p-convergent to  L) if for every ε>0 there exists an integer N such that |xjk-L|<ε whenever j,k>N. In this case, L is called the p-limit of x.A double sequencex=(xjk) of real or complex numbers is said to be bounded if ∥x∥∞=supj,k|xjk|<∞. The space of all bounded double sequences is denoted by ℳu.Ifx∈ℳu and is p-convergent to L, then x is said to be boundedlyp-convergent toL (shortly, bp-convergent toL). In this case, L is called the bp-limit of x. The assumption of p-convergent was made because a double sequence on which p-convergent is not necessarily bounded.In general, for any notion of convergenceν, the space of all ν-convergent double sequences will be denoted by 𝒞ν, the space of all ν-convergent to 0 double sequences by 𝒞ν0 and the limit of a ν-convergent double sequence x by ν-limj,kxjk, where ν∈{p,bp}.LetΩ denote the vector space of all double sequences with the vector space operations defined coordinatewise. Vector subspaces of Ω are called double sequence spaces. In addition to the above-mentioned double sequence spaces, we consider the double sequence space as
(1)ℒu∶={x∈Ω∣∥x∥1:=∑j,k‍|xjk|<∞}
of all absolutely summable double sequences.All considered double sequence spaces are supposed to contain(2)Φ∶=span{ejk∣j,k∈ℕ},
where
(3)eiljk={1;if(j,k)=(i,ℓ),0;otherwise.We denote the pointwise sums∑j,k‍ejk,  ∑j‍ejk(k∈ℕ), and ∑k‍ejk(j∈ℕ) by e,  ek, and ej, respectively.Letσ be a one-to-one mapping from the set ℕ of natural numbers into itself. A continuous linear functional φ on the space ℓ∞ of bounded single sequences is said to be an invariant  mean or a σ-mean if and only if (i) φ(x)≥0 when the sequence x=(xk) has xk≥0 for all k, (ii) φ(e)=1, where e=(1,1,1,…), and (iii) φ(x)=φ((xσ(k))) for all x∈ℓ∞.Throughout this paper we consider the mappingσ which has no finite orbits, that is, σp(k)≠k for all integer k≥0 and p≥1, where σp(k) denotes the pth iterate of σ at k. Note that a σ-mean extends the limit functional on the space c of convergent single sequences in the sense that φ(x)=limx for all x∈c, (see [2]). Consequently, c⊂Vσ the set of bounded sequences all of whose σ-means are equal. We say that a sequence x=(xk) is σ-convergent if and only if x∈Vσ. Using this concept, Schaefer [3] defined and characterized σ-conservative, σ-regular, and σ-coercive matrices for single sequences. If σ is translation then Vσ is reduced to the set f of almost convergent sequences [4]. Recently, Mohiuddine [5] has obtained an application of almost convergence for single sequences in approximation theorems and proved some related results.In 2006, Çakan et al. [6] presented the following definition of σ-convergence for double sequences and further studied by Mursaleen and Mohiuddine [7–9]. A double sequence x=(xjk) of real numbers is said to be σ-convergent to a number L if and only if x∈𝒱σ, where
(4)𝒱σ={x∈ℳu:limp,q→∞ζpqst(x)=Luniformlyins,t;L=σ-limxlimp,q→∞},ζpqst(x)=1(p+1)(q+1)∑j=0p∑k=0q‍xσj(s),σk(t),
while here the limit means bp-limit. Let us denote by 𝒱σ the space of σ-convergent double sequences x=(xjk). For σ(n)=n+1, the set 𝒱σ is reduced to the set ℱ of almost convergent double sequences [10]. Note that 𝒞bp⊂𝒱σ⊂ℳu.Maddox [11] has defined the concepts of strong almost convergence and M-convergence for single sequences and established inclusion relation between strong almost convergence, M-convergence, and almost convergence for single sequence. Başarir [12] extended the notion of strong almost convergence from single sequences to double sequences and proved some interesting results involving this idea and the notion of almost convergence for double sequences. In the recent past, Mursaleen and Mohiuddine [13] presented the notions of absolute and strong σ-convergence for double sequences. A bounded double sequence x=(xjk) is said to be stronglyσ-convergent if there exists a number ℓ such that
(5)1(p+1)(q+1)∑j=0p‍∑k=0q‍|xσj(s),σk(t)-ℓ|⟶0asp,q⟶∞uniformlyins,t,
while here the limit means bp-limit. In this case, we write [𝒱σ]-limx=ℓ. Let us denote by [𝒱σ] the set of all strongly σ-convergent sequences x=(xjk). If σ is translation then [𝒱σ] is reduced to the set [ℱ] of strong almost convergence double sequences due to Başarir [12].For more details of spaces for single and double sequences and related concepts, we refer to [14–31] and references therein.In this paper, we define and study some new spaces involving the idea of invariant mean andσ-convergence for double sequences and establish a relation between these spaces. Further, we extend above spaces to more general spaces by considering the double sequences α=(αjk) such that αjk>0 for all j,  k and supj,kαjk=H<∞ and prove some topological results.


## 2. The Double Sequence Spaces

We construct the following spaces involving the idea of invariant mean andσ-convergence for double sequences:
(6)𝒲σ={x=(xjk):1(m+1)(n+1)×∑p=0m‍∑q=0n‍ζpqst(x-ℓE)⟶0asm,n⟶∞,uniformlyins,t,forsomeℓ1(m+1)(n+1)},[𝒲σ]={x=(xjk):1(m+1)(n+1)×∑p=0m‍∑q=0n‍|ζpqst(x-ℓE)|⟶0asm,n⟶∞,uniformlyins,t,forsomeℓ1(m+1)(n+1)},𝒲σ′={∑p=0∞‍x=(xjk):∑p=0∞‍∑q=0∞‍|ξpqst-ξp-1,q,s,t-ξp,q-1,s,t+ξp-1,q-1,s,t|convergesuniformlyins,t1(m+1)(n+1)∑p=0∞‍},𝒲σ′′={∑q=0∞‍x=(xjk):sups,t∑p=0∞‍∑q=0∞‍|ξpqst-ξp-1,q,s,t-ξp,q-1,s,tsups,t∑p=0∞‍∑q=0∞‍+ξp-1,q-1,s,t|<∞∑q=0∞‍},
where E=(ejk) with ejk=1 for all j,k;
(7)ξmnst=ξmnst(x)=1(m+1)(n+1)∑p=0m∑q=0n‍ζpqst(x),ξ0,0,s,t(x)=ζ0,0,s,t=xs,t,ξ-1,0,s,t(x)=ζ-1,0,s,t(x)=xs-1,t,ξ0,-1,s,t(x)=ζ0,-1,s,t(x)=xs,t-1,ξ-1,-1,s,t(x)=ζ-1,-1,s,t(x)=xs-1,t-1.Remark 1.
If[𝒲σ]-limx=ℓ, that is,
(8)1(m+1)(n+1)∑p=0m‍∑q=0n‍|ζpqst-ℓ|⟶0
as m,n→∞, uniformly in s,  t; then
(9)1(m+1)(n+1)∑p=0m∑q=0n‍|1p+1∑j=0p‍ζjqst-ℓ|⟶0,1(m+1)(n+1)∑p=0m‍∑q=0n‍|1q+1∑k=0q‍ζpkst-ℓ|⟶0.We remark that by using Abel’s transformation for single series(10)∑i=1m‍vi(ui∓ui+1)=∑i=1m‍ui(vi∓vi-1)∓um+1vm.
We get Abel’s transformation for double series
(11)∑j=1p∑k=1q‍vjk(ujk-uj+1,k-uj,k+1+uj+1,k+1)=∑j=1p∑k=1q‍ujk(Δ11vjk)-∑j=1p‍uj,q+1(Δ10vjq)-∑k=1q‍up+1,k(Δ01vpk)+up+1,q+1vpq,
where
(12)Δ10vjq=vjq-vj-1,q,Δ01vpk=vpk-vp,k-1Δ11vjk=vjk-vj-1,k-vj,k-1+vj-1,k-1.In the recent past, Altay and Başar [32] also presented another form of Abel’s transformation for double series.


## 3. Inclusion Relations

In the following theorem, we establish a relationship between spaces defined in Section2. Before proceeding further, first we prove the following lemmas which will be used to prove our inclusion relations.Lemma 2.
Consider that[𝒲σ]-limx=ℓ if and only if (L1)
𝒲σ-limx=ℓ;(L2)
(1/uv)∑m=1u∑n=1v|Ωmnst†-ℓ|→0  as  u,v→∞ (uniformly in s, t);(L3)
(1/uv)∑m=1u∑n=1v|Ωmnst‡-ℓ|→0  as  u,v→∞ (uniformly in s, t);(L4)
(1/uv)∑m=1u∑n=1v|ζmnst+ξmnst-Ωmnst†-Ωmnst‡|→0  as  u,v→∞ (uniformly in s, t),
where(13)Ωmnst†=1(m+1)∑p=0m‍ζpnst,Ωmnst‡=1(n+1)∑q=0n‍ζmqst.Proof.
Suppose that[𝒲σ]-limx=ℓ. Thus, we have 𝒲σ-limx=ℓ, that is, (L1) holds. We see that conditions (L2) and (L3) follows from the Remark 1. Write
(14)1uv∑m=1u‍∑n=1v‍|ζmnst+ξmnst-Ωmnst†-Ωmnst‡|=1uv∑m=1u‍∑n=1v‍|ζmnst-ℓ+ξmnst-ℓ-Ωmnst†+ℓ-Ωmnst‡+ℓ|≤1uv∑m=1u‍∑n=1v‍|ζmnst-ℓ|+1uv∑m=1u‍∑n=1v‍|ξmnst-ℓ|+1uv∑m=1u‍∑n=1v‍|Ωmnst†-ℓ|+1uv∑m=1u‍∑n=1v‍|Ωmnst‡-ℓ|=Σ1+Σ2+Σ3+Σ4(say).
By our assumption, that is, [𝒲σ]-limx=ℓ,  Σ1→0 as u,v→∞ uniformly in s,  t. The condition (L1) implies that ξmnst tends to zero as m,n tending to ∞ uniformly in s,t; therefore Σ2→0 as u,v→∞ uniformly in s,t and Σ3,Σ4→0 as u,v→∞ uniformly in s,t by the conclusion (L2) and (L3), respectively. Thus, (14) tends to zero as u,v→∞ uniformly in s,t, that is, (L4) holds.
Conversely, let (L1)–(L4) hold. Then,(15)1uv∑m=1u‍∑n=1v‍|ζmnst-ℓ|≤1uv∑m=1u‍∑n=1v‍|ζmnst+ξmnst-Ωmnst†-Ωmnst‡|+1uv∑m=1u‍∑n=1v‍|ξmnst-ℓ|+1uv∑m=1u‍∑n=1v‍|Ωmnst†-ℓ|+1uv∑m=1u‍∑n=1v‍|Ωmnst‡-ℓ|→0 (u,v→∞) uniformly in s,t.Lemma 3.
One has(16)ζmnst+ξmnst-Ωmnst†-Ωmnst‡=mn[ξmnst-ξm-1,n,s,t-ξm,n-1,s,t+ξm-1,n-1,s,t].Proof.
Since(17)ξmnst-ξm-1,n,s,t-ξm,n-1,s,t+ξm-1,n-1,s,t=[1(m+1)(n+1)∑p=0m‍∑q=0n‍ζpqst-1m(n+1)∑p=0m-1‍∑q=0n‍ζpqst]-[1(m+1)n∑p=0m‍∑q=0n-1‍ζpqst-1mn∑p=0m-1‍∑q=0n-1‍ζpqst].
First, we solve the expression in the first bracket
(18)[1(m+1)(n+1)∑p=0m∑q=0n‍ζpqst-1m(n+1)∑p=0m-1∑q=0n‍ζpqst]=1m(m+1)(n+1)×[∑q=0n‍(m∑p=0m‍ζpqst-(m+1)∑p=0m-1‍ζpqst)]=1m(m+1)(n+1)∑q=0n‍[mζmqst-∑p=0m-1‍ζpqst]=1m(m+1)(n+1)∑q=0n‍[(m+1)ζmqst-∑p=0m‍ζpqst]=1m(n+1)∑q=0n‍[ζmqst-1(m+1)∑p=0m‍ζpqst]=1m(n+1)∑q=0n‍ζmqst-1m(m+1)(n+1)∑p=0m∑q=0n‍ζpqst=1m(n+1)∑q=0n‍ζmqst-1mξmnst.
Now, the expression in the second bracket
(19)1(m+1)n∑p=0m‍∑q=0n-1‍ζpqst-1mn∑p=0m-1‍∑q=0n-1‍ζpqst=1n∑q=0n-1‍[1m(m+1){m∑p=0m‍ζpqst-(m+1)∑p=0m-1‍ζpqst}]=1n∑q=0n-1‍[1m(m+1){mζmqst-∑p=0m-1‍ζpqst}]=1n∑q=0n-1‍[1m(m+1){(m+1)ζmqst-∑p=0m‍ζpqst}]=1mn∑q=0n-1‍ζmqst-1m(m+1)n∑p=0m‍∑q=0n-1‍ζpqst=1mn∑q=0n-1‍ζmqst-1mξm,n-1,s,t.
Substituting (18) and (19) in (17), we get
(20)ξmnst-ξm-1,n,s,t-ξm,n-1,s,t+ξm-1,n-1,s,t=1m(n+1)∑q=0n‍ζmqst-1mn∑q=0n-1‍ζmqst-1mξmnst+1mξm,n-1,s,t=1mn(n+1)[n∑q=0n‍ζmqst-(n+1)∑q=0n-1‍ζmqst]-1m(ξmnst-ξm,n-1,s,t)=1mn(n+1)[nζmnst-∑q=0n-1‍ζmqst]-1m(ξmnst-ξm,n-1,s,t)=1mn(n+1)[(n+1)ζmnst-∑q=0n‍ζmqst]-1m(ξmnst-ξm,n-1,s,t)=1mn[ζmnst-1(n+1)∑q=0n‍ζmqst]-1m(ξmnst-ξm,n-1,s,t).
We know that
(21)ξmnst=1(m+1)(n+1)∑p=0m‍∑q=0n‍ζpqst=1(m+1)(n+1)[∑p=0m-1‍∑q=0n‍ζpqst+∑q=0n‍ζmqst],ξm-1,n,s,t=1m(n+1)∑p=0m-1‍∑q=0n‍ζpqst.
From (21), we have
(22)(m+1)ξmnst-mξm-1,n,s,t=1(n+1)∑q=0n‍ζmqst.
Thus, (20) becomes
(23)ξmnst-ξm-1,n,s,t-ξm,n-1,s,t+ξm-1,n-1,s,t=1mn[ζmnst-(m+1)ξmnst+mξm-1,n,s,t]-1m(ξmnst-ξm,n-1,s,t)=1mn[ζmnst-ξmnst-m(ξmnst-ξm-1,n,s,t)-n(ξmnst-ξm,n-1,s,t)].
Also (22) can be written as
(24)m(ξmnst-ξm-1,n,s,t)=1(n+1)∑q=0n‍ζmqst-ξmnst.
Similarly, we can write
(25)n(ξmnst-ξm,n-1,s,t)=1(m+1)∑q=0m‍ζpnst-ξmnst.
Using (24) and (25) in (23), we get
(26)ξmnst-ξm-1,n,s,t-ξm,n-1,s,t+ξm-1,n-1,s,t=1mn[ζmnst+ξmnst-1(m+1)∑p=0m‍ζpnst-1(n+1)∑q=0n‍ζmqst].
This implies that
(27)ζmnst+ξmnst-Ωmnst†-Ωmnst‡=mn[ξmnst-ξm-1,n,s,t-ξm,n-1,s,t+ξm-1,n-1,s,t].Theorem 4.
One has the following inclusions and the limit is preserved in each case:(i)
[𝒱σ]⊂[𝒲σ]⊂𝒲σ,(ii)
𝒲σ′⊂[𝒲σ] if the conditions (L2) and (L3) of Lemma 2 hold,(iii)
𝒲σ′⊂𝒲σ′′.Proof.
(i) Letx∈[𝒱σ] with [𝒱σ]-limx=ℓ, say. Then,
(28)ζpqst(|x-ℓ|)⟶0asp,q⟶∞,uniformlyins,t.
This implies that
(29)1(m+1)(n+1)∑p=0m‍∑q=0n‍ζpqst(|x-ℓ|)⟶0asp,q⟶∞,uniformlyins,t.
Also, we have
(30)1(m+1)(n+1)|∑p=0m‍∑q=0n‍ζpqst(x-ℓ)|≤1(m+1)(n+1)∑p=0m‍∑q=0n‍|ζpqst(x-ℓ)|≤1(m+1)(n+1)∑p=0m‍∑q=0n‍ζpqst(|x-ℓ|).
Hence, [𝒱σ]⊂[𝒲σ]⊂𝒲σ and
(31)[𝒲σ]-limx=𝒲σ-limx=ℓ.
(ii) We have to show that 𝒲σ′⊂[𝒲σ]. If x∈𝒲σ′, then we have
(32)ℑmnst=∑p=m∞‍∑q=n∞‍|ξpqst-ξp-1,q,s,t-ξp,q-1,s,t+ξp-1,q-1,s,t|⟶0
as m,n→∞, uniformly in s, t; and
(33)ξpqst⟶ℓ(say)asp,q⟶∞,uniformlyins,t,
that is, 𝒲σ-limx=ℓ.
In order to prove thatx∈[𝒲σ], it is enough to show that condition (L4) of Lemma 2 holds. Now,
(34)ℑmnst-ℑm,n+1,s,t-ℑm+1,n,s,t+ℑm+1,n+1,s,t=∑p=m∞‍∑q=n∞‍|ξpqst-ξp-1,q,s,t-ξp,q-1,s,t+ξp-1,q-1,s,t|-∑p=m∞‍∑q=n+1∞‍|ξpqst-ξp-1,q,s,t-ξp,q-1,s,t+ξp-1,q-1,s,t|-∑p=m+1∞‍∑q=n∞‍|ξpqst-ξp-1,q,s,t-ξp,q-1,s,t+ξp-1,q-1,s,t|+∑p=m+1∞‍∑q=n+1∞‍|ξpqst-ξp-1,q,s,t-ξp,q-1,s,t+ξp-1,q-1,s,t|=∑p=m∞‍[∑q=n∞‍-∑q=n+1∞‍]×|ξpqst-ξp-1,q,s,t-ξp,q-1,s,t+ξp-1,q-1,s,t|-∑p=m+1∞‍[∑q=n∞‍-∑q=n+1∞‍]×|ξpqst-ξp-1,q,s,t-ξp,q-1,s,t+ξp-1,q-1,s,t|=∑p=m∞‍|ξpnst-ξp-1,n,s,t-ξp,n-1,s,t+ξp-1,n-1,s,t|-∑p=m+1∞‍|ξpnst-ξp-1,n,s,t-ξp,n-1,s,t+ξp-1,n-1,s,t|=[∑p=m∞‍-∑p=m+1∞‍]|ξpnst-ξp-1,n,s,t-ξp,n-1,s,t+ξp-1,n-1,s,t|=|ξmnst-ξm-1,n,s,t-ξm,n-1,s,t+ξm-1,n-1,s,t|.
Replacing m and n by p and q, respectively, we have
(35)ℑpqst-ℑp,q+1,s,t-ℑp+1,q,s,t+ℑp+1,q+1,s,t=|ξpqst-ξp-1,q,s,t-ξp,q-1,s,t+ξp-1,q-1,s,t|.
By Lemma 3, we have
(36)ζpqst+ξpqst-Ωpqst†-Ωpqst‡=pq[ξpqst-ξp-1,q,s,t-ξp,q-1,s,t+ξp-1,q-1,s,t].
So that we have
(37)1mn∑p=1m‍∑q=1n‍|ζpqst+ξpqst-Ωpqst†-Ωpqst‡|=1mn∑p=1m‍∑q=1n‍pq[ℑpqst-ℑp,q+1,s,t-ℑp+1,q,s,t+ℑp+1,q+1,s,t].
By using Abel’s transformation for double series in the right hand side of above equation, we have
(38)1mn∑p=1m‍∑q=1n‍|ζpqst+ξpqst-Ωpqst†-Ωpqst‡|=1mn[∑p=1m‍∑q=1n‍ℑpqst-m∑q=1n‍ℑm+1,q,s,t-n∑p=1m‍ℑp,n+1,s,t+mnℑm+1,n+1,s,t]→0  as  m,n→∞, uniformly in s,t (by (32)). Hence, by Lemma 2, x∈[𝒲σ].
(iii) Letx∈𝒲σ′, and we have to show that
(39)sups,t∑p=0∞‍∑q=0∞‍|ξpqst-ξp-1,q,s,t-ξp,q-1,s,t+ξp-1,q-1,s,t|≤K,
where K is an absolute constant. Since x∈𝒲σ′, there exist integers p0,q0 such that
(40)∑p>p0‍∑q>q0‍|ξpqst-ξp-1,q,s,t-ξp,q-1,s,t+ξp-1,q-1,s,t|<1,∀s,t.
Hence, it is left to show that for fixed p,q(41)|ξpqst-ξp-1,q,s,t-ξp,q-1,s,t+ξp-1,q-1,s,t|≤K,∀s,t.
From (40), we have
(42)|ξpqst-ξp-1,q,s,t-ξp,q-1,s,t+ξp-1,q-1,s,t|<1,
for every fixed p>p0,  q>q0 and for all s,  t. Since
(43)m(m+1)n(n+1)(ξmnst-ξm-1,n,s,t-ξm,n-1,s,t+ξm-1,n-1,s,t)=∑p=1m∑q=1n‍pq(ζpqst-ζp-1,q,s,t-ζp,q-1,s,t+ζp-1,q-1,s,t).
Accordingly,
(44)mn[Te(m+1)(n+1)×(ξmnst-ξm-1,n,s,t-ξm,n-1,s,t+ξm-1,n-1,s,t)-(m-1)(n+1)×(ξm-1,n,s,t-ξm-2,n,s,t-ξm-1,n-1,s,t+ξm-2,n-1,s,t)-(m+1)(n-1)×(ξm,n-1,s,t-ξm-1,n-1,s,t-ξm,n-2,s,t+ξm-1,n-2,s,t)+(m-1)(n-1)×(ξm-1,n-1,s,t-ξm-2,n-1,s,t-ξm-1,n-2,s,t+ξm-2,n-2,s,t)Te]=∑q=1n‍q[∑p=1m‍p(ζpqst-ζp-1,q,s,t-ζp,q-1,s,t+ζp-1,q-1,s,t)-∑p=1m-1‍p(ζpqst-ζp-1,q,s,t-ζp,q-1,s,t+ζp-1,q-1,s,t)]-∑q=1n-1‍q[∑p=1m‍p(ζpqst-ζp-1,q,s,t-ζp,q-1,s,t+ζp-1,q-1,s,t)-∑p=1m-1‍p(ζpqst-ζp-1,q,s,t-ζp,q-1,s,t+ζp-1,q-1,s,t)∑p=1m-1‍]=∑q=1n‍q[m(ζmqst-ζm-1,q,s,t-ζm,q-1,s,t+ζm-1,q-1,s,t)]-∑q=1n-1‍q[m(ζmqst-ζm-1,q,s,t-ζm,q-1,s,t+ζm-1,q-1,s,t)].
This implies that
(45)(m+1)(n+1)(ξmnst-ξm-1,n,s,t-ξm,n-1,s,t+ξm-1,n-1,s,t)-(m-1)(n+1)×(ξm-1,n,s,t-ξm-2,n,s,t-ξm-1,n-1,s,t+ξm-2,n-1,s,t)-(m+1)(n-1)×(ξm,n-1,s,t-ξm-1,n-1,s,t-ξm,n-2,s,t+ξm-1,n-2,s,t)+(m-1)(n-1)×(ξm-1,n-1,s,t-ξm-2,n-1,s,t-ξm-1,n-2,s,t+ξm-2,n-2,s,t)=[ζmnst-ζm-1,n,s,t-ζm,n-1,s,t+ζm-1,n-1,s,t].
Using (42) and (45), we have
(46)|ζmnst-ζm-1,n,s,t-ζm,n-1,s,t+ζm-1,n-1,s,t|≤K(m,n)
for every fixed m>p0,  n>q0 and for all s,  t, where K(m,n) is a constant depending upon m,n.
Now, for any given infinite double series∑s∑tast denoted as “a”, let us write
(47)xjk=∑s=0j‍∑t=0k‍ast,j,k=1,2,…
and σ be monotonically increasing. For simplicity in notation, we denote
(48)ϕmnst=ζmnst(a)-ζm-1,n,s,t(a)-ζm,n-1,s,t(a)+ζm-1,n-1,s,t(a).
Again from the definition of ζmnst, it is easy to obtain
(49)ϕmnst=1m(m+1)n(n+1)∑j=1m‍∑k=1n‍jk(∑u=djdj′‍∑v=hkhk′‍auv)
for all m,n≥1 and ϕ0,0,s,t(a)=ast with dj=σj-1(s)+1,  dj′=σj(s), hk=σk-1(t)+1, hk′=σk(t). Further, we calculate
(50)m(m+1)n(n+1)ϕmnst-m(m-1)n(n+1)ϕm-1,n,s,t-m(m+1)n(n-1)ϕm,n-1,s,t+m(m-1)n(n-1)ϕm-1,n-1,s,t={∑j=1m‍∑k=1n‍jk-∑j=1m-1‍∑k=1n‍jk-∑j=1m‍∑k=1n-1‍jk+∑j=1m-1‍∑k=1n-1‍jk}×(∑u=djdj′‍∑v=hkhk′‍auv)={∑k=1n‍k[∑j=1m‍j-∑j=1m-1‍j]-∑k=1n-1‍k[∑j=1m‍j+∑j=1m-1‍j]}×(∑u=djdj′‍∑v=hkhk′‍auv)={m∑k=1n‍k-m∑k=1n-1‍k}(∑u=dmdm′‍∑v=hkhk′‍auv)=mn(∑u=dmdm′‍∑v=hnhn′‍auv).
Thus, we have(51)∑u=dmdm′∑v=hnhn′auv=(m+1)(n+1)ϕm,n,s,t-(m-1)(n+1)ϕm-1,n,s,t-(m+1)(n-1)ϕm,n-1,s,t+(m-1)(n-1)ϕm-1,n-1,s,t.
Hence, it follows from (46) that for each fixed m>p0,  n>q0,
(52)|∑u=dmdm′‍∑v=hnhn′‍auv|≤K(m,n),∀s,t.
Hence, it follows from (52) that
(53)|auv|≤K∀u,v,
where K is independent of u,v. By (49), we have
(54)|ζmnst-ζm-1,n,s,t-ζm,n-1,s,t+ζm-1,n-1,s,t|≤K,∀m,n,s,t.
Also from (43) and (54), we have
(55)|ξpqst-ξp-1,q,s,t-ξp,q-1,s,t+ξp-1,q-1,s,t|≤K,∀p,q,s,t.


## 4. Topological Results

Here, we extend the spaces[𝒲σ],  𝒲σ′,    𝒲σ′′ to more general spaces, respectively, denoted by [𝒲σ(α)],  𝒲σ′(α),  𝒲σ′′(α), where α=(αjk) is a double sequence of positive real numbers for all j, k and supj,kαjk=H<∞. First, we recall the notion of paranorm as follows.Aparanorm is a function g:X→ℝ defined on a linear space X such that for all x,y,z∈X(P1)
g(x)=0 if x=θ(P2)
g(-x)=g(x)(P3)
g(x+y)≤g(x)+g(y)(P4)
If(αn) is a sequence of scalars with αn→α0(n→∞) and xn,a∈X with xn→a(n→∞) in the sense that g(xn-a)→0(n→∞), then αnxn→α0a  (n→∞), in the sense that g(αnxn-α0a)→0(n→∞).A paranormg for which g(x)=0 implies x=θ is called a total paranormon X, and the pair (X,g) is called a total paranormed space. Note that each seminorm (norm) p on X is a paranorm (total) but converse needs not be true.A paranormed space(X,g) is a topological linear space with the topology given by the paranorm g.Now, we define the following spaces:(56)[𝒲σ(α)]={∑q=0n‍x=(xjk):1(m+1)(n+1)×∑p=0m‍∑q=0n‍|ζpqst(x-ℓE)|αpq⟶0,asm,n⟶∞,uniformlyins,t∑q=0n‍},𝒲σ′(α)={∑q=0n‍x=(xjk):∑p=0∞‍∑q=0∞‍|ξpqst-ξp-1,q,s,t-ξp,q-1,s,t+ξp-1,q-1,s,t|αpqconvergesuniformlyins,t∑q=0n‍},𝒲σ′′(α)={∑q=0n‍x=(xjk):sups,t∑p=0∞‍∑q=0∞‍|ξpqst-ξp-1,q,s,t-ξp,q-1,s,t+ξp-1,q-1,s,t|αpq<∞∑q=0n‍}.We remark that ifα=(αjk) is a constant sequence, then we write [𝒲σ,α] in place of [𝒲σ(α)].Theorem 5.
Letα=(αpq) be a bounded double sequence of strictly positive real numbers. Then, [𝒲σ(α)] is a complete linear topological space paranormed by
(57)g(x)=supm,n,s,t(1(m+1)(n+1)∑p=0m‍∑q=0n‍|ζpqst(x)|αpq)1/M,
where M=max(1,supp,qαpq). If α≥1 then [𝒲σ,α] is a Banach space and [𝒲σ,α] is a α-normed space if 0<α<1.Proof.
Let(xpq) and (ypq) be two double sequences. Then,
(58)|xpq+ypq|αpq≤K(|xpq|αpq+|ypq|αpq),
where K=max(1,2H-1) and H=supp,qαpq. Since
(59)|λ|αpq≤max(1,|λ|H),
therefore
(60)1(m+1)(n+1)∑p=0m‍∑q=0n‍|ζpqst(λx+μy)|αpq≤KK11(m+1)(n+1)∑p=0m‍∑q=0n‍|ζpqst(x)|αpq+KK21(m+1)(n+1)∑p=0m‍∑q=0n‍|ζpqst(y)|αpq,
where K1=max(1,|λ|H) and K2=max(1,|μ|H). From (60), we have that if x,y∈[𝒲σ(α)], then λx+μy∈[𝒲σ(α)]. Thus, [𝒲σ(α)] is a linear space. Without loss of generality, we can take
(61)[𝒲σ(α)]-limx=0.
Clearly,g(0)=0,  g(-x)=g(x). From (58) and Minkowski’s inequality, we have
(62)(∑p=0m‍∑q=0n‍|ζpqst(x+y)|αpq)1/M≤[∑p=0m‍∑q=0n‍(|ζpqst(x)|αpq/M+|ζpqst(y)|αpq/M)]1/M≤(∑p=0m‍∑q=0n‍|ζpqst(x)|αpq)1/M+(∑p=0m‍∑q=0n‍|ζpqst(y)|αpq)1/M.
Hence,
(63)g(x+y)≤g(x)+g(y).
Since α=(αmn) is bounded away from zero, there exists a constant δ>0 such that αmn≥δ for all m,  n. Now for |λ|≤1,  |λ|αst≤|λ|δ and so
(64)g(λx)≤|λ|δ/Mg(x),
that is, the scaler multiplication is continuous. Hence, g is a paranorm on [𝒲σ(α)].
Let(xb) be a Cauchy sequence in [𝒲σ(α)], that is,
(65)g(xd-xb)⟶0asb,d⟶∞.
Since
(66)1(m+1)(n+1)∑p=0m‍∑q=0n‍|ζpqst(xd-xb)|αpq≤[g(xd-xb)]M,
it follows that
(67)|ζpqst(xd-xb)|αpq⟶0asb,d⟶∞∀p,q,s,t.
In particular,
(68)|ζ0,0,s,t(xd-xb)|=|xstd-xstb|⟶0asb,d⟶∞forfixeds,t.
Hence, (xd) is a Cauchy sequence in ℂ. Since ℂ is complete, there exists x=(xjk)∈ℂ such that xd→x coordinatewise as d→∞. It follows from (66) that given ϵ>0, there exists d0 such that
(69)(1(m+1)(n+1)∑p=0m‍∑q=0n‍|ζpqst(xd-xb)|αpq)1/M<ϵ,
for b,d>d0. Now taking b→∞ and sups,t,m,n in (69), we have g(xd-x)≤ϵ for d>d0. This proves that xd→x and x=(xjk)∈[𝒲σ(α)]. Hence [𝒲σ(α)] is complete. If α is a constant then it is easy to prove the rest of the theorem.Theorem 6.
One has the following:
𝒲
σ
′
(
α
) is a complete paranormed space, paranormed by
(70)h(x)=sups,t(∑p=0∞‍∑q=0∞‍|ξpqst-ξp-1,q,s,t-ξp,q-1,s,t+ξp-1,q-1,s,t|αpq∑p=0∞‍)1/M
which is defined on 𝒲σ′(α). If α≥1 then 𝒲σ,α′ is a Banach space and if 0<α<1, 𝒲σ,α is α-normed space.Proof.
In order for the paranorm in (70) be defined, we require that
(71)𝒲σ′(α)⊂𝒲σ′′(α)
which is proved in the next theorem (i.e., Theorem 7). Using the standard technique as in the previous theorem, we can prove that h is subadditive.
Now, we have to prove the continuity of scalar multiplication. Suppose thatx=(xjk)∈𝒲σ′(α). Then, for ϵ>0 there exist integers M,N>0 such that
(72)∑p=M∞∑q=N∞‍|ξpqst(x)-ξp-1,q,s,t(x)∑p=M∞‍∑q=N∞‍-ξp,q-1,s,t(x)+ξp-1,q-1,s,t(x)|αpq<ϵ.
If |λ|≤1, then by (72) we have
(73)∑p=M∞‍∑q=N∞‍|ξpqst(λx)-ξp-1,q,s,t(λx)∑p=M∞‍-ξp,q-1,s,t(λx)+ξp-1,q-1,s,t(λx)|αpq≤∑p=M∞‍∑q=N∞‍|ξpqst(x)-ξp-1,q,s,t(x)≤∑p=M∞‍-ξp,q-1,s,t(x)+ξp-1,q-1,s,t(x)|αpq<ϵ.
Since for fixed M,N(74)∑p=0M-1‍∑q=0N-1‍|ξpqst(λx)-ξp-1,q,s,t(λx)-ξp,q-1,s,t(λx)∑p=0M-1‍∑q=0N-1‍+ξp-1,q-1,s,t(λx)|αpq⟶0
as λ→0, it follows from (73) and (74) that for fixed x=(xjk),  h(λx)→0 as λ→0. Also, since |λ|αpq≤max(1,|λ|H) implies that
(75)h(λx)≤(sup|λ|αpq)1/Mh(x).
It follows that for fixed λ, h(λx)→0 as x→0. This proves the continuity of scalar multiplication. Hence, h is a paranorm. The proof of the completeness of 𝒲σ′(α) can be achieved by using the same technique as in Theorem 5.Theorem 7.
Suppose thatα=(αpq) is bounded double sequence of strictly positive real numbers. Then,(i)
𝒲σ′′(α) is a complete linear space paranormed by the function h defined in (70). In particular, if α is a constant, 𝒲σ,α′′ is a Banach space for α≥1 and α-normed space for 0<α<1,(ii)
𝒲σ′(α) is a closed subspace of 𝒲σ′′(α).Proof.
(i) Proceeding along the same lines as in Theorem6, except for the proof of continuity of scalar multiplication. If x=(xjk)∈𝒲σ′′(α), we cannot assert (72) as in the case when x=(xjk)∈𝒲σ′(α). Since α is bounded away from zero, there exists δ>0 such that αmn≥δ for all m, n. Hence, |λ|≤1 implies |λ|αpq≤|λ|δ. Since
(76)h(λx)≤|λ|αpqh(x),
continuity of scalar multiplication follows.
(ii) For this, first we have to show that(77)𝒲σ′(α)⊂𝒲σ′′(α).
Let x=(xjk)∈𝒲σ′(α). Then, there exist integers M,  N such that
(78)∑p=M∞‍∑q=N∞|ξpqst(x)-ξp-1,q,s,t(x)-ξp,q-1,s,t(x)∑p=M∞‍∑p=M∞‍+ξp-1,q-1,s,t(x)|αpq<1.
By an argument similar to Theorem4(iii), we obtain (77). Since 𝒲σ′(α) and 𝒲σ′′(α) are complete with the same metric, we have (ii).



---

*Source: 507950-2013-04-11.xml*