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

Abstract

We introduce some double sequences spaces involving the notions of invariant mean (or 𝜎-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.

1. Preliminaries, Background, and Notation

In 1900, Pringsheim [1] presented the following notion of convergence for double sequences. A double sequence 𝑥=(𝑥_𝑗𝑘) is said to converge to the limit 𝐿  in Pringsheim’s  sense (shortly, 𝑝-convergent to  𝐿) if for every 𝜀>0 there exists an integer 𝑁 such that |𝑥_𝑗𝑘−𝐿|<𝜀 whenever 𝑗,𝑘>𝑁. In this case, 𝐿 is called the 𝑝-limit of 𝑥.

A double sequence 𝑥=(𝑥_𝑗𝑘) of real or complex numbers is said to be bounded if ‖𝑥‖_∞=sup_𝑗,𝑘|𝑥_𝑗𝑘|<∞. The space of all bounded double sequences is denoted by ℳ_𝑢.

If 𝑥∈ℳ_𝑢 and is 𝑝-convergent to 𝐿, then 𝑥 is said to be boundedly 𝑝-convergent to 𝐿 (shortly, 𝑏𝑝-convergent to 𝐿). 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 convergence 𝜈, the space of all 𝜈-convergent double sequences will be denoted by 𝒞_𝜈, the space of all 𝜈-convergent to 0 double sequences by 𝒞_𝜈0 and the limit of a 𝜈-convergent double sequence 𝑥 by 𝜈-lim_𝑗,𝑘𝑥_𝑗𝑘, where 𝜈∈{𝑝,𝑏𝑝}.

Let Ω 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 ℒ_𝑢∶=𝑥∈Ω∣‖𝑥‖₁∶=_𝑗,𝑘𝑥_𝑗𝑘<∞ (1) of all absolutely summable double sequences.

All considered double sequence spaces are supposed to contain Φ∶=span𝐞^𝐣𝐤∣𝑗,𝑘∈ℕ, (2) where 𝐞^𝐣𝐤_𝐢𝐥=1;if𝑗,𝑘=(𝑖,ℓ),0;otherwise. (3)

We denote the pointwise sums ∑_𝑗,𝑘𝐞^𝐣𝐤,  ∑_𝑗𝐞^𝐣𝐤(𝑘∈ℕ), and ∑_𝑘𝐞^𝐣𝐤(𝑗∈ℕ) by 𝐞,  𝐞^𝐤, and 𝐞_𝐣, respectively.

Let 𝜎 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=(𝑥_𝑘) has 𝑥_𝑘≥0 for all 𝑘, (ii) 𝜑(𝑒)=1, where 𝑒=(1,1,1,…), and (iii) 𝜑(𝑥)=𝜑((𝑥_𝜎(𝑘))) for all 𝑥∈ℓ_∞.

Throughout this paper we consider the mapping 𝜎 which has no finite orbits, that is, 𝜎^𝑝(𝑘)≠𝑘 for all integer 𝑘≥0 and 𝑝≥1, where 𝜎^𝑝(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⊂𝑉_𝜎 the set of bounded sequences all of whose 𝜎-means are equal. We say that a sequence 𝑥=(𝑥_𝑘) is 𝜎-convergent if and only if 𝑥∈𝑉_𝜎. Using this concept, Schaefer [3] defined and characterized 𝜎-conservative, 𝜎-regular, and 𝜎-coercive matrices for single sequences. If 𝜎 is translation then 𝑉_𝜎 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=(𝑥_𝑗𝑘) of real numbers is said to be 𝜎-convergent to a number 𝐿 if and only if 𝑥∈𝒱_𝜎, where 𝒱_𝜎=𝑥∈ℳ_𝑢∶lim_{𝑝,𝑞→∞}𝜁_{𝑝𝑞𝑠𝑡}(𝑥),𝜁=𝐿uniformlyin𝑠,𝑡;𝐿=𝜎-lim𝑥_{𝑝𝑞𝑠𝑡}(𝑥)=1𝑝+1𝑞+1^𝑝𝑞_𝑗=0𝑘=0𝑥_{𝜎𝑗(𝑠),𝜎𝑘(𝑡)}, (4) while here the limit means 𝑏𝑝-limit. Let us denote by 𝒱_𝜎 the space of 𝜎-convergent double sequences 𝑥=(𝑥_𝑗𝑘). For 𝜎(𝑛)=𝑛+1, the set 𝒱_𝜎 is reduced to the set ℱ of almost convergent double sequences [10]. Note that 𝒞_𝑏𝑝⊂𝒱_𝜎⊂ℳ_𝑢.

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=(𝑥_𝑗𝑘) is said to be strongly 𝜎-convergent if there exists a number ℓ such that 1𝑝+1𝑞+1^𝑝𝑞_𝑗=0𝑘=0𝑥_{𝜎𝑗(𝑠),𝜎𝑘(𝑡)}−ℓ⟶0as𝑝,𝑞⟶∞uniformlyin𝑠,𝑡, (5) while here the limit means 𝑏𝑝-limit. In this case, we write [𝒱_𝜎]-lim𝑥=ℓ. Let us denote by [𝒱_𝜎] the set of all strongly 𝜎-convergent sequences 𝑥=(𝑥_𝑗𝑘). If 𝜎 is translation then [𝒱_𝜎] 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 and 𝜎-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=(𝛼_𝑗𝑘) such that 𝛼_𝑗𝑘>0 for all 𝑗,  𝑘 and sup_𝑗,𝑘𝛼_𝑗𝑘=𝐻<∞ and prove some topological results.

2. The Double Sequence Spaces

We construct the following spaces involving the idea of invariant mean and 𝜎-convergence for double sequences: 𝒲_𝜎=𝑥=𝑥_𝑗𝑘∶1()×𝑚+1)(𝑛+1_{𝑚𝑛𝑝=0𝑞=0}𝜁_{𝑝𝑞𝑠𝑡}(),𝒲𝑥−ℓ𝐸⟶0as𝑚,𝑛⟶∞,uniformlyin𝑠,𝑡,forsomeℓ_𝜎=𝑥=𝑥_𝑗𝑘∶1()×𝑚+1)(𝑛+1_{𝑚𝑛𝑝=0𝑞=0}𝜁_{𝑝𝑞𝑠𝑡}(,𝒲𝑥−ℓ𝐸)⟶0as𝑚,𝑛⟶∞,uniformlyin𝑠,𝑡,forsomeℓ_′𝜎=𝑥=𝑥_𝑗𝑘∶_{∞∞𝑝=0𝑞=0}𝜉_{𝑝𝑞𝑠𝑡}−𝜉_{𝑝−1,𝑞,𝑠,𝑡}−𝜉_{𝑝,𝑞−1,𝑠,𝑡}+𝜉_{𝑝−1,𝑞−1,𝑠,𝑡},𝒲convergesuniformlyin𝑠,𝑡_𝜎′′=𝑥=𝑥_𝑗𝑘∶sup^∞_𝑠,𝑡^∞_𝑝=0𝑞=0𝜉_{𝑝𝑞𝑠𝑡}−𝜉_{𝑝−1,𝑞,𝑠,𝑡}−𝜉_{𝑝,𝑞−1,𝑠,𝑡}+𝜉_{𝑝−1,𝑞−1,𝑠,𝑡}<∞, (6) where 𝐸=(𝑒_𝑗𝑘) with 𝑒_𝑗𝑘=1 for all 𝑗,𝑘; 𝜉_{𝑚𝑛𝑠𝑡}=𝜉_{𝑚𝑛𝑠𝑡}(𝑥)=1(𝑚+1)(𝑛+1)^𝑚𝑛_𝑝=0𝑞=0𝜁_{𝑝𝑞𝑠𝑡}(𝑥),𝜉_{0,0,𝑠,𝑡}(𝑥)=𝜁_{0,0,𝑠,𝑡}=𝑥_𝑠,𝑡,𝜉_{−1,0,𝑠,𝑡}(𝑥)=𝜁_{−1,0,𝑠,𝑡}(𝑥)=𝑥_{𝑠−1,𝑡},𝜉_{0,−1,𝑠,𝑡}(𝑥)=𝜁_{0,−1,𝑠,𝑡}(𝑥)=𝑥_{𝑠,𝑡−1},𝜉_{−1,−1,𝑠,𝑡}(𝑥)=𝜁_{−1,−1,𝑠,𝑡}(𝑥)=𝑥_{𝑠−1,𝑡−1}. (7)

Remark 1. If [𝒲_𝜎]-lim𝑥=ℓ, that is, 1(𝑚+1)(𝑛+1)^𝑚𝑛_𝑝=0𝑞=0𝜁_{𝑝𝑞𝑠𝑡}−ℓ⟶0 (8) as 𝑚,𝑛→∞, uniformly in 𝑠,  𝑡; then 1(𝑚+1)(𝑛+1)^𝑚𝑛_𝑝=0𝑞=01𝑝+1^𝑝_𝑗=0𝜁_{𝑗𝑞𝑠𝑡}1−ℓ⟶0,(𝑚+1)(𝑛+1)^𝑚𝑛_𝑝=0𝑞=01𝑞+1^𝑞_𝑘=0𝜁_{𝑝𝑘𝑠𝑡}−ℓ⟶0. (9)

We remark that by using Abel’s transformation for single series ^𝑚_𝑖=1𝑣_𝑖𝑢_𝑖∓𝑢_𝑖+1=^𝑚_𝑖=1𝑢_𝑖𝑣_𝑖∓𝑣_𝑖−1∓𝑢_𝑚+1𝑣_𝑚. (10) We get Abel’s transformation for double series ^𝑝𝑞_𝑗=1𝑘=1𝑣_𝑗𝑘𝑢_𝑗𝑘−𝑢_𝑗+1,𝑘−𝑢_𝑗,𝑘+1+𝑢_{𝑗+1,𝑘+1}=^𝑝𝑞_𝑗=1𝑘=1𝑢_𝑗𝑘Δ₁₁𝑣_𝑗𝑘−^𝑝_𝑗=1𝑢_𝑗,𝑞+1Δ₁₀𝑣_𝑗𝑞−^𝑞_𝑘=1𝑢_𝑝+1,𝑘Δ₀₁𝑣_𝑝𝑘+𝑢_{𝑝+1,𝑞+1}𝑣_𝑝𝑞, (11) where Δ₁₀𝑣_𝑗𝑞=𝑣_𝑗𝑞−𝑣_{𝑗−1,𝑞},Δ₀₁𝑣_𝑝𝑘=𝑣_𝑝𝑘−𝑣_{𝑝,𝑘−1}Δ₁₁𝑣_𝑗𝑘=𝑣_𝑗𝑘−𝑣_{𝑗−1,𝑘}−𝑣_{𝑗,𝑘−1}+𝑣_{𝑗−1,𝑘−1}. (12)

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 Section 2. Before proceeding further, first we prove the following lemmas which will be used to prove our inclusion relations.

Lemma 2. Consider that [𝒲_𝜎]-lim𝑥=ℓ if and only if (L1)𝒲_𝜎-lim𝑥=ℓ; (L2)(1/𝑢𝑣)∑^𝑢_𝑚=1∑^𝑣_𝑛=1|Ω^†_{𝑚𝑛𝑠𝑡}−ℓ|→0  as  𝑢,𝑣→∞ (uniformly in 𝑠, 𝑡); (L3)(1/𝑢𝑣)∑^𝑢_𝑚=1∑^𝑣_𝑛=1|Ω^‡_{𝑚𝑛𝑠𝑡}−ℓ|→0  as  𝑢,𝑣→∞ (uniformly in 𝑠, 𝑡); (L4)(1/𝑢𝑣)∑^𝑢_𝑚=1∑^𝑣_𝑛=1|𝜁_{𝑚𝑛𝑠𝑡}+𝜉_{𝑚𝑛𝑠𝑡}−Ω^†_{𝑚𝑛𝑠𝑡}−Ω^‡_{𝑚𝑛𝑠𝑡}|→0  as  𝑢,𝑣→∞ (uniformly in 𝑠, 𝑡),
where Ω^†_{𝑚𝑛𝑠𝑡}=1(𝑚+1)^𝑚_𝑝=0𝜁_{𝑝𝑛𝑠𝑡},Ω^‡_{𝑚𝑛𝑠𝑡}=1(𝑛+1)^𝑛_𝑞=0𝜁_{𝑚𝑞𝑠𝑡}. (13)

Proof. Suppose that [𝒲_𝜎]-lim𝑥=ℓ. Thus, we have 𝒲_𝜎-lim𝑥=ℓ, that is, (L1) holds. We see that conditions (L2) and (L3) follows from the Remark 1. Write 1𝑢𝑣^𝑢𝑣_𝑚=1𝑛=1𝜁_{𝑚𝑛𝑠𝑡}+𝜉_{𝑚𝑛𝑠𝑡}−Ω^†_{𝑚𝑛𝑠𝑡}−Ω^‡_{𝑚𝑛𝑠𝑡}=1𝑢𝑣^𝑢𝑣_𝑚=1𝑛=1𝜁_{𝑚𝑛𝑠𝑡}−ℓ+𝜉_{𝑚𝑛𝑠𝑡}−ℓ−Ω^†_{𝑚𝑛𝑠𝑡}+ℓ−Ω^‡_{𝑚𝑛𝑠𝑡}≤1+ℓ𝑢𝑣^𝑢𝑣_𝑚=1𝑛=1𝜁_{𝑚𝑛𝑠𝑡}+1−ℓ𝑢𝑣^𝑢𝑣_𝑚=1𝑛=1𝜉_{𝑚𝑛𝑠𝑡}+1−ℓ𝑢𝑣^𝑢𝑣_𝑚=1𝑛=1Ω^†_{𝑚𝑛𝑠𝑡}+1−ℓ𝑢𝑣^𝑢𝑣_𝑚=1𝑛=1Ω^‡_{𝑚𝑛𝑠𝑡}−ℓ=Σ₁+Σ₂+Σ₃+Σ₄say. (14) By our assumption, that is, [𝒲_𝜎]-lim𝑥=ℓ,  Σ₁→0 as 𝑢,𝑣→∞ uniformly in 𝑠,  𝑡. The condition (L1) implies that 𝜉_{𝑚𝑛𝑠𝑡} 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, 1𝑢𝑣^𝑢𝑣_𝑚=1𝑛=1𝜁_{𝑚𝑛𝑠𝑡}≤1−ℓ𝑢𝑣^𝑢𝑣_𝑚=1𝑛=1𝜁_{𝑚𝑛𝑠𝑡}+𝜉_{𝑚𝑛𝑠𝑡}−Ω^†_{𝑚𝑛𝑠𝑡}−Ω^‡_{𝑚𝑛𝑠𝑡}+1𝑢𝑣^𝑢𝑣_𝑚=1𝑛=1𝜉_{𝑚𝑛𝑠𝑡}+1−ℓ𝑢𝑣^𝑢𝑣_𝑚=1𝑛=1Ω^†_{𝑚𝑛𝑠𝑡}+1−ℓ𝑢𝑣^𝑢𝑣_𝑚=1𝑛=1Ω^‡_{𝑚𝑛𝑠𝑡}−ℓ (15)→0 (𝑢,𝑣→∞) uniformly in 𝑠,𝑡.

Lemma 3. One has 𝜁_{𝑚𝑛𝑠𝑡}+𝜉_{𝑚𝑛𝑠𝑡}−Ω^†_{𝑚𝑛𝑠𝑡}−Ω^‡_{𝑚𝑛𝑠𝑡}=𝑚𝑛𝜉_{𝑚𝑛𝑠𝑡}−𝜉_{𝑚−1,𝑛,𝑠,𝑡}−𝜉_{𝑚,𝑛−1,𝑠,𝑡}+𝜉_{𝑚−1,𝑛−1,𝑠,𝑡}. (16)

Proof. Since 𝜉_{𝑚𝑛𝑠𝑡}−𝜉_{𝑚−1,𝑛,𝑠,𝑡}−𝜉_{𝑚,𝑛−1,𝑠,𝑡}+𝜉_{𝑚−1,𝑛−1,𝑠,𝑡}=1(𝑚+1)(𝑛+1)^𝑚𝑛_𝑝=0𝑞=0𝜁_{𝑝𝑞𝑠𝑡}−1𝑚(𝑛+1)^𝑚−1𝑛_𝑝=0𝑞=0𝜁_{𝑝𝑞𝑠𝑡}−1(𝑚+1)𝑛^𝑚_𝑝=0^𝑛−1_𝑞=0𝜁_{𝑝𝑞𝑠𝑡}−1𝑚𝑛^𝑚−1_𝑝=0^𝑛−1_𝑞=0𝜁_{𝑝𝑞𝑠𝑡}. (17) First, we solve the expression in the first bracket 1(𝑚+1)(𝑛+1)^𝑚𝑛_𝑝=0𝑞=0𝜁_{𝑝𝑞𝑠𝑡}−1𝑚(𝑛+1)^𝑚−1𝑛_𝑝=0𝑞=0𝜁_{𝑝𝑞𝑠𝑡}=1𝑚()×𝑚+1)(𝑛+1_𝑛𝑞=0𝑚^𝑚_𝑝=0𝜁_{𝑝𝑞𝑠𝑡}−(𝑚+1)^𝑚−1_𝑝=0𝜁_{𝑝𝑞𝑠𝑡}=1𝑚(𝑚+1)(𝑛+1)^𝑛_𝑞=0𝑚𝜁_{𝑚𝑞𝑠𝑡}−^𝑚−1_𝑝=0𝜁_{𝑝𝑞𝑠𝑡}=1𝑚(𝑚+1)(𝑛+1)^𝑛_𝑞=0(𝑚+1)𝜁_{𝑚𝑞𝑠𝑡}−^𝑚_𝑝=0𝜁_{𝑝𝑞𝑠𝑡}=1𝑚(𝑛+1)^𝑛_𝑞=0𝜁_{𝑚𝑞𝑠𝑡}−1(𝑚+1)^𝑚_𝑝=0𝜁_{𝑝𝑞𝑠𝑡}=1𝑚(𝑛+1)^𝑛_𝑞=0𝜁_{𝑚𝑞𝑠𝑡}−1𝑚(𝑚+1)(𝑛+1)^𝑚𝑛_𝑝=0𝑞=0𝜁_{𝑝𝑞𝑠𝑡}=1𝑚(𝑛+1)^𝑛_𝑞=0𝜁_{𝑚𝑞𝑠𝑡}−1𝑚𝜉_{𝑚𝑛𝑠𝑡}. (18) Now, the expression in the second bracket 1(𝑚+1)𝑛^𝑚_𝑝=0^𝑛−1_𝑞=0𝜁_{𝑝𝑞𝑠𝑡}−1𝑚𝑛^𝑚−1_𝑝=0^𝑛−1_𝑞=0𝜁_{𝑝𝑞𝑠𝑡}=1𝑛^𝑛−1_𝑞=01𝑚()𝑚𝑚+1^𝑚_𝑝=0𝜁_{𝑝𝑞𝑠𝑡}−(𝑚+1)^𝑚−1_𝑝=0𝜁_{𝑝𝑞𝑠𝑡}=1𝑛^𝑛−1_𝑞=01𝑚(𝑚+1)𝑚𝜁_{𝑚𝑞𝑠𝑡}−^𝑚−1_𝑝=0𝜁_{𝑝𝑞𝑠𝑡}=1𝑛^𝑛−1_𝑞=01𝑚()()𝜁𝑚+1𝑚+1_{𝑚𝑞𝑠𝑡}−^𝑚_𝑝=0𝜁_{𝑝𝑞𝑠𝑡}=1𝑚𝑛^𝑛−1_𝑞=0𝜁_{𝑚𝑞𝑠𝑡}−1𝑚(𝑚+1)𝑛^𝑚_𝑝=0^𝑛−1_𝑞=0𝜁_{𝑝𝑞𝑠𝑡}=1𝑚𝑛^𝑛−1_𝑞=0𝜁_{𝑚𝑞𝑠𝑡}−1𝑚𝜉_{𝑚,𝑛−1,𝑠,𝑡}. (19) Substituting (18) and (19) in (17), we get 𝜉_{𝑚𝑛𝑠𝑡}−𝜉_{𝑚−1,𝑛,𝑠,𝑡}−𝜉_{𝑚,𝑛−1,𝑠,𝑡}+𝜉_{𝑚−1,𝑛−1,𝑠,𝑡}=1𝑚(𝑛+1)^𝑛_𝑞=0𝜁_{𝑚𝑞𝑠𝑡}−1𝑚𝑛^𝑛−1_𝑞=0𝜁_{𝑚𝑞𝑠𝑡}−1𝑚𝜉_{𝑚𝑛𝑠𝑡}+1𝑚𝜉_{𝑚,𝑛−1,𝑠,𝑡}=1()𝑛𝑚𝑛𝑛+1^𝑛_𝑞=0𝜁_{𝑚𝑞𝑠𝑡}−(𝑛+1)^𝑛−1_𝑞=0𝜁_{𝑚𝑞𝑠𝑡}−1𝑚𝜉_{𝑚𝑛𝑠𝑡}−𝜉_{𝑚,𝑛−1,𝑠,𝑡}=1𝑚𝑛(𝑛+1)𝑛𝜁_{𝑚𝑛𝑠𝑡}−^𝑛−1_𝑞=0𝜁_{𝑚𝑞𝑠𝑡}−1𝑚𝜉_{𝑚𝑛𝑠𝑡}−𝜉_{𝑚,𝑛−1,𝑠,𝑡}=1()()𝜁𝑚𝑛𝑛+1𝑛+1_{𝑚𝑛𝑠𝑡}−^𝑛_𝑞=0𝜁_{𝑚𝑞𝑠𝑡}−1𝑚𝜉_{𝑚𝑛𝑠𝑡}−𝜉_{𝑚,𝑛−1,𝑠,𝑡}=1𝜁𝑚𝑛^{𝑚𝑛𝑠𝑡}−1(𝑛+1)^𝑛_𝑞=0𝜁_{𝑚𝑞𝑠𝑡}−1𝑚𝜉_{𝑚𝑛𝑠𝑡}−𝜉_{𝑚,𝑛−1,𝑠,𝑡}. (20) We know that 𝜉_{𝑚𝑛𝑠𝑡}=1(𝑚+1)(𝑛+1)^𝑚𝑛_𝑝=0𝑞=0𝜁_{𝑝𝑞𝑠𝑡}=1(𝑚+1)(𝑛+1)^𝑚−1𝑛_𝑝=0𝑞=0𝜁_{𝑝𝑞𝑠𝑡}+^𝑛_𝑞=0𝜁_{𝑚𝑞𝑠𝑡},𝜉_{𝑚−1,𝑛,𝑠,𝑡}=1𝑚(𝑛+1)^𝑚−1𝑛_𝑝=0𝑞=0𝜁_{𝑝𝑞𝑠𝑡}. (21) From (21), we have (𝑚+1)𝜉_{𝑚𝑛𝑠𝑡}−𝑚𝜉_{𝑚−1,𝑛,𝑠,𝑡}=1(𝑛+1)^𝑛_𝑞=0𝜁_{𝑚𝑞𝑠𝑡}. (22)
Thus, (20) becomes 𝜉_{𝑚𝑛𝑠𝑡}−𝜉_{𝑚−1,𝑛,𝑠,𝑡}−𝜉_{𝑚,𝑛−1,𝑠,𝑡}+𝜉_{𝑚−1,𝑛−1,𝑠,𝑡}=1𝜁𝑚𝑛^{𝑚𝑛𝑠𝑡}−(𝑚+1)𝜉_{𝑚𝑛𝑠𝑡}+𝑚𝜉_{𝑚−1,𝑛,𝑠,𝑡}−1𝑚𝜉_{𝑚𝑛𝑠𝑡}−𝜉_{𝑚,𝑛−1,𝑠,𝑡}=1𝜁𝑚𝑛^{𝑚𝑛𝑠𝑡}−𝜉_{𝑚𝑛𝑠𝑡}−𝑚𝜉_{𝑚𝑛𝑠𝑡}−𝜉_{𝑚−1,𝑛,𝑠,𝑡}−𝑛𝜉_{𝑚𝑛𝑠𝑡}−𝜉_{𝑚,𝑛−1,𝑠,𝑡}. (23) Also (22) can be written as 𝑚𝜉_{𝑚𝑛𝑠𝑡}−𝜉_{𝑚−1,𝑛,𝑠,𝑡}=1(𝑛+1)^𝑛_𝑞=0𝜁_{𝑚𝑞𝑠𝑡}−𝜉_{𝑚𝑛𝑠𝑡}. (24) Similarly, we can write 𝑛𝜉_{𝑚𝑛𝑠𝑡}−𝜉_{𝑚,𝑛−1,𝑠,𝑡}=1(𝑚+1)^𝑚_𝑞=0𝜁_{𝑝𝑛𝑠𝑡}−𝜉_{𝑚𝑛𝑠𝑡}. (25) Using (24) and (25) in (23), we get 𝜉_{𝑚𝑛𝑠𝑡}−𝜉_{𝑚−1,𝑛,𝑠,𝑡}−𝜉_{𝑚,𝑛−1,𝑠,𝑡}+𝜉_{𝑚−1,𝑛−1,𝑠,𝑡}=1𝜁𝑚𝑛^{𝑚𝑛𝑠𝑡}+𝜉_{𝑚𝑛𝑠𝑡}−1(𝑚+1)^𝑚_𝑝=0𝜁_{𝑝𝑛𝑠𝑡}−1(𝑛+1)^𝑛_𝑞=0𝜁_{𝑚𝑞𝑠𝑡}. (26) This implies that 𝜁_{𝑚𝑛𝑠𝑡}+𝜉_{𝑚𝑛𝑠𝑡}−Ω^†_{𝑚𝑛𝑠𝑡}−Ω^‡_{𝑚𝑛𝑠𝑡}=𝑚𝑛𝜉_{𝑚𝑛𝑠𝑡}−𝜉_{𝑚−1,𝑛,𝑠,𝑡}−𝜉_{𝑚,𝑛−1,𝑠,𝑡}+𝜉_{𝑚−1,𝑛−1,𝑠,𝑡}. (27)

Theorem 4. One has the following inclusions and the limit is preserved in each case:(i)[𝒱_𝜎]⊂[𝒲_𝜎]⊂𝒲_𝜎, (ii)𝒲^′_𝜎⊂[𝒲_𝜎] if the conditions (L2) and (L3) of Lemma 2 hold, (iii)𝒲^′_𝜎⊂𝒲_𝜎^′′.

Proof. (i) Let 𝑥∈[𝒱_𝜎] with [𝒱_𝜎]-lim𝑥=ℓ, say. Then, 𝜁_{𝑝𝑞𝑠𝑡}(|𝑥−ℓ|)⟶0as𝑝,𝑞⟶∞,uniformlyin𝑠,𝑡. (28) This implies that 1(𝑚+1)(𝑛+1)^𝑚𝑛_𝑝=0𝑞=0𝜁_{𝑝𝑞𝑠𝑡}(|𝑥−ℓ|)⟶0as𝑝,𝑞⟶∞,uniformlyin𝑠,𝑡. (29) Also, we have 1(𝑚+1)(𝑛+1)^𝑚𝑛_𝑝=0𝑞=0𝜁_{𝑝𝑞𝑠𝑡}()≤1𝑥−ℓ(𝑚+1)(𝑛+1)^𝑚𝑛_𝑝=0𝑞=0𝜁_{𝑝𝑞𝑠𝑡}(≤1𝑥−ℓ)(𝑚+1)(𝑛+1)^𝑚𝑛_𝑝=0𝑞=0𝜁_{𝑝𝑞𝑠𝑡}(|𝑥−ℓ|). (30) Hence, [𝒱_𝜎]⊂[𝒲_𝜎]⊂𝒲_𝜎 and 𝒲_𝜎-lim𝑥=𝒲_𝜎-lim𝑥=ℓ. (31) (ii) We have to show that 𝒲^′_𝜎⊂[𝒲_𝜎]. If 𝑥∈𝒲^′_𝜎, then we have ℑ_{𝑚𝑛𝑠𝑡}=^∞∞_{𝑝=𝑚𝑞=𝑛}𝜉_{𝑝𝑞𝑠𝑡}−𝜉_{𝑝−1,𝑞,𝑠,𝑡}−𝜉_{𝑝,𝑞−1,𝑠,𝑡}+𝜉_{𝑝−1,𝑞−1,𝑠,𝑡}⟶0 (32) as 𝑚,𝑛→∞, uniformly in 𝑠, 𝑡; and 𝜉_{𝑝𝑞𝑠𝑡}⟶ℓsayas𝑝,𝑞⟶∞,uniformlyin𝑠,𝑡, (33) that is, 𝒲_𝜎-lim𝑥=ℓ.
In order to prove that 𝑥∈[𝒲_𝜎], it is enough to show that condition (L4) of Lemma 2 holds. Now, ℑ_{𝑚𝑛𝑠𝑡}−ℑ_{𝑚,𝑛+1,𝑠,𝑡}−ℑ_{𝑚+1,𝑛,𝑠,𝑡}+ℑ_{𝑚+1,𝑛+1,𝑠,𝑡}=^∞∞_{𝑝=𝑚𝑞=𝑛}𝜉_{𝑝𝑞𝑠𝑡}−𝜉_{𝑝−1,𝑞,𝑠,𝑡}−𝜉_{𝑝,𝑞−1,𝑠,𝑡}+𝜉_{𝑝−1,𝑞−1,𝑠,𝑡}−^∞∞_{𝑝=𝑚𝑞=𝑛+1}𝜉_{𝑝𝑞𝑠𝑡}−𝜉_{𝑝−1,𝑞,𝑠,𝑡}−𝜉_{𝑝,𝑞−1,𝑠,𝑡}+𝜉_{𝑝−1,𝑞−1,𝑠,𝑡}−^∞∞_{𝑝=𝑚+1𝑞=𝑛}𝜉_{𝑝𝑞𝑠𝑡}−𝜉_{𝑝−1,𝑞,𝑠,𝑡}−𝜉_{𝑝,𝑞−1,𝑠,𝑡}+𝜉_{𝑝−1,𝑞−1,𝑠,𝑡}+^∞∞_{𝑝=𝑚+1𝑞=𝑛+1}𝜉_{𝑝𝑞𝑠𝑡}−𝜉_{𝑝−1,𝑞,𝑠,𝑡}−𝜉_{𝑝,𝑞−1,𝑠,𝑡}+𝜉_{𝑝−1,𝑞−1,𝑠,𝑡}=^∞_𝑝=𝑚^∞_𝑞=𝑛−^∞_𝑞=𝑛+1×𝜉_{𝑝𝑞𝑠𝑡}−𝜉_{𝑝−1,𝑞,𝑠,𝑡}−𝜉_{𝑝,𝑞−1,𝑠,𝑡}+𝜉_{𝑝−1,𝑞−1,𝑠,𝑡}−^∞_𝑝=𝑚+1^∞_𝑞=𝑛−^∞_𝑞=𝑛+1×𝜉_{𝑝𝑞𝑠𝑡}−𝜉_{𝑝−1,𝑞,𝑠,𝑡}−𝜉_{𝑝,𝑞−1,𝑠,𝑡}+𝜉_{𝑝−1,𝑞−1,𝑠,𝑡}=^∞_𝑝=𝑚𝜉_{𝑝𝑛𝑠𝑡}−𝜉_{𝑝−1,𝑛,𝑠,𝑡}−𝜉_{𝑝,𝑛−1,𝑠,𝑡}+𝜉_{𝑝−1,𝑛−1,𝑠,𝑡}−^∞_𝑝=𝑚+1𝜉_{𝑝𝑛𝑠𝑡}−𝜉_{𝑝−1,𝑛,𝑠,𝑡}−𝜉_{𝑝,𝑛−1,𝑠,𝑡}+𝜉_{𝑝−1,𝑛−1,𝑠,𝑡}=^∞_𝑝=𝑚−^∞_𝑝=𝑚+1𝜉_{𝑝𝑛𝑠𝑡}−𝜉_{𝑝−1,𝑛,𝑠,𝑡}−𝜉_{𝑝,𝑛−1,𝑠,𝑡}+𝜉_{𝑝−1,𝑛−1,𝑠,𝑡}=𝜉_{𝑚𝑛𝑠𝑡}−𝜉_{𝑚−1,𝑛,𝑠,𝑡}−𝜉_{𝑚,𝑛−1,𝑠,𝑡}+𝜉_{𝑚−1,𝑛−1,𝑠,𝑡}. (34) Replacing 𝑚 and 𝑛 by 𝑝 and 𝑞, respectively, we have ℑ_{𝑝𝑞𝑠𝑡}−ℑ_{𝑝,𝑞+1,𝑠,𝑡}−ℑ_{𝑝+1,𝑞,𝑠,𝑡}+ℑ_{𝑝+1,𝑞+1,𝑠,𝑡}=𝜉_{𝑝𝑞𝑠𝑡}−𝜉_{𝑝−1,𝑞,𝑠,𝑡}−𝜉_{𝑝,𝑞−1,𝑠,𝑡}+𝜉_{𝑝−1,𝑞−1,𝑠,𝑡}. (35) By Lemma 3, we have 𝜁_{𝑝𝑞𝑠𝑡}+𝜉_{𝑝𝑞𝑠𝑡}−Ω^†_{𝑝𝑞𝑠𝑡}−Ω^‡_{𝑝𝑞𝑠𝑡}=𝑝𝑞𝜉_{𝑝𝑞𝑠𝑡}−𝜉_{𝑝−1,𝑞,𝑠,𝑡}−𝜉_{𝑝,𝑞−1,𝑠,𝑡}+𝜉_{𝑝−1,𝑞−1,𝑠,𝑡}. (36) So that we have 1𝑚𝑛^𝑚𝑛_𝑝=1𝑞=1𝜁_{𝑝𝑞𝑠𝑡}+𝜉_{𝑝𝑞𝑠𝑡}−Ω^†_{𝑝𝑞𝑠𝑡}−Ω^‡_{𝑝𝑞𝑠𝑡}=1𝑚𝑛^𝑚𝑛_𝑝=1𝑞=1𝑝𝑞ℑ_{𝑝𝑞𝑠𝑡}−ℑ_{𝑝,𝑞+1,𝑠,𝑡}−ℑ_{𝑝+1,𝑞,𝑠,𝑡}+ℑ_{𝑝+1,𝑞+1,𝑠,𝑡}. (37) By using Abel’s transformation for double series in the right hand side of above equation, we have 1𝑚𝑛^𝑚𝑛_𝑝=1𝑞=1𝜁_{𝑝𝑞𝑠𝑡}+𝜉_{𝑝𝑞𝑠𝑡}−Ω^†_{𝑝𝑞𝑠𝑡}−Ω^‡_{𝑝𝑞𝑠𝑡}=1𝑚𝑛^𝑚𝑛_𝑝=1𝑞=1ℑ_{𝑝𝑞𝑠𝑡}−𝑚^𝑛_𝑞=1ℑ_{𝑚+1,𝑞,𝑠,𝑡}−𝑛^𝑚_𝑝=1ℑ_{𝑝,𝑛+1,𝑠,𝑡}+𝑚𝑛ℑ_{𝑚+1,𝑛+1,𝑠,𝑡} (38)→0  as  𝑚,𝑛→∞, uniformly in 𝑠,𝑡 (by (32)). Hence, by Lemma 2, 𝑥∈[𝒲_𝜎].
(iii) Let 𝑥∈𝒲^′_𝜎, and we have to show that sup^∞_𝑠,𝑡^∞_𝑝=0𝑞=0𝜉_{𝑝𝑞𝑠𝑡}−𝜉_{𝑝−1,𝑞,𝑠,𝑡}−𝜉_{𝑝,𝑞−1,𝑠,𝑡}+𝜉_{𝑝−1,𝑞−1,𝑠,𝑡}≤𝐾, (39) where 𝐾 is an absolute constant. Since 𝑥∈𝒲^′_𝜎, there exist integers 𝑝₀,𝑞₀ such that _{𝑝>𝑝0𝑞>𝑞0}𝜉_{𝑝𝑞𝑠𝑡}−𝜉_{𝑝−1,𝑞,𝑠,𝑡}−𝜉_{𝑝,𝑞−1,𝑠,𝑡}+𝜉_{𝑝−1,𝑞−1,𝑠,𝑡}<1,∀𝑠,𝑡. (40) Hence, it is left to show that for fixed 𝑝,𝑞𝜉_{𝑝𝑞𝑠𝑡}−𝜉_{𝑝−1,𝑞,𝑠,𝑡}−𝜉_{𝑝,𝑞−1,𝑠,𝑡}+𝜉_{𝑝−1,𝑞−1,𝑠,𝑡}≤𝐾,∀𝑠,𝑡. (41) From (40), we have 𝜉_{𝑝𝑞𝑠𝑡}−𝜉_{𝑝−1,𝑞,𝑠,𝑡}−𝜉_{𝑝,𝑞−1,𝑠,𝑡}+𝜉_{𝑝−1,𝑞−1,𝑠,𝑡}<1, (42) for every fixed 𝑝>𝑝₀,  𝑞>𝑞₀ and for all 𝑠,  𝑡. Since 𝑚(𝑚+1)𝑛(𝑛+1)𝜉_{𝑚𝑛𝑠𝑡}−𝜉_{𝑚−1,𝑛,𝑠,𝑡}−𝜉_{𝑚,𝑛−1,𝑠,𝑡}+𝜉_{𝑚−1,𝑛−1,𝑠,𝑡}=^𝑚𝑛_𝑝=1𝑞=1𝑝𝑞𝜁_{𝑝𝑞𝑠𝑡}−𝜁_{𝑝−1,𝑞,𝑠,𝑡}−𝜁_{𝑝,𝑞−1,𝑠,𝑡}+𝜁_{𝑝−1,𝑞−1,𝑠,𝑡}. (43) Accordingly, )×𝜉𝑚𝑛(𝑚+1)(𝑛+1_{𝑚𝑛𝑠𝑡}−𝜉_{𝑚−1,𝑛,𝑠,𝑡}−𝜉_{𝑚,𝑛−1,𝑠,𝑡}+𝜉_{𝑚−1,𝑛−1,𝑠,𝑡}−()×𝜉𝑚−1)(𝑛+1_{𝑚−1,𝑛,𝑠,𝑡}−𝜉_{𝑚−2,𝑛,𝑠,𝑡}−𝜉_{𝑚−1,𝑛−1,𝑠,𝑡}+𝜉_{𝑚−2,𝑛−1,𝑠,𝑡}−()×𝜉𝑚+1)(𝑛−1_{𝑚,𝑛−1,𝑠,𝑡}−𝜉_{𝑚−1,𝑛−1,𝑠,𝑡}−𝜉_{𝑚,𝑛−2,𝑠,𝑡}+𝜉_{𝑚−1,𝑛−2,𝑠,𝑡}+()×𝜉𝑚−1)(𝑛−1_{𝑚−1,𝑛−1,𝑠,𝑡}−𝜉_{𝑚−2,𝑛−1,𝑠,𝑡}−𝜉_{𝑚−1,𝑛−2,𝑠,𝑡}+𝜉_{𝑚−2,𝑛−2,𝑠,𝑡}=^𝑛_𝑞=1𝑞^𝑚_𝑝=1𝑝𝜁_{𝑝𝑞𝑠𝑡}−𝜁_{𝑝−1,𝑞,𝑠,𝑡}−𝜁_{𝑝,𝑞−1,𝑠,𝑡}+𝜁_{𝑝−1,𝑞−1,𝑠,𝑡}−^𝑚−1_𝑝=1𝑝𝜁_{𝑝𝑞𝑠𝑡}−𝜁_{𝑝−1,𝑞,𝑠,𝑡}−𝜁_{𝑝,𝑞−1,𝑠,𝑡}+𝜁_{𝑝−1,𝑞−1,𝑠,𝑡}−^𝑛−1_𝑞=1𝑞^𝑚_𝑝=1𝑝𝜁_{𝑝𝑞𝑠𝑡}−𝜁_{𝑝−1,𝑞,𝑠,𝑡}−𝜁_{𝑝,𝑞−1,𝑠,𝑡}+𝜁_{𝑝−1,𝑞−1,𝑠,𝑡}−^𝑚−1_𝑝=1𝑝𝜁_{𝑝𝑞𝑠𝑡}−𝜁_{𝑝−1,𝑞,𝑠,𝑡}−𝜁_{𝑝,𝑞−1,𝑠,𝑡}+𝜁_{𝑝−1,𝑞−1,𝑠,𝑡}=^𝑛_𝑞=1𝑞𝑚𝜁_{𝑚𝑞𝑠𝑡}−𝜁_{𝑚−1,𝑞,𝑠,𝑡}−𝜁_{𝑚,𝑞−1,𝑠,𝑡}+𝜁_{𝑚−1,𝑞−1,𝑠,𝑡}−^𝑛−1_𝑞=1𝑞𝑚𝜁_{𝑚𝑞𝑠𝑡}−𝜁_{𝑚−1,𝑞,𝑠,𝑡}−𝜁_{𝑚,𝑞−1,𝑠,𝑡}+𝜁_{𝑚−1,𝑞−1,𝑠,𝑡}. (44) This implies that (𝑚+1)(𝑛+1)𝜉_{𝑚𝑛𝑠𝑡}−𝜉_{𝑚−1,𝑛,𝑠,𝑡}−𝜉_{𝑚,𝑛−1,𝑠,𝑡}+𝜉_{𝑚−1,𝑛−1,𝑠,𝑡}−()×𝜉𝑚−1)(𝑛+1_{𝑚−1,𝑛,𝑠,𝑡}−𝜉_{𝑚−2,𝑛,𝑠,𝑡}−𝜉_{𝑚−1,𝑛−1,𝑠,𝑡}+𝜉_{𝑚−2,𝑛−1,𝑠,𝑡}−()×𝜉𝑚+1)(𝑛−1_{𝑚,𝑛−1,𝑠,𝑡}−𝜉_{𝑚−1,𝑛−1,𝑠,𝑡}−𝜉_{𝑚,𝑛−2,𝑠,𝑡}+𝜉_{𝑚−1,𝑛−2,𝑠,𝑡}+()×𝜉𝑚−1)(𝑛−1_{𝑚−1,𝑛−1,𝑠,𝑡}−𝜉_{𝑚−2,𝑛−1,𝑠,𝑡}−𝜉_{𝑚−1,𝑛−2,𝑠,𝑡}+𝜉_{𝑚−2,𝑛−2,𝑠,𝑡}=𝜁_{𝑚𝑛𝑠𝑡}−𝜁_{𝑚−1,𝑛,𝑠,𝑡}−𝜁_{𝑚,𝑛−1,𝑠,𝑡}+𝜁_{𝑚−1,𝑛−1,𝑠,𝑡}. (45) Using (42) and (45), we have 𝜁_{𝑚𝑛𝑠𝑡}−𝜁_{𝑚−1,𝑛,𝑠,𝑡}−𝜁_{𝑚,𝑛−1,𝑠,𝑡}+𝜁_{𝑚−1,𝑛−1,𝑠,𝑡}≤𝐾(𝑚,𝑛) (46) 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∑_𝑡𝑎_𝑠𝑡 denoted as “𝑎”, let us write 𝑥_𝑗𝑘=^𝑗𝑘_𝑠=0𝑡=0𝑎_𝑠𝑡,𝑗,𝑘=1,2,… (47) and 𝜎 be monotonically increasing. For simplicity in notation, we denote 𝜙_{𝑚𝑛𝑠𝑡}=𝜁_{𝑚𝑛𝑠𝑡}(𝑎)−𝜁_{𝑚−1,𝑛,𝑠,𝑡}(𝑎)−𝜁_{𝑚,𝑛−1,𝑠,𝑡}(𝑎)+𝜁_{𝑚−1,𝑛−1,𝑠,𝑡}(𝑎). (48) Again from the definition of 𝜁_{𝑚𝑛𝑠𝑡}, it is easy to obtain 𝜙_{𝑚𝑛𝑠𝑡}=1𝑚(𝑚+1)𝑛(𝑛+1)^𝑚𝑛_𝑗=1𝑘=1𝑗𝑘^𝑑′𝑗_{𝑢=𝑑𝑗}^ℎ′𝑘_𝑣=ℎ𝑘𝑎_𝑢𝑣 (49) for all 𝑚,𝑛≥1 and 𝜙_{0,0,𝑠,𝑡}(𝑎)=𝑎_𝑠𝑡 with 𝑑_𝑗=𝜎^𝑗−1(𝑠)+1,  𝑑^′_𝑗=𝜎^𝑗(𝑠), ℎ_𝑘=𝜎^𝑘−1(𝑡)+1, ℎ^′_𝑘=𝜎^𝑘(𝑡). Further, we calculate 𝑚(𝑚+1)𝑛(𝑛+1)𝜙_{𝑚𝑛𝑠𝑡}−𝑚(𝑚−1)𝑛(𝑛+1)𝜙_{𝑚−1,𝑛,𝑠,𝑡}−𝑚(𝑚+1)𝑛(𝑛−1)𝜙_{𝑚,𝑛−1,𝑠,𝑡}+𝑚(𝑚−1)𝑛(𝑛−1)𝜙_{𝑚−1,𝑛−1,𝑠,𝑡}=^𝑚𝑛_𝑗=1𝑘=1𝑗𝑘−^𝑚−1𝑛_𝑗=1𝑘=1𝑗𝑘−^𝑚_𝑗=1^𝑛−1_𝑘=1𝑗𝑘+^𝑚−1_𝑗=1^𝑛−1_𝑘=1×𝑗𝑘_{𝑑′𝑗𝑢=𝑑𝑗ℎ′𝑘𝑣=ℎ𝑘}𝑎_𝑢𝑣=^𝑛_𝑘=1𝑘^𝑚_𝑗=1𝑗−^𝑚−1_𝑗=1𝑗−^𝑛−1_𝑘=1𝑘^𝑚_𝑗=1𝑗+^𝑚−1_𝑗=1𝑗×^𝑑′𝑗_{𝑢=𝑑𝑗}^ℎ′𝑘_𝑣=ℎ𝑘𝑎_𝑢𝑣=𝑚^𝑛_𝑘=1𝑘−𝑚^𝑛−1_𝑘=1𝑘^𝑑′𝑚_{𝑢=𝑑𝑚}^ℎ′𝑘_𝑣=ℎ𝑘𝑎_𝑢𝑣=𝑚𝑛^𝑑′𝑚_{𝑢=𝑑𝑚}^ℎ′𝑛_𝑣=ℎ𝑛𝑎_𝑢𝑣. (50)
Thus, we have ^𝑑′𝑚_{𝑢=𝑑𝑚}^ℎ′𝑛_𝑣=ℎ𝑛𝑎_𝑢𝑣=(𝑚+1)(𝑛+1)𝜙_{𝑚,𝑛,𝑠,𝑡}−(𝑚−1)(𝑛+1)𝜙_{𝑚−1,𝑛,𝑠,𝑡}−(𝑚+1)(𝑛−1)𝜙_{𝑚,𝑛−1,𝑠,𝑡}+(𝑚−1)(𝑛−1)𝜙_{𝑚−1,𝑛−1,𝑠,𝑡}. (51) Hence, it follows from (46) that for each fixed 𝑚>𝑝₀,  𝑛>𝑞₀, ^𝑑′𝑚_{𝑢=𝑑𝑚}^ℎ′𝑛_𝑣=ℎ𝑛𝑎_𝑢𝑣≤𝐾(𝑚,𝑛),∀𝑠,𝑡. (52) Hence, it follows from (52) that 𝑎_𝑢𝑣≤𝐾∀𝑢,𝑣, (53) where 𝐾 is independent of 𝑢,𝑣. By (49), we have 𝜁_{𝑚𝑛𝑠𝑡}−𝜁_{𝑚−1,𝑛,𝑠,𝑡}−𝜁_{𝑚,𝑛−1,𝑠,𝑡}+𝜁_{𝑚−1,𝑛−1,𝑠,𝑡}≤𝐾,∀𝑚,𝑛,𝑠,𝑡. (54) Also from (43) and (54), we have 𝜉_{𝑝𝑞𝑠𝑡}−𝜉_{𝑝−1,𝑞,𝑠,𝑡}−𝜉_{𝑝,𝑞−1,𝑠,𝑡}+𝜉_{𝑝−1,𝑞−1,𝑠,𝑡}≤𝐾,∀𝑝,𝑞,𝑠,𝑡. (55)

4. Topological Results

Here, we extend the spaces [𝒲_𝜎],  𝒲^′_𝜎,    𝒲_𝜎^′′ to more general spaces, respectively, denoted by [𝒲_𝜎(𝛼)],  𝒲^′_𝜎(𝛼),  𝒲_𝜎^′′(𝛼), where 𝛼=(𝛼_𝑗𝑘) is a double sequence of positive real numbers for all 𝑗, 𝑘 and sup_𝑗,𝑘𝛼_𝑗𝑘=𝐻<∞. First, we recall the notion of paranorm as follows.

A paranorm 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 (𝛼_𝑛) is a sequence of scalars with 𝛼_𝑛→𝛼₀(𝑛→∞) and 𝑥_𝑛,𝑎∈𝑋 with 𝑥_𝑛→𝑎(𝑛→∞) in the sense that 𝑔(𝑥_𝑛−𝑎)→0(𝑛→∞), then 𝛼_𝑛𝑥_𝑛→𝛼₀𝑎  (𝑛→∞), in the sense that 𝑔(𝛼_𝑛𝑥_𝑛−𝛼₀𝑎)→0(𝑛→∞).

A paranorm 𝑔 for which 𝑔(𝑥)=0 implies 𝑥=𝜃 is called a total paranorm on 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: 𝒲_𝜎(𝛼)=𝑥=𝑥_𝑗𝑘∶1()×𝑚+1)(𝑛+1_{𝑚𝑛𝑝=0𝑞=0}𝜁_{𝑝𝑞𝑠𝑡}(𝑥−ℓ𝐸)^𝛼𝑝𝑞,𝒲⟶0,as𝑚,𝑛⟶∞,uniformlyin𝑠,𝑡_′𝜎(𝛼)=𝑥=𝑥_𝑗𝑘∶_{∞∞𝑝=0𝑞=0}𝜉_{𝑝𝑞𝑠𝑡}−𝜉_{𝑝−1,𝑞,𝑠,𝑡}−𝜉_{𝑝,𝑞−1,𝑠,𝑡}+𝜉_{𝑝−1,𝑞−1,𝑠,𝑡}^𝛼𝑝𝑞,𝒲convergesuniformlyin𝑠,𝑡_𝜎′′(𝛼)=𝑥=𝑥_𝑗𝑘∶sup^∞_𝑠,𝑡^∞_𝑝=0𝑞=0𝜉_{𝑝𝑞𝑠𝑡}−𝜉_{𝑝−1,𝑞,𝑠,𝑡}−𝜉_{𝑝,𝑞−1,𝑠,𝑡}+𝜉_{𝑝−1,𝑞−1,𝑠,𝑡}^𝛼𝑝𝑞<∞. (56)

We remark that if 𝛼=(𝛼_𝑗𝑘) is a constant sequence, then we write [𝒲_𝜎,𝛼] in place of [𝒲_𝜎(𝛼)].

Theorem 5. Let 𝛼=(𝛼_𝑝𝑞) be a bounded double sequence of strictly positive real numbers. Then, [𝒲_𝜎(𝛼)] is a complete linear topological space paranormed by 𝑔(𝑥)=sup_{𝑚,𝑛,𝑠,𝑡}1(𝑚+1)(𝑛+1)^𝑚𝑛_𝑝=0𝑞=0𝜁_{𝑝𝑞𝑠𝑡}(𝑥)^{𝛼𝑝𝑞1/𝑀}, (57) where 𝑀=max(1,sup_𝑝,𝑞𝛼_𝑝𝑞). If 𝛼≥1 then [𝒲_𝜎,𝛼] is a Banach space and [𝒲_𝜎,𝛼] is a 𝛼-normed space if 0<𝛼<1.

Proof. Let (𝑥_𝑝𝑞) and (𝑦_𝑝𝑞) be two double sequences. Then, 𝑥_𝑝𝑞+𝑦_𝑝𝑞^𝛼𝑝𝑞≤𝐾𝑥_𝑝𝑞^𝛼𝑝𝑞+𝑦_𝑝𝑞^𝛼𝑝𝑞, (58) where 𝐾=max(1,2^𝐻−1) and 𝐻=sup_𝑝,𝑞𝛼_𝑝𝑞. Since |𝜆|^𝛼𝑝𝑞≤max1,|𝜆|^𝐻, (59) therefore 1(𝑚+1)(𝑛+1)^𝑚𝑛_𝑝=0𝑞=0𝜁_{𝑝𝑞𝑠𝑡}𝜆𝑥+𝜇𝑦^𝛼𝑝𝑞≤𝐾𝐾₁1(𝑚+1)(𝑛+1)^𝑚𝑛_𝑝=0𝑞=0𝜁_{𝑝𝑞𝑠𝑡}(𝑥)^𝛼𝑝𝑞+𝐾𝐾₂1(𝑚+1)(𝑛+1)^𝑚𝑛_𝑝=0𝑞=0𝜁_{𝑝𝑞𝑠𝑡}𝑦^𝛼𝑝𝑞, (60) where 𝐾₁=max(1,|𝜆|^𝐻) and 𝐾₂=max(1,|𝜇|^𝐻). From (60), we have that if 𝑥,𝑦∈[𝒲_𝜎(𝛼)], then 𝜆𝑥+𝜇𝑦∈[𝒲_𝜎(𝛼)]. Thus, [𝒲_𝜎(𝛼)] is a linear space. Without loss of generality, we can take 𝒲_𝜎(𝛼)-lim𝑥=0. (61)
Clearly, 𝑔(0)=0,  𝑔(−𝑥)=𝑔(𝑥). From (58) and Minkowski’s inequality, we have ^𝑚𝑛_𝑝=0𝑞=0𝜁_{𝑝𝑞𝑠𝑡}𝑥+𝑦^{𝛼𝑝𝑞1/𝑀}≤^𝑚𝑛_𝑝=0𝑞=0𝜁_{𝑝𝑞𝑠𝑡}(𝑥)^{𝛼𝑝𝑞/𝑀}+𝜁_{𝑝𝑞𝑠𝑡}𝑦^{𝛼𝑝𝑞/𝑀1/𝑀}≤^𝑚𝑛_𝑝=0𝑞=0𝜁_{𝑝𝑞𝑠𝑡}(𝑥)^{𝛼𝑝𝑞1/𝑀}+^𝑚𝑛_𝑝=0𝑞=0𝜁_{𝑝𝑞𝑠𝑡}𝑦^{𝛼𝑝𝑞1/𝑀}. (62) Hence, 𝑔𝑥+𝑦≤𝑔(𝑥)+𝑔𝑦. (63) Since 𝛼=(𝛼_𝑚𝑛) is bounded away from zero, there exists a constant 𝛿>0 such that 𝛼_𝑚𝑛≥𝛿 for all 𝑚,  𝑛. Now for |𝜆|≤1,  |𝜆|^𝛼𝑠𝑡≤|𝜆|^𝛿 and so 𝑔(𝜆𝑥)≤|𝜆|^𝛿/𝑀𝑔(𝑥), (64) that is, the scaler multiplication is continuous. Hence, 𝑔 is a paranorm on [𝒲_𝜎(𝛼)].
Let (𝑥^𝑏) be a Cauchy sequence in [𝒲_𝜎(𝛼)], that is, 𝑔𝑥^𝑑−𝑥^𝑏⟶0as𝑏,𝑑⟶∞. (65) Since 1(𝑚+1)(𝑛+1)^𝑚𝑛_𝑝=0𝑞=0𝜁_{𝑝𝑞𝑠𝑡}𝑥^𝑑−𝑥^{𝑏𝛼𝑝𝑞}≤𝑔𝑥^𝑑−𝑥^𝑏𝑀, (66) it follows that 𝜁_{𝑝𝑞𝑠𝑡}𝑥^𝑑−𝑥^{𝑏𝛼𝑝𝑞}⟶0as𝑏,𝑑⟶∞∀𝑝,𝑞,𝑠,𝑡. (67) In particular, 𝜁_{0,0,𝑠,𝑡}𝑥^𝑑−𝑥^𝑏=𝑥^𝑑_𝑠𝑡−𝑥^𝑏_𝑠𝑡⟶0as𝑏,𝑑⟶∞forfixed𝑠,𝑡. (68) Hence, (𝑥^𝑑) is a Cauchy sequence in ℂ. Since ℂ is complete, there exists 𝑥=(𝑥_𝑗𝑘)∈ℂ such that 𝑥^𝑑→𝑥 coordinatewise as 𝑑→∞. It follows from (66) that given 𝜖>0, there exists 𝑑₀ such that 1(𝑚+1)(𝑛+1)^𝑚𝑛_𝑝=0𝑞=0𝜁_{𝑝𝑞𝑠𝑡}𝑥^𝑑−𝑥^{𝑏𝛼𝑝𝑞1/𝑀}<𝜖, (69) for 𝑏,𝑑>𝑑₀. Now taking 𝑏→∞ and sup_{𝑠,𝑡,𝑚,𝑛} in (69), we have 𝑔(𝑥^𝑑−𝑥)≤𝜖 for 𝑑>𝑑₀. This proves that 𝑥^𝑑→𝑥 and 𝑥=(𝑥_𝑗𝑘)∈[𝒲_𝜎(𝛼)]. Hence [𝒲_𝜎(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 ℎ(𝑥)=sup_𝑠,𝑡^∞∞_𝑝=0𝑞=0𝜉_{𝑝𝑞𝑠𝑡}−𝜉_{𝑝−1,𝑞,𝑠,𝑡}−𝜉_{𝑝,𝑞−1,𝑠,𝑡}+𝜉_{𝑝−1,𝑞−1,𝑠,𝑡}^{𝛼𝑝𝑞1/𝑀} (70) which is defined on 𝒲^′_𝜎(𝛼). If 𝛼≥1 then 𝒲^′_𝜎,𝛼 is a Banach space and if 0<𝛼<1, 𝒲_𝜎,𝛼 is 𝛼-normed space.

Proof. In order for the paranorm in (70) be defined, we require that 𝒲^′_𝜎(𝛼)⊂𝒲_𝜎^′′(𝛼) (71) 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 that 𝑥=(𝑥_𝑗𝑘)∈𝒲^′_𝜎(𝛼). Then, for 𝜖>0 there exist integers 𝑀,𝑁>0 such that ^∞∞_{𝑝=𝑀𝑞=𝑁}𝜉_{𝑝𝑞𝑠𝑡}(𝑥)−𝜉_{𝑝−1,𝑞,𝑠,𝑡}(𝑥)−𝜉_{𝑝,𝑞−1,𝑠,𝑡}(𝑥)+𝜉_{𝑝−1,𝑞−1,𝑠,𝑡}(𝑥)^𝛼𝑝𝑞<𝜖. (72) If |𝜆|≤1, then by (72) we have ^∞∞_{𝑝=𝑀𝑞=𝑁}𝜉_{𝑝𝑞𝑠𝑡}(𝜆𝑥)−𝜉_{𝑝−1,𝑞,𝑠,𝑡}(𝜆𝑥)−𝜉_{𝑝,𝑞−1,𝑠,𝑡}(𝜆𝑥)+𝜉_{𝑝−1,𝑞−1,𝑠,𝑡}(𝜆𝑥)^𝛼𝑝𝑞≤^∞∞_{𝑝=𝑀𝑞=𝑁}𝜉_{𝑝𝑞𝑠𝑡}(𝑥)−𝜉_{𝑝−1,𝑞,𝑠,𝑡}(𝑥)−𝜉_{𝑝,𝑞−1,𝑠,𝑡}(𝑥)+𝜉_{𝑝−1,𝑞−1,𝑠,𝑡}(𝑥)^𝛼𝑝𝑞<𝜖. (73) Since for fixed 𝑀,𝑁^𝑀−1_𝑝=0^𝑁−1_𝑞=0𝜉_{𝑝𝑞𝑠𝑡}(𝜆𝑥)−𝜉_{𝑝−1,𝑞,𝑠,𝑡}(𝜆𝑥)−𝜉_{𝑝,𝑞−1,𝑠,𝑡}(𝜆𝑥)+𝜉_{𝑝−1,𝑞−1,𝑠,𝑡}(𝜆𝑥)^𝛼𝑝𝑞⟶0 (74) as 𝜆→0, it follows from (73) and (74) that for fixed 𝑥=(𝑥_𝑗𝑘),  ℎ(𝜆𝑥)→0 as 𝜆→0. Also, since |𝜆|^𝛼𝑝𝑞≤max(1,|𝜆|^𝐻) implies that ℎ(𝜆𝑥)≤sup|𝜆|^{𝛼𝑝𝑞1/𝑀}ℎ(𝑥). (75) 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 𝒲^′_𝜎(𝛼) can be achieved by using the same technique as in Theorem 5.

Theorem 7. Suppose that 𝛼=(𝛼_𝑝𝑞) is bounded double sequence of strictly positive real numbers. Then,(i)𝒲_𝜎^′′(𝛼) is a complete linear space paranormed by the function ℎ defined in (70). In particular, if 𝛼 is a constant, 𝒲^′′_𝜎,𝛼 is a Banach space for 𝛼≥1 and 𝛼-normed space for 0<𝛼<1, (ii)𝒲^′_𝜎(𝛼) is a closed subspace of 𝒲_𝜎^′′(𝛼).

Proof. (i) Proceeding along the same lines as in Theorem 6, except for the proof of continuity of scalar multiplication. If 𝑥=(𝑥_𝑗𝑘)∈𝒲_𝜎^′′(𝛼), we cannot assert (72) as in the case when 𝑥=(𝑥_𝑗𝑘)∈𝒲^′_𝜎(𝛼). Since 𝛼 is bounded away from zero, there exists 𝛿>0 such that 𝛼_𝑚𝑛≥𝛿 for all 𝑚, 𝑛. Hence, |𝜆|≤1 implies |𝜆|^𝛼𝑝𝑞≤|𝜆|^𝛿. Since ℎ(𝜆𝑥)≤|𝜆|^𝛼𝑝𝑞ℎ(𝑥), (76) continuity of scalar multiplication follows.
(ii) For this, first we have to show that 𝒲^′_𝜎(𝛼)⊂𝒲_𝜎^′′(𝛼). (77) Let 𝑥=(𝑥_𝑗𝑘)∈𝒲^′_𝜎(𝛼). Then, there exist integers 𝑀,  𝑁 such that ^∞∞_{𝑝=𝑀𝑞=𝑁}𝜉_{𝑝𝑞𝑠𝑡}(𝑥)−𝜉_{𝑝−1,𝑞,𝑠,𝑡}(𝑥)−𝜉_{𝑝,𝑞−1,𝑠,𝑡}(𝑥)+𝜉_{𝑝−1,𝑞−1,𝑠,𝑡}(𝑥)^𝛼𝑝𝑞<1. (78)
By an argument similar to Theorem 4(iii), we obtain (77). Since 𝒲^′_𝜎(𝛼) and 𝒲_𝜎^′′(𝛼) are complete with the same metric, we have (ii).

Acknowledgement

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. (130-068-D1433). The authors, therefore, acknowledge with thanks DSR technical and financial support.