On a New I-Convergent Double-Sequence Space

Abstract

The sequence space BV_𝜎 was introduced and studied by Mursaleen (1983). In this article we introduce the sequence space ₂BV^𝐼_𝜎 and study some of its properties and inclusion relations.

1. Introduction and Preliminaries

Let ℕ, ℝ, and ℂ be the sets of all natural, real, and complex numbers, respectively. We write 𝜔=𝑥=𝑥_𝑘∶𝑥_𝑘∈ℂ, (1)showing the space of all real or complex sequences.

Definition 1. A double sequence of complex numbers is defined as a function 𝑥∶ℕ×ℕ→ℂ. We denote a double sequence as (𝑥_𝑖𝑗) where the two subscripts run through the sequence of natural numbers independent of each other [1]. A number 𝑎∈ℂ is called a double limit of a double sequence (𝑥_𝑖𝑗) if for every 𝜖>0 there exists some 𝑁=𝑁(𝜖)∈ℕ such that 𝑥_𝑖𝑗−𝑎<𝜖,∀𝑖,𝑗≥𝑁, (2)(see  [2]).

Let 𝑙_∞ and 𝑐 denote the Banach spaces of bounded and convergent sequences, respectively, with norm ‖𝑥‖_∞=sup_𝑘|𝑥_𝑘|. Let 𝑣 denote the space of sequences of bounded variation; that is, 𝑣=𝑥=𝑥_𝑘∶^∞_𝑘=0𝑥_𝑘−𝑥_𝑘−1<∞,𝑥₋₁=0, (3) where 𝑣is a Banach space normed by ‖𝑥‖=^∞_𝑘=0𝑥_𝑘−𝑥_𝑘−1, (4)(see  [3]).

Definition 2. Let 𝜎 be a mapping of the set of the positive integers into itself having no finite orbits. A continuous linear functional 𝜙 on 𝑙_∞ is said to be an invariant mean or 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,…};(iii)𝜙(𝑥_𝜎(𝑛))=𝜙(𝑥) for all 𝑥∈𝑙_∞.

In case 𝜎 is the translation mapping 𝑛→𝑛+1, a 𝜎-mean is often called a Banach limit (see [4]), and 𝑉_𝜎, the set of bounded sequences all of whose invariant means are equal, is the set of almost convergent sequences (see [5]).

If 𝑥=(𝑥_𝑘), then 𝑇𝑥=(𝑇𝑥_𝑘)=(𝑥_𝜎(𝑘)). Then it can be shown that 𝑉_𝜎=𝑥=𝑥_𝑘∶^∞_𝑚=1𝑡_𝑚,𝑘(𝑥),=𝐿uniformlyin𝑘,𝐿=𝜎−lim𝑥 (5) where 𝑚≥0, 𝑘>0. Consider 𝑡_𝑚,𝑘(𝑥)=𝑥_𝑘+𝑥_𝜎(𝑘)+⋯+𝑥_{𝜎𝑚(𝑘)}𝑚+1,𝑡_−1,𝑘=0, (6) where 𝜎^𝑚(𝑘) denote the 𝑚th iterate of 𝜎(𝑘) at 𝑘. The special case of (5) in which 𝜎(𝑛)=𝑛+1 was given by Lorentz [5, Theorem 1], and that the general result can be proved in a similar way. It is familiar that a Banach limit extends the limit functional on 𝑐.

Theorem 3. A 𝜎-mean extends the limit functional on 𝑐 in the sense that 𝜙(𝑥)=lim𝑥 for all 𝑥∈𝑐 if and only if 𝜎 has no finite orbits; that is to say, if and only if, for all 𝑘≥0, 𝑗≥1, (see [3]) 𝜎^𝑗(𝑘)≠𝑘. (7) Put 𝜙_𝑚,𝑘(𝑥)=𝑡_𝑚,𝑘(𝑥)−𝑡_{𝑚−1,𝑘}(𝑥), (8) assuming that 𝑡_−1,𝑘=0. A straight forward calculation shows (see [6]) that 𝜙_𝑚,𝑘(𝑥)=1𝑚(𝑚+1)^𝑚_𝑗∑₌₁𝐽𝑥_{𝜎𝑗(𝑘)}−𝑥_{𝜎𝑗−1(𝑘)}),𝑥(𝑚≥1_𝑘,(𝑚=0). (9)

For any sequence 𝑥, 𝑦, and scalar 𝜆, we have 𝜙_𝑚,𝑘𝑥+𝑦=𝜙_𝑚,𝑘(𝑥)+𝜙_𝑚,𝑘𝑦,𝜙_𝑚,𝑘(𝜆𝑥)=𝜆𝜙_𝑚,𝑘(𝑥). (10)

Definition 4. A sequence 𝑥∈𝑙_∞ is of 𝜎-bounded variation if and only if (i)∑^∞_𝑘=0|𝜙_𝑚,𝑘(𝑥)| converges uniformly in 𝑛;(ii)lim_𝑚→∞𝑡_𝑚,𝑘(𝑥), which must exist, should take the same value for all 𝑘.

We denote by BV_𝜎, the space of all sequences of 𝜎-bounded variation (see [7]): BV_𝜎=𝑥∈𝑙_∞∶_𝑚𝜙_𝑚,𝑘(𝑥)<∞,uniformlyin𝑛. (11)

Theorem 5. BV_𝜎 is a Banach space normed by ‖𝑥‖=sup_𝑘^∞_𝑚=0𝜙_𝑚,𝑘(𝑥), (12) (see [8]).

Subsequently, invariant means have been studied by Ahmad and Mursaleen [9], Mursaleen et al. [3, 6, 8, 10–14], Raimi [15], Schaefer [16], Savas and Rhoades [17], Vakeel et al. [18–20], and many others [21–23]. For the first time, I-convergence was studied by Kostyrko et al. [24]. Later on, it was studied by Šalát et al. [25, 26], Tripathy and Hazarika [27], Ebadullah et al. [18–20, 28], and Vakeel et al. [1, 29].

Definition 6 (see [30, 31]). Let 𝑋 be a nonempty set. Then, a family of sets 𝐼⊆2^𝑋 (2^𝑋 denoting the power set of 𝑋) is said to be an ideal in 𝑋 if(i)∅∈𝐼;(ii)𝐼 is additive; that is, 𝐴,𝐵∈𝐼⇒𝐴∪𝐵∈𝐼;(iii)𝐼 is hereditary that is, 𝐴∈𝐼, 𝐵⊆𝐴⇒𝐵∈𝐼;

An Ideal 𝐼⊆2^𝑋 is called nontrivial if 𝐼≠2^𝑋. A non-trivial ideal 𝐼⊆2^𝑋 is called admissible if {{𝑥}∶𝑥∈𝑋}⊆𝐼.

A non-trivial ideal 𝐼 is maximal if there cannot exist any non-trivial ideal 𝐽≠𝐼 containing 𝐼 as a subset.

For each ideal 𝐼, there is a filter £(𝐼) corresponding to 𝐼. That is, £(𝐼)=𝐾⊆𝑁∶𝐾^𝑐∈𝐼,where𝐾^𝑐=𝑁−𝐾. (13)

Definition 7 (see [24, 31, 32]). A double sequence (𝑥_𝑖𝑗)∈𝜔 is said to be 𝐼-convergent to a number 𝐿 if for every 𝜖>0, 𝑖,𝑗∈ℕ∶𝑥_𝑖𝑗−𝐿≥𝜖∈𝐼. (14) In this case, we write 𝐼−lim𝑥_𝑖𝑗=𝐿.

Definition 8 (see [2]). A double sequence (𝑥_𝑖𝑗)∈𝜔 is said to be 𝐼-null if 𝐿=0.  In this case, we write 𝐼−lim𝑥_𝑖𝑗=0. (15)

Definition 9. A double sequence (𝑥_𝑖𝑗)∈𝜔 is said to be 𝐼-cauchy if for every 𝜖>0 there exist numbers 𝑚=𝑚(𝜖), 𝑛=𝑛(𝜖) such that 𝑖,𝑗∈ℕ∶𝑥_𝑖𝑗−𝑥_𝑚𝑛≥𝜖∈𝐼. (16)

Definition 10. A double sequence (𝑥_𝑖𝑗)∈𝜔 is said to be 𝐼-bounded if there exists 𝑀>0 such that 𝑖,𝑗∈ℕ∶𝑥_𝑖𝑗>𝑀. (17)

Definition 11. A double-sequence space 𝐸 is said to be solid or normal if (𝑥_𝑖𝑗)∈𝐸 implies (𝛼_𝑖𝑗𝑥_𝑖𝑗)∈𝐸 for all sequence of scalars (𝛼_𝑖𝑗) with |𝛼_𝑖𝑗|<1 for all 𝑖,𝑗∈ℕ.

Definition 12 (see [24, 33]). A nonempty family of sets £(𝐼)⊆2^𝑋 is said to be filter on 𝑋 if and only if(i)Φ∉£(𝐼);(ii)for 𝐴,𝐵∈£(𝐼), we have 𝐴∩𝐵∈£(𝐼);(iii)for each 𝐴∈£(𝐼) and 𝐴⊆𝐵 implies 𝐵∈£(𝐼).

Definition 13. Let 𝑋 be a linear space. A function 𝑔∶𝑋→𝑅 is called a paranorm, if for all 𝑥,𝑦,𝑧∈𝑋,(i)𝑔(𝑥)=0 if 𝑥=𝜃;(ii)𝑔(−𝑥)=𝑔(𝑥); (iii)𝑔(𝑥+𝑦)≤𝑔(𝑥)+𝑔(𝑦); (iv)if (𝜆_𝑛) is a sequence of scalars with 𝜆_𝑛→𝜆(𝑛→∞) and 𝑥_𝑛,𝑎∈𝑋 with 𝑥_𝑛→𝑎(𝑛→∞), in the sense that 𝑔(𝑥_𝑛−𝑎)→0(𝑛→∞), in the sense that 𝑔(𝜆_𝑛𝑥_𝑛−𝜆𝑎)→0(𝑛→∞).

The concept of paranorm is closely related to that of linear metric spaces. It is a generalization of that of absolute value (see [34, 35]).

2. Main Results

In this paper, we introduce the sequence space₂BV^𝐼_𝜎∶=𝑥=𝑥_𝑖𝑗𝜙∈𝜔∶𝑖,𝑗∈ℕ∶_{𝑚𝑛𝑖𝑗}(𝑥).−𝐿≥𝜖∈𝐼forsome𝐿∈ℂ (18)

Theorem 14. ₂BV^𝐼_𝜎 is a linear space.

Proof. Let (𝑥_𝑖𝑗),(𝑦_𝑖𝑗)∈₂BV^𝐼_𝜎 and 𝛼, 𝛽 be two scalars in ℂ. Then for a given 𝜖>0, we have 𝑖,𝑗∈ℕ∶𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)−𝐿₁≥𝜖2∈𝐼,forsome𝐿₁𝜙∈ℂ,𝑖,𝑗∈ℕ∶_{𝑚𝑛,𝑖𝑗}𝑦−𝐿₂≥𝜖2∈𝐼,forsome𝐿₂∈ℂ. (19) Now let, 𝐴₁=𝑖,𝑗∈ℕ∶𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)−𝐿₁<𝜖2∈𝐼,forsome𝐿₁𝐴∈ℂ,₂=𝑖,𝑗∈ℕ∶𝜙_{𝑚𝑛,𝑖𝑗}𝑦−𝐿₂<𝜖2∈𝐼,forsome𝐿₂∈ℂ (20) be such that 𝐴^𝑐₁,𝐴^𝑐₂∈𝐼. Now consider 𝜙_{𝑚𝑛,𝑖𝑗}𝛼𝑥+𝛽𝑦−𝛼𝐿₁+𝛽𝐿₂=𝜙_{𝑚𝑛,𝑖𝑗}(𝛼𝑥)+𝜙_{𝑚𝑛,𝑖𝑗}𝛽𝑦−𝛼𝐿₁−𝛽𝐿₂=𝜙_{𝑚𝑛,𝑖𝑗}(𝛼𝑥)−𝛼𝐿₁+𝜙_{𝑚𝑛,𝑖𝑗}𝛽𝑦−𝛽𝐿₂≤𝜙_{𝑚𝑛,𝑖𝑗}(𝛼𝑥)−𝛼𝐿₁+𝜙_{𝑚𝑛,𝑖𝑗}𝛽𝑦−𝛽𝐿₂=|𝛼|𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)−𝐿₁+𝛽𝜙_{𝑚𝑛,𝑖𝑗}𝑦−𝐿₂≤|𝛼|𝜖2+𝛽𝜖2=|𝛼|+𝛽𝜖2≤𝜖^′say, (21) this implies that the sequence space 𝐴₃=𝑖,𝑗∈ℕ∶𝜙_{𝑚𝑛,𝑖𝑗}𝛼𝑥+𝛽𝑦−𝛼𝐿₁+𝛽𝐿₂<𝜖^′∈𝐼,forsome𝐿₁,𝐿₂∈ℂ. (22) Hence, (𝛼𝑥+𝛽𝑦)∈₂BV^𝐼_𝜎. Therefore, ₂BV^𝐼_𝜎 is a linear space.

Theorem 15. The space ₂BV^𝐼_𝜎 is a paranormed space, paranormed by 𝑔𝑥_𝑖𝑗=sup_𝑖𝑗𝜙_{𝑚𝑛,𝑖𝑗}(𝑥). (23)

Proof. For 𝑥=(𝑥_𝑖𝑗)=0, 𝑔(𝑥_𝑖𝑗)=0 is trivial.
For 𝑥=(𝑥_𝑖𝑗)≠0, 𝑔(𝑥_𝑖𝑗)≠0, we have(i)𝑔(𝑥)=sup_𝑖𝑗|𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)|≥ 0 for all 𝑥∈₂BV^𝐼_𝜎.(ii)𝑔(−𝑥)=sup_𝑖𝑗|𝜙_{𝑚𝑛,𝑖𝑗}(−𝑥)|=sup_𝑖𝑗|−𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)|=sup_𝑖𝑗|𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)|=𝑔(𝑥) for all 𝑥∈₂BV^𝐼_𝜎.(iii)𝑔(𝑥+𝑦)=sup_𝑖𝑗|𝜙_{𝑚𝑛,𝑖𝑗}(𝑥+𝑦)|≤sup_𝑖𝑗|𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)|+sup_𝑖𝑗|𝜙_{𝑚𝑛,𝑖𝑗}(𝑦)|=𝑔(𝑥)+𝑔(𝑦).(iv)Let (𝜆_𝑖𝑗) be a sequence of scalars with 𝜆_𝑖𝑗→𝜆(𝑖𝑗→∞) and (𝑥)∈₂BV^𝐼_𝜎 such that 𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)→𝐿𝑖𝑗→∞, (24)in the sense that 𝑔𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)−𝐿→0𝑖𝑗→∞. (25) Therefore, 𝑔𝜆_𝑖𝑗𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)𝜆−𝜆𝐿≤𝑔_𝑖𝑗𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)−𝑔(𝜆𝐿)=𝜆_𝑖𝑗𝑔𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)−𝜆𝑔(𝐿)→0as𝑖𝑗→∞. (26) Hence,   ₂BV^𝐼_𝜎 is a paranormed space.

Theorem 16. ₂BV^𝐼_𝜎 is a closed subspace of ₂𝑙^𝐼_∞.

Proof. Let (𝑥^(𝑝𝑞)_𝑖𝑗) be a cauchy sequence in ₂BV^𝐼_𝜎 such that 𝑥^(𝑝𝑞)→𝑥. We show that 𝑥∈₂BV^𝐼_𝜎. Since (𝑥^(𝑝𝑞)_𝑖𝑗)∈₂BV^𝐼_𝜎, then there exists 𝑎_𝑝𝑞 such that 𝑖,𝑗∈ℕ∶𝜙_{𝑚𝑛,𝑖𝑗}𝑥^(𝑝𝑞)−𝑎_𝑝𝑞≥𝜖∈𝐼. (27)
We need to show that (i)(𝑎_𝑝𝑞) converges to 𝑎.(ii)If 𝑈={𝑖,𝑗∈ℕ∶|𝑥_𝑖𝑗−𝑎|<𝜖}, then 𝑈^𝑐∈𝐼.
Since (𝑥^(𝑝𝑞)_𝑖𝑗) is a cauchy sequence in ₂BV^𝐼_𝜎, then for a given 𝜖>0, there exists 𝑘₀∈ℕ such that sup_𝑖𝑗𝜙_{𝑚𝑛,𝑖𝑗}𝑥^(𝑝𝑞)_𝑖𝑗−𝜙_{𝑚𝑛,𝑖𝑗}𝑥^(𝑟𝑠)_𝑖𝑗<𝜖3,∀𝑝,𝑞,𝑟,𝑠≥𝑘₀. (28)
For a given 𝜖>0, we have 𝐵_{𝑝𝑞𝑟𝑠}=𝑖,𝑗∈ℕ∶𝜙_{𝑚𝑛,𝑖𝑗}𝑥^(𝑝𝑞)_𝑖𝑗−𝜙_{𝑚𝑛,𝑖𝑗}𝑥^(𝑟𝑠)_𝑖𝑗<𝜖3,𝐵_𝑝𝑞=𝑖,𝑗∈ℕ∶𝜙_{𝑚𝑛,𝑖𝑗}𝑥^(𝑝𝑞)_𝑖𝑗−𝑎_𝑝𝑞<𝜖3,𝐵_𝑟𝑠=𝑖,𝑗∈ℕ∶𝜙_{𝑚𝑛,𝑖𝑗}𝑥^(𝑟𝑠)_𝑖𝑗−𝑎_𝑟𝑠<𝜖3. (29) Then 𝐵^𝑐_{𝑝𝑞𝑟𝑠},𝐵^𝑐_𝑝𝑞, and 𝐵^𝑐_𝑟𝑠∈𝐼. Let 𝐵^𝑐=𝐵^𝑐_{𝑝𝑞𝑟𝑠}∩𝐵^𝑐_𝑝𝑞∩𝐵^𝑐_𝑟𝑠,where 𝐵={𝑖,𝑗∈ℕ∶|𝑎_𝑝𝑞−𝑎_𝑟𝑠|<𝜖}. Then 𝐵^𝑐∈𝐼. We choose 𝑘₀∈𝐵^𝑐, then for each 𝑝,𝑞,𝑟,𝑠≥𝑘₀, we have 𝑖,𝑗∈ℕ∶𝑎_𝑝𝑞−𝑎_𝑟𝑠⊇𝜙<𝜖𝑖,𝑗∈ℕ∶_{𝑚𝑛,𝑖𝑗}𝑥^(𝑝𝑞)_𝑖𝑗−𝑎_𝑝𝑞<𝜖3∩𝑖,𝑗∈ℕ∶𝜙_{𝑚𝑛,𝑖𝑗}𝑥^(𝑝𝑞)_𝑖𝑗−𝜙_{𝑚𝑛,𝑖𝑗}𝑥^(𝑟𝑠)_𝑖𝑗<𝜖3∩𝑖,𝑗∈ℕ∶𝜙_{𝑚𝑛,𝑖𝑗}𝑥^(𝑟𝑠)_𝑖𝑗−𝑎_𝑟𝑠<𝜖3. (30) Then (𝑎_𝑝𝑞) is a cauchy sequence of scalars in ℂ, so there exists a scalar 𝑎∈ℂ such that (𝑎_𝑝𝑞)→𝑎, as 𝑝,𝑞→∞.
For the next step, let 0<𝛿<1 be given. Then, we show that if 𝑈=𝑖,𝑗∈ℕ∶𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)−𝑎<𝛿, (31) then 𝑈^𝑐∈𝐼. Since 𝜙_{𝑚𝑛,𝑖𝑗}(𝑥^(𝑝𝑞))→𝜙_{𝑚𝑛,𝑖𝑗}(𝑥), then there exists 𝑝₀,𝑞₀∈ℕ such that 𝑃=𝑖,𝑗∈ℕ∶𝜙_{𝑚𝑛,𝑖𝑗}𝑥^(𝑝0𝑞0)_𝑖𝑗−𝜙_{𝑚𝑛,𝑖𝑗}(𝑥<𝛿)3 (32) which implies that 𝑃^𝑐∈𝐼. The number 𝑝₀,  𝑞₀ can be so chosen that together with (32), we have 𝑄=𝑖,𝑗∈ℕ∶𝑎_𝑝0𝑞0<𝛿−𝑎3 (33) such that 𝑄^𝑐∈𝐼. Since {𝑖,𝑗∈ℕ∶|𝜙_{𝑚𝑛,𝑖𝑗}(𝑥^(𝑝0𝑞0)_𝑖𝑗)−𝑎_𝑝0𝑞0|≥𝛿}∈𝐼, then we have a subset 𝑆 of ℕ such that 𝑆^𝑐∈𝐼, where 𝑆=𝑖,𝑗∈ℕ∶𝜙_{𝑚𝑛,𝑖𝑗}𝑥^(𝑝0𝑞0)_𝑖𝑗−𝑎_𝑝0𝑞0<𝛿3. (34) Let 𝑈^𝑐=𝑃^𝑐∩𝑄^𝑐∩𝑆^𝑐, where 𝑈={𝑖,𝑗∈ℕ∶|𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)−𝑎|<𝛿}.
Therefore, for each 𝑖,𝑗∈𝑈^𝑐, we have 𝑖,𝑗∈ℕ∶𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)⊇𝜙−𝑎<𝛿𝑖,𝑗∈ℕ∶_{𝑚𝑛,𝑖𝑗}𝑥^(𝑝0𝑞0)_𝑖𝑗−𝜙_{𝑚𝑛,𝑖𝑗}(𝑥<𝛿)3∩𝑖,𝑗∈ℕ∶𝜙_{𝑚𝑛,𝑖𝑗}𝑥^(𝑝0𝑞0)_𝑖𝑗−𝑎_𝑝0𝑞0<𝛿3∩𝑖,𝑗∈ℕ∶𝑎_𝑝0𝑞0<𝛿−𝑎3. (35) Hence, the result ₂BV^𝐼_𝜎⊂₂𝑙^𝐼_∞ follows.

Theorem 17. The space   ₂BV^𝐼_𝜎 is nowhere dense subsets of ₂𝑙^𝐼_∞.

Proof. Proof of the result follows from the previous theorem.

Theorem 18. The space ₂BV^𝐼_𝜎 is solid and monotone.

Proof. Let (𝑥_𝑖𝑗)∈₂BV^𝐼_𝜎 and 𝛼_𝑖𝑗 be a sequence of scalars with |𝛼_𝑖𝑗|≤1 for all 𝑖,𝑗∈ℕ. Then, we have 𝛼_𝑖𝑗𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)≤𝛼_𝑖𝑗𝜙_{𝑚𝑛,𝑖𝑗}(𝑥≤𝜙_{𝑚𝑛,𝑖𝑗})(𝑥),∀𝑖,𝑗∈ℕ. (36) The space   ₂BV^𝐼_𝜎 is solid follows from the following inclusion relation: 𝑖,𝑗∈ℕ∶𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)≥𝜖⊇𝑖,𝑗∈ℕ∶𝛼_𝑖𝑗𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)≥𝜖. (37) Also a sequence space is solid implies monotone. Hence, the space  ₂BV^𝐼_𝜎is monotone.

Theorem 19. ₂𝑐^𝐼₀⊂₂BV^𝐼_𝜎⊂₂𝑙^𝐼_∞ and the inclusions are proper.

Proof. Let 𝑥=(𝑥_𝑖𝑗)∈₂𝑐^𝐼₀. Then, we have {𝑖,𝑗∈ℕ∶|𝑥_𝑖𝑗|≥𝜖}∈𝐼. Since  ₂𝑐₀⊂₂BV_𝜎, 𝑥=(𝑥_𝑖𝑗)∈₂BV^𝐼_𝜎 implies 𝑖,𝑗∈ℕ∶𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)≥𝜖∈𝐼. (38)
Now let, 𝐴₁=𝑖,𝑗∈ℕ∶𝑥_𝑖𝑗,𝐴<𝜖₂=𝑖,𝑗∈ℕ∶𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)<𝜖 (39) be such that 𝐴^𝑐₁,𝐴^𝑐₂∈𝐼. As 𝑙_∞={𝑥=(𝑥_𝑖𝑗)∶sup_𝑖𝑗|𝑥_𝑖𝑗|<∞}, taking supremum over 𝑖,𝑗 we get 𝐴^𝑐₁⊂𝐴^𝑐₂. Hence,  ₂𝑐^𝐼₀⊂₂BV^𝐼_𝜎⊂₂𝑙^𝐼_∞.
Next we show that the inclusion is proper
(i) First for ₂𝑐^𝐼₀⊂₂BV^𝐼_𝜎. Consider 𝑥∈₂BV^𝐼_𝜎, then by the definition ₂BV^𝐼_𝜎∶=𝑥=𝑥_𝑖𝑗𝜙∈𝜔∶𝑖,𝑗∈ℕ∶_{𝑚𝑛𝑖𝑗}(𝑥),−𝐿≥𝜖∈𝐼forsome𝐿∈ℂ (40) we have 𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)=𝑡_{𝑚𝑛,𝑖𝑗}(𝑥)−𝑡_{(𝑚−1)(𝑛−1),𝑖𝑗}(𝑥), (41) where 𝑡_{𝑚𝑛,𝑖𝑗}(𝑥)=𝑥_𝑖𝑗+𝑥_{𝜎(𝑖𝑗)}+𝑥_{𝜎2(𝑖𝑗)}+⋯+𝑥_{𝜎𝑚𝑛(𝑖𝑗)}.𝑚𝑛 (42) Therefore, 𝑡_{𝑚𝑛,𝑖𝑗}(𝑥)−𝑡_{(𝑚−1)(𝑛−1),𝑖𝑗}(𝑥)=𝑥_𝑖𝑗+𝑥_{𝜎(𝑖𝑗)}+𝑥_{𝜎2(𝑖𝑗)}+⋯+𝑥_{𝜎𝑚𝑛(𝑖𝑗)}−𝑥𝑚𝑛_𝑖𝑗+𝑥_{𝜎(𝑖𝑗)}+𝑥_{𝜎2(𝑖𝑗)}+⋯+𝑥_{𝜎(𝑚−1)(𝑛−1)(𝑖𝑗)}()=()𝑥𝑚−1)(𝑛−1𝑚−1)(𝑛−1_𝑖𝑗+𝑥_{𝜎(𝑖𝑗)}+⋯+𝑥_{𝜎𝑚𝑛(𝑖𝑗)}()−𝑥𝑚𝑛𝑚−1)(𝑛−1𝑚𝑛_𝑖𝑗+𝑥_{𝜎(𝑖𝑗)}+⋯+𝑥_{𝜎(𝑚−1)(𝑛−1)(𝑖𝑗)}().𝑚𝑛𝑚−1)(𝑛−1 (43) On solving, we get 𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)=𝑚𝑛𝑥_{𝜎𝑚𝑛(𝑖𝑗)}()+()𝑥𝑚𝑛𝑚−1)(𝑛−11−𝑚−𝑛_𝑖𝑗+𝑥_{𝜎(𝑖𝑗)}+𝑥_{𝜎2(𝑖𝑗)}+⋯+𝑥_{𝜎𝑚𝑛(𝑖𝑗)}().𝑚𝑛𝑚−1)(𝑛−1 (44) As 𝜎 is a translation map, that is, 𝜎(𝑛)=𝑛+1, we have 𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)=𝑚𝑛𝑥_{(𝑖+𝑚)(𝑗+𝑛)}()+()𝑥𝑚𝑛𝑚−1)(𝑛−11−𝑚−𝑛_𝑖𝑗+𝑥_{(𝑖+1)(𝑗+1)}+⋯+𝑥_{(𝑖+𝑚)(𝑗+𝑛)}().𝑚𝑛𝑚−1)(𝑛−1 (45) Taking lim𝑖,𝑗→∞, we have lim_{(𝑖,𝑗)→∞}𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)=lim_{(𝑖,𝑗)→∞}𝑚𝑛𝑥_{(𝑖+𝑚)(𝑗+𝑛)}+()×𝑥1−𝑚−𝑛_𝑖𝑗+𝑥_{(𝑖+1)(𝑗+1)}+⋯+𝑥_{(𝑖+𝑚)(𝑗+𝑛)}×(𝑚𝑛(𝑚−1)(𝑛−1))⁻¹,𝐿(𝑚𝑛(𝑚−1)(𝑛−1))=lim_{(𝑖,𝑗)→∞}𝑚𝑛𝑥_{(𝑖+𝑚)(𝑗+𝑛)}+()×𝑥1−𝑚−𝑛_𝑖𝑗+𝑥_{(𝑖+1)(𝑗+1)}+⋯+𝑥_{(𝑖+𝑚)(𝑗+𝑛)}. (46) Since 𝑚,𝑛,𝐿≠0, therefore lim_{𝑖,𝑗→∞}𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)≠0 which implies that 𝑥∉(₂𝑐^𝐼₀). Hence, we get that the inclusion is proper.
(ii) Second for   ₂BV^𝐼_𝜎⊂₂𝑙^𝐼_∞.
The result of this part follows from the proof of Theorem 18.

Theorem 20. ₂𝑐^𝐼⊂₂BV^𝐼_𝜎⊂₂𝑙^𝐼_∞ and the inclusions are proper.

Proof. Let 𝑥=(𝑥_𝑖𝑗)∈₂𝑐^𝐼. Then, we have 𝑖,𝑗∈ℕ∶𝑥_𝑖𝑗−𝐿≥𝜖∈𝐼. (47) Since ₂𝑐⊂₂BV_𝜎⊂₂𝑙_∞, which implies 𝑥=(𝑥_𝑖𝑗)∈₂BV^𝐼_𝜎 implies 𝑖,𝑗∈ℕ∶𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)−𝐿≥𝜖∈𝐼. (48)
Now let, 𝐵₁=𝑖,𝑗∈ℕ∶𝜙_𝑖,𝑗,𝐵−𝐿<𝜖₂=𝑖,𝑗∈ℕ∶𝜙_{𝑚𝑛,𝑖𝑗}(𝑥)−𝐿<𝜖 (49) be such that 𝐵^𝑐₁,𝐵^𝑐₂∈𝐼. As ₂𝑙_∞=𝑥=𝑥_𝑖𝑗∶sup_𝑖𝑗𝑥_𝑖𝑗<∞, (50) taking limsup over 𝑖,𝑗, we get 𝐵^𝑐₁⊂𝐵^𝑐₂. Hence, ₂𝑐^𝐼⊂₂BV^𝐼_𝜎⊂₂𝑙^𝐼_∞.
Next, we show that the inclusion is proper
(i) First for ₂𝑐^𝐼⊂₂BV^𝐼_𝜎. We show that ₂𝑐^𝐼⊊₂BV^𝐼_𝜎.
Let 𝑥=(𝑥_𝑖𝑗)∈₂BV^𝐼_𝜎, then by the definition ₂BV^𝐼_𝜎∶=𝑥=𝑥_𝑖𝑗𝜙∈𝜔∶𝑖,𝑗∈ℕ∶_{𝑚𝑛𝑖𝑗}(𝑥).−𝐿≥𝜖∈𝐼forsome𝐿∈ℂ (51) We have, |𝜙_{𝑚𝑛𝑖𝑗}(𝑥)−𝐿|≥𝜖. We say that the 𝐼−lim(𝜙_{𝑚𝑛𝑖𝑗}(𝑥))=𝐿.
Now considering the case when |𝜙_{𝑚𝑛𝑖𝑗}(𝑥)−𝐿|<𝜖, then 𝑡_{𝑚𝑛,𝑖𝑗}(𝑥)−𝑡_{(𝑚−1)(𝑛−1),𝑖𝑗}(𝑥)−𝐿<𝜖, (52) when 𝑚,𝑛=0, then we have 𝜙_{𝑚𝑛𝑖𝑗}(𝑥)=𝑡_𝑖𝑗(𝑥)=𝑥_𝑖𝑗. Therefore we get, 𝑥_𝑖𝑗−𝐿<𝜖∀𝑖,𝑗∈ℕ. (53) Hence, 𝑥∉₂𝑐^𝐼={𝑖,𝑗∈ℕ∶|𝑥_𝑖𝑗−𝐿|≥𝜖}∈𝐼. Hence, the inclusion is proper.
(ii) Second for ₂BV^𝐼_𝜎⊂₂𝑙^𝐼_∞.
The result follows from the proof of Theorem 18.

Acknowledgment

The authors would like to record their gratitude to the reviewer for his careful reading and making some useful corrections which improved the presentation of the paper.