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

()

of all absolutely summable double sequences.

All considered double sequence spaces are supposed to contain

()

where

()

We denote the pointwise sums ∑_j,k e^jk, ∑_j e^jk (k ∈ ℕ), and ∑_k e^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) = lim x 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

()

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

()

while here the limit means bp-limit. In this case, we write [𝒱_σ]-lim x = 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:

()

where E = (e_jk) with e_jk = 1 for all j, k;

()

Remark 1. If [𝒲_σ]-lim x = ℓ, that is,

()

as m, n → ∞, uniformly in s, t; then

()

We remark that by using Abel’s transformation for single series

()

We get Abel’s transformation for double series

()

where

()

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]-lim x = ℓ if and only if

(L1)
𝒲_σ-lim x = ℓ;
(L2)
as u, v → ∞ (uniformly in s, t);
(L3)
as u, v → ∞ (uniformly in s, t);
(L4)
as u, v → ∞ (uniformly in s, t),

where

()

Proof. Suppose that [𝒲_σ]-lim x = ℓ. Thus, we have 𝒲_σ-lim x = ℓ, that is, (L1) holds. We see that conditions (L2) and (L3) follows from the Remark 1. Write

()

By our assumption, that is, [𝒲_σ]-lim x = 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,

()

→0 (u, v → ∞) uniformly in s, t.

Lemma 3. One has

()

Proof. Since

()

First, we solve the expression in the first bracket

()

Now, the expression in the second bracket

()

Substituting (18) and (19) in (17), we get

()

We know that

()

From (21), we have

()

Thus, (20) becomes

()

Also (22) can be written as

()

Similarly, we can write

()

Using (24) and (25) in (23), we get

()

This implies that

()

Theorem 4. One has the following inclusions and the limit is preserved in each case:

(i)
[𝒱_σ]⊂[𝒲_σ] ⊂ 𝒲_σ,
(ii)
if the conditions (L2) and (L3) of Lemma 2 hold,
(iii)
.

Proof. (i) Let x ∈ [𝒱_σ] with [𝒱_σ]-lim x = ℓ, say. Then,

()

This implies that

()

Also, we have

()

Hence, [𝒱_σ]⊂[𝒲_σ] ⊂ 𝒲_σ and

()

(ii) We have to show that

. If

, then we have

()

as m, n → ∞, uniformly in s, t; and

()

that is, 𝒲_σ-lim x = ℓ.

In order to prove that x ∈ [𝒲_σ], it is enough to show that condition (L4) of Lemma 2 holds. Now,

()

Replacing m and n by p and q, respectively, we have

()

By Lemma 3, we have

()

So that we have

()

By using Abel’s transformation for double series in the right hand side of above equation, we have

()

→0 as m, n → ∞, uniformly in s, t (by (32)). Hence, by Lemma 2, x ∈ [𝒲_σ].

(iii) Let , and we have to show that

()

where K is an absolute constant. Since

, there exist integers p₀, q₀ such that

()

Hence, it is left to show that for fixed p, q

()

From (40), we have

()

for every fixed p > p₀, q > q₀ and for all s, t. Since

()

Accordingly,

()

This implies that

()

Using (42) and (45), we have

()

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 denoted as “a”, let us write

()

and σ be monotonically increasing. For simplicity in notation, we denote

()

Again from the definition of ζ_mnst, it is easy to obtain

()

for all m, n ≥ 1 and ϕ_0,0,s,t(a) = a_st with d_j = σ^j−1(s) + 1,

, h_k = σ^k−1(t) + 1,

. Further, we calculate

()

Thus, we have

()

Hence, it follows from (46) that for each fixed m > p₀, n > q₀,

()

Hence, it follows from (52) that

()

where K is independent of u, v. By (49), we have

()

Also from (43) and (54), we have

()

4. Topological Results

Here, we extend the spaces [𝒲_σ], , to more general spaces, respectively, denoted by [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:

()

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

()

where M = max (1, sup _p,q α_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,

()

where K = max (1, 2^H−1) and H = sup _p,qα_pq. Since

()

therefore

()

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

()

Clearly, g(0) = 0, g(−x) = g(x). From (58) and Minkowski’s inequality, we have

()

Hence,

()

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,

and so

()

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,

()

Since

()

it follows that

()

In particular,

()

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

()

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:

is a complete paranormed space, paranormed by

()

which is defined on

. If α ≥ 1 then

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

()

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 . Then, for ϵ > 0 there exist integers M, N > 0 such that

()

If |λ | ≤ 1, then by (72) we have

()

Since for fixed M, N

()

as λ → 0, it follows from (73) and (74) that for fixed x = (x_jk), h(λx) → 0 as λ → 0. Also, since

implies that

()

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

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)
is a complete linear space paranormed by the function h defined in (70). In particular, if α is a constant, is a Banach space for α ≥ 1 and α-normed space for 0 < α < 1,
(ii)
is a closed subspace of .

Proof. (i) Proceeding along the same lines as in Theorem 6, except for the proof of continuity of scalar multiplication. If , we cannot assert (72) as in the case when . Since α is bounded away from zero, there exists δ > 0 such that α_mn ≥ δ for all m, n. Hence, |λ | ≤ 1 implies . Since

()

continuity of scalar multiplication follows.

(ii) For this, first we have to show that

()

Let

. Then, there exist integers M, N such that

()

By an argument similar to Theorem 4(iii), we obtain (77). Since and 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.

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Citing Literature

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

Abstract

1. Preliminaries, Background, and Notation

2. The Double Sequence Spaces

3. Inclusion Relations

4. Topological Results

Acknowledgement

References

Citing Literature

References

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