The generalized χ² sequence spaces over p- metric spaces defined by Musielak

Abstract

In this paper, we introduce generalized χ² sequence spaces over p- metric spaces defined by Musielak function f = (f _mn ) and study some topological properties.

MSC

40A05; 40C05; 40D05

Introduction

Throughout this paper, w, χ, and Λ denote the classes of all, gai, and analytic scalar valued single sequences, respectively. We write w² for the set of all complex sequences (x _mn ), where$m,n\in \mathbb{N}$, the set of positive integers. Then, w² is a linear space under the coordinatewise addition and scalar multiplication.

Some initial works on double sequence spaces is found in Bromwich[1]. Later on, they were investigated by Hardy[2], Moricz[3], Moricz and Rhoades[4], Basarir and Solankan[5], Tripathy[6], Turkmenoglu[7], and many others. We procure the following sets of double sequences:

$$\begin{array}{ll}{\mathcal{M}}_u(t)& :=\left\{\left(x_{\mathit{\text{mn}}}\right)\in w^2:{\mathit{\text{sup}}}_{m,n\in N}{\left|x_{\mathit{\text{mn}}}\right|}^{t_{\mathit{\text{mn}}}}<\infty \right\},\\ {\mathcal{C}}_p(t)& :=\left\{\left(x_{\mathit{\text{mn}}}\right)\in w^2:p-{\mathit{\text{lim}}}_{m,n\to \infty }{\left|x_{\mathit{\text{mn}}}-\right|}^{t_{\mathit{\text{mn}}}}\right.\\ =1\left.\text{for some}\in \mathbb{C}\right\},\\ {\mathcal{C}}_{0p}(t)& :=\left\{\left(x_{\mathit{\text{mn}}}\right)\in w^2:p-{\mathit{\text{lim}}}_{m,n\to \infty }{\left|x_{\mathit{\text{mn}}}\right|}^{t_{\mathit{\text{mn}}}}=1\right\},\\ {\mathcal{L}}_u(t)& :=\left\{\left(x_{\mathit{\text{mn}}}\right)\in w^2:{\sum }_{m=1}^{\infty }{\sum }_{n=1}^{\infty }{\left|x_{\mathit{\text{mn}}}\right|}^{t_{\mathit{\text{mn}}}}<\infty \right\},\\ {\mathcal{C}}_{\mathit{\text{bp}}}(t)& :={\mathcal{C}}_p(t)\bigcap {\mathcal{M}}_u(t)\text{and}{\mathcal{C}}_{0\mathit{\text{bp}}}(t)={\mathcal{C}}_{0p}(t)\bigcap {\mathcal{M}}_u(t),\end{array}$$

where t = (t _mn ) is the sequence of strictly positive reals t _mn for all$m,n\in \mathbb{N}$ and p − lim_m,n→∞ denotes the limit in the Pringsheim’s sense. In the case where t _mn  = 1 for all$m,n\in \mathbb{N},{\mathcal{M}}_u(t),{\mathcal{C}}_p(t),{\mathcal{C}}_{0p}(t),{\mathcal{L}}_u(t),{\mathcal{C}}_{\mathit{\text{bp}}}(t)$, and${\mathcal{C}}_{0\mathit{\text{bp}}}(t)$ reduce to the sets${\mathcal{M}}_u,{\mathcal{C}}_p,{\mathcal{C}}_{0p},{\mathcal{L}}_u,{\mathcal{C}}_{\mathit{\text{bp}}}$, and${\mathcal{C}}_{0\mathit{\text{bp}}}$, respectively. Now, we may summarize the knowledge given in some document related to the double sequence spaces. Gö khan and Colak[8, 9] have proved that${\mathcal{M}}_u(t)$ and${\mathcal{C}}_p(t),{\mathcal{C}}_{\mathit{\text{bp}}}(t)$ are complete paranormed spaces of double sequences and gave the α−, β−, γ− duals of the spaces${\mathcal{M}}_u(t)$ and${\mathcal{C}}_{\mathit{\text{bp}}}(t).$ Quite recently, in her PhD thesis, Zeltser[10] has essentially studied both the theory of topological double sequence spaces and the theory of summability of double sequences. Mursaleen and Edely[11], and Tripathy[6] have independently introduced the statistical convergence and Cauchy for double sequences and given the relation between statistical convergent and strongly Cesà ro summable double sequences. Altay and BaŞar[12] have defined the spaces$\mathcal{B}\mathcal{S},\mathcal{B}\mathcal{S}(t),{\mathcal{\text{CS}}}_p,{\mathcal{\text{CS}}}_{\mathit{\text{bp}}},{\mathcal{\text{CS}}}_r$, and$\mathcal{B}\mathcal{V}$ of double sequences consisting of all double series whose sequence of partial sums is in the spaces${\mathcal{M}}_u,{\mathcal{M}}_u(t),{\mathcal{C}}_p,{\mathcal{C}}_{\mathit{\text{bp}}},{\mathcal{C}}_r$, and${\mathcal{L}}_u$, respectively, and also examined some properties of those sequence spaces and determined the α− duals of the spaces$\mathcal{B}\mathcal{S},\mathcal{B}\mathcal{V},{\mathcal{\text{CS}}}_{\mathit{\text{bp}}}$, and the β(ϑ)− duals of the spaces${\mathcal{\text{CS}}}_{\mathit{\text{bp}}}$ and${\mathcal{\text{CS}}}_r$ of double series. Basar and Sever[13] have introduced the Banach space${\mathcal{L}}_q$ of double sequences corresponding to the well-known space ℓ _q of single sequences and examined some properties of the space${\mathcal{L}}_q$. Quite recently, Subramanian and Misra[14] have studied the space${\chi }_M^2\left(p,q,u\right)$ of double sequences and gave some inclusion relations.

The class of sequences which is strongly Cesà ro summable with respect to a modulus was introduced by Maddox[15] as an extension of the definition of strongly Cesà ro summable sequences. Connor[16] further extended this definition to a definition of strong A- summability with respect to a modulus, where A = (a_n,k) is a non-negative regular matrix, and established some connections between strong A- summability, strong A- summability with respect to a modulus, and A- statistical convergence. In[17], the notion of convergence of double sequences was presented by Pringsheim. Also, in[18, 19], and[20], the four-dimensional matrix transformation${(\mathit{\text{Ax}})}_{k,\ell }=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{a}_{k\ell }^{\mathit{\text{mn}}}{x}_{\mathit{\text{mn}}}$ was studied extensively by Hamilton.

We need the following inequality in the sequel of the paper. For a, b, ≥ 0 and 0 < p < 1, we have

$${(a+b)}^p\le a^p+b^p.$$

(1)

The double series$\sum _{m,n=1}^{\infty }{x}_{\mathit{\text{mn}}}$ is called convergent if and only if the double sequence (s _mn ) is convergent, where$s_{\mathit{\text{mn}}}=\sum _{i,j=1}^{m,n}{x}_{\mathit{\text{ij}}}(m,n\in \mathbb{N})$. A sequence x = (x _mn ) is said to be double analytic if sup _mn |x _mn |^1/m+n < ∞. The vector space of all double analytic sequences will be denoted by Λ². A sequence x = (x _mn ) is called double gai sequence if ((m + n)!|x _mn |)^1/m+n → 0 as m,n→∞. The double gai sequences will be denoted by χ². Let ϕ = {all finite sequences}.

Consider a double sequence x = (x _ij ). The (m,n)^th section x^[m,n] of the sequence is defined by$x^{[m,n]}=\sum _{i,j=0}^{m,n}{x}_{\mathit{\text{ij}}}{\Im }_{\mathit{\text{ij}}}$ for all$m,n\in \mathbb{N}$, where ℑ _ij denotes the double sequence whose only non-zero term is a$\frac{1}{\left(i+j\right)!}$ in the (i,j)th place for each$i,j\in \mathbb{N}$.

A Fréchet coordinate space (FK-space or a metric space) X is said to have an AK property if (ℑ _mn ) is a Schauder basis for X, or equivalently x^[m,n] → x. An FDK-space is a double sequence space endowed with a complete metrizable space, locally convex topology under which the coordinate mappings$x=(x_k)\to (x_{\mathit{\text{mn}}})(m,n\in \mathbb{N})$ are also continuous.

Let M and Φ be mutually complementary modulus functions. Then, we have

• (1) For all u,y ≥ 0,

  $$\mathit{\text{uy}}\le M(u)+\mathrm{\Phi }(y),(\text{Young’s inequality; see [21]}).$$

  (2)

• (2) For all u ≥ 0,

  $$u\eta (u)=M(u)+\mathrm{\Phi }\left(\eta (u)\right).$$

  (3)

• (3) For all u ≥ 0 and 0 < λ < 1,

  $$M\left(\lambda u\right)\le \lambda M(u).$$

  (4)

Lindenstrauss and Tzafriri[22] used the idea of Orlicz function to construct Orlicz sequence space

$${\ell }_M=\left\{x\in w:{\sum }_{k=1}^{\infty }M\left(\frac{\left|x_k\right|}{\rho }\right)<\infty ,\text{for some}\rho >0\right\}.$$

The space ℓ _M with the norm

$$∥x∥=\mathit{\text{inf}}\left\{\rho >0:{\sum }_{k=1}^{\infty }M\left(\frac{\left|x_k\right|}{\rho }\right)\le 1\right\}$$

becomes a Banach space which is called an Orlicz sequence space. For M(t) = t^p(1 ≤ p < ∞), the spaces ℓ _M coincide with the classical sequence space ℓ _p .

A sequence f = (f _mn ) of modulus function is called a Musielak-modulus function. A sequence g = (g _mn ) defined by

$$g_{\mathit{\text{mn}}}(v)=\mathit{\text{sup}}\left\{\left|v\right|u-\left(f_{\mathit{\text{mn}}}\right)(u):u\ge 0\right\},m,n=1,2,\cdots$$

is called the complementary function of a Musielak-modulus function f. For a given Musielak modulus function f, the Musielak-modulus sequence space t _f and its subspace h _f are defined, respectively, as follows:

$$t_f=\left\{x\in w^2:I_f{\left(\left|x_{\mathit{\text{mn}}}\right|\right)}^{1/m+n}\to 0\text{as}m,n\to \infty \right\}$$

and

$$h_f=\left\{x\in w^2:I_f{\left(\left|x_{\mathit{\text{mn}}}\right|\right)}^{1/m+n}\to 0\text{as}m,n\to \infty \right\},$$

where I _f is a convex modular defined by

$$I_f(x)={\sum }_{m=1}^{\infty }{\sum }_{n=1}^{\infty }{f}_{\mathit{\text{mn}}}{\left(\left|x_{\mathit{\text{mn}}}\right|\right)}^{1/m+n},x=\left(x_{\mathit{\text{mn}}}\right)\in t_f.$$

We consider that t _f is equipped with the Luxemburg metric

$$\begin{array}{ll}d\left(x,y\right)& ={\mathit{\text{sup}}}_{\mathit{\text{mn}}}\left\{\mathit{\text{inf}}\left({\sum }_{m=1}^{\infty }{\sum }_{n=1}^{\infty }{f}_{\mathit{\text{mn}}}\right.\right.\\ \left.\left.\times \left(\frac{{\left|x_{\mathit{\text{mn}}}\right|}^{1/m+n}}{\mathit{\text{mn}}}\right)\right)\le 1\right\}.\end{array}$$

If X is a sequence space, we give the following definitions:

• (1) X ^′ = the continuous dual of X;

• (2)
  $$X^{\alpha }=\left\{a=(a_{\mathit{\text{mn}}}):\sum _{m,n=1}^{\infty }\left|a_{\mathit{\text{mn}}}{x}_{\mathit{\text{mn}}}\right|<\infty ,\text{for each}x\in X\right\}$$

  ;

• (3)
  $$X^{\beta }=\left\{a=(a_{\mathit{\text{mn}}}):\sum _{m,n=1}^{\infty }{a}_{\mathit{\text{mn}}}{x}_{\mathit{\text{mn}}}\text{is convergent, for each}x\in X\right\}$$

  ;

• (4)
  $$X^{\gamma }=\left\{a=(a_{\mathit{\text{mn}}}):{\mathit{\text{sup}}}_{\mathit{\text{mn}}}\ge 1\left|\sum _{m,n=1}^{M,N}{a}_{\mathit{\text{mn}}}{x}_{\mathit{\text{mn}}}\right|<\infty ,\text{for each}x\in X\right\}$$

  ;

• (5) let X be an FK-space  ⊃ ϕ, then X ^f = {f(ℑ _mn ) : f ∈ X ^′};

• (6) X ^δ = {a = (a _mn ) : sup _mn  |a _mn x _mn |^1/m+n < ∞, for each x ∈ X},

where X^α, X^β, and X^γ are called α− (or Kö the-Toeplitz) dual of X, β− (or generalized Kö the-Toeplitz) dual of X, γ− dual of X, and δ− dual of X, respectively. X^α is defined by Kantham and Gupta[21]. It is clear that X^α ⊂ X^β and X^α ⊂ X^γ, but X^β ⊂ X^γ does not hold since the sequence of partial sums of a double convergent series needs not to be bounded.

The notion of difference sequence spaces (for single sequences) was introduced by Kizmaz[23] as follows:

$$Z\left(\mathrm{\Delta }\right)=\left\{x=\left(x_k\right)\in w:\left(\mathrm{\Delta }{x}_k\right)\in Z\right\}$$

for Z = c,c₀ and ℓ _∞ , where Δx _k  = x _k −x_k+1 for all$k\in \mathbb{N}$.

Here, c, c₀, and ℓ _∞ denote the classes of convergent, null, and bounded scalar valued single sequences, respectively. The difference sequence space bv _p of the classical space ℓ _p is introduced and studied in the case 1 ≤ p ≤ ∞ and in the case 0 < p < 1 by Altay and BaŞar in[12]. The spaces c(Δ), c₀(Δ), ℓ _∞ (Δ), and bv _p are Banach spaces normed by

$$\begin{array}{ll}∥x∥& =\left|x_1\right|+{\mathit{\text{sup}}}_{k\ge 1}\left|\mathrm{\Delta }{x}_k\right|\text{and}{∥x∥}_{{\mathit{\text{bv}}}_p}\\ ={\left({\sum }_{k=1}^{\infty }{\left|x_k\right|}^p\right)}^{1/p},\left(1\le p<\infty \right).\end{array}$$

Later on, the notion was further investigated by many others. We now introduce the following difference double sequence spaces defined by

$$Z\left(\mathrm{\Delta }\right)=\left\{x=\left(x_{\mathit{\text{mn}}}\right)\in w^2:\left(\mathrm{\Delta }{x}_{\mathit{\text{mn}}}\right)\in Z\right\},$$

where Z = Λ²,χ² and Δx _mn  = (x _mn  − x_mn+1) − (x_m+1n − x_m+1n+1) = x _mn  − x_mn+1 − x_m+1n + x_m+1n+1 for all$m,n\in \mathbb{N}$.

Definition and preliminaries

Let$n\in \mathbb{N}$ and X be a real vector space of dimension w, where n ≤ w. A real valued function$d_p(x_1,\dots ,x_n)=\parallel (d_1(x_1),\dots ,d_n(x_n)){\parallel }_p$ on X satisfying the following four conditions:

• (1)
  $$\parallel (d_1(x_1),\dots ,d_n(x_n)){\parallel }_p=0$$

  if and and only if $d_1(x_1),\dots ,d_n(x_n)$ are linearly dependent,

• (2)
  $$\parallel (d_1(x_1),\dots ,d_n(x_n)){\parallel }_p$$

  is invariant under permutation,

• (3)
  $$\parallel (\alpha d_1(x_1),\dots ,d_n(x_n)){\parallel }_p=\left|\alpha \right|\parallel (d_1(x_1),\dots ,d_n(x_n)){\parallel }_p,\alpha \in ℝ$$

• (4) d _p ((x ₁,y ₁),(x ₂,y ₂)⋯(x _n ,y _n )) = (d _X (x ₁,x ₂,⋯x _n )^p + d _Y (y ₁,y ₂,⋯y _n )^p)^1/p for 1 ≤ p < ∞; (or)

• (5) d((x ₁,y ₁),(x ₂,y ₂),⋯(x _n ,y _n )) := sup{d _X (x ₁,x ₂,⋯x _n ),d _Y (y ₁,y ₂,⋯y _n )}, for x ₁,x ₂,⋯x _n  ∈ X,y ₁,y ₂,⋯y _n  ∈ Y which is called the p product metric of the Cartesian product of n metric spaces is the p norm of the n-vector of the norms of the n subspaces.

A trivial example of the p product metric of the n metric space is the p norm space which is$X=\mathbb{R}$ equipped with the following Euclidean metric in the product space:

$$\begin{array}{l}\parallel (d_1(x_1),\dots ,d_n(x_n)){\parallel }_E=\mathit{\text{sup}}\left(|\mathit{\text{det}}(d_{\mathit{\text{mn}}}\left(x_{\mathit{\text{mn}}}\right))|\right)=\\ \mathit{\text{sup}}\left(\left|\begin{array}{cccc}{d}_{11}\left(x_{11}\right)& d_{12}\left(x_{12}\right)& \cdots & d_{1n}\left(x_{1n}\right)\\ d_{21}\left(x_{21}\right)& d_{22}\left(x_{22}\right)& \cdots & d_{2n}\left(x_{1n}\right)\\ ⋮\\ d_{n1}\left(x_{n1}\right)& d_{n2}\left(x_{n2}\right)& \cdots & d_{\mathit{\text{nn}}}\left(x_{\mathit{\text{nn}}}\right)\end{array}\right|\right),\end{array}$$

where$x_i=\left(x_{i1},\cdots x_{\mathit{\text{in}}}\right)\in {\mathbb{R}}^n$ for each i = 1,2,⋯n.

If every Cauchy sequence in X converges to some L ∈ X, then X is said to be complete with respect to the p- metric. Any complete p- metric space is said to be p- Banach metric space.

Let X be a linear metric space. A function$w:X\to \mathbb{R}$ is called paranorm if

• (1) w(x) ≥ 0 for all x ∈ X;

• (2) w(−x) = w(x) for all x ∈ X,

• (3) w(x + y) ≤ w(x) + w(y) for all x,y ∈ X;

• (4) If (σ _mn ) is a sequence of scalars with σ _mn  → σ as m,n → ∞, and (x _mn ) is a sequence of vectors with w(x _mn  − x) → 0 as m,n → ∞, then w(σ _mn x _mn  − σ x) → 0 as m,n → ∞.

A paranorm w for which w(x) = 0 implies x = 0 is called a total paranorm, and the pair (X,w) is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see[24], Theorem 10.4.2, p.183).

The notion of λ− double gai and double analytic sequences is as follows: Let$\lambda ={\left({\lambda }_{\mathit{\text{mn}}}\right)}_{m,n=0}^{\infty }$ be a strictly increasing sequence of positive real numbers tending to infinity, that is,

$$0<{\lambda }_0<{\lambda }_1<\cdots \text{and}{\lambda }_{\mathit{\text{mn}}}\to \infty \text{as}m,n\to \infty$$

and that a sequence x = (x _mn ) ∈ w² is λ− convergent to 0, called a the λ− limit of x, if μ _mn (x) → 0 as m,n → ∞, where

$$\begin{array}{ll}{\mu }_{\mathit{\text{mn}}}(x)=& \frac{1}{{\phi }_{\mathit{\text{rs}}}}{\sum }_{m\in \sigma ,\sigma \in P_{\mathit{\text{rs}}}}{\sum }_{n\in \sigma ,\sigma \in P_{\mathit{\text{rs}}}}\\ \times \left({\lambda }_{m,n}-{\lambda }_{m,n+1}-{\lambda }_{m+1,n}+{\lambda }_{m+1,n+1}\right){\left|x_{\mathit{\text{mn}}}\right|}^{1/m+n}.\end{array}$$

The sequence x = (x _mn ) ∈ w² is λ− double analytic if sup _uv  |μ _mn (x)| < ∞. If lim _mn x _mn  = 0 in the ordinary sense of convergence, then

$$\begin{array}{ll}{\mathit{\text{lim}}}_{\mathit{\text{mn}}}& \left(\frac{1}{{\phi }_{\mathit{\text{rs}}}}\sum _{m\in \sigma ,\sigma \in P_{\mathit{\text{rs}}}}\sum _{n\in \sigma ,\sigma \in P_{\mathit{\text{rs}}}}\left({\lambda }_{m,n}-{\lambda }_{m,n+1}-{\lambda }_{m+1,n}\right.\right.\\ \left.\left.+{\lambda }_{m+1,n+1}\right){\left(\left(m+n\right)!\left|x_{\mathit{\text{mn}}}-0\right|\right)}^{1/m+n}\right)=0.\end{array}$$

This implies that it yields lim _uv μ _mn (x) = 0, and hence, x = (x _mn ) ∈ w² is λ− convergent to 0. Let f = (f _mn ) be a Musielak-modulus function, (X,∥(d(x₁),d(x₂),⋯ ,d(x_n−1))∥ _p ) be a p-metric space, and q = (q _mn ) be double analytic sequence of strictly positive real numbers. By w²(p − X), we denote the space of all sequences as (X,∥(d(x₁),d(x₂),⋯ ,d(x_n−1))∥ _p ). The following inequality will be used throughout the paper. If 0 ≤ q _mn  ≤ supq _mn  = H,K = m a x(1,2^H−1), then

$${\left|a_{\mathit{\text{mn}}}+b_{\mathit{\text{mn}}}\right|}^{q_{\mathit{\text{mn}}}}\le K\left\{{\left|a_{\mathit{\text{mn}}}\right|}^{q_{\mathit{\text{mn}}}}+{\left|b_{\mathit{\text{mn}}}\right|}^{q_{\mathit{\text{mn}}}}\right\}$$

(5)

for all m,n and$a_{\mathit{\text{mn}}},b_{\mathit{\text{mn}}}\in \mathbb{C}$. Also,${\left|a\right|}^{q_{\mathit{\text{mn}}}}\le \max \left(1,{\left|a\right|}^H\right)$ for all$a\in \mathbb{C}$.

In the present paper, we define the following sequence spaces:

$$\begin{array}{l}\left[{\chi }_{f\mu }^{2q},{∥\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ ={lim}_{\mathit{\text{mn}}}\left\{\left[f_{\mathit{\text{mn}}}\left(∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\\ \left.{\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p\right)\right]}^{q_{\mathit{\text{mn}}}}=0\right\},\\ \left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ ={\mathit{\text{sup}}}_{\mathit{\text{mn}}}\left\{\left[f_{\mathit{\text{mn}}}\left(∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\\ \left.{\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p\right)\right]}^{q_{\mathit{\text{mn}}}}<\infty \right\}.\end{array}$$

If we take f _mn (x) = x, we get

$$\begin{array}{l}\left[{\chi }_{\mu }^{2q},{∥\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ ={lim}_{\mathit{\text{mn}}}\left\{\left[\left(∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\\ \left.{\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p\right)\right]}^{q_{\mathit{\text{mn}}}}=0\right\},\\ \left[{\mathrm{\Lambda }}_{\mu }^{2q},{∥\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ ={\mathit{\text{sup}}}_{\mathit{\text{mn}}}\left\{\left[\left(∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\\ \left.{\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p\right)\right]}^{q_{\mathit{\text{mn}}}}<\infty \right\}.\end{array}$$

If we take q = (q _mn ) = 1

$$\begin{array}{l}\left[{\chi }_{f\mu }^2,{∥\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ ={lim}_{\mathit{\text{mn}}}\left\{\left[f_{\mathit{\text{mn}}}\left(∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\\ \left.\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p\right)\right]=0\right\},\\ \left[{\mathrm{\Lambda }}_{f\mu }^2,{∥\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ ={\mathit{\text{sup}}}_{\mathit{\text{mn}}}\left\{\left[f_{\mathit{\text{mn}}}\left(∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\\ \left.\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p\right)\right]<\infty \right\}.\end{array}$$

In the present paper, we plan to study some topological properties and inclusion relation between the above defined sequence spaces,$\left[{\chi }_{f\mu }^{2q},{∥\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$ and$\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$, which we shall discuss in this paper.

Main results

Theorem 1

Let f = ( f _mn ) be a Musielak-modulus function and q = (q _mn ) be a double analytic sequence of strictly positive real numbers; the sequence spaces$\left[{\chi }_{f\mu }^{2q},{∥\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$and$\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$are linear spaces.

Proof

It is routine verification. Therefore, the proof is omitted. □

Theorem 2

Let f = ( f _mn ) be a Musielak-modulus function and q = (q _mn ) be a double analytic sequence of strictly positive real numbers; the sequence space$\left[{\chi }_{f\mu }^{2q},{∥\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$is a paranormed space with respect to the paranorm defined by

$$\begin{array}{ll}g(x)& =\mathit{\text{inf}}\left\{\left(\left[f_{\mathit{\text{mn}}}\left(∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\right.\\ \left.{\left.{\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\le 1\right\},\end{array}$$

where H = max(1,sup _mn q _mn  < ∞).

Proof

Clearly, g(x) ≥ 0 for$x=\left(x_{\mathit{\text{mn}}}\right)\in \left[{\chi }_{f\mu }^{2q},{∥\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{V_2}\right]$. Since f _mn (0) = 0, we get g(0) = 0.

Conversely, suppose that g(x) = 0, then

$$\begin{array}{ll}\mathit{\text{inf}}& \left\{\left(\left[f_{\mathit{\text{mn}}}\left(∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\right.\\ \left.{\left.{\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\right\}\le 1=0.\end{array}$$

Suppose that μ _mn (x) ≠ 0 for each$m,n\in \mathbb{N}$. Then,${∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\to \infty$. It follows that${\left({\left[f_{\mathit{\text{mn}}}\left({∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{V_2}\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\to \infty$ which is a contradiction. Therefore, μ _mn (x) = 0. Let

$$\begin{array}{c}{\left({\left[f_{\mathit{\text{mn}}}\left({∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\le 1\end{array}$$

and

$$\begin{array}{c}{\left({\left[f_{\mathit{\text{mn}}}\left({∥{\mu }_{\mathit{\text{mn}}}(y),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\le 1.\end{array}$$

Then, by using Minkowski’s inequality, we have

$$\begin{array}{l}{\left({\left[f_{\mathit{\text{mn}}}\left({∥{\mu }_{\mathit{\text{mn}}}\left(x+y\right),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\\ \le {\left({\left[f_{\mathit{\text{mn}}}\left({∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\\ +{\left({\left[f_{\mathit{\text{mn}}}\left({∥{\mu }_{\mathit{\text{mn}}}(y),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}.\end{array}$$

So, we have

$$\begin{array}{ll}g\left(x+y\right)& =\mathit{\text{inf}}\left\{\left(\left[f_{\mathit{\text{mn}}}\left(∥{\mu }_{\mathit{\text{mn}}}\left(x+y\right),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\right.\\ \left.{\left.{\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\le 1\right\}\\ \le \mathit{\text{inf}}\left\{\left(\left[f_{\mathit{\text{mn}}}\left(∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\right.\\ \left.{\left.{\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\le 1\right\}\\ +\mathit{\text{inf}}\left\{\left(\left[f_{\mathit{\text{mn}}}\left(∥{\mu }_{\mathit{\text{mn}}}(y),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\right.\\ \left.{\left.{\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\le 1\right\}.\end{array}$$

Therefore,

$$g\left(x+y\right)\le g(x)+g(y).$$

Finally, to prove that the scalar multiplication is continuous, let λ be any complex number. By definition,

$$\begin{array}{l}g\left(\lambda x\right)=\mathit{\text{inf}}\left\{\left(\left[f_{\mathit{\text{mn}}}\left(∥{\mu }_{\mathit{\text{mn}}}\left(\lambda x\right),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\right.\\ \left.{\left.{\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\le 1\right\}.\end{array}$$

Then,

$$\begin{array}{l}g\left(\lambda x\right)=\mathit{\text{inf}}\left\{({(\left|\lambda \right|t)}^{q_{\mathit{\text{mn}}}/H}:\left(\left[f_{\mathit{\text{mn}}}\left(∥{\mu }_{\mathit{\text{mn}}}\left(x\right),\left(d\left(x_1\right),d\left(x_2\right),\right.\right.\right.\right.\right.\\ \left.{\left.{\left.\left.{\left.\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\le 1\right\},\end{array}$$

where$t=\frac{1}{\left|\lambda \right|}$. Since${\left|\lambda \right|}^{q_{\mathit{\text{mn}}}}\le \max \left(1,{\left|\lambda \right|}^{\mathit{\text{sup}}{p}_{\mathit{\text{mn}}}}\right)$, we have

$$\begin{array}{ll}g\left(\lambda x\right)& \le \max \left(1,{\left|\lambda \right|}^{\mathit{\text{sup}}{p}_{\mathit{\text{mn}}}}\right)\\ \times \mathit{\text{inf}}\left\{t^{q_{\mathit{\text{mn}}}/H}:\left(\left[f_{\mathit{\text{mn}}}\left(∥{\mu }_{\mathit{\text{mn}}}\left(\lambda x\right),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\right.\\ \left.{\left.{\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\le 1\right\}.\end{array}$$

□

Theorem 3

The β− dual space of${\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\beta }=\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$.

Proof

First, we observe that

$$\begin{array}{l}{\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\beta }\\ \subset \left[{\mathrm{\Gamma }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}\left(x\right),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

Therefore,

$$\begin{array}{l}{\left[{\mathrm{\Gamma }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\beta }\\ \subset {\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\beta }.\end{array}$$

But

$$\begin{array}{l}{\left[{\mathrm{\Gamma }}_{f\mu }^{2q}\right]}^{\beta }\stackrel{\subset }{\ne }\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

Hence,

$$\begin{array}{l}\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ \subset {\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\beta }.\end{array}$$

(6)

Next, we show that

$$\begin{array}{l}{\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\beta }\\ \subset \left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

Let$y=\left(y_{\mathit{\text{mn}}}\right)\in {\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\beta }$. Consider$f(x)={\sum }_{m=1}^{\infty }{\sum }_{n=1}^{\infty }{x}_{\mathit{\text{mn}}}{y}_{\mathit{\text{mn}}}$ with

$$\begin{array}{l}x=\left(x_{\mathit{\text{mn}}}\right)\in \left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ x=\left[\left({\lambda }_{\mathit{\text{mn}}}-{\lambda }_{\mathit{\text{mn}}+1}\right)-\left({\lambda }_{m+1n}-{\lambda }_{m+1n+1}\right)\right]\\ =\left(\begin{array}{cccccc}0& 0& \dots 0& 0& \dots & 0\\ 0& 0& \dots 0& 0& \dots & 0\\ ⋮\\ 0& 0& \dots \frac{{\phi }_{\mathit{\text{rs}}}}{\mathrm{\Delta }{\lambda }_{\mathit{\text{mn}}}\left(m+n\right)!}& \frac{-{\phi }_{\mathit{\text{rs}}}}{\mathrm{\Delta }{\lambda }_{\mathit{\text{mn}}}\left(m+n\right)!}& \dots & 0\\ 0& 0& \dots 0& 0& \dots & 0\end{array}\right)\\ -\left(\begin{array}{cccccc}0& 0& \dots 0& 0& \dots & 0\\ 0& 0& \dots 0& 0& \dots & 0\\ ⋮\\ 0& 0& \dots \frac{{\phi }_{\mathit{\text{rs}}}}{\mathrm{\Delta }{\lambda }_{\mathit{\text{mn}}}\left(m+n\right)!}& \frac{-{\phi }_{\mathit{\text{rs}}}}{\mathrm{\Delta }{\lambda }_{\mathit{\text{mn}}}\left(m+n\right)!}& \dots & 0\\ 0& 0& \dots 0& 0& \dots & 0\end{array}\right)\end{array}$$

$$\begin{array}{l}\left[f_{\mathit{\text{mn}}}\left({∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]\\ =\left(\begin{array}{cccccc}0& 0& \dots 0& 0& \dots & 0\\ 0& 0& \dots 0& 0,& \dots & 0\\ ⋮\\ 0& 0& \dots f_{\mathit{\text{mn}}}\left(\frac{{\phi }_{\mathit{\text{rs}}}}{\mathrm{\Delta }{\lambda }_{\mathit{\text{mn}}}\left(m+n\right)!}\right)& f_{\mathit{\text{mn}}}\left(\frac{-{\phi }_{\mathit{\text{rs}}}}{\mathrm{\Delta }{\lambda }_{\mathit{\text{mn}}}\left(m+n\right)!}\right)& \dots & 0\\ 0& 0& \dots f_{\mathit{\text{mn}}}\left(\frac{-{\phi }_{\mathit{\text{rs}}}}{\mathrm{\Delta }{\lambda }_{\mathit{\text{mn}}}\left(m+n\right)!}\right)& f_{\mathit{\text{mn}}}\left(\frac{{\phi }_{\mathit{\text{rs}}}}{\mathrm{\Delta }{\lambda }_{\mathit{\text{mn}}}\left(m+n\right)!}\right)& \dots & 0\\ 0& 0& \dots 0& 0,& \dots & 0\end{array}\right).\end{array}$$

Hence, it converges to zero.

Therefore,

$$\begin{array}{l}\left[\left({\lambda }_{\mathit{\text{mn}}}-{\lambda }_{\mathit{\text{mn}}+1}\right)-\left({\lambda }_{m+1n}-{\lambda }_{m+1n+1}\right)\right]\\ \in \left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

Hence, d((λ _mn  − λ_mn+1) − (λ_m+1n − λ_m+1n+1),0) = 1. But

$$\begin{array}{ll}\left|y_{\mathit{\text{mn}}}\right|& \le ∥f∥d\left(\left({\lambda }_{\mathit{\text{mn}}}-{\lambda }_{\mathit{\text{mn}}+1}\right)-\left({\lambda }_{m+1n}-{\lambda }_{m+1n+1}\right),0\right)\\ \le ∥f∥·1<\infty \end{array}$$

for each m,n. Thus, (y _mn ) is a p- metric paranormed space of double analytic sequence and, hence, an p- metric double analytic sequence.

In other words.$y\in \left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$. But y = (y _mn ) is arbitrary in${\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\beta }$.Therefore,

$$\begin{array}{l}{\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\beta }\\ \subset \left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

(7)

From (6) and (7), we get

$$\begin{array}{l}{\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\beta }\\ =\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

□

Theorem 4

The dual space of$\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$is$\left[{\mathrm{\Lambda }}_{f\mu }^{2q},∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\right.\right.$$\left.{\left.\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$. In other words.${\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\ast }=\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$.

Proof

We recall that

$${\lambda }_{\mathit{\text{mn}}}=\left(\begin{array}{ccccc}0& 0& \dots 0& 0& \dots \\ 0& 0& \dots 0& 0& \dots \\ ⋮\\ 0& 0& \dots \frac{{\phi }_{\mathit{\text{rs}}}}{\mathrm{\Delta }{\lambda }_{\mathit{\text{mn}}}\left(m+n\right)!}& 0& \dots \\ 0& 0& \dots 0& 0& \dots \end{array}\right)$$

with$\frac{{\phi }_{\mathit{\text{rs}}}}{\mathrm{\Delta }{\lambda }_{\mathit{\text{mn}}}\left(m+n\right)!}$ in the (m,n)th position and zeros elsewhere,

$$\begin{array}{l}\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ =\left(\begin{array}{cccc}0.& .& .& 0\\ .\\ .\\ .\\ 0& f{\left(\frac{{\phi }_{\mathit{\text{rs}}}}{\mathrm{\Delta }{\lambda }_{\mathit{\text{mn}}}\left(m+n\right)!}\right)}^{1/m+n}& .& 0\\ {(m,n)}^{\text{th}}\\ 0& .& .& 0\end{array}\right)\end{array}$$

which is a p- metric of double gai sequence. Hence,

$$\begin{array}{l}x\in \left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\mathrm{.f}(x)\\ ={\sum }_{m,n=1}^{\infty }{x}_{\mathit{\text{mn}}}{y}_{\mathit{\text{mn}}}\end{array}$$

with$x\in \left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$ and$f\in {\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\ast }$, where${\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\ast }$ is the dual space of$\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$.

Take$x=(x_{\mathit{\text{mn}}})\in \left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$. Then,

$$\left|y_{\mathit{\text{mn}}}\right|\le ∥f∥d({\phi }_{\mathit{\text{rs}}},0)<\infty \forall m,n.$$

(8)

Thus, (y _mn ) is a p- metric of the double analytic sequence and an p- metric of double analytic sequence. In other words,$y\in \left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$. Therefore,

$$\begin{array}{l}{\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\ast }\\ =\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

This completes the proof. □

Theorem 5

(1) If the sequence (f _mn ) satisfies uniform Δ₂− condition, then

$$\begin{array}{l}{\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\alpha }\\ =\left[{\chi }_g^{2q\mu },{∥{\mu }_{\mathit{\text{uv}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

• (2) If the sequence (g _mn ) satisfies uniform Δ₂− condition, then

  $$\begin{array}{l}{\left[{\chi }_g^{2q\mu },{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\alpha }\\ =\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

Proof

Let the sequence ( f _mn ) satisfies uniform Δ₂− condition; we get

$$\begin{array}{l}\left[{\chi }_g^{2q\mu },{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ \subset {\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\alpha }.\end{array}$$

(9)

To prove the inclusion

$$\begin{array}{l}{\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\alpha }\\ \subset \left[{\chi }_g^{2q\mu },{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right],\end{array}$$

let$a\in {\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\alpha }$. Then, for all {x _mn } with$\left(x_{\mathit{\text{mn}}}\right)\in \left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$, we have

$$\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }\left|x_{\mathit{\text{mn}}}{a}_{\mathit{\text{mn}}}\right|<\infty .$$

(10)

Since the sequence ( f _mn ) satisfies the uniform Δ₂− condition and then

$$\left(y_{\mathit{\text{mn}}}\right)\in \left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right],$$

we get$\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }\left|\frac{{\phi }_{\mathit{\text{rs}}}{y}_{\mathit{\text{mn}}}{a}_{\mathit{\text{mn}}}}{\mathrm{\Delta }{\lambda }_{\mathit{\text{mn}}}\left(m+n\right)!}\right|<\infty .$ by (10). Thus,$\left({\phi }_{\mathit{\text{rs}}}{a}_{\mathit{\text{mn}}}\right)\in \left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]=\left[{\chi }_g^{2q\mu },{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$, and hence,$\left(a_{\mathit{\text{mn}}}\right)\in \left[{\chi }_g^{2q\mu },{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$. This gives that

$$\begin{array}{l}{\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\alpha }\\ \subset \left[{\chi }_g^{2q\mu },{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

(11)

We are granted with (9) and (11) that

$$\begin{array}{l}{\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\alpha }\\ =\left[{\chi }_g^{2q\mu },{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

• (3) Similarly, one can prove that

  $$\begin{array}{l}{\left[{\chi }_g^{2q\mu },{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\alpha }\\ \subset \left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}\left(x\right),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\end{array}$$

if the sequence (g _mn ) satisfies the uniform Δ₂− condition. □

Proposition 1

If 0 < q _mn  < p _mn  < ∞ for each m and m, then

$$\begin{array}{l}\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ \subseteq \left[{\mathrm{\Lambda }}_{f\mu }^{2p},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

Proof

Let$x=\left(x_{\mathit{\text{mn}}}\right)\in \left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$. We have

$${\mathit{\text{sup}}}_{\mathit{\text{mn}}}\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]<\infty .$$

This implies that

$$\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]<1$$

for sufficiently large value of m and n. Since f _mn s are non-decreasing, we get

$$\begin{array}{l}{\mathit{\text{sup}}}_{\mathit{\text{mn}}}\left[{\mathrm{\Lambda }}_{f\mu }^{2p},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ \le {\mathit{\text{sup}}}_{\mathit{\text{mn}}}\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

Thus,$x=\left(x_{\mathit{\text{mn}}}\right)\in \left[{\mathrm{\Lambda }}_{f\mu }^{2p},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$. □

Proposition 2

(1) If 0 < infq _mn  ≤ q _mn  < 1, then

$$\begin{array}{l}\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ \subset \left[{\mathrm{\Lambda }}_{f\mu }^2,{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

(2) If 1 ≤ q _mn  ≤ supq _mn  < ∞, then$\left[{\mathrm{\Lambda }}_{f\mu }^2,{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\subset \left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$.

Proof

Let$x=\left(x_{\mathit{\text{mn}}}\right)\in \left[{\mathrm{\Lambda }}_{f\mu }^2,{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$. Since 0 < infq _mn  ≤ 1, we have

$$\begin{array}{l}{\mathit{\text{sup}}}_{\mathit{\text{uv}}}\left[{\mathrm{\Lambda }}_{f\mu }^2,{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ \le \left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right],\end{array}$$

and hence

$$x=\left(x_{\mathit{\text{mn}}}\right)\in \left[{\mathrm{\Lambda }}_{f\mu }^2,{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].$$

• (3) Let q _mn for each (m,n) and sup _mn q _mn  < ∞.

Let$x=\left(x_{\mathit{\text{mn}}}\right)\in \left[{\mathrm{\Lambda }}_{f\mu }^2,{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$. Then, for each 0 < ε < 1, there exists a positive integer$\mathbb{N}$ such that

$${\mathit{\text{sup}}}_{\mathit{\text{uv}}}\left[{\mathrm{\Lambda }}_{f\mu }^2,{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\le \epsilon <1,$$

for all m,n ≥ N. This implies that

$$\begin{array}{l}{\mathit{\text{sup}}}_{\mathit{\text{mn}}}\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ \le {\mathit{\text{sup}}}_{\mathit{\text{mn}}}\left[{\mathrm{\Lambda }}_{f\mu }^2,{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

Thus,$x=\left(x_{\mathit{\text{mn}}}\right)\in \left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$. □

Proposition 3

Let $f^{\prime }=\left(f_{\mathit{\text{mn}}}^{\prime }\right)$ and $f^{\prime \prime }=\left(f_{\mathit{\text{mn}}}^{\prime \prime }\right)$ be sequences of Musielak functions; we have

$$\begin{array}{l}\left[{\mathrm{\Lambda }}_{f^{\prime }\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ \times \bigcap \left[{\mathrm{\Lambda }}_{f^{\prime \prime }\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ \times \subseteq \left[{\mathrm{\Lambda }}_{f^{\prime }+f^{\prime \prime }\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

Proof

The proof is easy, so we omit it. □

Proposition 4

For any sequence of Musielak functions f = (f _mn ) and q = (q _mn ) be double analytic sequence of strictly positive real numbers. Then,

$$\begin{array}{l}\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ \subset \left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

Proof

The proof is easy, so we omit it. □

Proposition 5

The sequence space $\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$ is solid.

Proof

Let$x=\left(x_{\mathit{\text{mn}}}\right)\in \left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$, i.e.,

$${\mathit{\text{sup}}}_{\mathit{\text{mn}}}\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]<\infty .$$

Let (α _mn ) be double sequence of scalars such that |α _mn | ≤ 1 for all m,n ∈ N × N. Then, we get

$$\begin{array}{l}{\mathit{\text{sup}}}_{\mathit{\text{mn}}}\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}\left(\alpha x\right),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ \le {\mathit{\text{sup}}}_{\mathit{\text{mn}}}\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

□

Proposition 6

The sequence space $\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$ is monotone.

Proof

The proof follows from Proposition 5. □

Proposition 7

If f = (f _mn ) is any Musielak function, then

$$\begin{array}{l}\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{{\phi }^{\ast }}\right]\\ \subset \left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{{\phi }^{\ast \ast }}\right]\end{array}$$

if and only if${\mathit{\text{sup}}}_{r,s\ge 1}\frac{{\phi }_{\mathit{\text{rs}}}^{\ast }}{{\phi }_{\mathit{\text{rs}}}^{\ast \ast }}<\infty$.

Proof

Let$x\in \left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{{\phi }^{\ast }}\right]$ and$N={\mathit{\text{sup}}}_{r,s\ge 1}\frac{{\phi }_{\mathit{\text{rs}}}^{\ast }}{{\phi }_{\mathit{\text{rs}}}^{\ast \ast }}<\infty$. Then, we get

$$\begin{array}{l}\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{{\phi }_{\mathit{\text{rs}}}^{\ast \ast }}\right]\\ =N\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{{\phi }_{\mathit{\text{rs}}}^{\ast }}\right]\\ =0.\end{array}$$

Thus,$x\in \left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{{\phi }^{\ast \ast }}\right]$. Conversely, suppose that

$$\begin{array}{l}\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{{\phi }^{\ast }}\right]\\ \subset \left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{{\phi }^{\ast \ast }}\right]\end{array}$$

and$x\in \left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{{\phi }^{\ast }}\right]$.

Then,$\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{{\phi }^{\ast }}\right]<\epsilon$for every ε > 0. Suppose that${\mathit{\text{sup}}}_{r,s\ge 1}\frac{{\phi }_{\mathit{\text{rs}}}^{\ast }}{{\phi }_{\mathit{\text{rs}}}^{\ast \ast }}=\infty$, then there exists a sequence of members (rs _jk ) such that${\mathit{\text{lim}}}_{j,k\to \infty }\frac{{\phi }_{\mathit{\text{jk}}}^{\ast }}{{\phi }_{\mathit{\text{jk}}}^{\ast \ast }}=\infty$. Hence, we have$\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{{\phi }_{\mathit{\text{rs}}}^{\ast }}\right]=\infty$. Therefore,$x\notin \left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{{\phi }^{\ast \ast }}\right]$, which is a contradiction. □

Proposition 8

If f = (f _mn ) is any Musielak function, then

$$\begin{array}{l}\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{{\phi }^{\ast }}\right]\\ =\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{{\phi }^{\ast \ast }}\right]\end{array}$$

if and only if${\mathit{\text{sup}}}_{r,s\ge 1}\frac{{\phi }_{\mathit{\text{rs}}}^{\ast }}{{\phi }_{\mathit{\text{rs}}}^{\ast \ast }}<\infty ,{\mathit{\text{sup}}}_{r,s\ge 1}\frac{{\phi }_{\mathit{\text{rs}}}^{\ast \ast }}{{\phi }_{\mathit{\text{rs}}}^{\ast }}>\infty$.

Proof

It is easy to prove, so we omit it. □

Proposition 9

The sequence space $\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$ is not solid.

Proof

The result follows from the following example. Consider

$$\begin{array}{l}x=\left(x_{\mathit{\text{mn}}}\right)=\left(\begin{array}{llll}1& 1& \dots & 1\\ 1& 1& \dots & 1\\ ⋮\\ 1& 1& \dots & 1\end{array}\right)\\ \in \left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

Let

$${\alpha }_{\mathit{\text{mn}}}=\left(\begin{array}{cccc}-1^{m+n}& -1^{m+n}& \dots & -1^{m+n}\\ -1^{m+n}& -1^{m+n}& \dots & -1^{m+n}\\ ⋮\\ -1^{m+n}& -1^{m+n}& \dots & -1^{m+n}\end{array}\right),$$

for all$m,n\in \mathbb{N}$. Then,${\alpha }_{\mathit{\text{mn}}}{x}_{\mathit{\text{mn}}}\notin \left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$. Hence,$\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$ is not solid. □

Proposition 10

The sequence space $\left[{\chi }_{f\mu }^{2q},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$ is not monotone.

Proof

The proof follows from Proposition 9. □

Generalized four-dimensional infinite matrix sequence spaces

Let$A=\left(a_{k\ell }^{\mathit{\text{mn}}}\right)$ be a four-dimensional infinite matrix of complex numbers. Then, we have$A(x)={(\mathit{\text{Ax}})}_{k\ell }=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{a}_{k\ell }^{\mathit{\text{mn}}}{x}_{\mathit{\text{mn}}}$ which converges for each k,ℓ.

In this section, we introduce the following sequence spaces:

$$\begin{array}{l}\left[{\chi }_{f\mu }^{2\mathit{\text{qA}}},{∥\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ ={\mathit{\text{lim}}}_{\mathit{\text{mn}}}\left\{\left[f_{\mathit{\text{mn}}}\left(∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\\ \left.{\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p\right)\right]}^{q_{\mathit{\text{mn}}}}=0\right\},\\ \left[{\mathrm{\Lambda }}_{f\mu }^{2\mathit{\text{qA}}},{∥\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ ={\mathit{\text{sup}}}_{\mathit{\text{mn}}}\left\{\left[f_{\mathit{\text{mn}}}\left(∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\\ \left.{\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p\right)\right]}^{q_{\mathit{\text{mn}}}}<\infty \right\}.\end{array}$$

If we take f _mn (x) = x, we get

$$\begin{array}{l}\left[{\chi }_{\mu }^{2\mathit{\text{qA}}},{∥\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ ={\mathit{\text{lim}}}_{\mathit{\text{mn}}}\left\{\left[\left(∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\\ \left.{\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p\right)\right]}^{q_{\mathit{\text{mn}}}}=0\right\},\\ \left[{\mathrm{\Lambda }}_{\mu }^{2\mathit{\text{qA}}},{∥\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ ={\mathit{\text{sup}}}_{\mathit{\text{mn}}}\left\{\left[\left(∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\\ \left.{\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p\right)\right]}^{q_{\mathit{\text{mn}}}}<\infty \right\}.\end{array}$$

If we take q = (q _mn ) = 1,

$$\begin{array}{l}\left[{\chi }_{f\mu }^{2A},{∥\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ ={\mathit{\text{lim}}}_{\mathit{\text{mn}}}\left\{\left[f_{\mathit{\text{mn}}}\left(∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\\ \left.\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p\right)\right]=0\right\},\\ \left[{\mathrm{\Lambda }}_{f\mu }^{2A},{∥\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ ={\mathit{\text{sup}}}_{\mathit{\text{mn}}}\left\{\left[f_{\mathit{\text{mn}}}\left(∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}\left(x\right),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\\ \left.\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p\right)\right]<\infty \right\}.\end{array}$$

Theorem 6

For a Musielak-modulus function, f = (f _mn ). Then, the sequence spaces$\left[{\chi }_{f\mu }^{2\mathit{\text{qA}}},{∥\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$and$\left[{\mathrm{\Lambda }}_{f\mu }^{2\mathit{\text{qA}}},{∥\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$are linear spaces over the set of complex numbers$\mathbb{C}$.

Proof

It is routine verification. Therefore, the proof is omitted. □

Theorem 7

For any Musielak-modulus function f = (f _mn ) and a double analytic sequence q = (q _mn ) of strictly positive real numbers, the space$\left[{\chi }_{f\mu }^{2\mathit{\text{qA}}},{∥\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$is a topological linear space paranormed by

$$\begin{array}{ll}g(x)=& \mathit{\text{inf}}\left\{\left(\left[f_{\mathit{\text{mn}}}\left(∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\right.\\ \left.{\left.{\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\le 1\right\},\end{array}$$

where H = max(1,sup _mn q _mn  < ∞).

Proof

Clearly, g(x) ≥ 0 for$x=\left(x_{\mathit{\text{mn}}}\right)\in \left[{\chi }_{f\mu }^{2\mathit{\text{qA}}},{∥\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{V_2}\right]$. Since f _mn (0) = 0, we get g(0) = 0. Conversely, suppose that g(x) = 0, then$\mathit{\text{inf}}\left\{{\left({\left[f_{\mathit{\text{mn}}}\left({∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\right\}\le 1=0$. Suppose that A _mn μ _mn (x) ≠ 0 for each$m,n\in \mathbb{N}$, then

$${∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\to \infty .$$

(12)

It follows that${\left({\left[f_{\mathit{\text{mn}}}\left({∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{V_2}\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\to \infty$ which is a contradiction. Therefore, A _mn μ _mn (x) = 0. Let${\left({\left[f_{\mathit{\text{mn}}}\left({∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\le 1$ and${\left({\left[f_{\mathit{\text{mn}}}\left({∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(y),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\le 1$.

Then, by using Minkowski’s inequality, we have

$$\begin{array}{l}\left(\left[f_{\mathit{\text{mn}}}\left(∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}\left(x+y\right),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\\ {\left.{\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\\ \le \left(\left[f_{\mathit{\text{mn}}}\left(∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\\ {\left.{\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\\ +\left(\left[f_{\mathit{\text{mn}}}\left(∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(y),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\\ {\left.{\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}.\end{array}$$

So, we have

$$\begin{array}{ll}g\left(x+y\right)& =\mathit{\text{inf}}\left\{\left(\left[f_{\mathit{\text{mn}}}\left(∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}\left(x+y\right),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\right.\\ \left.{\left.{\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\le 1\right\}\\ \le \mathit{\text{inf}}\left\{\left(\left[f_{\mathit{\text{mn}}}\left(∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\right.\\ \left.{\left.{\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\le 1\right\}\\ +\mathit{\text{inf}}\left\{\left(\left[f_{\mathit{\text{mn}}}\left(∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(y),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\right.\\ \left.{\left.{\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\le 1\right\}.\end{array}$$

Therefore,

$$g\left(x+y\right)\le g(x)+g(y).$$

Finally, to prove that the scalar multiplication is continuous, let λ be any complex number. By definition,

$$\begin{array}{ll}g\left(\lambda x\right)& =\mathit{\text{inf}}\left\{\left(\left[f_{\mathit{\text{mn}}}\left(∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}\left(\lambda x\right),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,\right.\right.\right.\right.\right.\\ \left.{\left.{\left.\left.{\left.d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\le 1\right\}.\end{array}$$

Then,

$$\begin{array}{ll}g\left(\lambda x\right)& =\mathit{\text{inf}}\left\{({(\left|\lambda \right|t)}^{q_{\mathit{\text{mn}}}/H}:\left(\left[f_{\mathit{\text{mn}}}\left(∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),\right.\right.\right.\right.\right.\\ \left.{\left.{\left.\left.{\left.d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\le 1\right\},\end{array}$$

where$t=\frac{1}{\left|\lambda \right|}$. Since${\left|\lambda \right|}^{q_{\mathit{\text{mn}}}}\le \max \left(1,{\left|\lambda \right|}^{{\mathit{\text{supp}}}_{\mathit{\text{mn}}}}\right)$, we have

$$\begin{array}{ll}g\left(\lambda x\right)& \le \max \left(1,{\left|\lambda \right|}^{{\mathit{\text{supp}}}_{\mathit{\text{mn}}}}\right)\mathit{\text{inf}}\left\{t^{q_{\mathit{\text{mn}}}/H}:\left(\left[f_{\mathit{\text{mn}}}\left(∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}\left(\lambda x\right),\right.\right.\right.\right.\\ \left.{\left.{\left.\left.{\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]}^{q_{\mathit{\text{mn}}}}\right)}^{1/H}\le 1\right\}.\end{array}$$

□

Theorem 8

The β− dual space of${\left[{\chi }_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\beta }=\left[{\mathrm{\Lambda }}_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$.

Proof

First, we observe that

$$\begin{array}{l}{\left[{\chi }_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\beta }\\ \subset \left[{\mathrm{\Gamma }}_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

Therefore,

$$\begin{array}{l}{\left[{\mathrm{\Gamma }}_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\beta }\\ \subset {\left[{\chi }_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\beta }.\end{array}$$

But

$$\begin{array}{ll}{\left[{\mathrm{\Gamma }}_{f\mu }^{2\mathit{\text{qA}}}\right]}^{\beta }\stackrel{\subset }{\ne }& \left[{\mathrm{\Lambda }}_{f\mu }^{2\mathit{\text{qA}}},∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),\right.\right.d\left(x_2\right),\cdots ,\\ \left.{\left.d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

Hence,

$$\begin{array}{l}\left[{\mathrm{\Lambda }}_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ \subset {\left[{\chi }_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\beta }.\end{array}$$

Next, we show that

$$\begin{array}{l}{\left[{\chi }_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\beta }\\ \subset \left[{\mathrm{\Lambda }}_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

Let$y=\left(y_{\mathit{\text{mn}}}\right)\in {\left[{\chi }_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\beta }$. Consider$f(x)=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{x}_{\mathit{\text{mn}}}{y}_{\mathit{\text{mn}}}$ with

$$\begin{array}{l}x=\left(x_{\mathit{\text{mn}}}\right)\in \left[{\chi }_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ x=\left[\left({\lambda }_{\mathit{\text{mn}}}-{\lambda }_{\mathit{\text{mn}}+1}\right)-\left({\lambda }_{m+1n}-{\lambda }_{m+1n+1}\right)\right]\\ =\left(\begin{array}{cccccc}0& 0& \cdots 0& 0& \cdots & 0\\ 0& 0& \cdots 0& 0& \cdots & 0\\ \dots \\ 0& 0& \cdots \frac{{\phi }_{\mathit{\text{rs}}}}{a_{k\ell }^{\mathit{\text{mn}}}\mathrm{\Delta }{\lambda }_{\mathit{\text{mn}}}\left(m+n\right)!}& \frac{-{\phi }_{\mathit{\text{rs}}}}{a_{k\ell }^{\mathit{\text{mn}}}\mathrm{\Delta }{\lambda }_{\mathit{\text{mn}}}\left(m+n\right)!}& \mathrm{...}& 0\\ 0& 0& \cdots 0& 0& \cdots & 0\end{array}\right)\\ -\left(\begin{array}{cccccc}0& 0& \cdots 0& 0& \cdots & 0\\ 0& 0& \cdots 0& 0& \cdots & 0\\ ⋮\\ 0& 0& \cdots \frac{{\phi }_{\mathit{\text{rs}}}}{a_{k\ell }^{\mathit{\text{mn}}}\mathrm{\Delta }{\lambda }_{\mathit{\text{mn}}}\left(m+n\right)!}& \frac{-{\phi }_{\mathit{\text{rs}}}}{a_{k\ell }^{\mathit{\text{mn}}}\mathrm{\Delta }{\lambda }_{\mathit{\text{mn}}}\left(m+n\right)!}& \mathrm{...}& 0\\ 0& 0& \cdots 0& 0& \cdots & 0\end{array}\right)\\ \left[f_{\mathit{\text{mn}}}\left({∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right)\right]\\ =\left(\begin{array}{cccccc}0& 0& \mathrm{...}0& 0& \mathrm{...}& 0\\ 0& 0& \mathrm{...}0& 0,& \mathrm{...}& 0\\ .\\ .\\ .\\ 0& 0& \mathrm{...}{f}_{\mathit{\text{mn}}}\left(\frac{{\phi }_{\mathit{\text{rs}}}}{a_{k\ell }^{\mathit{\text{mn}}}\mathrm{\Delta }{\lambda }_{\mathit{\text{mn}}}\left(m+n\right)!}\right)& f_{\mathit{\text{mn}}}\left(\frac{-{\phi }_{\mathit{\text{rs}}}}{a_{k\ell }^{\mathit{\text{mn}}}\mathrm{\Delta }{\lambda }_{\mathit{\text{mn}}}\left(m+n\right)!}\right)& \mathrm{...}& 0\\ 0& 0& \mathrm{...}{f}_{\mathit{\text{mn}}}\left(\frac{-{\phi }_{\mathit{\text{rs}}}}{a_{k\ell }^{\mathit{\text{mn}}}\mathrm{\Delta }{\lambda }_{\mathit{\text{mn}}}\left(m+n\right)!}\right)& f_{\mathit{\text{mn}}}\left(\frac{{\phi }_{\mathit{\text{rs}}}}{a_{k\ell }^{\mathit{\text{mn}}}\mathrm{\Delta }{\lambda }_{\mathit{\text{mn}}}\left(m+n\right)!}\right)& \mathrm{...}& 0\\ 0& 0& \mathrm{...}0& 0,& \mathrm{...}& 0\end{array}\right).\end{array}$$

Hence, converges to zero.

Therefore,

$$\begin{array}{l}\left[\left({\lambda }_{\mathit{\text{mn}}}-{\lambda }_{\mathit{\text{mn}}+1}\right)-\left({\lambda }_{m+1n}-{\lambda }_{m+1n+1}\right)\right]\\ \times \in \left[{\chi }_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

Hence,$d\left(a_{k\ell }^{\mathit{\text{mn}}}\left({\lambda }_{\mathit{\text{mn}}}-{\lambda }_{\mathit{\text{mn}}+1}\right)-\left({\lambda }_{m+1n}-{\lambda }_{m+1n+1}\right),0\right)=1$. However,$\left|y_{\mathit{\text{mn}}}\right|\le \parallel f\parallel d\left(a_{k\ell }^{\mathit{\text{mn}}}\left({\lambda }_{\mathit{\text{mn}}}-{\lambda }_{\mathit{\text{mn}}+1}\right)-\left({\lambda }_{m+1n}-{\lambda }_{m+1n+1}\right),0\right)\le \parallel f\parallel ·1<\infty$ for each m,n. Thus, (y _mn ) is a p- metric paranormed space of double analytic sequence and, hence, an p- metric double analytic sequence.

In other words,$y\in \left[{\mathrm{\Lambda }}_{f\mu }^{2\mathit{\text{qA}}},{∥{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$. However, y = (y _mn ) is arbitrary in${\left[{\chi }_{f\mu }^{2q},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\beta }$. Therefore,

$$\begin{array}{l}{\left[{\chi }_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\beta }\\ \subset \left[{\mathrm{\Lambda }}_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

(13)

From (12) and (13), we get

$$\begin{array}{l}{\left[{\chi }_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\beta }\\ =\left[{\mathrm{\Lambda }}_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

□

Theorem 9

The dual space of$\left[{\chi }_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$is$\left[{\mathrm{\Lambda }}_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$. In other words,

$$\begin{array}{l}{\left[{\chi }_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\ast }\\ =\left[{\mathrm{\Lambda }}_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

Proof

We recall that

$${\lambda }_{\mathit{\text{mn}}}=\left(\begin{array}{ccccc}0& 0& \cdots 0& 0& \cdots \\ 0& 0& \cdots 0& 0& \cdots \\ ⋮\\ 0& 0& \cdots \frac{{\phi }_{\mathit{\text{rs}}}}{a_{k\ell }^{\mathit{\text{mn}}}\mathrm{\Delta }{\lambda }_{\mathit{\text{mn}}}(m+n)!}& 0& \cdots \\ 0& 0& \cdots 0& 0& \cdots \end{array}\right)$$

with$\frac{{\phi }_{\mathit{\text{rs}}}}{a_{k\ell }^{\mathit{\text{mn}}}\mathrm{\Delta }{\lambda }_{\mathit{\text{mn}}}(m+n)!}$ in the (m,n)th position and zero elsewhere,

$$\begin{array}{l}\left[{\chi }_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\\ =\left(\begin{array}{cccc}0.& .& .& 0\\ ⋮\\ 0& f{\left(\frac{{\phi }_{\mathit{\text{rs}}}}{a_{k\ell }^{\mathit{\text{mn}}}\mathrm{\Delta }{\lambda }_{\mathit{\text{mn}}}(m+n)!}\right)}^{1/m+n}& .& 0\\ {(m,n)}^{\text{th}}\\ 0& .& .& 0\end{array}\right)\end{array}$$

which is a p- metric of double gai sequence. Hence,

$$\begin{array}{l}x\in \left[{\chi }_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\mathrm{.f}(x)\\ ={\sum }_{m,n=1}^{\infty }{x}_{\mathit{\text{mn}}}{y}_{\mathit{\text{mn}}}\end{array}$$

with

$$\begin{array}{l}x\in \left[{\chi }_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]\end{array}$$

and

$$\begin{array}{l}f\in {\left[{\chi }_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\ast },\end{array}$$

where

$${\left[{\chi }_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\ast }$$

is the dual space of$\left[{\chi }_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$.

Take$x=(x_{\mathit{\text{mn}}})\in \left[{\chi }_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$. Then,

$$\left|y_{\mathit{\text{mn}}}\right|\le ∥f∥d({\phi }_{\mathit{\text{rs}}},0)<\infty \forall m,n.$$

(14)

Thus, (y _mn ) is a p- metric of double analytic sequence and, hence, an p- metric of double analytic sequence. In other words,$y\in \left[{\mathrm{\Lambda }}_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]$. Therefore,

$$\begin{array}{l}{\left[{\chi }_{f\mu }^{2\mathit{\text{qA}}},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right]}^{\ast }\\ =\left[{\mathrm{\Lambda }}_{f\mu }^{2q},{∥A_{\mathit{\text{mn}}}{\mu }_{\mathit{\text{mn}}}(x),\left(d\left(x_1\right),d\left(x_2\right),\cdots ,d\left(x_{n-1}\right)\right)∥}_p^{\phi }\right].\end{array}$$

□