Double sequence spaces over n-normed spaces defined by a sequence of Orlicz functions

Abstract

In the present paper we introduce double sequence space $m^2(\mathcal{M},A,\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )$ defined by a sequence of Orlicz functions over n-normed space. We examine some of its topological properties and establish some inclusion relations.

MSC:40A05, 46A45.

1 Introduction and preliminaries

The initial works on double sequences is found in Bromwich [1]. Later on, it was studied by Hardy [2], Moricz [3], Moricz and Rhoades [4], Başarır and Sonalcan [5] and many others. Hardy [2] introduced the notion of regular convergence for double sequences. Quite recently, Zeltser [6] in her PhD thesis has essentially studied both the theory of topological double sequence spaces and the theory of summability of double sequences. Mursaleen and Edely [7] have recently introduced the statistical convergence which was further studied in locally solid Riesz spaces [8]. Nextly, Mursaleen [9] and Mursaleen and Savas [10] have defined the almost regularity and almost strong regularity of matrices for double sequences and applied these matrices to establish core theorems and introduced the M-core for double sequences and determined those four dimensional matrices transforming every bounded double sequences $x=(x_{k,l})$ into one whose core is a subset of the M-core of x. More recently, Altay and Başar [11] have defined the spaces $\mathcal{B}\mathcal{S}$, $\mathcal{B}\mathcal{S}(t)$, $\mathcal{C}{\mathcal{S}}_p$, $\mathcal{C}{\mathcal{S}}_{bp}$, $\mathcal{C}{\mathcal{S}}_r$ and $\mathcal{B}\mathcal{V}$ of double sequences consisting of all double series whose sequence of partial sums are in the spaces ${\mathcal{M}}_u$, ${\mathcal{M}}_u(t)$, ${\mathcal{C}}_p$, ${\mathcal{C}}_{bp}$, ${\mathcal{C}}_r$ and ${\mathcal{L}}_u$, respectively and also examined some properties of these sequence spaces and determined the α-duals of the spaces $\mathcal{B}\mathcal{S}$, $\mathcal{B}\mathcal{V}$, $\mathcal{C}{\mathcal{S}}_{bp}$ and the $\beta (v)$-duals of the spaces $\mathcal{C}{\mathcal{S}}_{bp}$ and $\mathcal{C}{\mathcal{S}}_r$ of double series. Recently Başar and Sever [12] have introduced the Banach space ${\mathcal{L}}_q$ of double sequences corresponding to the well known space ${\ell }_q$ of single sequences and examined some properties of the space ${\mathcal{L}}_q$. Now, recently Raj and Sharma [13] have introduced entire double sequence spaces. By the convergence of a double sequence we mean the convergence in the Pringsheim sense i.e. a double sequence $x=(x_{k,l})$ has Pringsheim limit L (denoted by $P\text{-}limx=L$) provided that given $ϵ>0$ there exists $n\in N$ such that $|x_{k,l}-L|<ϵ$ whenever $k,l>n$, see [14]. The double sequence $x=(x_{k,l})$ is bounded if there exists a positive number M such that $|x_{k,l}|<M$ for all k and l.

Throughout this paper, ℕ and ℂ denote the set of positive integers and complex numbers, respectively. A complex double sequence is a function x from $N\times \mathbb{N}$ into ℂ and briefly denoted by $\{x_{k,l}\}$. If for all $ϵ>0$, there is $n_{ϵ}\in \mathbb{N}$ such that $|x_{k,l}-a|<ϵ$ where $k>n_{ϵ}$ and $l>n_{ϵ}$, then a double sequence $\{x_{k,l}\}$ is said to be convergent to $a\in \mathbb{C}$. A real double sequence $\{x_{k,l}\}$ is non-decreasing, if $x_{k,l}\le x_{p,q}$ for $(k,l)<(p,q)$. A double series is infinite sum ${\sum }_{k,l=1}^{\mathrm{\infty }}{x}_{k,l}$ and its convergence implies the convergence of partial sums sequence $\{S_{n,m}\}$, where $S_{n,m}={\sum }_{k=1}^m{\sum }_{l=1}^n{x}_{k,l}$ (see [15]). For recent development on double sequences, we refer to [16–20] and [21–23].

A double sequence space E is said to be solid if $\{x_{k,l}{y}_{k,l}\}\in E$ for all double sequences $\{y_{k,l}\}$ of scalars such that $|y_{k,l}|<1$ for all $k,l\in N$ whenever $\{x_{k,l}\}\in E$.

Let $x=\{x_{k,l}\}$ be a double sequence. A set $S(x)$ is defined by

$$S(x)=\{\{X_{{\pi }_1(k),{\pi }_2(k)}\}:{\pi }_1\text{ and }{\pi }_2\text{ are permutation of }\mathbb{N}\}.$$

If $S(x)\subseteq E$ for all $x\in E$, then E is said to be symmetric. Now let ${\mathcal{P}}_s$ be a family of subsets σ having at most elements s in ℕ. Also ${\mathcal{P}}_{s,t}$ denotes the class of subsets $\sigma ={\sigma }_1\times {\sigma }_2$ in $\mathbb{N}\times \mathbb{N}$ such that the element numbers of ${\sigma }_1$ and ${\sigma }_2$ are at most s and t, respectively. Besides $\{{\varphi }_{k,l}\}$ is taken as a non-decreasing double sequence of the positive real numbers such that

$$\begin{array}{c}k{\varphi }_{k+1,l}\le (k+1){\varphi }_{k,l},\hfill \\ l{\varphi }_{k,l+1}\le (l+1){\varphi }_{k,l}.\hfill \end{array}$$

An Orlicz function $M:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is a continuous, non-decreasing, and convex function such that $M(0)=0$, $M(x)>0$ for $x>0$ and $M(x)\to \mathrm{\infty }$ as $x\to \mathrm{\infty }$.

Lindenstrauss and Tzafriri [24] used the idea of Orlicz function to define the following sequence space:

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

which is called an Orlicz sequence space. Also ${\ell }_M$ is a Banach space with the norm

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

Also, it was shown that every Orlicz sequence space ${\ell }_M$ contains a subspace isomorphic to ${\ell }_p$ ($p\ge 1$). The ${\mathrm{\Delta }}_2$-condition is equivalent to $M(Lx)\le LM(x)$, for all L with $0<L<1$. An Orlicz function M can always be represented in the following integral form:

$$M(x)={\int }_0^x\eta (t)dt,$$

where η, known as the kernel of M, is right differentiable for $t\ge 0$, $\eta (0)=0$, $\eta (t)>0$, η is non-decreasing and $\eta (t)\to \mathrm{\infty }$ as $t\to \mathrm{\infty }$.

For further reading on Orlicz spaces, we refer to [25–29].

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

• (1) $p(x)\ge 0$ for all $x\in X$,

• (2) $p(-x)=p(x)$ for all $x\in X$,

• (3) $p(x+y)\le p(x)+p(y)$ for all $x,y\in X$,

• (4) if $({\lambda }_n)$ is a sequence of scalars with ${\lambda }_n\to \lambda$ as $n\to \mathrm{\infty }$ and $(x_n)$ is a sequence of vectors with $p(x_n-x)\to 0$ as $n\to \mathrm{\infty }$, then $p({\lambda }_n{x}_n-\lambda x)\to 0$ as $n\to \mathrm{\infty }$.

A paranorm p for which $p(x)=0$ implies $x=0$ is called a total paranorm and the pair $(X,p)$ 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 [30], Theorem 10.4.2, p.183).

The concept of 2-normed spaces was initially developed by Gähler [31] in the mid-1960s, while that of n-normed spaces one can see in Misiak [32]. Since then, many others have studied this concept and obtained various results; see Gunawan [33, 34] and Gunawan and Mashadi [35] and references therein. Let $n\in \mathbb{N}$ and X be a linear space over the field , where is the field of real or complex numbers of dimension d, where $d\ge n\ge 2$. A real valued function $\parallel \cdot ,\dots ,\cdot \parallel$ on $X^n$ satisfying the following four conditions:

• (1) $\parallel x_1,x_2,\dots ,x_n\parallel =0$ if and only if $x_1,x_2,\dots ,x_n$ are linearly dependent in X;

• (2) $\parallel x_1,x_2,\dots ,x_n\parallel$ is invariant under permutation;

• (3) $\parallel \alpha x_1,x_2,\dots ,x_n\parallel =|\alpha |\parallel x_1,x_2,\dots ,x_n\parallel$ for any $\alpha \in \mathbb{K}$, and

• (4) $\parallel x+x^{\prime },x_2,\dots ,x_n\parallel \le \parallel x,x_2,\dots ,x_n\parallel +\parallel x^{\prime },x_2,\dots ,x_n\parallel$

is called a n-norm on X, and the pair $(X,\parallel \cdot ,\dots ,\cdot \parallel )$ is called a n-normed space over the field . For example, we may take $X={\mathbb{R}}^n$ being equipped with the Euclidean n-norm ${\parallel x_1,x_2,\dots ,x_n\parallel }_E$, the volume of the n-dimensional parallelepiped spanned by the vectors $x_1,x_2,\dots ,x_n$ which may be given explicitly by the formula

$${\parallel x_1,x_2,\dots ,x_n\parallel }_E=|det(x_{ij})|,$$

where $x_i=(x_{i1},x_{i2},\dots ,x_{in})\in {\mathbb{R}}^n$ for each $i=1,2,\dots ,n$. Let $(X,\parallel \cdot ,\dots ,\cdot \parallel )$ be a n-normed space of dimension $d\ge n\ge 2$ and $\{a_1,a_2,\dots ,a_n\}$ be linearly independent set in X. Then the function ${\parallel \cdot ,\dots ,\cdot \parallel }_{\mathrm{\infty }}$ on $X^{n-1}$ defined by

$${\parallel x_1,x_2,\dots ,x_{n-1}\parallel }_{\mathrm{\infty }}=max\{\parallel x_1,x_2,\dots ,x_{n-1},a_i\parallel :i=1,2,\dots ,n\}$$

defines an $(n-1)$-norm on X with respect to $\{a_1,a_2,\dots ,a_n\}$.

A sequence $(x_k)$ in a n-normed space $(X,\parallel \cdot ,\dots ,\cdot \parallel )$ is said to converge to some $L\in X$ if

$$\underset{k\to \mathrm{\infty }}{lim}\parallel x_k-L,z_1,\dots ,z_{n-1}\parallel =0\text{for every }{z}_1,\dots ,z_{n-1}\in X.$$

A sequence $(x_k)$ in a n-normed space $(X,\parallel \cdot ,\dots ,\cdot \parallel )$ is said to be Cauchy if

$$\underset{p\to \mathrm{\infty }}{\underset{k\to \mathrm{\infty }}{lim}}\parallel x_k-x_p,z_1,\dots ,z_{n-1}\parallel =0\text{for every }{z}_1,\dots ,z_{n-1}\in X.$$

If every Cauchy sequence in X converges to some $L\in X$, then X is said to be complete with respect to the n-norm. Any complete n-normed space is said to be n-Banach space.

The space $m(\varphi )$ was introduced by Sargent [36]:

$$m(\varphi )=\{x=(x_k)\in w:{\parallel x\parallel }_{m(\varphi )}=\underset{s\ge 1,\sigma \in {\mathcal{P}}_s}{sup}\frac{1}{{\varphi }_s}\sum _{k\in \sigma }|x_k|<\mathrm{\infty }\},$$

which was further studied in [37, 38] and [39]. Recently, Duyar and Oǧur [40] introduced the sequence space $m^2(M,A,\varphi ,p)$ and studied some of its properties.

Let $A=(a_{ijkl})$ be an infinite double matrix of complex numbers, $\mathcal{M}=(M_{k,l})$ be a sequence of Orlicz functions, and $p=(p_{k,l})$ be a bounded double sequence of positive real numbers. In the present paper we define the following sequence space:

$$\begin{array}{c}{m}^2(\mathcal{M},A,\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )\hfill \\ =\{x=(x_{k,l})\in w^2(X):sup\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(\parallel \frac{A_{ij}(x)}{\rho },z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:\hfill \\ (s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}<\mathrm{\infty }\text{ for some }\rho >0\},\hfill \end{array}$$

where $A(x)=(A_{ij}(x))$ if $A_{ij}(x)={\sum }_{k,l=1}^{\mathrm{\infty }}{a}_{ijkl}{x}_{k,l}$ converges for each $(i,j)\in \mathbb{N}\times \mathbb{N}$.

If $p=(p_{ij})=1$, we have

$$\begin{array}{c}{m}^2(\mathcal{M},A,\varphi ,\parallel \cdot ,\dots ,\cdot \parallel )\hfill \\ =\{x=(x_{k,l})\in w^2(X):sup\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}\left(\parallel \frac{A_{ij}(x)}{\rho },z_1,\dots ,z_{n-1}\parallel \right):\hfill \\ (s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}<\mathrm{\infty }\text{ for some }\rho >0\}.\hfill \end{array}$$

The following inequality will be used throughout the paper:

$${|a+b|}^{p_{ij}}\le max(1,2^{H-1})({|a|}^{p_{ij}}+{|b|}^{p_{ij}}),$$

(1.1)

where $a,b\in \mathbb{C}$ and $H=sup\{p_{ij}:(i,j)\in \mathbb{N}\times \mathbb{N}\}$.

We examine some topological properties of $m^2(\mathcal{M},A,\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )$ and establish some inclusion relations.

2 Main results

Theorem 2.1 Let $\mathcal{M}=(M_{k,l})$ be a sequence of Orlicz functions and $p=(p_{k,l})$ be a bounded sequence of positive real numbers, then the space $m^2(\mathcal{M},A,\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )$ is linear space over the field of complex number ℂ.

Proof Let $x=(x_{k,l}),y=(y_{k,l})\in m^2(\mathcal{M},A,\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )$ and $\alpha ,\beta \in \mathbb{C}$. Then there exist positive numbers ${\rho }_1$ and ${\rho }_2$ such that

$$sup\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(\parallel \frac{A_{ij}(x)}{{\rho }_1},z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:(s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}<\mathrm{\infty }$$

and

$$sup\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(\parallel \frac{A_{ij}(y)}{{\rho }_2},z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:(s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}<\mathrm{\infty }.$$

Let ${\rho }_3=max(2|\alpha |{\rho }_1,2|\beta |{\rho }_2)$. Since ℳ is a non-decreasing and convex function, we have

$$\begin{array}{c}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(\parallel \frac{A_{ij}(\alpha x+\beta y)}{{\rho }_3},z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}\hfill \\ \le \sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(\parallel \frac{\alpha a_{ijkl}{x}_{k,l}+\beta a_{ijkl}{y}_{k,l}}{{\rho }_3},z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}\hfill \\ \le \sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{(\parallel \frac{\alpha a_{ijkl}{x}_{k,l}}{2|\alpha |{\rho }_1},z_1,\dots ,z_{n-1}\parallel +\parallel \frac{\beta a_{ijkl}{y}_{k,l}}{2|\beta |{\rho }_2},z_1,\dots ,z_{n-1}\parallel )}^{p_{ij}}\hfill \\ =\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{(\parallel \frac{a_{ijkl}{x}_{k,l}}{2{\rho }_1},z_1,\dots ,z_{n-1}\parallel +\parallel \frac{a_{ijkl}{y}_{k,l}}{2{\rho }_2},z_1,\dots ,z_{n-1}\parallel )}^{p_{ij}}\hfill \\ \le max(1,2^{H-1})(\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(\parallel \frac{a_{ijkl}{x}_{k,l}}{2{\rho }_1},z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}\hfill \\ +\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(\parallel \frac{a_{ijkl}{y}_{k,l}}{2{\rho }_2},z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}).\hfill \end{array}$$

Thus, we have

$$\begin{array}{c}sup\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(\parallel \frac{A_{ij}(\alpha x+\beta y)}{{\rho }_3},z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:\hfill \\ (s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}\hfill \\ \le max(1,2^{H-1})\{sup\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(\parallel \frac{a_{ijkl}{x}_{k,l}}{2{\rho }_1},z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:\hfill \\ (s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}\hfill \\ +sup\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}{M}_{k,l}{\left(\parallel \frac{a_{ijkl}{y}_{k,l}}{2{\rho }_2},z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:\hfill \\ (s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\left\}\right\}.\hfill \end{array}$$

This proves that $\alpha x+\beta y\in m^2(\mathcal{M},A,\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )$. Hence $m^2(\mathcal{M},A,\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )$ is a linear space. This completes the proof of the theorem. □

Theorem 2.2 $\mathcal{M}=(M_{k,l})$ be a sequence of Orlicz functions and $p=(p_{k,l})$ be a bounded sequence of positive real numbers, then the space $m^2(\mathcal{M},A,\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )$ is a paranormed space with the paranorm defined by

$$\begin{array}{rcl}g(x)& =& inf\{{\rho }^{\frac{p_{qr}}{H}}:[sup\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(\parallel \frac{A_{ij}(x)}{\rho },z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:\\ {(s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}\le 1]}^{1/H},q\in \mathbb{N},r\in \mathbb{N}\}.\end{array}$$

Proof It is clear that $g(x)=g(-x)$ and $g(x)=0$ if $x=0$. Then there exist positive numbers ${\rho }_1$ and ${\rho }_2$ such that

$$sup\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(\parallel \frac{A_{ij}(x)}{{\rho }_1},z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:(s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}<1$$

and

$$sup\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(\parallel \frac{A_{ij}(y)}{{\rho }_2},z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:(s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}<1.$$

Then, by using Minkowski’s inequality, we have

$$\begin{array}{c}sup\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(\parallel \frac{A_{ij}(x+y)}{{\rho }_1+{\rho }_2},z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:\hfill \\ (s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}\hfill \\ \le sup\left\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}\right(\parallel \frac{A_{ij}(x)}{{\rho }_1+{\rho }_2},z_1,\dots ,z_{n-1}\parallel \hfill \\ {+\parallel \frac{A_{ij}(y)}{{\rho }_1+{\rho }_2},z_1,\dots ,z_{n-1}\parallel )}^{p_{ij}}:(s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}\hfill \\ \le {\left(\frac{{\rho }_1}{{\rho }_1+{\rho }_2}\right)}^h\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(\parallel \frac{A_{ij}(x)}{{\rho }_1},z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:\hfill \\ (s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}\hfill \\ +{\left(\frac{{\rho }_2}{{\rho }_1+{\rho }_2}\right)}^h\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(\parallel \frac{A_{ij}(y)}{{\rho }_2},z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:\hfill \\ (s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\},\hfill \end{array}$$

where $h=inf{p}_{ij}$. This shows that $g(x+y)\le g(x)+g(y)$. Using this triangle inequality we can write

$$g({\lambda }^n{x}^n-\lambda x)\le g({\lambda }^n{x}^n-{\lambda }^{n}x)+g({\lambda }^{n}x-\lambda x).$$

Thus we have

$$\begin{array}{c}g({\lambda }^n{x}^n-{\lambda }^{n}x)\hfill \\ =inf\{{\rho }_n^{\frac{p_{qr}}{H}}:[sup\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(\parallel \frac{A_{ij}({\lambda }^n{x}^n-{\lambda }^{n}x)}{{\rho }_n},z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:\hfill \\ (s,t)\ge (1,1),{\sigma }_1{\sigma }_2\in {\mathcal{P}}_{s,t}\}\le 1]{}^{1/H}\le 1,q\in \mathbb{N},r\in \mathbb{N}\}\hfill \\ =inf\{{\rho }_n^{p_{qr}/H}:[sup\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(\parallel \frac{A_{ij}(x^n-x)}{({\rho }_n/|{\lambda }^n|)},z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:\hfill \\ (s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}\le 1]{}^{1/H}\le 1,q\in \mathbb{N},r\in \mathbb{N}\}\hfill \\ =inf\{{\left({\lambda }^n{\rho }_n\right)}^{p_{qr}/H}:[sup\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(\parallel \frac{A_{ij}(x^n-x)}{{\rho }_n},z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:\hfill \\ (s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}]{}^{1/H}\le 1,q\in \mathbb{N},r\in \mathbb{N}\}\hfill \\ \le max\{{\left|{\lambda }^n\right|}^{h/H},\left|{\lambda }^n\right|\}\times inf\{{\left(\left|{\lambda }^n\right|{\rho }_n\right)}^{p_{qr}/H}:\hfill \\ [sup\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(\parallel \frac{A_{ij}(x^n-x)}{{\rho }_n},z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:\hfill \\ (s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}]{}^{1/H}\le 1,q\in \mathbb{N},r\in \mathbb{N}\}\hfill \\ =max\left\{\right|{\lambda }^n{|}^{h/H},\left|{\lambda }^n\right|\}\cdot g(x^n-x).\hfill \end{array}$$

Thus

$$\begin{array}{c}g({\lambda }^{n}x-\lambda x)\hfill \\ =inf\{{\rho }_n^{p_{qr}/H}:[sup\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(\parallel \frac{A_{ij}(({\lambda }^n-\lambda )x)}{{\rho }_n},z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:\hfill \\ (s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}]{}^{1/H}\le 1,q\in \mathbb{N},r\in \mathbb{N}\}\hfill \\ =inf\{{\rho }_n^{p_{qr}/H}:[sup\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(\parallel \frac{A_{ij}(x)}{{\rho }_n/|{\lambda }^n-\lambda |},z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:\hfill \\ (s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}]{}^{1/H}\le 1,q\in \mathbb{N},r\in \mathbb{N}\}\hfill \\ =inf\{{\left(\right|{\lambda }^n-\lambda \left|{\rho }_n\right)}^{p_{qr}/H}:[sup\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(\parallel \frac{A_{ij}(x)}{{\rho }_n},z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:\hfill \\ (s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}]{}^{1/H}\le 1,q\in \mathbb{N},r\in \mathbb{N}\}\hfill \\ \le max\{{|{\lambda }^n-\lambda |}^{h/H},|{\lambda }^n-\lambda |\}g(x).\hfill \end{array}$$

Hence $g({\lambda }^n{x}^n-\lambda x)\to 0$ where ${\lambda }^n\to \lambda$ and $x^n\to x$ as $n\to \mathrm{\infty }$. This proves that $m^2(\mathcal{M},A,\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )$ is a paranormed space with the paranorm defined by g. This completes the proof of the theorem. □

Theorem 2.3 Let ϕ and ψ be two double sequences then $m^2(\mathcal{M},A,\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )\subseteq m^2(\mathcal{M},A,\psi ,p,\parallel \cdot ,\dots ,\cdot \parallel )$ if and only if ${sup}_{(s,t)\ge (1,1)}({\varphi }_{s,t}/{\psi }_{s,t})<\mathrm{\infty }$.

Proof Let $K={sup}_{(s,t)\ge (1,1)}({\varphi }_{s,t}/{\psi }_{s,t})<\mathrm{\infty }$. Then ${\varphi }_{s,t}\le K\cdot {\psi }_{s,t}$ for all $(s,t)\ge (1,1)$. If $x=\{x_{k,l}\}\in m^2(\mathcal{M},A,\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )$, then

$$\begin{array}{c}sup\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(\parallel \frac{A_{ij}(x)}{\rho },z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:(s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}\hfill \\ <\mathrm{\infty }\text{for some }\rho >0.\hfill \end{array}$$

Thus

$$\begin{array}{c}sup\{\frac{1}{K{\psi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(\parallel \frac{A_{ij}(x)}{\rho },z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:(s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}\hfill \\ <\mathrm{\infty }\text{for some }\rho >0,\hfill \end{array}$$

and hence $x=\{x_{k,l}\}\in m^2(\mathcal{M},A,\psi ,p,\parallel \cdot ,\dots ,\cdot \parallel )$. This shows that

$$m^2(\mathcal{M},A,\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )\subseteq m^2(\mathcal{M},A,\psi ,p,\parallel \cdot ,\dots ,\cdot \parallel ).$$

Conversely, let $m^2(\mathcal{M},A,\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )\subseteq m^2(\mathcal{M},A,\psi ,p,\parallel \cdot ,\dots ,\cdot \parallel )$ and ${\alpha }_{s,t}=\frac{{\varphi }_{s,t}}{{\psi }_{s,t}}$ for all $(s,t)\ge (1,1)$, and suppose ${sup}_{(s,t)\ge (1,1)}{\alpha }_{s,t}=\mathrm{\infty }$. Then there exists a subsequence $\{{\alpha }_{s_i,t_i}\}$ of $\{{\alpha }_{s,t}\}$ such that ${lim}_{i\to \mathrm{\infty }}{\alpha }_{s_i,t_i}=\mathrm{\infty }$. If $x=\{x_{k,l}\}\in m^2(\mathcal{M},A,\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )$, then we have

$$\begin{array}{c}sup\{\frac{1}{{\psi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}^{\mathrm{\infty }}{\left(\parallel \frac{A_{ij}(x)}{\rho },z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:(s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}\hfill \\ =sup\{{\alpha }_{s,t}\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}^{\mathrm{\infty }}{\left(\parallel \frac{A_{ij}(x)}{\rho },z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:\hfill \\ (s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}\hfill \\ \ge \left\{\underset{m\ge 1}{sup}{\alpha }_{s_m,t_m}\right\}sup\{\frac{1}{{\varphi }_{s_m,t_m}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}^{\mathrm{\infty }}{\left(\parallel \frac{A_{ij}(x)}{\rho },z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:\hfill \\ (s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}=\mathrm{\infty }.\hfill \end{array}$$

This is a contradiction as $x=\{x_{k,l}\}\notin m^2(\mathcal{M},A,\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )$. This completes the proof of the theorem. □

Corollary 2.4 Let ϕ and ψ be two double sequences then $m^2(\mathcal{M},A,\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )=m^2(\mathcal{M},A,\psi ,p,\parallel \cdot ,\dots ,\cdot \parallel )$ if and only if ${sup}_{(s,t)\ge (1,1)}{\alpha }_{s,t}<\mathrm{\infty }$ and ${sup}_{(s,t)\ge (1,1)}{\alpha }_{s,t}^{-1}<\mathrm{\infty }$.

Proof It is easy to prove so we omit the details. □

Theorem 2.5 Let $\mathcal{M}=(M_{k,l})$, ${\mathcal{M}}^{\prime }=(M_{k,l}^{\prime })$ and ${\mathcal{M}}^{″}=(M_{k,l}^{\prime \prime })$ be sequences of Orlicz functions satisfying ${\mathrm{\Delta }}_2$-condition. Then

• (i) $m^2(\mathcal{M},\varphi ,\parallel \cdot ,\dots ,\cdot \parallel )\subseteq m^2(\mathcal{M}\circ {\mathcal{M}}^{\prime },\varphi ,\parallel \cdot ,\dots ,\cdot \parallel )$,

• (ii) $m^2({\mathcal{M}}^{\prime },A,\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )\cap m^2({\mathcal{M}}^{″},A,\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )\subseteq m^2({\mathcal{M}}^{\prime }+{\mathcal{M}}^{″},A,\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )$.

Proof (i) Let $x=\{x_{k,l}\}\in m^2(\mathcal{M},A,\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )$. Then there exists $\rho >0$ such that

$$sup\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(\parallel \frac{A_{ij}(x)}{\rho },z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:(s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}<\mathrm{\infty }.$$

By the continuity of ℳ, we can take a number δ with $0<\delta <1$ such that $M_{k,l}(t)<ϵ$, whenever $0\le t<\delta$, for arbitrary $0<ϵ<1$. Now let

$$y_{i,j}=\left(\parallel \frac{A_{i,j}(x)}{\rho },z_1,\dots ,z_{n-1}\parallel \right).$$

Thus we have

$$\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{(y_{i,j})}^{p_{i,j}}=\sum _{y_{i,j}\le \delta }\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{(y_{i,j})}^{p_{i,j}}+\sum _{y_{i,j}>\delta }\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{(y_{i,j})}^{p_{i,j}}.$$

By the properties of the Orlicz function we have

$$\sum _{y_{i,j}\le \mathrm{\infty }}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{(y_{i,j})}^{p_{i,j}}\le max\{1,M_{k,l}{(1)}^H\}\sum _{y_{i,j}\le \mathrm{\infty }}{(y_{i,j})}^{p_{i,j}}.$$

Again, we have

$$M_{k,l}(y_{i,j})<M_{k,l}(1+\frac{y_{i,j}}{\delta })<\frac{1}{2}{M}_{k,l}(2)+\frac{1}{2}{M}_{k,l}\left(\frac{2y_{i,j}}{\delta }\right)$$

for $y_{i,j}>\delta$. If ℳ satisfies the ${\mathrm{\Delta }}_2$-condition, then we have

$$M_{k,l}(y_{i,j})<\frac{1}{2}T\frac{y_{i,j}}{\delta }{M}_{k,l}(2)+\frac{1}{2}T\frac{y_{i,j}}{\delta }{M}_{k,l}(2),$$

and so

$$\sum _{y_{i,j}>\delta }{M}_{k,l}{(y_{i,j})}^{p_{i,j}}\le max(1,{(\frac{T}{\delta }{M}_{k,l}(2))}^H)\sum _{y_{i,j}>\delta }{y}_{i,j}.$$

Hence, we have

$$\begin{array}{c}sup\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{\left(M_{k,l}^{\prime }\left(\parallel \frac{A_{ij}(x)}{\rho },z_1,\dots ,z_{n-1}\parallel \right)\right)}^{p_{ij}}:\hfill \\ (s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}\hfill \\ <\mathrm{\infty }\hfill \\ \le max\{1,M_{k,l}{(1)}^H\}sup\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{(y_{i,j})}^{p_{ij}}:(s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}\hfill \\ <\mathrm{\infty }+max(1,{(\frac{T}{\delta }{M}_{k,l}(2))}^H)\hfill \\ \times sup\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}{(y_{i,j})}^{p_{ij}}:(s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}<\mathrm{\infty }.\hfill \end{array}$$

Thus, we have $x=\{x_{k,l}\}\in m^2(\mathcal{M}\circ {\mathcal{M}}^{\prime },\varphi ,\parallel \cdot ,\dots ,\cdot \parallel )$ and hence $m^2(\mathcal{M},\varphi ,\parallel \cdot ,\dots ,\cdot \parallel )\subseteq m^2(\mathcal{M}\circ {\mathcal{M}}^{\prime },\varphi ,\parallel \cdot ,\dots ,\cdot \parallel )$.

• (ii) Let $x=\{x_{k,l}\}\in m^2({\mathcal{M}}^{\prime },A,\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )\cap m^2({\mathcal{M}}^{″},A,\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )$. Then there exists a $\rho >0$ such that

  $$sup\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}^{\prime }{\left(\parallel \frac{A_{i,j}(x)}{\rho },z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:(s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}<\mathrm{\infty }$$

and

$$sup\{\frac{1}{{\varphi }_{s,t}}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}^{\prime \prime }{\left(\parallel \frac{A_{i,j}(x)}{\rho },z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:(s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}<\mathrm{\infty }.$$

By the inequality, we have

$$\begin{array}{c}\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}(M_{k,l}^{\prime }+M_{k,l}^{\prime \prime }){\left(\parallel \frac{A_{i,j}(x)}{\rho },z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}\hfill \\ \le max(1,2^{H-1})\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}^{\prime }{\left(\parallel \frac{A_{i,j}(x)}{\rho },z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}\hfill \\ +max(1,2^{H-1})\sum _{i\in {\sigma }_1}\sum _{j\in {\sigma }_2}\sum _{k,l=1}^{\mathrm{\infty }}{M}_{k,l}^{\prime \prime }{\left(\parallel \frac{A_{i,j}(x)}{\rho },z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}.\hfill \end{array}$$

Hence

$$\begin{array}{c}{m}^2({\mathcal{M}}^{\prime },A,\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )\cap m^2({\mathcal{M}}^{″},A,\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )\hfill \\ \subseteq m^2({\mathcal{M}}^{\prime }+{\mathcal{M}}^{″},A,\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel ).\hfill \end{array}$$

This completes the proof of the theorem. □

Theorem 2.6 The sequence space $m^2(\mathcal{M},\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )$ is solid.

Proof Let $\alpha =\{{\alpha }_{k,l}\}$ be a double sequence of scalars such that $|{\alpha }_{k,l}|\le 1$ and $y=\{y_{k,l}\}\in m^2(\mathcal{M},\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )$. Then we have

$$\begin{array}{c}sup\{\frac{1}{{\varphi }_{s,t}}\sum _{k\in {\sigma }_1}\sum _{l\in {\sigma }_2}{M}_{k,l}{\left(\parallel \frac{{\alpha }_{k,l}{x}_{k,l}}{\rho },z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:(s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}\hfill \\ \le sup\{\frac{1}{{\varphi }_{s,t}}\sum _{k\in {\sigma }_1}\sum _{l\in {\sigma }_2}{M}_{k,l}{\left(\parallel \frac{{\alpha }_{k,l}{y}_{k,l}}{\rho },z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:(s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}\hfill \\ \le sup\{\frac{1}{{\varphi }_{s,t}}\sum _{k\in {\sigma }_1}\sum _{l\in {\sigma }_2}{M}_{k,l}{\left(\parallel \frac{y_{k,l}}{\rho },z_1,\dots ,z_{n-1}\parallel \right)}^{p_{ij}}:(s,t)\ge (1,1),{\sigma }_1\times {\sigma }_2\in {\mathcal{P}}_{s,t}\}.\hfill \end{array}$$

This implies that $\{{\alpha }_{k,l}{y}_{k,l}\}\in m^2(\mathcal{M},\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )$. This proves that the space $m^2(\mathcal{M},\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )$ is a solid. □

Corollary 2.7 The sequence space $m^2(\mathcal{M},\varphi ,p,\parallel \cdot ,\dots ,\cdot \parallel )$ is monotone.

Proof It is trivial so we omit the details. □