We recall that the concept of a 2-normed space was first given in the works of Gähler ([1, 2]) as an interesting nonlinear generalization of a normed linear space which was subsequently studied by many authors (see, [3, 4]). Recently, a lot of activities have started to study summability, sequence spaces, and related topics in these nonlinear spaces (see, e.g., [5–9]). In particular, Savaş [10] combined Orlicz function and ideal convergence to define some sequence spaces using 2-norm.

In this paper, we introduce and study some new double-sequence spaces, whose elements are form -normed spaces, using an Orlicz function, which may be considered as an extension of various sequence spaces to -normed spaces. We begin with recalling some notations and backgrounds.

Recall in [11] that an Orlicz function is continuous, convex, and nondecreasing function such that and for , and as .

Subsequently, Orlicz function was used to define sequence spaces by Parashar and Choudhary [12] and others. An Orlicz function can always be represented in the following integral form: , where is the known kernel of , right differential for , , for , is nondecreasing, and as .

If convexity of Orlicz function is replaced by , then this function is called Modulus function, which was presented and discussed by Ruckle [13] and Maddox [14].

Remark 1.1.

If is a convex function and , then for all with .

Let and be real vector space of dimension , where . An -norm on is a function which satisfies the following four conditions:

(i) if and only if are linearly dependent,

(ii) are invariant under permutation,

(iii), ,

(iv).

The pair is then called an -normed space [3].

Let be equipped with the -norm, then the volume of the -dimensional parallelepiped spanned by the vectors, which may be given explicitly by the formula

where denotes inner product. Let be an -normed space of dimension and a linearly independent set in . Then, the function on is defined by

is defines an norm on with respect to (see, [15]).

Definition 1.2 (see [7]).

A sequence in -normed space is aid to be convergent to an in (in the -norm) if

for every .

Definition 1.3 (see [16]).

Let be a linear space. Then, a map is called a paranorm (on ) if it is satisfies the following conditions for all and scalar:

(i)  ,

(ii),

(iii),

(iv) and imply .

2. Main Results

Let be any -normed space, and let denote -valued sequence spaces. Clearly is a linear space under addition and scalar multiplication.

Definition 2.1.

Let be an Orlicz function and any -normed space. Further, let be a bounded sequence of positive real numbers. Now, we define the following new double sequence space as follows:

for each .

The following inequalities will be used throughout the paper. Let be a double sequence of positive real numbers with , and let . Then, for the factorable sequences and in the complex plane, we have as in Maddox [16]

Theorem 2.2.

sequences space is a linear space.

Proof.

Now, assume that and . Then,

Since is a -norm on , and is an Orlicz function, we get

where

and this completes the proof.

Theorem 2.3.

space is a paranormed space with the paranorm defined by

where , .

Proof.

(i) Clearly, and (ii) . (iii) Let , then there exists such that

So, we have

and thus

(iv) Now, let and . Since

This gives us .

Theorem 2.4.

If for each and , then .

Proof.

If , then there exists some such that

This implies that

for sufficiently large values of and . Since is nondecreasing, we are granted

Thus, . This completes the proof.

The following result is a consequence of the above theorem.

Corollary 2.5.

(i) If for each and , then

(ii) If for each and , then

Theorem 2.6.

, where is the double space of bounded sequences and .

Proof.

. Then, there exists an such that for each , . We want to show . But

and this completes the proof.

Theorem 2.7.

Let and be Orlicz function. Then, we have

Proof.

We have

Let ; when adding the above inequality from to we get and this completes the proof.

Definition 2.8 (see [10]).

Let be a sequence space. Then, is called solid if whenever for all sequences of scalars with for all .

Definition 2.9.

Let be a sequence space. Then, is called monotone if it contains the canonical preimages of all its step spaces (see, [17]).

Theorem 2.10.

The sequence space is solid.

Proof.

Let ; that is,

Let () be double sequence of scalars such that for all . Then, the result follows from the following inequality:

and this completes the proof.

We have the following result in view of Remark 1.1 and Theorem 2.10.

Corollary 2.11.

The sequence space is monotone.

Definition 2.12 (see [18]).

Let denote a four-dimensional summability method that maps the complex double sequences into the double-sequence , where the th term to is as follows:

Such transformation is said to be nonnegative if is nonnegative for all , and .

Definition 2.13.

Let be a nonnegative matrix. Let be an Orlicz function and a factorable double sequence of strictly positive real numbers. Then, we define the following sequence spaces:

for each . If , then we say is summable to , where .

If we take and for all , then we have

Theorem 2.14.

is linear spaces.

Proof.

This can be proved by using the techniques similar to those used in Theorem 2.2.

Theorem 2.15.

(1) If , then

(2) If , then

Proof.

(1) Let ; since , we have

and hence .

(2) Let for each and . Let .

Then, for each , there exists a positive integer such that

for all . This implies that

Thus, , and this completes the proof.