Since the proofs of the following theorems are not hard we omit them.

Theorem 3.1.

The sequence spaces are linear spaces over the complex field .

Theorem 3.2.

The space is a linear topological space paranormed by

In what follows, we will show inclusion theorems between spaces .

Theorem 3.3.

if and only if

Proof.

Let and . Then we get

hence . Conversely, let us suppose that and . Then there exists a such that

for every . Suppose that , then there exists a sequence of natural numbers such that . Hence we can write

Therefore, , which is contradiction.

Corollary 3.4.

Let be an Orlicz function. Then if and only if

Theorem 3.5.

Let , , be Orlicz functions which satisfy the -condition and , and seminorms. Then

(1),

(2),

(3),

(4)If is stronger than , then , and

(5)If is equivalent to , then .

Proof.

Proof is similar to [14, Theorem 2.5].

Corollary 3.6.

Let be an Orlicz function which satisfy the -condition. Then .

Theorem 3.7.

Let be a sequence of Orlicz functions. Then the sequence space is solid and monotone.

Proof.

Let , then there exists such that

for every . Let be a sequence of scalars with for all . Then from properties of Orlicz functions and seminorm, we get

which proves that is solid space and monotone.