Strongly Summable Vector Valued Sequence Spaces Defined by 2 Modular


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 6(4) (2021), Pages 279-285 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: January 23, 2021 Reviewed: March 29, 2021 Accepted: April 07, 2021 
DOI: http://dx.doi.org/10.18860/ca.v6i4.11484    

Strongly Summable Vector Valued Sequence Spaces Defined by 2 
Modular 

B. A. Nurnugroho1, P.W. Prasetyo2 

1,2Departement of Mathematical Education, Universitas Ahmad Dahlan, Yogyakarta, 
Indonesia 

 

Email: burhanudin@pmat.uad.ac.id 

ABSTRACT 

Summability is an important concept in sequence spaces. One summability concept is strongly 
Cesaro summable. In this paper, we study a subset of the set of all vector-valued sequence in 2-
modular space. Some facts that we investigated in this paper include linearity, the existence of 
modular and completeness with respect to these modular.  

Keywords: Strongly; Summable;  Sequence Spaces; 2-modular 

INTRODUCTION 

Summability is an important concept in sequence spaces. The familiar example of 
sequence spaces that using the summability concept is ℓ𝑝 spaces.  In [1], it is explained 
that Kutner discusses spaces of strongly Cesaro summable sequences, and furthermore, 
Maddox generalizes this concept.   If 𝜔 denote the set of all  infinite sequence of 
real/complex numbers, then the set  

𝑤 = {(𝑥𝑘 ) ∈ 𝜔: ∃𝐿, ∋ lim
𝑛→∞

1

𝑛
∑|𝑥𝑘 − 𝐿| = 0

𝑛

𝑘=1

}, 

denote the space of strongly Cesaro summable sequence [2] [3].  
Let 𝑋 be a real linear space of dimension 𝑑 ≥ 2.  A 2-norm on 𝑋 is a function 

‖. , . ‖: 𝑋 × 𝑋 → ℝ , where for all 𝑥, 𝑦, 𝑧 ∈ 𝑋, satisfy  
(i) ‖𝑥, 𝑦‖ = 0 if and only if 𝑥  and 𝑦 are linearly dependent 
(ii) ‖𝑥, 𝑦‖ = ‖𝑦, 𝑥‖ 
(iii) ‖𝛼𝑥, 𝑦‖ = |𝛼|‖𝑥, 𝑦‖, 𝛼 ∈ ℝ 
(iv) ‖𝑥 + 𝑦, 𝑧‖ ≤ ‖𝑥, 𝑧‖ + ‖𝑦, 𝑧‖. 

The pair (𝑋, ‖. , . ‖) is then called a 2-normed space [4].  The concept is initially introduced 
by Gahler [5] in the middle of 1963. Furthermore, in 1989, Misiak generalized the 2-
normed concept to be n-normed [6]. Since then, many kinds research on 2-normed (n-
normed) spaces, include research on strongly Cesaro summable vector-valued sequences 
or the generalize in 2-normed (n-normed) spaces [7] [8] [9] [10] [11].  

In 1950, Nakano developed modular function and it was generalized by Musielak 
and Orlicz [12] [13].  Modular is the generalization of the norm. Let 𝑌 be a real linear space, 
a functional  𝑔: 𝑌 → ℝ∗ is said tobe modular if it satisfies the following conditions: 

(i) 𝑔(𝑥) = 0 if and if 𝑥 = 0 
(ii) 𝑔(−𝑥) = 𝑔(𝑥) 
(iii) 𝑔(𝛼𝑥 + 𝛽𝑦) ≤ 𝑔(𝑥) + 𝑔(𝑦), every 𝑥, 𝑦 ∈ 𝑌, 𝛼, 𝛽 ≥ 0, 𝛼 + 𝛽 = 1. 

http://dx.doi.org/10.18860/ca.v6i4.11484
mailto:burhanudin@pmat.uad.ac.id


Strongly Summable Vector Valued Sequence Spaces Defined by 2 Modular 

B. A. Nurnugroho 280 

The pair (𝑌, 𝑔) is then called a modular space.  Following the 2-norm (n-norm) 
concept,  K. Nourouzi and S. Shabanian in 2009  initially introduced the n-modular concept 
[14] [15]. Let 𝑋 be a real linear space of dimension 𝑑 ≥ 2.  A 2-modular on 𝑋 is a function 
𝜌(. , . ): 𝑋 × 𝑋 → ℝ∗ where for all 𝑥, 𝑦, 𝑧 ∈ 𝑋, satisfy  

(i) 𝜌(𝑥, 𝑦) = 0 if and only if 𝑥  and 𝑦 are linearly dependent 
(ii) 𝜌(𝑥, 𝑦) = 𝜌(𝑦, 𝑥) 
(iii) 𝜌(−𝑥, 𝑦) = 𝜌(𝑥, 𝑦), 
(iv) 𝜌(𝛼𝑥 + 𝛽𝑦, 𝑧) ≤  𝜌(𝑥, 𝑧) + 𝜌(𝑦, 𝑧),  every   𝛼, 𝛽 ≥ 0, 𝛼 + 𝛽 = 1. 

The pair (𝑋, ‖. , . ‖) is then called a 2-modular  space. The 2-modular space, with 𝜌 satisfies 
Δ2-condition, if there exist 𝐿 > 0, such that 

𝜌(2𝑥, 𝑦) ≤ 𝐿𝜌(𝑥, 𝑦), 
for all 𝑥, 𝑦 ∈ 𝑋. A  sequence (𝑥𝑘 ) in 𝑋 is said to be 2-modular convergent to 𝑥0 ∈ 𝑋 if  

lim
𝑘→∞

𝜌(𝑥𝑘 − 𝑥0, 𝑦) = 0, ∀𝑦 ∈ 𝑋. 

It means that for every 𝜖 > 0, there exists an 𝑘0 ∈ ℕ, such that for any 𝑘 ∈ ℕ, 𝑘 ≥ 𝑘0, we 
have 

𝜌(𝑥𝑘 − 𝑥0, 𝑦) < 𝜖, ∀𝑦 ∈ 𝑋.  
Furthermore, a  sequence (𝑥𝑘 ) in 𝑋 is called 2-modular Cauchy sequence  if, for all 𝑦 ∈ 𝑋, 
we have 

lim
𝑘,𝑙→∞

𝜌(𝑥𝑘 − 𝑥𝑙 , 𝑦) = 0. 

The standard example of a 2-modular space is 𝑋 = ℝ2, with 2-modular on ℝ2 
define by 

𝜌(�̅�, �̅�) = √|det (
𝑥1 𝑥2
𝑦1 𝑦2

)|, 

Where �̅� = (𝑥1, 𝑥2), �̅� = (𝑦1, 𝑦2) ∈ ℝ
2. Clearly that 𝜌 satisfies Δ2-condition and the 

sequence ((
1

𝑛
, 0)) in ℝ2 is 2-modular convergent to (0,0) ∈ ℝ2.  

This paper will be constructed t spaces of strongly Cesaro summable vector-valued 
sequences in 2-modular spaces based on the facts presented above.   
 

METHODS 

Let  (𝑋, 𝜌) be a  2-modular space, with 𝜌 satisfies Δ2-condition and the dimension of 𝑋 
greater than one. We define  

𝑋𝜌 = {𝑥 ∈ 𝑋: 𝜌(𝑥, 𝑦) < ∞, ∀𝑦 ∈ 𝑋}. 

Because 𝜌 satisfies Δ2-condition, then there exists 𝐾 > 0, such that for all 𝑥, 𝑦 ∈ 𝑋𝜌, 𝑧 ∈ 𝑋 

and 𝛼 ∈ ℝ, we have  

𝜌(𝑥 + 𝑦, 𝑧) = 𝜌 (
2𝑥 + 2𝑦

2
, 𝑧) 

                              ≤  𝜌(2𝑥, 𝑧) + 𝜌(2𝑦, 𝑧) 
                             ≤ 𝐾𝜌(𝑥, 𝑧) + 𝐾𝜌(𝑦, 𝑧) 

< ∞     
Based on Archimedean property, there exists 𝑛0 ∈ ℕ, such that 𝛼 ≤ 2

𝑛0  
𝜌(𝛼𝑥, 𝑧) ≤ 𝜌(2𝑛0 𝑥, 𝑧)          

        ≤ 𝐾𝑛0 𝜌(𝑥, 𝑧) 
< ∞.          

Hence, we have that 𝑋𝜌 is a subspace linear of 𝑋. Furthermore (𝑋𝜌, 𝜌) is a 2-modular space 

too. 
The notation 𝜔(𝑋𝜌) will donate as the set of all sequences in 𝑋𝜌 



Strongly Summable Vector Valued Sequence Spaces Defined by 2 Modular 

B. A. Nurnugroho 281 

𝜔(𝑋𝜌) = {(𝑥𝑘 ): 𝑥𝑘 ∈ 𝑋, 𝑘 ∈ ℕ}                                                      (1) 

  
where linear space operations are defined coordinatewise,  

(𝑥𝑘 ) + (𝑦𝑘 ) = (𝑥𝑘 + 𝑦𝑘 ), 𝛼(𝑥𝑘 ) = (𝛼𝑥𝑘 ) 

for all (𝑥𝑘 ), (𝑦𝑘 ) ∈ 𝜔(𝑋𝜌) and 𝛼 ∈ ℝ.  

The goal of this paper is that we want to extend the concept of strongly Cesaro 
summable to 2-modular spaces valued sequences, defined as 

𝑤0
𝜌

(𝑋𝜌) = {(𝑥𝑘 ) ∈ 𝜔(𝑋𝜌): 𝑙𝑖𝑚
𝑛→∞

1

𝑛
∑ 𝜌(𝑥𝑘 , 𝑦)

𝑛

𝑘=1

= 0, ∀𝑦 ∈ 𝑋𝜌 }                                         (2) 

𝑤 𝜌(𝑋𝜌 ) = {(𝑥𝑘 ) ∈ 𝜔(𝑋𝜌): ∃𝑥0 ∈ 𝑋𝜌, 𝑙𝑖𝑚
𝑛→∞

1

𝑛
∑ 𝜌(𝑥𝑘 − 𝑥0, 𝑦)

𝑛

𝑘=1

= 0, ∀𝑦 ∈ 𝑋𝜌 }                      (3) 

Furthermore, we also studied  the properties of  𝑤0
𝜌

(𝑋𝜌) and  𝑤
𝜌(𝑋𝜌). 

RESULTS AND DISCUSSION  

Henceforth, if not specified then 𝑋 is a 2-modular space with 2-modular 𝜌,  that 
satisfies the  Δ2-conditions.  

First,  we will prove that the mean Cesaro theorem applies to 2-modular space. 
 
Theorem 1. Let  sequence (𝑥𝑘 ) in 𝑋𝜌 2-modular convergent to 𝑥0 ∈ 𝑋𝜌, then  

 

lim
𝑛→∞

1

𝑛
∑ 𝜌(𝑥𝑘 − 𝑥0, 𝑦)

𝑛

𝑘=1

= 0, ∀𝑦 ∈ 𝑋𝜌 

 
Proof.  Since the sequence (𝑥𝑘 ) in 𝑋𝜌 2-modular convergent to 𝑥0 ∈ 𝑋𝜌, then for all 𝜖 > 0, 

there exists 𝑛𝜖 ∈ ℕ, such that for all 𝑘 ≥ 𝑛𝜖 , we have 

𝜌(𝑥𝑘 − 𝑥0, 𝑦) <
𝜖

2
, 

for all 𝑦 ∈ 𝑋.  Note that, for all 𝑛 ≥ 𝑛𝜖 , we have 

1

𝑛
∑ 𝜌(𝑥𝑘 − 𝑥0, 𝑦) =

1

𝑛
∑ 𝜌(𝑥𝑘 − 𝑥0, 𝑦)

𝑛𝜖

𝑘=1

+
1

𝑛
∑ 𝜌(𝑥𝑘 − 𝑥0, 𝑦)

𝑛

𝑘=𝑛𝜖+1

𝑛

𝑘=1

                                                

               ≤
1

𝑛
∑ 𝜌(𝑥𝑘 − 𝑥0, 𝑦)1≤𝑘≤𝑛𝜖

𝑚𝑎𝑥 

𝑛𝜖

𝑘=1

+
1

𝑛
∑ 𝜌(𝑥𝑘 − 𝑥0, 𝑦)𝑛𝜖+1≤𝑘≤𝑛

𝑚𝑎𝑥 

𝑛

𝑘=𝑛𝜖+1

 

                  =
𝜌(𝑥𝑘 − 𝑥0, 𝑦)1≤𝑘≤𝑛𝜖

𝑚𝑎𝑥 

𝑛
∑ 1

𝑛𝜖

𝑘=1

+
𝜌(𝑥𝑘 − 𝑥0, 𝑦)𝑛𝜖+1≤𝑘≤𝑛

𝑚𝑎𝑥 

𝑛
∑ 1

𝑛

𝑘=𝑛𝜖+1

        

            = 𝜌(𝑥𝑘 − 𝑥0, 𝑦)1≤𝑘≤𝑛𝜖
𝑚𝑎𝑥 

𝑛𝜖
𝑛

+ 𝜌(𝑥𝑘 − 𝑥0, 𝑦)𝑛𝜖+1≤𝑘≤𝑛
𝑚𝑎𝑥 

𝑛 − 𝑛𝜖
𝑛

          

= 𝜌(𝑥𝑘 − 𝑥0, 𝑦)1≤𝑘≤𝑛𝜖
𝑚𝑎𝑥 

𝑛𝜖
𝑛

+ 𝜌(𝑥𝑘 − 𝑥0, 𝑦)𝑛𝜖+1≤𝑘≤𝑛
𝑚𝑎𝑥             

= 𝜌(𝑥𝑘 − 𝑥0, 𝑦)1≤𝑘≤𝑛𝜖
𝑚𝑎𝑥 

𝑛𝜖
𝑛

+
𝜖

2
 .                                                   



Strongly Summable Vector Valued Sequence Spaces Defined by 2 Modular 

B. A. Nurnugroho 282 

 
By Archimedean property, there exists 𝑛′ ≥ 𝑛𝜖  , such that  for all 𝑛 ≥ 𝑛′, we have  

 

𝜌(𝑥𝑘 − 𝑥0, 𝑦)1≤𝑘≤𝑛𝜖
𝑚𝑎𝑥 

𝑛𝜖
𝑛

<
𝜖

2
. 

Hence,  for all 𝑛 ≥ 𝑛′, we have 

1

𝑛
∑ 𝜌(𝑥𝑘 − 𝑥0, 𝑦)

𝑛

𝑘=1

< 𝜖.        

In other words, the proof is complete. ∎ 
 

Based on Theorem 1, we can say that for all 2-modular convergent sequence (𝑥𝑘 ) in 
𝑋𝜌 is an element of 𝑤

𝜌(𝑋𝜌).   

 
Theorem 2.  The set  𝑤 𝜌(𝑋𝜌) is a linear subspace of 𝜔(𝑋𝜌). 

 
Proof.  Note  that for all  (𝑥𝑘 ), (𝑦𝑘 ) ∈ 𝑤

𝜌(𝑋𝜌) and 𝛼 ∈ ℝ, there exsist 𝑥0, 𝑦0 ∈ 𝑋𝜌 so that 

for all 𝑦 ∈ 𝑋𝜌, we have  

lim
𝑛→∞

1

𝑛
∑ 𝜌(𝑥𝑘 − x0, 𝑦)

𝑛

𝑘=1

= 0, and lim
𝑛→∞

1

𝑛
∑ 𝜌(𝑥𝑘 − y0, 𝑦)

𝑛

𝑘=1

= 0. 

Therefore, 𝜌 satisfy  Δ2-condition, then there exists 𝐿 > 0 and 𝑛0 ∈ ℕ so that 
 

    0 ≤  𝜌((𝑥𝑘 + 𝑦𝑘 ) − (𝑥0 + 𝑦0), 𝑦) = 𝜌((𝑥𝑘 − 𝑥0) + (𝑦𝑘 − 𝑦0), 𝑦)                                  

                                                         ≤   𝜌(2(𝑥𝑘 − 𝑥0), 𝑦) + 𝜌(2(𝑦𝑘 − 𝑦0), 𝑦)      
                                                        ≤   𝐿𝜌((𝑥𝑘 − 𝑥0), 𝑦) + 𝐿𝜌((𝑦𝑘 − 𝑦0), 𝑦)      

and  
         0 ≤ 𝜌(𝛼𝑥𝑘 − 𝛼𝑙, 𝑦) = 𝜌(𝛼(𝑥𝑘 − 𝑙), 𝑦)                                     
                                              ≤   𝜌(2𝑛0 (𝑥𝑘 − 𝑥0), 𝑦)                              

            ≤ 𝐿𝑛0 𝜌(𝑥𝑘 − 𝑥0, 𝑦). 
Hence, we have  

lim
𝑛→∞

1

𝑛
∑ 𝜌((𝑥𝑘 + 𝑦𝑘 ) − (𝑥0 + 𝑦0), 𝑦)

𝑛

𝑘=1

= 0 

and  

lim
𝑛→∞

1

𝑛
∑ 𝜌(𝛼𝑥𝑘 − 𝛼𝑥0, 𝑦)

𝑛

𝑘=1

= 0. 

In other words (𝑥𝑘 ) + (𝑦𝑘 ), 𝛼(𝑥𝑘 ) ∈ 𝑤
𝜌(𝑋𝜌), and we proof that 𝑤

𝜌(𝑋𝜌)  is a subspace 

linear of  𝜔(𝑋𝜌).∎ 

 

Theorem 3. If (𝑥𝑘 ) ∈ 𝑤
𝜌(𝑋𝜌), then for all 𝑦 ∈ 𝑋𝜌,  (

1

𝑛
∑ 𝜌(𝑥𝑘 , 𝑦)

𝑛
𝑘=1 )  is a bounded 

sequence of real numbers. 
 
Proof.  If (𝑥𝑘 ) ∈ 𝑤

𝜌(𝑋𝜌), then there exist  𝑥0 ∈ 𝑋𝜌, such that for all 𝑦 ∈ 𝑋𝜌, we have 

lim
n→∞

1

n
∑ ρ(xk − 𝑥0, y)

n

k=1

= 0. 

Hence, there exist 𝑛0 ∈ ℕ, such that for all 𝑛 ∈ ℕ, with 𝑛 ≥ 𝑛0 we have  



Strongly Summable Vector Valued Sequence Spaces Defined by 2 Modular 

B. A. Nurnugroho 283 

1

n
∑ ρ(xk − 𝑥0, y)

n

k=1

≤ 1. 

Since 𝜌 satisfies the  Δ2-conditions, there exist 𝐿 > 0,  for all 𝑦 ∈ 𝑋𝜌,  we have 

𝜌(𝑥𝑘 , 𝑦) = 𝜌 (
2(𝑥𝑘 − 𝑥0)

2
+

2𝑥0
2

, 𝑦) ≤ 𝐿𝜌(𝑥𝑘 − 𝑥0, 𝑦) + 𝐿𝜌(𝑥0, 𝑦). 

It implies,  

1

𝑛
∑ 𝜌(𝑥𝑘 , 𝑦)

𝑛

𝑘=1

≤
𝐿

𝑛
∑ 𝜌(𝑥𝑘 − 𝑥0, 𝑦)

𝑛

𝑘=1

+ 𝐿𝜌(𝑥0, 𝑦). 

 
If we set 

𝑀 = sup {𝜌(𝑥1 − 𝑥0, 𝑦),
1

2
∑ 𝜌(xk − 𝑥0, y), ⋯ ,

1

n0 − 1
∑ ρ(x1 − 𝑥0, y)

n0−1

k=1

, 1 

2

k=1

} 

 
then it follows that we have 𝐾 = 𝐿(𝑀 + 𝜌(𝑥0, 𝑦)), such that 

1

𝑛
∑ 𝜌(𝑥𝑘 , 𝑦) ≤ 𝐾,

𝑛

𝑘=1

 

for all 𝑛 ∈ ℕ. This implies that for all 𝑦 ∈ 𝑋𝜌,  (
1

𝑛
∑ 𝜌(𝑥𝑘 , 𝑦)

𝑛
𝑘=1 )  is a bounded sequence. ∎  

Theorem 4. Function  

𝑔((𝑥𝑘 )) = sup {
1

𝑛
∑ 𝜌(𝑥𝑘 , 𝑧)

𝑛

𝑘=1

, ∀𝑧 ∈ 𝑋𝜌}                                         (5) 

is a modular on 𝑤 𝜌(𝑋𝜌).   

 

Proof.  If (𝑥𝑘 ) = 𝟎 is the zero sequence. Then it is clear that 𝑔((𝑥𝑘 )) = 0. Conversely, if 

((𝑥𝑘 )) = 0, then we have  

sup {
1

𝑛
∑ 𝜌(𝑥𝑘 , 𝑧)

𝑛

𝑘=1

, ∀𝑧 ∈ 𝑋𝜌} = 0. 

Hence, it implies  for all 𝑛 ∈ ℕ and 𝑧 ∈ 𝑋𝜌, we have 

1

𝑛
∑ 𝜌(𝑥𝑘 , 𝑦𝑘 )

𝑛

𝑘=1

= 0 ⇔ 𝜌(𝑥𝑘 , 𝑧) = 0 ⇔ 𝑥𝑘 = 0, ∀𝑘 ∈ ℕ. 

Thus, it is evident that (𝑥𝑘 ) = 𝟎.   
 
Since 𝜌(−𝑥, 𝑦) = 𝜌(𝑥, 𝑦) applies, for all 𝑥, 𝑦 ∈ 𝑋𝜌,  consequently, it is clear that 

𝑔(−(𝑥𝑘 )) = 𝑔((𝑥𝑘 )).  Finally, for all 𝛼, 𝛽 ≥ 0  with 𝛼 + 𝛽 = 1, the for all  (𝑥𝑘 ), (𝑦𝑘 ) ∈

𝑤 𝜌(𝑋𝜌) we have, 

𝑔(𝛼(𝑥𝑘 ) + 𝛽(𝑦𝑘 )) = sup {
1

𝑛
∑ 𝜌(𝛼𝑥𝑘 + 𝛽 𝑦𝑘 , 𝑧)

𝑛

𝑘=1

, ∀𝑧 ∈ 𝑋𝜌}                                                            

= sup {
1

𝑛
∑( 𝜌(𝑥𝑘 , z)  + 𝜌(𝑦𝑘 , 𝑧))

𝑛

𝑘=1

, ∀𝑧 ∈ 𝑋𝜌}                  



Strongly Summable Vector Valued Sequence Spaces Defined by 2 Modular 

B. A. Nurnugroho 284 

               ≤ sup {
1

𝑛
∑ 𝜌(𝑥𝑘 , 𝑧)

𝑛

𝑘=1

, ∀𝑧 ∈ 𝑋𝜌} + sup {
1

𝑛
∑ 𝜌(𝑦𝑘 , 𝑧 )

𝑛

𝑘=1

, ∀𝑧 ∈ 𝑋𝜌} 

= 𝑔((𝑥𝑘 )) + 𝑔((𝑦𝑘 )).                                                                 

This completes the proof. ∎ 
          
Theorem 5.   If 𝑋𝜌 2-modular complete, then (𝑤

𝜌(𝑋𝜌), 𝑔) is a modular complete.  

Proof.   Let 𝑛 ∈ ℕ and  (𝑥𝑖 ) be a 2-modular Cauchy sequence  in 𝑤 𝜌(𝑋𝜌), where 𝑥
𝑖 = (𝑥𝑘

𝑖 ), 

for all 𝑖 ∈ ℕ. Hence, for all 𝜖 > 0, there exists  𝑛0 ∈ ℕ, such that for all 𝑖, 𝑗 ∈ ℕ, with 𝑖, 𝑗 ≥
𝑛0, we have 

𝑔(𝑥𝑖 − 𝑥 𝑗 ) = sup {
1

𝑛
∑ 𝜌(𝑥𝑘

𝑖 − 𝑥𝑘
𝑗
, 𝑧)

𝑛

𝑘=1

, ∀𝑧 ∈ 𝑋𝜌} < 𝜖.  

It implies that, for all 𝑖, 𝑗 ≥ 𝑛0, we have 

1

𝑛
∑ 𝜌(𝑥𝑘

𝑖 − 𝑥𝑘
𝑗
, 𝑧)

𝑛

𝑘=1

< 𝜖, ∀𝑧 ∈ 𝑋𝜌, 

or 

∑ 𝜌(𝑥𝑘
𝑖 − 𝑥𝑘

𝑗
, 𝑧)

𝑛

𝑘=1

< 𝑛𝜖, ∀𝑧 ∈ 𝑋𝜌, 

such that, 

𝜌(𝑥𝑘
𝑖 − 𝑥𝑘

𝑗
, 𝑧) < 𝑛𝜖, ∀𝑧 ∈ 𝑋𝜌. 

Hence, for all 𝑘 ∈ ℕ,  (𝑥𝑘
𝑖 ) is a 𝜌-Cauchy sequence in 𝑋𝜌.  Since 𝑋𝜌 complete 2-modular, 

then (𝑥𝑘
𝑖 ) is 2-modular convergent in 𝑋𝜌, for all 𝑘 ∈ ℕ.  Therefore, for 𝑘 ∈ ℕ, there exist 

𝑥𝑘 ∈ 𝑋𝜌 , such that for all 𝑧 ∈ 𝑋𝜌, we have 

lim
𝑖→∞

𝜌(𝑥𝑘
𝑖 − 𝑥𝑘 , 𝑧) = 0. 

Since, for all 𝑖, 𝑗 ≥ 𝑛0, we have 

1

𝑛
∑ 𝜌(𝑥𝑘

𝑖 − 𝑥𝑘 , 𝑧)

𝑛

𝑘=1

= lim
𝑗→∞

1

𝑛
∑ 𝜌(𝑥𝑘

𝑖 − 𝑥𝑘
𝑗
, 𝑧)

𝑛

𝑘=1

< 𝜖, ∀𝑧 ∈ 𝑋𝜌, 

then 

𝑔 ((𝑥𝑘
𝑖 ) − (𝑥𝑘 )) = sup (

1

𝑛
∑ 𝜌(𝑥𝑘

𝑖 − 𝑥𝑘 , 𝑧)

𝑛

𝑘=1

) < 𝜖, for all 𝑖 ≥ 𝑛0, 

such that  

𝜌(𝑥𝑘
𝑖 − 𝑥𝑘 , 𝑧) < 𝑛𝜖, for all 𝑖 ≥ 𝑛0 

Therefore (𝑥𝑖 ) modular convergent to (𝑥𝑘 ), and (𝑥𝑘
𝑖 − 𝑥𝑘 ) ∈ 𝑤(𝑋𝜌). Since (𝑥𝑘

𝑖 ) ∈ 𝑤(𝑋𝜌) and 

𝑤(𝑋𝜌) is a linear spaces, so we have  

(𝑥𝑘 ) = (𝑥𝑘
𝑖 ) − (𝑥𝑘

𝑖 − 𝑥𝑘 ) ∈ 𝑤(𝑋𝜌). 

This complete the proof that (𝑤 𝜌(𝑋𝜌), 𝑔) is a complete modular (𝜌-complete). ∎ 

 
CONCLUSIONS 

If (𝑋, 𝜌) is a 2-modular space, with 𝜌 satisfies Δ2-condition, then we can construct 
𝑤 𝜌(𝑋𝜌) ⊂ 𝑤(𝑋𝜌) is the space of  strongly Cesaro summable vector-valued sequences in 

2-modular (𝑋𝜌, 𝜌).  It certainly can be shown that 𝑤
𝜌(𝑋𝜌) is a linear space. Furthermore, 

if (𝑥𝑘 ) ∈ 𝑤
𝜌(𝑋𝜌), then we can prove that for all 𝑦 ∈ 𝑋𝜌,  (

1

𝑛
∑ 𝜌(𝑥𝑘 , 𝑦)

𝑛
𝑘=1 )  is a bounded 



Strongly Summable Vector Valued Sequence Spaces Defined by 2 Modular 

B. A. Nurnugroho 285 

sequence of real numbers. This fact provides a guarantee for us to be able to build a 
modular 𝑔 on 𝑤 𝜌(𝑋𝜌). Finally, we proved that (𝑤

𝜌(𝑋𝜌), 𝑔) is modular complete, if  (𝑋𝜌, 𝜌) 

is a 2-modular complete. 
 
ACKNOWLEDGMENTS  
The author would like to thank LPPM UAD for funding this research 
 
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