# Classes ofI-Convergent Double Sequences over n-Normed Spaces

Authors: Nazneen Khan
Journal: Journal of Function Spaces
 (2016)

---

## Abstract

I introduce some new classes ofI-convergent double sequences defined by a sequence of moduli over n-normed space. Study of their algebraic and topological properties and some inclusion relations has also been done.


---

## Body

## 1. Introduction

The notion ofI-convergence was introduced by Kostyrko et al. in [1]. It is known that I-convergence is generalization of the statistical convergence which was introduced by Fast [2]. It was further studied by Demirci [3], Das et al. [4], Šalát et al. [5], and many others.For a nonempty setX, the family of sets I
⊂
2
X, the power set of X, is said to be an ideal if(1)
ϕ
∈
I;
(2)
I is additive; that is, A
,
B
∈
I
⇒
A
∪
B
∈
I;
(3)
I is hereditary; that is, A
∈
I
,
B
⊆
A
⇒
B
∈
I.A nontrivial idealI is called admissible if {
{
x
}
:
x
∈
X
}
⊆
I. I is maximal if there cannot exist any nontrivial ideal J
≠
I containing I as a subset.LetN
,
R, and C denote the set of natural, real, and complex numbers, respectively. A double sequence of complex numbers is defined as a function from N
×
N to C. A number a
∈
C is called a limit of a double sequence (
x
i
j
) if for every ϵ
>
0 there exists some N
=
N
(
ϵ
)
∈
N such that |
x
i
j
-
a
|
<
ϵ
,
∀
i
,
j
≥
N
. The set of all double sequences is denoted by ω
2. Any subset of the ω
2 is called double sequence space. A sequence (
x
i
j
)
∈
ω
2 is said to be I-convergent to a number L if, for every ϵ
>
0, {
(
i
,
j
)
∈
N
×
N
:
x
i
j
-
L
≥
ϵ
}
∈
I. In this case we write I
-
lim
⁡
x
i
j
=
L.A double sequence spaceE is said to be solid or normal if (
x
i
j
)
∈
E implies (
α
i
j
x
i
j
)
∈
E for all sequences of scalars (
α
i
j
) with |
α
i
j
|
<
1 for all i
,
j
∈
N. For more details please see [6–8].Example 1.
LetI
2
(
P
) be the class of all subsets of N
×
N such that D
∈
I
2
(
P
) implies that there exists n
0
,
k
0
∈
N such that D
⊂
N
×
N
∣
{
(
n
,
k
)
∈
N
×
N
:
n
≥
n
0
,
k
≥
k
0
}.
ThenI
2
(
P
) is an ideal of N
×
N in the usual Pringsheim sense of convergence of double sequences. If I
2
(
P
) is replaced by I
(
f
), the class of finite subsets of N, then we get the usual regular convergence of double sequences.The theory of 2-normed spaces was first introduced by Gähler [9] in 1964. Later on it was extended to n-normed spaces by Misiak [10]. Since then many mathematicians have worked in this field and obtained many interesting results; for instance see Gunawan [11, 12], Gunawan and Mashadi [13], Mursaleen and Mohiuddine [14], Şahiner et al. [15, 16], and Yamancı and Gürdal [17]. Let n
∈
N and let X
n be a linear metric space over the field K of real or complex numbers of dimension d, where d
≥
n
≥
2. A real valued function ·
,
…
,
· on X
n satisfying the following four conditions:(1)
x
1
,
x
2
,
…
,
x
n
=
0 if and only if x
1
,
x
2
,
…
,
x
n are linearly dependent;
(2)
x
1
,
x
2
,
…
,
x
n is invariant under permutation;
(3)
α
x
1
,
x
2
,
…
,
x
n
=
α
x
1
,
x
2
,
…
,
x
n for any α
∈
K;
(4)
x
+
x
′
,
x
2
,
…
,
x
n
≤
x
,
x
2
,
…
,
x
n
+
x
′
,
x
2
,
…
,
x
nis called an n-norm on X and the pair (
X
,
·
,
…
,
·
) is called an n-normed space over the field K.Example 2.
If we takeX
=
R
n, equipped with Euclidean n-norm x
1
,
x
2
,
…
,
x
n
E
=
V
o
l
(
n
-
d
i
m
p
a
r
a
l
l
e
l
o
p
i
p
e
d
) spanned by vectors (
x
1
,
x
2
,
…
,
x
n
), then the n-norm may be given by the formula x
1
,
x
2
,
…
,
x
n
E
=
x
i
j, where (
x
i
j
)
=
(
x
i
1
,
x
i
2
,
…
,
x
i
n
) for i
=
1,2
,
3
,
…
,
n.The standardn-norm on X is defined as (1)
x
1
,
x
2
,
…
,
x
n
E
=
x
1
,
x
1
⋯
x
1
,
x
n
⋮
⋯
⋮
x
1
,
x
1
⋯
x
1
,
x
1
1
/
2
,where·
,
· denotes the inner product on X. If X
=
R
n, then this n-norm is exactly the same as the Euclidean n-norm x
1
,
x
2
,
…
,
x
n
E mentioned earlier. For n
=
1 this n-norm is the usual norm x
=
〈
x
1
,
x
1
〉
1
/
2.A sequence(
x
k
) in an n-normed space (
X
,
·
,
…
,
·
) is said to converge to some L
∈
K if (2)
l
i
m
k
→
∞
x
k
-
L
,
z
1
,
…
,
z
n
-
1
=
0
for  every
z
1
,
z
2
,
…
,
z
n
-
1
∈
X
.A sequence(
x
k
) in an n-normed space (
X
,
·
,
…
,
·
) is said to be Cauchy if (3)
l
i
m
k
,
p
→
∞
x
k
-
x
p
,
z
1
,
…
,
z
n
-
1
=
0
for  every
z
1
,
z
2
,
…
,
z
n
-
1
∈
X
.If every Cauchy sequence inX converges to some L
∈
X, then X is said to be complete with respect to the n-norm. Any complete n- normed space is said to be an n-Banach space.The concept of modulus function was introduced by Nakano [18] in the year 1953. It was further studied by [7, 8, 19–21] and many more. It is defined as a function f: [
0
,
∞
)
→
[
0
,
∞
) satisfying the following conditions:(1)
f
(
t
)
=
0 if and only if t
=
0,
(2)
f
(
t
+
u
)
≤
f
(
t
)
+
f
(
u
) for all t
,
u
≥
0,
(3)
f is increasing,
(4)
f is continuous from the right at zero.Ruckle [22] used the idea of a modulus function f to construct the sequence space (4)
X
f
=
x
=
x
k
:
∑
k
=
1
∞
f
x
k
<
∞
.The space X
(
f
) is closely related to the space l
1 which is an X
(
f
) space with f
(
x
)
=
x for all real x
≥
0. Thus Ruckle [23, 24] proved that, for any modulus f,(5)
X
f
⊂
l
1
,
X
f
α
=
l
∞
.The space X
(
f
) is a Banach space with respect to the norm (6)
x
=
∑
k
=
1
∞
f
x
k
<
∞
.After then Kolk [25, 26] gave an extension of X
(
f
) by considering a sequence of modulus functions called the sequence of moduli F
=
(
f
k
) and defined the sequence space: (7)
X
F
=
x
=
x
k
:
f
k
x
k
∈
X
.From the above four properties of modulus function it can be clearly seen that f must be continuous everywhere on [
0
,
∞). For a sequence of moduli, we have further two properties:(5)
sup
k
⁡
f
k
(
t
)
<
∞ for all t
>
0;
(6)
lim
t
→
0
⁡
f
k
(
t
)
=
0 uniformly in X and for k
≥
1.Example 3.
Letf be a function from [
0
,
∞
) to [
0
,
∞
). If we take f
(
x
)
=
x
/
(
x
+
1
), then the function f is a bounded modulus function and if we take f
(
x
)
=
x
p
,
0
<
p
<
1, then f is an unbounded modulus function.By a lacunary sequenceθ
=
(
k
r
);  r
=
0,1
,
2
,
…,  where k
0
=
0, we mean an increasing sequence of nonnegative integers h
r
=
(
k
r
-
k
r
-
1
)
→
∞ as (
r
→
∞
). The intervals determined by θ are denoted by I
r
=
(
k
r
-
1
,
k
r
] and the ratio k
r
/
(
k
r
-
1
) will be denoted by q
r. The space of lacunary strongly convergent sequence N
θ was defined by Freedman et al. [27] as follows: (8)
N
θ
=
x
=
x
k
:
lim
r
→
∞
⁡
1
h
r
∑
k
∈
I
r
x
k
-
L
=
0
,
f
o
r
s
o
m
e
L
.The double lacunary sequence was defined by Savaş and Patterson [28]. A double sequence θ
r
;
s
=
{
(
k
r
,
l
s
)
} is called double lacunary if there exist two increasing sequences of integers such that (9)
k
0
=
0
,
h
r
=
k
r
-
k
r
-
1
⟶
∞
as
r
⟶
∞
,
l
0
=
0
,
h
s
=
l
s
-
l
s
-
1
⟶
∞
as
s
⟶
∞
.The following interval is determined by θ: (10)
I
r
,
s
=
k
,
l
:
k
r
-
1
<
k
<
k
r
,
l
s
-
1
<
l
<
l
s
.The space of double lacunary strongly convergent sequence is defined as follows: (11)
N
θ
r
,
s
=
x
=
x
k
l
:
lim
r
,
s
⁡
1
h
r
,
s
∑
k
,
l
∈
I
r
,
s
x
k
l
-
L
=
0
,
for  some
L
.


## 2. New Classes of Double Sequences

Now, we will define the new classes of double sequences.LetI be an admissible ideal, let F
=
(
f
i
j
) be a sequence of moduli, let (
X
,
·
,
…
,
·
) be an n-normed space, let p
=
(
p
i
j
) be a sequence of positive real numbers, let u
=
(
u
i
j
) be a sequence of strictly positive real numbers, and let w
2
(
n
-
X
) be the space of all double sequences defined over the n-normed space (
X
,
·
,
…
,
·
); then for some L
∈
K and every z
i
∈
X, we define(i)
[
2
N
θ
,
F
,
u
,
p
,
·
,
…
,
·
,
X
s
]
I
=
{
x
=
(
x
i
j
)
∈
w
2
(
n
-
X
)
:
[
(
r
,
s
)
∈
N
×
N : (
1
/
h
r
,
s
)
∑
(
i
,
j
)
∈
I
r
,
s
u
i
j
[
F
(
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
)
]
p
i
j
≥
ϵ
]
∈
I
},
(ii)
[
2
N
θ
,
F
,
u
,
p
,
·
,
…
,
·
,
X
s
]
0
I
=
{
x
=
(
x
i
j
)
∈
w
2
(
n
-
X
)
:
[
(
r
,
s
)
∈
N
×
N : (
1
/
h
r
,
s
)
∑
(
i
,
j
)
∈
I
r
,
s
u
i
j
[
F
(
x
i
j
,
z
1
,
z
2
,
…
,
z
n
-
1
)
]
p
i
j
≥
ϵ
]
∈
I
}
.Case 1.
IfF
(
x
)
=
x, then we get(i)
[
2
N
θ
,
u
,
p
,
·
,
…
,
·
,
X
s
]
I
=
{
x
=
(
x
i
j
)
∈
w
2
(
n
-
X
)
:
[
(
r
,
s
)
∈
N
×
N : (
1
/
h
r
,
s
)
∑
(
i
,
j
)
∈
I
r
,
s
u
i
j
(
x
i
j
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
)
p
i
j
≥
ϵ
]
∈
I
},
(ii)
[
2
N
θ
,
u
,
p
,
·
,
…
,
·
,
X
s
]
0
I
=
{
x
=
(
x
i
j
)
∈
w
2
(
n
-
X
)
:
[
(
r
,
s
)
∈
N
×
N : (
1
/
h
r
,
s
)
∑
(
i
,
j
)
∈
I
r
,
s
u
i
j
(
x
i
j
,
z
1
,
z
2
,
…
,
z
n
-
1
)
p
i
j
≥
ϵ
]
∈
I
}
.Case 2.
Ifp
=
(
p
i
j
)
=
1, then we get(i)
[
2
N
θ
,
F
,
u
,
·
,
…
,
·
,
X
s
]
I
=
{
x
=
(
x
i
j
)
∈
2
w
(
n
-
X
)
:
[
(
r
,
s
)
∈
N
×
N : (
1
/
h
r
,
s
)
∑
(
i
,
j
)
∈
I
r
,
s
u
i
j
[
F
(
x
i
j
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
)
]
≥
ϵ
]
∈
I
},
(ii)
[
2
N
θ
,
F
,
u
,
·
,
…
,
·
,
X
s
]
0
I
=
{
x
=
x
i
j
∈
2
w
(
n
-
X
)
:
[
(
r
,
s
)
∈
N
×
N : (
1
/
h
r
,
s
)
∑
(
i
,
j
)
∈
I
r
,
s
u
i
j
[
F
(
x
i
j
,
z
1
,
z
2
,
…
,
z
n
-
1
)
]
≥
ϵ
]
∈
I
}
.Case 3.
Ifp
=
(
p
i
j
)
=
1 and u
=
(
u
i
j
)
=
1, then we get(i)
[
2
N
θ
,
F
,
·
,
…
,
·
,
X
s
]
I
=
{
x
=
(
x
i
j
)
∈
2
w
(
n
-
X
)
:
[
(
r
,
s
)
∈
N
×
N : (
1
/
h
r
,
s
)
∑
(
i
,
j
)
∈
I
r
,
s
[
F
(
x
i
j
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
)
]
≥
ϵ
]
∈
I
},
(ii)
[
2
N
θ
,
F
,
·
,
…
,
·
,
X
s
]
0
I
=
{
x
=
(
x
i
j
)
∈
2
w
(
n
-
X
)
:
[
(
r
,
s
)
∈
N
×
N : (
1
/
h
r
,
s
)
∑
(
i
,
j
)
∈
I
r
,
s
[
F
(
x
i
j
,
z
1
,
z
2
,
…
,
z
n
-
1
)
]
≥
ϵ
]
∈
I
}
.The following inequality will be used throughout the paper. If 0
≤
p
i
j
≤
sup
⁡
p
i
j
=
H
,
K
=
max
⁡
(
1
,
2
H
-
1
), then we have (12)
a
i
j
+
b
i
j
p
i
j
≤
K
a
i
j
p
i
j
+
b
i
j
p
i
jfor all a
i
j
,
b
i
j
∈
C and (
i
,
j
)
∈
N
×
N. Also |
a
|
p
i
j
≤
max
⁡
(
1
,
|
a
|
H
) for all a
∈
C
.


## 3. Main Results

Theorem 4.
The sets[
2
N
θ
,
F
,
u
,
p
,
·
,
…
,
·
,
X
s
]
I and [
2
N
θ
,
F
,
u
,
p
,
·
,
…
,
·
,
X
s
]
0
I are linear spaces over the field of complex numbers C.Proof.
Letx
=
x
i
j
,
y
=
y
i
j
∈
[
2
N
θ
,
F
,
u
,
p
,
·
,
…
,
·
,
X
s
]
I, α
,
β
∈
C, and C
=
max
⁡
1
,
α
/
(
α
+
β
)
H
,
β
/
(
α
+
β
)
H; then for every z
i
∈
X we can write (13)
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
F
x
i
j
-
L
1
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
≤
ϵ
2
K
C
,
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
F
y
i
j
-
L
2
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
≤
ϵ
2
K
C
.By the use of inequality (12), we have the following inequality: (14)
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
F
α
x
i
j
+
β
y
i
j
-
α
L
1
+
β
L
2
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
≤
K
C
ϵ
2
K
C
<
ϵ
.
This inequality says to us that the inclusion(15)
r
,
s
∈
N
×
N
:
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
F
α
x
+
β
y
-
α
L
1
+
β
L
2
,
z
1
,
z
2
,
…
,
z
n
-
1
≥
ϵ
⊆
r
,
s
∈
N
×
N
:
K
C
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
F
x
-
L
1
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
≥
ϵ
∪
r
,
s
∈
N
×
N
:
K
C
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
F
y
-
L
2
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
≥
ϵholds. From here, since the right side belongs to I, the left side also belongs to I. This completes the proof.Lemma 5.
Letf be a modulus function and let 0
<
δ
<
1. Then for each x
>
δ, one has (16)
f
x
≤
2
f
1
δ
-
1
x
.Theorem 6.
LetF
=
(
f
i
j
) be a sequence of moduli and 0
≤
inf
i
,
j
⁡
p
i
j
=
h
≤
sup
i
,
j
⁡
p
i
j
=
H
<
∞
. Then the following statements hold:(i)
[
2
N
θ
,
u
,
p
,
·
,
…
,
·
,
X
s
]
I
⊂
[
2
N
θ
,
F
,
u
,
p
,
·
,
…
,
·
,
X
s
]
I,
(ii)
[
2
N
θ
,
u
,
p
,
·
,
…
,
·
,
X
s
]
0
I
⊂
[
2
N
θ
,
F
,
u
,
p
,
·
,
…
,
·
,
X
s
]
0
I.Proof.
For someδ
>
0, choose δ
0
>
0 such that max
⁡
δ
0
h
,
δ
0
H
<
δ. By the continuity of F
=
(
f
i
j
) for all (
i
,
j
)
∈
N
×
N, we can choose some ϵ
∈
(
0,1
) such that for every t with 0
<
t
≤
ϵ we have (17)
F
t
<
δ
0
∀
i
,
j
∈
N
×
N
.Let x
=
x
i
j
∈
[
2
N
θ
,
u
,
p
,
·
,
…
,
·
,
X
s
]
I; then for some L
>
0
,
δ
>
0 and for every z
i
∈
X
,
i
=
1,2
,
3
,
…
,
(
n
-
1
), we have (18)
A
=
r
,
s
∈
N
×
N
:
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
≥
ϵ
H
∈
I
.Therefore for (
r
,
s
)
∉
A, we have (19)
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
<
ϵ
H
,
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
≤
ϵ
.So, by inequality (17), we can write (20)
F
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
≤
δ
0
,
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
F
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
<
max
⁡
δ
0
h
,
δ
0
H
<
δ
,
r
,
s
∈
N
×
N
:
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
F
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
≥
δ
∈
I
.This implies that x
∈
[
2
N
θ
,
F
,
u
,
p
,
·
,
…
,
·
,
X
s
]
I. Hence [
2
N
θ
,
u
,
p
,
·
,
…
,
·
,
X
s
]
I
⊂
[
2
N
θ
,
F
,
u
,
p
,
·
,
…
,
·
,
X
s
]
I
. The inclusion is strict as for the reverse inclusion we need the condition given in the next theorem. The other part can be proved similarly.Theorem 7.
LetF be a sequence of moduli. If lim
t
⁡
sup
⁡
(
f
i
j
t
/
t
)
=
A
>
0 for all (
i
,
j
)
∈
N
×
N, then(i)
[
2
N
θ
,
u
,
p
,
·
,
…
,
·
,
X
s
]
I
=
[
2
N
θ
,
F
,
u
,
p
,
·
,
…
,
·
,
X
s
]
I,
(ii)
[
2
N
θ
,
u
,
p
,
·
,
…
,
·
,
X
s
]
0
I
=
[
2
N
θ
,
F
,
u
,
p
,
·
,
…
,
·
,
X
s
]
0
I
.Proof.
(i) To prove[
2
N
θ
,
u
,
p
,
·
,
…
,
·
,
X
s
]
I
=
[
2
N
θ
,
F
,
u
,
p
,
·
,
…
,
·
,
X
s
]
I, it is sufficient to show that [
2
N
θ
,
F
,
u
,
p
,
·
,
…
,
·
,
X
s
]
I
⊂
[
2
N
θ
,
u
,
p
,
·
,
…
,
·
,
X
s
]
I. Let x
∈
[
2
N
θ
,
F
,
u
,
p
,
·
,
…
,
·
,
X
s
]
I; then, by the definition, we get (21)
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
F
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
<
δ
.By the given condition lim
t
⁡
sup
⁡
(
f
i
j
t
/
t
)
=
A
>
0 for all (
i
,
j
)
∈
N
×
N, we have f
i
j
(
t
)
≥
A
t for all (
i
,
j
); that is, (22)
F
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
≥
A
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
,
(23)
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
F
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
≥
A
H
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
.From inequalities (21) and (23), we get (24)
δ
>
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
F
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
≥
A
H
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
jwhich consequently implies that (25)
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
<
δ
.That is, (26)
r
,
s
∈
N
×
N
:
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
≥
δ
∈
I
.
This implies thatx
∈
[
2
N
θ
,
u
,
p
,
·
,
…
,
·
,
X
s
]
I. (27)
2
N
θ
,
u
,
p
,
·
,
…
,
·
,
X
s
I
⊆
2
N
θ
,
F
,
u
,
p
,
·
,
…
,
·
,
X
s
I
.
Hence, from the previous theorem and inclusion (27), we get the required result. The other part can be proved similarly.Corollary 8.
LetF
′
=
(
f
i
j
′
) and F
′
′
=
(
f
i
j
′
′
) be sequences of moduli. If lim
t
⁡
sup
⁡
(
f
i
j
′
t
/
f
i
j
′
′
t
)
<
∞ for all (
i
,
j
)
∈
N
×
N, then(i)
[
2
N
θ
,
F
′
,
u
,
p
,
·
,
…
,
·
,
X
s
]
I
=
[
2
N
θ
,
F
′
′
,
u
,
p
,
·
,
…
,
·
,
X
s
]
I,
(ii)
[
2
N
θ
,
F
′
,
u
,
p
,
·
,
…
,
·
,
X
s
]
0
I
=
[
2
N
θ
,
F
′
′
,
u
,
p
,
·
,
…
,
·
,
X
s
]
0
I.Theorem 9.
Let(
X
,
·
,
…
,
·
X
S
) and (
X
,
·
,
…
,
·
X
E
) be the standard and the Euclidean n-norm spaces, respectively. Then (28)
2
N
θ
,
F
,
u
,
p
,
·
,
…
,
·
,
X
S
I
∩
2
N
θ
,
F
,
u
,
p
,
·
,
…
,
·
,
X
E
I
⊂
2
N
θ
,
F
,
u
,
p
,
·
,
…
,
·
X
S
+
·
,
…
,
·
X
E
I
.Proof.
The proof of this result is easy, so it is omitted.Theorem 10.
LetF
′
=
(
f
i
j
′
) and F
′
′
=
(
f
i
j
′
′
) be sequences of moduli; then(i)
[
2
N
θ
,
F
′
′
,
u
,
p
,
·
,
…
,
·
,
X
s
]
I
⊆
[
2
N
θ
,
F
′
∘
F
′
′
,
u
,
p
,
·
,
…
,
·
,
X
s
]
I,
(ii)
[
2
N
θ
,
F
′
′
,
u
,
p
,
·
,
…
,
·
,
X
s
]
0
I
⊆
[
2
N
θ
,
F
′
∘
F
′
′
,
u
,
p
,
·
,
…
,
·
,
X
s
]
0
I,
(iii)
[
2
N
θ
,
F
′
,
u
,
p
,
·
,
…
,
·
,
X
S
]
I
∩
[
2
N
θ
,
F
′
′
,
u
,
p
,
·
,
…
,
·
,
X
S
]
I
⊆
[
2
N
θ
,
F
′
+
F
′
′
,
u
,
p
,
·
,
…
,
·
,
X
S
]
I,
(iv)
[
2
N
θ
,
F
′
,
u
,
p
,
·
,
…
,
·
,
X
S
]
0
I
∩
[
2
N
θ
,
F
′
′
,
u
,
p
,
·
,
…
,
·
,
X
S
]
0
I
⊆
[
2
N
θ
,
F
′
+
F
′
′
,
u
,
p
,
·
,
…
,
·
,
X
S
]
0
I
.Proof.
(i) For someϵ
>
0, we choose ϵ
0
>
0 such that max
⁡
ϵ
0
H
<
ϵ. Now as F
′ is a sequence of modulus functions which are always continuous, we can choose δ
∈
(
0,1
) such that, for every t
∈
(
0
,
δ
), we get F
′
(
t
)
<
ϵ
0: (29)
x
=
x
i
j
∈
2
N
θ
,
F
′
′
,
u
,
p
,
·
,
…
,
·
,
X
s
I
.Then, by the definition, we have (30)
A
=
r
,
s
∈
N
×
N
:
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
F
′
′
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
≥
δ
H
∈
I
.Thus, for (
r
,
s
)
∉
A, we get (31)
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
F
′
′
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
<
δ
H
,
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
F
′
′
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
≤
δ
.Now, by the continuity of F
′, we have (32)
F
′
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
F
′
′
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
<
ϵ
,which further implies that (33)
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
F
′
F
′
′
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
<
max
⁡
ϵ
H
<
ϵ
,which implies (34)
r
,
s
∈
N
×
N
:
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
F
′
∘
F
′
′
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
≥
ϵ
∈
I
.Therefore x
∈
[
2
N
θ
,
F
′
∘
F
′
′
,
u
,
p
,
·
,
…
,
·
,
X
s
]
I. This completes the proof.(iii) Again consider(35)
x
∈
2
N
θ
,
F
′
,
u
,
p
,
·
,
…
,
·
,
X
s
I
∩
2
N
θ
,
F
′
′
,
u
,
p
,
·
,
…
,
·
,
X
s
I
.Then by the definition of both the spaces, we get (36)
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
F
′
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
<
ϵ
,
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
F
′
′
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
<
ϵ
.Using the fact that (
F
′
+
F
′
′
)
(
x
)
≤
K
F
′
(
x
)
+
K
F
′
′
(
x
), we have (37)
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
F
′
+
F
′
′
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
≤
K
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
F
′
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
+
K
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
F
′
′
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
≤
K
ϵ
+
ϵ
=
2
K
ϵ
=
ϵ
′
say
.So we get (38)
r
,
s
∈
N
×
N
:
1
h
r
,
s
∑
i
,
j
∈
I
r
,
s
u
i
j
F
′
+
F
′
′
x
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
p
i
j
≥
ϵ
∈
I
.Therefore we have (39)
x
∈
2
N
θ
,
F
′
+
F
′
′
,
u
,
p
,
·
,
…
,
·
,
X
s
I
.Hence we have (40)
2
N
θ
,
F
′
,
u
,
p
,
·
,
…
,
·
,
X
S
I
∩
2
N
θ
,
F
′
′
,
u
,
p
,
·
,
…
,
·
,
X
S
I
⊆
2
N
θ
,
F
′
+
F
′
′
,
u
,
p
,
·
,
…
,
·
,
X
S
I
.


## 4. Conclusion

A detailed study of some new classes ofI-convergent double sequences over n-normed spaces has been done. Some algebraic and topological properties and inclusion relations have been proved with supported examples.



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*Source: 7594031-2016-05-09.xml*