# On the Ideal Convergence of Double Sequences in Locally Solid Riesz Spaces

Authors: A. Alotaibi; B. Hazarika; S. A. Mohiuddine
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The aim of this paper is to define the notions of ideal convergence,I-bounded for double sequences in setting of locally solid Riesz spaces and study some results related to these notions. We also define the notion of I
*-convergence for double sequences in locally solid Riesz spaces and establish its relationship with ideal convergence.


---

## Body

## 1. Introduction and Preliminaries

In 1951, Fast [1] and Steinhaus [2] introduced the concept of statistical convergence for single sequences, independently. Some basic and important properties of this concept were studied by Buck [3], Šalát [4], Schoenberg [5], and Fridy [6]. Later, the notion of statistical convergence for single sequences was further defined in various spaces; see Çakalli and Khan [7–9], Di Maio et al. [10, 11], Hazarika [12–14], Maddox [15], Mohiuddine et al. [16–19], and so forth. Some application of statistical summability methods is presented in [20, 21]. In 2003, the notion of statistical convergence for single sequences has been extended to double sequences by Mursaleen and Edely [22]. Recently, the statistical convergence and statistical Cauchy for double sequences have been defined in the framework fuzzy and intuitionistic normed spaces by Mohiuddine et al. [23] and Mursaleen and Mohiuddine [24], respectively, and established some interesting results related to the concept of statistical convergence and statistical Cauchy double sequences. Recently, it was defined and studied by Mohiuddine et al. [25] in the setting of locally solid Riesz spaces while for single sequences this concept was first studied by Albayrak and Pehlivan [26] (also see [27–29]). An application of locally solid Riesz spaces in economics can be found in [30].The notion of ideal convergence for single sequences, which is a generalization of the concept of statistical convergence, was first defined and studied by Kostyrko et al. [31]. Let us recall the notion of ideal convergence and related concepts by Kostyrko et al. [31] as follows. Let ℕ be a nonempty set. Then a family of sets I
⊆
P
(
ℕ
) (power set of ℕ) is said to be an ideal if I is additive; that is, A
,
B
∈
I
⇒
A
∪
B
∈
I and A
∈
I
,
B
⊆
A
⇒
B
∈
I. A family of sets I
⊂
P
(
ℕ
) (power sets of ℕ) is called anideal if and only if, for each A
,
B
∈
I, we have A
∪
B
∈
I and, for each A
∈
I and each B
⊂
A, we have B
∈
I. A nonempty family of sets ℱ
⊂
P
(
ℕ
) is afilter on ℕ if and only if Φ
∉
ℱ; for each A
,
B
∈
ℱ, we have A
∩
B
∈
ℱ and for each A
∈
ℱ and each A
⊂
B, we have B
∈
ℱ. An ideal I is called nontrivial ideal if I
≠
Φ and ℕ
∉
I. Clearly I
⊂
P
(
ℕ
) is a nontrivial ideal if and only if ℱ
=
ℱ
(
I
)
=
{
ℕ
-
A
:
A
∈
I
} is a filter on ℕ. A nontrivial ideal I
⊂
P
(
ℕ
) is calledadmissible if and only if {
{
x
}
:
x
∈
ℕ
}
⊂
I. A nontrivial ideal I ismaximal if there cannot exist any nontrivial ideal J
≠
I containing I as a subset.We remark that ifI
=
I
f
=
{
A
⊆
ℕ
:
A
is
a
finite
subset
}, then the corresponding convergence coincides with the usual convergence. Also, if I
=
I
δ
=
{
A
⊆
ℕ
:
δ
(
A
)
=
0
}, then the corresponding convergence coincides with the statistical convergence (where δ
(
A
) denotes the natural density of the set A). In the above cases, both I
f and I
δ are nontrivial admissible ideals of ℕ.Kumar [32] defined the notions of I and I
*-convergence of double sequence and studied some properties of these notions. Recently, Das et al. [33] introduced the concepts of I and I
*-convergence of double sequences in the setting of metric space and established some relationship between these types of convergence. Quite recently, Mursaleen and Mohiuddine defined and studied the notion of I-convergence, I
*-convergence, I-limit points, and I-cluster points for single and double sequences, in [34, 35], respectively, in probabilistic normed spaces. Şahiner et al. [36] and Gürdal and Açik [37] introduced the notion of ideal convergence and I-Cauchy sequence in 2-normed spaces, respectively. Mursaleen and Alotaibi [38] introduced the notion of ideal convergence in random 2-normed spaces and later on it was extended by Mohiuddine et al. [39] from single to double sequences. For more details on these concepts, one can be referred to [40–52].Now we recall the definition of locally solid Riesz spaces and some related concepts as follows. LetX be a real vector space and let ≤ be a partial order on this space. X is said to be anordered vector space if it satisfies the following properties:(1)
ifx
,
y
∈
X and y
≤
x, then y
+
z
≤
x
+
z for each z
∈
X;
(2)
ifx
,
y
∈
X and y
≤
x, then a
y
≤
a
x for each a
≥
0.If, in addition, X is a lattice with respect to the partial ordering, then X is said to be aRiesz space (or avector lattice) (see [53]).For an elementx of a Riesz space X, the positive part of x is defined by x
+
=
x
∨
θ
¯
=
sup
⁡
{
x
,
θ
¯
}, the negative part of x by x
-
=
(
-
x
)
∨
θ
¯, and the absolute value of x by |
x
|
=
x
∨
(
-
x
), where θ
¯ is the zero element of X.A subsetS of X is said to besolid if y
∈
S and |
x
|
≤
|
y
| implies x
∈
S.A topologyτ on a real vector space X that makes the addition and scalar multiplication continuous is said to be a linear topology, that is, when the mappings
(1)
(
x
,
y
)
⟶
(
x
+
y
)
(
from
(
X
×
X
,
τ
×
τ
)
⟶
(
X
,
τ
)
)
,
(
λ
,
x
)
⟶
(
λ
x
)
(
from
(
ℝ
×
X
,
τ
′
×
τ
)
⟶
(
X
,
τ
)
)
are continuous, where τ
′ is the usual topology on ℝ. In this case the pair (
X
,
τ
) is called atopological vector space.Every linear topologyτ on a vector space X has a base N for the neighborhoods of θ
¯ satisfying the following properties.(1)
EachY
∈
N is abalanced set; that is, a
x
∈
Y holds for all x
∈
Y and every a
∈
ℝ with |
a
|
≤
1.
(2)
EachY
∈
N is anabsorbing set; that is, for every x
∈
X, there exists a
>
0 such that a
x
∈
Y.
(3)
For eachY
∈
N there exists some E
∈
N with E
+
E
⊆
Y.A linear topologyτ on a Riesz space X is said to belocally solid (see [54]) if τ has a base at zero consisting of solid sets. Alocally solid Riesz space
(
X
,
τ
) is a Riesz space X equipped with a locally solid topology τ. For more details on these concepts, one can be referred to [55–57].Throughout the paper, the symbolN
sol will stand for a base at zero consisting of solid sets and satisfying conditions (1), (2), and (3) in a locally solid topology. Also we assume that I
2 is a nontrivial admissible ideal of ℕ
×
ℕ.


## 2. Ideal Convergence of Double Sequences in LSR-Spaces

Throughout the paperX will denote the Hausdorff locally solid Riesz space, which satisfies the first axiom of countability. For our convenience, here and in what follows, we will write an LSR-space instead of a locally solid Riesz space.The notion of convergence for double sequence was first introduced by Pringsheim [58] as follows. We say that a double sequence x
=
(
x
j
,
k
)
j
,
k
∈
ℕ of reals is convergent to L in Pringsheim’s sense (briefly, P-convergent) provided that given ϵ
>
0 there exists a positive integer N such that |
x
j
,
k
-
L
|
<
ϵ whenever j
,
k
≥
N.LetK
⊂
ℕ
×
ℕ and K
(
m
,
n
) denotes the number of (
i
,
j
) in K such that i
≤
m and j
≤
n (see [22]). Then the lower natural density of K is defined by δ
_
2
(
K
)
=
lim
⁡
inf
⁡
m
,
n
→
∞
(
|
K
(
m
,
n
)
|
/
m
n
). In this case, the sequence (
K
(
m
,
n
)
/
m
n
) has a limit in Pringsheim’s sense; then we say that K has adouble natural density and is defined by P
-
lim
⁡
m
,
n
→
∞
(
|
K
(
m
,
n
)
|
/
m
n
)
=
δ
2
(
K
).In the recent past, Mohiuddine et al. [25] introduced the notion of statistical convergence of double sequences in LSR-space as follows. Let (
X
,
τ
) be a LSR-space. A double sequence (
x
k
,
l
) of points in X is said to be S
2
(
τ
)-convergent to an element x
0 if for each τ-neighborhood V of zero
(2)
δ
2
(
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
x
0
∉
V
}
)
=
0
.
Now we introduce the notions of I
2
(
τ
)-convergence and I
2
(
τ
)-bounded double sequences in LSR-spaces.Definition 1.
Let(
X
,
τ
) be a LSR-space. A double sequence (
x
k
,
l
) of points in X is said to be I
2
(
τ
)-convergent to an element x
0 of X if for each τ-neighborhood V of zero
(3)
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
x
0
∉
V
}
∈
I
2
.
That is,
(4)
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
x
0
∈
V
}
∈
ℱ
.
In this case, one writes I
2
(
τ
)
-
lim
⁡
k
,
l
→
∞
x
k
,
l
=
x
0 or (
x
k
,
l
)
→
I
2
(
τ
)
x
0.Definition 2.
Let(
X
,
τ
) be a LSR-space. Then, a double sequence (
x
k
,
l
) of points in X is said to be I
2
(
τ
)-bounded in X if, for each τ-neighborhood V of zero, there is some a
>
0,
(5)
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
a
x
k
,
l
∉
V
}
∈
I
2
.Definition 3.
Let(
X
,
τ
) be a LSR-space. One says that a double sequence x
=
(
x
k
,
l
) is I
2
(
τ
)-Cauchy in X if, for each τ-neighborhood V of zero, there exist p
,
q
∈
ℕ such that, for all k
,
m
≥
p and l
,
n
≥
q,
(6)
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
x
m
,
n
∉
V
}
∈
I
2
.Definition 4.
Let(
X
,
τ
) be a LSR-space. Then, a double sequence x
=
(
x
k
,
l
) in X is said to be I
2
*
(
τ
)-convergent to x
0 if there is a set K
=
{
(
k
,
l
)
}
⊆
ℕ
×
ℕ, k
,
l
=
1,2
,
…, with K
∈
F such that lim
⁡
k
,
l
x
k
,
l
=
x
0. In this case, one writes I
2
*
(
τ
)-lim
⁡
k
,
l
x
k
,
l
=
x
0.Theorem 5.
Let(
X
,
τ
) be a LSR-space. Every I
2
(
τ
)-convergent sequence in X has only one limit.Proof.
Suppose thatx
=
(
x
k
,
l
) is a double sequence in X such that I
2
(
τ
)-lim
⁡
k
,
l
x
k
,
l
=
x
0 and I
2
(
τ
)-lim
⁡
k
,
l
x
k
,
l
=
y
0. Let V be any τ-neighborhood of zero. Also for each τ-neighborhood V of zero there is a set Y
∈
N
sol such that Y
⊆
V. Let W in N
sol be such that W
+
W
⊆
Y. We define the sets A
1 and A
2 as follows:
(7)
A
1
=
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
x
0
∈
W
}
,
A
2
=
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
y
0
∈
W
}
.
Since I
2
(
τ
)
-lim
x
k
,
l
=
x
0 and I
2
(
τ
)
-lim
x
k
,
l
=
y
0, we get A
1
,
A
2
∈
ℱ. Now, let A
=
A
1
∩
A
2. Then we have
(8)
x
0
-
y
0
=
x
0
-
x
k
,
l
+
x
k
,
l
-
y
0
∈
W
+
W
⊆
Y
⊆
V
.
As we know, intersection of all τ-neighborhoods V of zero is the singleton set {
θ
¯
} because (
X
,
τ
) is Hausdorff. Hence x
0
-
y
0
=
0; that is, x
0
=
y
0.Theorem 6.
Let(
X
,
τ
) be a LSR-space and let (
x
k
,
l
) and (
y
k
,
l
) be two double sequences of points in X. Then,(i)
ifI
2
(
τ
)-lim
⁡
k
,
l
x
k
,
l
=
x
0 and I
2
(
τ
)-lim
⁡
k
,
l
y
k
,
l
=
y
0, then I
2
(
τ
)-lim
⁡
k
,
l
(
x
k
,
l
+
y
k
,
l
)
=
x
0
+
y
0;
(ii)
ifI
2
(
τ
)-lim
⁡
k
,
l
x
k
,
l
=
x
0, then I
2
(
τ
)-lim
⁡
k
,
l
a
x
k
,
l
=
a
x
0 for a
∈
ℝ.Proof.
Assume thatI
2
(
τ
)-lim
⁡
k
,
l
x
k
,
l
=
x
0 and I
2
(
τ
)-lim
⁡
k
,
l
y
k
,
l
=
y
0. Suppose that V is an arbitrary τ-neighborhood of zero. Then there exists Y
∈
N
sol such that Y
⊆
V. Let W
∈
N
sol such that W
+
W
⊆
Y. Thus, we can write
(9)
B
1
=
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
x
0
∈
W
}
,
B
2
=
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
y
k
,
l
-
y
0
∈
W
}
.
Then we have B
1
,
B
2
∈
ℱ.
LetB
=
B
1
∩
B
2. Hence we have B
∈
ℱ and
(10)
(
x
k
,
l
+
y
k
,
l
)
-
(
x
0
+
y
0
)
=
(
x
k
,
l
-
x
0
)
+
(
y
k
,
l
-
y
0
)
∈
W
+
W
⊆
Y
⊆
V
.
Therefore
(11)
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
(
x
k
,
l
+
y
k
,
l
)
-
(
x
0
+
y
0
)
∈
V
}
∈
ℱ
.
Since V is arbitrary, we have I
2
(
τ
)
-lim
(
x
k
,
l
+
y
k
,
l
)
=
x
0
+
y
0.
(ii) Suppose thatI
2
(
τ
)-lim
⁡
k
,
l
x
k
,
l
=
x
0 and also suppose that V is an arbitrary τ-neighborhood of zero. Then there exists Y
∈
N
sol such that Y
⊆
V, so we have
(12)
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
x
0
∈
Y
}
∈
ℱ
.
Since Y is balanced, a
(
x
k
,
l
-
x
0
)
∈
Y holds for all x
k
,
l
-
x
0
∈
Y and for every a
∈
ℝ with |
a
|
≤
1. Therefore
(13)
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
x
0
∈
Y
}
⊆
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
a
x
k
,
l
-
a
x
0
∈
Y
}
⊆
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
x
0
∈
V
}
.
Thus, we have
(14)
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
x
0
∈
V
}
∈
ℱ
for each τ-neighborhood V of zero. Now let |
a
|
>
1 and [
|
a
|
] be the smallest integer greater than or equal to |
a
|. Then there exists W
∈
N
sol such that [
|
a
|
]
W
⊆
Y. From our assumption that I
2
(
τ
)-lim
⁡
k
,
l
x
k
,
l
=
x
0, we obtain that
(15)
K
=
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
x
0
∈
W
}
∈
ℱ
.
Therefore
(16)
|
a
x
k
,
l
-
a
x
0
|
=
|
a
|
|
x
k
,
l
-
x
0
|
≤
[
|
a
|
]
|
x
k
,
l
-
x
0
|
∈
[
|
a
|
]
W
⊆
Y
⊆
V
.
Since Y is solid, a
x
k
-
a
x
0
∈
Y. It follows that a
x
k
,
l
-
a
x
0
∈
V. Thus,
(17)
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
a
x
k
,
l
-
a
x
0
∈
V
}
∈
ℱ
,
for each τ-neighborhood V of zero. We conclude that I
2
(
τ
)-lim
⁡
k
,
l
a
x
k
,
l
=
a
x
0.Theorem 7.
Let(
X
,
τ
) be a LSR-space. If a double sequence (
x
k
,
l
) in X is I
2
(
τ
)-convergent, then it is I
2
(
τ
)-bounded.Proof.
Assume thatI
2
(
τ
)-lim
⁡
k
,
l
→
∞
x
k
,
l
=
x
0. Suppose V is an arbitrary τ-neighborhood of zero. Then, there exists Y
∈
N
sol such that Y
⊆
V. Let W
∈
N
sol such that W
+
W
⊆
Y. Using our assumption, we obtain that
(18)
A
=
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
x
0
∉
W
}
∈
I
2
.
Since W is absorbing, there exists a
>
0 such that a
x
0
∈
W. Let b be such that |
b
|
≤
1 and b
≤
a. Since W is solid and |
b
x
0
|
≤
|
a
x
0
|, we have b
x
0
∈
W. Also, since W is balanced, x
k
,
l
-
x
0
∈
W implies b
(
x
k
,
l
-
x
0
)
∈
W. Then we have
(19)
b
x
k
,
l
=
b
(
x
k
,
l
-
x
0
)
+
b
x
0
∈
W
+
W
⊆
V
,
for
each
k
,
l
∈
ℕ
-
A
.
Thus
(20)
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
b
x
k
,
l
∉
W
}
∈
I
2
.
Hence (
x
k
,
l
) is I
2
(
τ
)-bounded.Theorem 8.
Let(
X
,
τ
) be a LSR-space and let (
x
k
,
l
), (
y
k
,
l
), and (
z
k
,
l
) be three double sequences of points in X such that(i)
x
k
,
l
≤
y
k
,
l
≤
z
k
,
l
, for all
k
,
l
∈
ℕ
,
(ii)
I
2
(
τ
)-lim
⁡
k
,
l
x
k
,
l
=
x
0
=
I
2
(
τ
)-lim
⁡
k
,
l
z
k
,
l.
Then
I
2
(
τ
)-lim
⁡
k
,
l
y
k
,
l
=
x
0
.Proof.
Suppose that the given conditions (i) and (ii) hold for the double sequences(
x
k
,
l
), (
y
k
,
l
), and (
z
k
,
l
). Suppose V is an arbitrary τ-neighborhood of zero. Then, there exists Y
∈
N
sol such that Y
⊆
V. Let W
∈
N
sol such that W
+
W
⊆
Y. It follows from (ii) that P
,
Q
∈
F, where
(21)
P
=
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
x
0
∈
W
}
,
Q
=
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
z
k
,
l
-
x
0
∈
W
}
.
Also from the given condition (i), we have
(22)
x
k
,
l
-
x
0
≤
y
k
,
l
-
x
0
≤
z
k
,
l
-
x
0
⟹
|
y
k
,
l
-
x
0
|
≤
|
x
k
,
l
-
x
0
|
+
|
z
k
,
l
-
x
0
|
∈
W
+
W
⊆
Y
.
Since Y is solid, we have y
k
,
l
-
x
0
∈
Y
⊆
V. Thus,
(23)
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
y
k
,
l
-
x
0
∈
V
}
∈
ℱ
,
for each τ-neighborhood V of zero. Thus I
2
(
τ
)-lim
⁡
k
,
l
y
k
,
l
=
x
0.Theorem 9.
Let(
X
,
τ
) be a LSR-space. A double sequence (
x
k
,
l
) is I
2
(
τ
)-convergent to x
0 in X if and only if for each τ-neighborhood V of zero there exists a subsequence (
x
k
′
(
r
)
,
l
′
(
s
)
) of (
x
k
,
l
) such that lim
⁡
r
,
s
→
∞
x
k
′
(
r
)
,
l
′
(
s
)
=
x
0 and
(24)
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
x
k
′
(
r
)
,
l
′
(
s
)
∉
V
}
∈
I
2
.Proof.
Suppose thatI
2
(
τ
)
-
lim
⁡
k
,
l
→
∞
x
k
,
l
=
x
0. Also, suppose that V is an arbitrary τ-neighborhood of zero. Let {
V
i
} be a sequence of nested base of τ-neighborhoods of zero. For each i
∈
ℕ, put
(25)
E
(
i
)
=
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
x
0
∉
V
i
}
.
Then, E
(
i
+
1
)
⊂
E
(
i
) and E
(
i
)
∈
F. Let m
(
1
) and n
(
1
) be such that r
>
m
(
1
) and s
>
n
(
1
), respectively. Then E
(
1
)
≠
ϕ. For r
,
s
∈
ℕ such that m
(
1
)
≤
r
<
m
(
2
) and n
(
1
)
≤
s
<
n
(
2
), choose k
′
(
r
)
,
l
′
(
s
)
∈
E
(
i
); that is, x
k
′
(
r
)
,
l
′
(
s
)
-
x
0
∈
V
1. In general, choose m
(
p
+
1
)
>
m
(
p
) and n
(
p
+
1
)
>
n
(
p
) such that r
>
m
(
p
+
1
) and s
>
n
(
p
+
1
) hold. Then E
(
p
+
1
)
≠
ϕ. Therefore for all r
,
s which satisfy m
(
p
)
≤
r
<
m
(
p
+
1
) and n
(
p
)
≤
s
<
n
(
p
+
1
), choose k
′
(
r
)
,
l
′
(
s
)
∈
E
(
p
); that is, x
k
′
(
r
)
,
l
′
(
s
)
-
x
0
∈
V
p. Hence, it follows that lim
⁡
r
,
s
x
k
′
(
r
)
,
l
′
(
s
)
=
x
0.
SinceV is an arbitrary τ-neighborhood of zero, there exists Y
∈
N
sol such that Y
⊆
V. Let W
∈
N
sol such that W
+
W
⊆
Y. Now
(26)
x
k
,
l
-
x
k
′
(
r
)
,
l
′
(
s
)
=
x
k
,
l
-
x
0
+
x
k
′
(
r
)
,
l
′
(
s
)
-
x
0
∈
W
+
W
⊆
Y
⊆
V
.
Also I
2
(
τ
)
-
lim
⁡
k
,
l
→
∞
x
k
,
l
=
x
0 and lim
⁡
r
→
∞
x
k
′
(
r
)
,
l
′
(
s
)
=
x
0 imply that
(27)
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
x
k
′
(
r
)
,
l
′
(
s
)
∉
V
}
∈
I
2
.
Next suppose for an arbitrary τ-neighborhood V of zero that there exists a subsequence (
x
k
′
(
r
)
,
l
′
(
s
)
) of (
x
k
,
l
) such that lim
⁡
r
,
s
→
∞
x
k
′
(
r
)
,
l
′
(
s
)
=
x
0 and
(28)
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
x
k
′
(
r
)
,
l
′
(
s
)
∉
V
}
∈
I
2
.
Since V is any τ-neighborhood of zero, we choose W
∈
N
sol such that W
+
W
⊆
V. Then we have
(29)
x
k
,
l
-
x
0
=
x
k
,
l
-
x
k
′
(
r
)
,
l
′
(
s
)
+
x
k
′
(
r
)
,
l
′
(
s
)
-
x
0
∈
W
+
W
⊆
V
.
That is,
(30)
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
x
0
∉
V
}
⊆
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
x
k
′
(
r
)
∉
W
}
∪
{
(
r
,
s
)
∈
ℕ
×
ℕ
:
x
k
′
(
r
)
,
l
′
(
s
)
-
x
0
∉
W
}
.
Therefore
(31)
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
x
0
∉
V
}
∈
I
2
.Theorem 10.
Iflim
⁡
k
,
l
→
∞
x
k
,
l
=
x
0 and I
2
(
τ
)
-
lim
⁡
k
,
l
→
∞
y
k
,
l
=
0, then I
2
(
τ
)
-
lim
⁡
k
,
l
→
∞
(
x
k
,
l
+
y
k
,
l
)
=
lim
⁡
k
,
l
→
∞
x
k
,
l.Proof.
LetV be any τ-neighborhood of 0. Then there exists Y
∈
N
sol such that Y
⊆
V. Let W
∈
N
sol such that W
+
W
⊆
Y. Since lim
⁡
k
,
l
→
∞
x
k
,
l
=
x
0, then there exist integers n
0
,
m
0 such that k
≥
n
0
,
l
≥
m
0 implies that x
k
,
l
-
x
0
∈
W. Hence
(32)
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
x
0
∉
W
}
⊆
ℕ
×
ℕ
-
{
(
n
0
,
m
0
)
}
.
By the assumption I
2
(
τ
)
-
lim
⁡
k
,
l
→
∞
y
k
,
l
=
0, {
(
k
,
l
)
∈
ℕ
×
ℕ
:
y
k
,
l
∉
W
}
∈
I
2. Thus
(33)
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
(
x
k
,
l
-
x
0
)
+
y
k
,
l
∉
V
}
⊆
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
x
0
∉
W
}
∪
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
y
k
,
l
∉
W
}
.
That is,
(34)
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
(
x
k
,
l
-
x
0
)
+
y
k
,
l
∉
V
}
∈
I
2
.
This implies that I
2
(
τ
)
-
lim
⁡
k
,
l
→
∞
(
x
k
,
l
+
y
k
,
l
)
=
lim
⁡
k
,
l
→
∞
x
k
,
l.Theorem 11.
Let(
X
,
τ
) be a LSR-space and let x
=
(
x
k
,
l
) be a double sequence in X. If there is a I
2
(
τ
)-convergent sequence y
=
(
y
k
,
l
) in X such that {
(
k
,
l
)
∈
ℕ
×
ℕ
:
y
k
,
l
≠
x
k
,
l
∉
V
}
∈
I
2 then x is also I
2
(
τ
)-convergent.Proof.
Suppose that{
(
k
,
l
)
∈
ℕ
×
ℕ
:
y
k
,
l
≠
x
k
,
l
∉
V
}
∈
I
2 and I
2
(
τ
)
-
lim
⁡
k
,
l
y
k
,
l
=
x
0. Then for an arbitrary τ-neighborhood V of zero, we have
(35)
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
y
k
,
l
-
x
0
∉
V
}
∈
I
2
.
Now,
(36)
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
x
0
∉
V
}
⊆
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
y
k
,
l
≠
x
k
,
l
∉
V
}
∪
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
y
k
,
l
-
x
0
∉
V
}
.
Therefore, we have
(37)
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
x
0
∉
V
}
∈
I
2
.Theorem 12.
Let(
X
,
τ
) be a LSR-space. If a double sequence x
=
(
x
k
,
l
) is I
2
*
(
τ
)-convergent to x
0, then it is I
2
(
τ
)-convergent to x
0.Proof.
Suppose thatI
2
*
(
τ
)-lim
⁡
k
x
k
,
l
=
x
0. Let V be an arbitrary τ-neighborhood V of zero. Since I
2
*
(
τ
)-lim
⁡
k
,
l
x
k
,
l
=
x
0, there is a set K
=
{
(
k
,
l
)
}
⊆
ℕ
×
ℕ, (
k
,
l
∈
ℕ
) with K
∈
F such that k
≥
n, l
≥
m and (
k
,
l
)
∈
K implies x
k
,
l
-
x
0
∈
V. Then
(38)
K
1
=
{
(
k
,
l
)
∈
ℕ
×
ℕ
:
x
k
,
l
-
x
0
∉
V
}
⊆
ℕ
×
ℕ
-
{
(
k
n
+
1
,
l
m
+
1
)
,
(
k
n
+
2
,
l
m
+
2
)
,
…
}
.
Therefore
(39)
K
1
∈
I
2
.
Hence x is I
2
(
τ
)-convergent to x
0.Theorem 13.
The sequential methodI
2
(
τ
) is regular.Proof of the theorem is straightforward, so it is omitted.From Theorem12, we can easily obtain the following useful result.Theorem 14.
The sequential methodI
2
(
τ
) is subsequential.



---

*Source: 396254-2014-04-02.xml*