# Spaces of Ideal Convergent Sequences

Authors: M. Mursaleen; Sunil K. Sharma
Journal: The Scientific World Journal
 (2014)

---

## Abstract

In the present paper, we introduce some sequence spaces using ideal convergence and Musielak-Orlicz functionℳ
=
M
k. We also examine some topological properties of the resulting sequence spaces.


---

## Body

## 1. Introduction and Preliminaries

The notion of ideal convergence was first introduced by Kostyrko et al. [1] as a generalization of statistical convergence which was further studied in topological spaces by Das et al. see [2]. More applications of ideals can be seen in [2, 3]. We continue in this direction and introduce I-convergence of generalized sequences with respect to Musielak-Orlicz function.A familyℐ
⊂
2
X of subsets of a nonempty set X is said to be an ideal in X if(1)
ϕ
∈
ℐ,
(2)
A
,
B
∈
ℐ imply A
∪
B
∈
ℐ,
(3)
A
∈
ℐ, B
⊂
A imply B
∈
ℐ,while an admissible ideal ℐ of X further satisfies {
x
}
∈
ℐ for each x
∈
X; see [1]. A sequence (
x
n
)
n
∈
ℕ in X is said to be I-convergent to x
∈
X. If for each ϵ
>
0, the set A
(
ϵ
)
=
{
n
∈
ℕ
:
∥
x
n
-
x
∥
≥
ϵ
} belongs to ℐ; see [1]. For more details about ideal convergent  sequence spaces, see [4–10] and references therein.Mursaleen and Noman [11] introduced the notion of λ-convergent and λ-bounded sequences as follows.Letλ
=
(
λ
k
)
k
=
1
∞ be a strictly increasing sequence of positive real numbers tending to infinity; that is,
(1)
0
<
λ
0
<
λ
1
<
⋯
,
λ
k
⟶
∞
as
k
⟶
∞
.
The sequence x
=
(
x
k
)
∈
w is λ-convergent to the number L, called the λ-limit of x, if Λ
m
(
x
)
→
L, as m
→
∞, where
(2)
Λ
m
(
x
)
=
1
λ
m
∑
k
=
1
m
‍
(
λ
k
-
λ
k
-
1
)
x
k
.The sequencex
=
(
x
k
)
∈
w is λ-bounded if sup
⁡
m
|
Λ
m
(
x
)
|
<
∞. It is well known [11] that if lim
⁡
m
x
m
=
a in the ordinary sense of convergence, then
(3)
lim
⁡
m
(
1
λ
m
(
∑
k
=
1
m
‍
(
λ
k
-
λ
k
-
1
)
|
x
k
-
a
|
)
)
=
0
.This implies that(4)
lim
⁡
m
|
Λ
m
(
x
)
-
a
|
=
lim
⁡
m
|
1
λ
m
∑
k
=
1
m
‍
(
λ
k
-
λ
k
-
1
)
(
x
k
-
a
)
|
=
0
,
which yields that lim
⁡
m
Λ
m
(
x
)
=
a and hence x
=
(
x
k
)
∈
w is λ-convergent to a.LetX be a linear metric space. A function p
:
X
→
ℝ is called paranorm if(1)
p
(
x
)
≥
0, for all x
∈
X,
(2)
p
(
-
x
)
=
p
(
x
), for all x
∈
X,
(3)
p
(
x
+
y
)
≤
p
(
x
)
+
p
(
y
), for all x
,
y
∈
X,
(4)
if(
λ
n
) is a sequence of scalars with λ
n
→
λ as n
→
∞ and (
x
n
) is a sequence of vectors with p
(
x
n
-
x
)
→
0 as n
→
∞, then p
(
λ
n
x
n
-
λ
x
)
→
0 as n
→
∞.A paranormp for which p
(
x
)
=
0 implies that x
=
0 is called total paranorm and the pair (
X
,
p
) is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see [12, Theorem 10.4.2, P-183]). For more details about sequence spaces, see [13–15] and references therein.An Orlicz functionM is a function which is continuous, nondecreasing, and convex with M
(
0
)
=
0, M
(
x
)
>
0 for x
>
0 and M
(
x
)
→
∞ as x
→
∞.Lindenstrauss and Tzafriri [16] used the idea of Orlicz function to define the following sequence space. Let w be the space of all real or complex sequences x
=
(
x
k
). Then,
(5)
ℓ
M
=
{
x
∈
w
:
∑
k
=
1
∞
‍
M
(
|
x
k
|
ρ
)
<
∞
}
which is called an Orlicz sequence space. The space ℓ
M is a Banach space with the norm
(6)
∥
x
∥
=
inf
⁡
{
ρ
>
0
:
∑
k
=
1
∞
‍
M
(
|
x
k
|
ρ
)
≤
1
}
.It is shown in [16] that every Orlicz sequence space ℓ
M contains a subspace isomorphic to ℓ
p
(
p
≥
1
). The Δ
2-condition is equivalent to M
(
L
x
)
≤
k
L
M
(
x
) for all values of x
≥
0 and for L
>
1.A sequenceℳ
=
(
M
k
) of Orlicz function is called a Musielak-Orlicz function see; [17, 18]. A sequence 𝒩
=
(
N
k
) defined by
(7)
N
k
(
v
)
=
sup
⁡
{
|
v
|
u
-
(
M
k
)
:
u
≥
0
}
,
k
=
1,2
,
…
,
is called the complementary function of a Musielak-Orlicz function ℳ. For a given Musielak-Orlicz function ℳ, the Musielak-Orlicz sequence space t
ℳ and its subspace h
ℳ are defined as follows:
(8)
t
ℳ
=
{
x
∈
w
:
I
ℳ
(
c
x
)
<
∞
for
some
c
>
0
}
,
h
ℳ
=
{
x
∈
w
:
I
ℳ
(
c
x
)
<
∞
for
all
c
>
0
}
,
where I
ℳ is a convex modular defined by
(9)
I
ℳ
(
x
)
=
∑
k
=
1
∞
‍
M
k
(
x
k
)
,
x
=
(
x
k
)
∈
t
ℳ
.We considert
ℳ equipped with the Luxemburg norm
(10)
∥
x
∥
=
inf
⁡
{
k
>
0
:
I
ℳ
(
x
k
)
≤
1
}
,
or equipped with the Orlicz norm
(11)
∥
x
∥
0
=
inf
⁡
{
1
k
(
1
+
I
ℳ
(
k
x
)
)
:
k
>
0
}
.Letℳ
=
(
M
k
) be a Musielak-Orlicz function and let p
=
(
p
k
) be a bounded sequence of positive real numbers. We define the following sequence spaces:
(12)
c
I
(
ℳ
,
Λ
,
p
)
=
{
x
=
(
x
k
)
∈
w
:
I
-
lim
⁡
k
M
k
(
|
Λ
k
(
x
)
-
L
|
ρ
)
p
k
=
0
,
H
h
h
H
for
some
L
and
ρ
>
0
(
|
Λ
k
(
x
)
-
L
|
ρ
)
p
k
}
,
c
0
I
(
ℳ
,
Λ
,
p
)
=
{
x
=
(
x
k
)
∈
w
:
I
-
lim
⁡
k
M
k
(
|
Λ
k
(
x
)
|
ρ
)
p
k
=
0
,
hhhh
for
some
ρ
>
0
(
|
Λ
k
(
x
)
|
ρ
)
p
k
}
,
l
∞
(
ℳ
,
Λ
,
p
)
=
{
x
=
(
x
k
)
∈
w
:
sup
⁡
k
M
k
(
|
Λ
k
(
x
)
|
ρ
)
p
k
<
∞
,
hhhhhhhhhhhh
for
some
ρ
>
0
(
|
Λ
k
(
x
)
|
ρ
)
p
k
}
.We can write(13)
m
I
(
ℳ
,
Λ
,
p
)
=
c
I
(
ℳ
,
Λ
,
p
)
∩
l
∞
(
ℳ
,
Λ
,
p
)
,
m
0
I
(
ℳ
,
Λ
,
p
)
=
c
0
I
(
ℳ
,
Λ
,
p
)
∩
l
∞
(
ℳ
,
Λ
,
p
)
.If we takep
=
(
p
k
)
=
1, for all k
∈
ℕ, we have
(14)
c
I
(
ℳ
,
Λ
)
=
{
x
=
(
x
k
)
∈
w
:
I
-
lim
⁡
k
M
k
(
|
Λ
k
(
x
)
-
L
|
ρ
)
=
0
,
h
h
h
h
.
h
for
some
L
and
ρ
>
0
|
Λ
k
(
x
)
-
L
|
ρ
}
,
c
0
I
(
ℳ
,
Λ
)
=
{
x
=
(
x
k
)
∈
w
:
I
-
lim
⁡
k
M
k
(
|
Λ
k
(
x
)
|
ρ
)
=
0
,
h
h
h
h
h
h
h
h
h
h
.
for
some
ρ
>
0
(
|
Λ
k
(
x
)
|
ρ
)
p
k
}
,
l
∞
(
ℳ
,
Λ
)
=
{
x
=
(
x
k
)
∈
w
:
sup
⁡
k
M
k
(
|
Λ
k
(
x
)
|
ρ
)
<
∞
,
h
h
h
h
h
h
h
h
h
h
.
for
some
ρ
>
0
(
|
Λ
k
(
x
)
|
ρ
)
p
k
}
.The following inequality will be used throughout the paper. If0
≤
p
k
≤
sup
⁡
p
k
=
H, D
=
max
⁡
(
1
,
2
H
-
1
), then
(15)
|
a
k
+
b
k
|
p
k
≤
D
{
|
a
k
|
p
k
+
|
B
k
|
p
k
}
,
for all k, and a
k
,
b
k
∈
ℂ. Also |
a
|
p
k
≤
max
⁡
(
1
,
|
a
|
H
) for all a
∈
ℂ.The main aim of this paper is to study some ideal convergent sequence spaces defined by a Musielak-Orlicz functionℳ
=
(
M
k
). We also make an effort to study some topological properties and prove some inclusion relations between these spaces.


## 2. Main Results

Theorem 1.
Letℳ
=
(
M
k
) be a Musielak-Orlicz function and let p
=
(
p
k
) be a bounded sequence of positive real numbers. Then, the spaces c
I
(
ℳ
,
Λ
,
p
), c
0
I
(
ℳ
,
Λ
,
p
), m
I
(
ℳ
,
Λ
,
p
), and m
0
I
(
ℳ
,
Λ
,
p
) are linear.Proof.
Letx
,
y
∈
c
I
(
ℳ
,
Λ
,
p
) and let α, β be scalars. Then, there exist positive numbers ρ
1 and ρ
2 such that
(16)
I
-
lim
⁡
k
M
k
(
|
Λ
k
(
x
)
-
L
1
|
ρ
1
)
p
k
=
0
,
for
some
L
1
∈
ℂ
,
I
-
lim
⁡
k
M
k
(
|
Λ
k
(
y
)
-
L
2
|
ρ
2
)
p
k
=
0
,
for
some
L
2
∈
ℂ
.
For a givenϵ
>
0, we have
(17)
D
1
=
{
k
∈
ℕ
:
M
k
(
|
Λ
k
(
x
)
-
L
1
|
ρ
1
)
p
k
}
,
D
2
=
{
k
∈
ℕ
:
M
k
(
|
Λ
k
(
y
)
-
L
2
|
ρ
2
)
p
k
}
.
Letρ
3
=
max
⁡
{
2
|
α
|
ρ
1
,
2
|
β
|
ρ
2
}. Since ℳ
=
(
M
k
) is nondecreasing convex function, so by using inequality (15), we have
(18)
lim
⁡
k
M
k
(
|
Λ
k
(
(
α
x
+
β
y
)
-
(
α
L
1
+
β
L
2
)
)
|
ρ
3
)
p
k
≤
lim
⁡
k
M
k
(
|
α
|
|
Λ
k
(
x
)
-
L
1
|
ρ
3
+
|
β
|
|
Λ
k
(
y
)
-
L
2
|
ρ
3
)
p
k
≤
lim
⁡
k
M
k
(
|
Λ
k
(
x
)
-
L
1
|
ρ
1
)
p
k
+
lim
⁡
k
M
k
(
|
Λ
k
(
y
)
-
L
2
|
ρ
2
)
p
k
.
Now, by (17), we have
(19)
{
k
∈
ℕ
:
lim
⁡
k
M
k
(
|
Λ
k
(
(
α
x
+
β
y
)
-
(
α
L
1
+
β
L
2
)
)
|
ρ
3
)
p
k
h
.
>
ϵ
(
|
Λ
k
(
(
α
x
+
β
y
)
-
(
α
L
1
+
β
L
2
)
)
|
ρ
3
)
p
k
}
⊂
D
1
∪
D
2
.
Therefore,α
x
+
β
y
∈
c
I
(
ℳ
,
Λ
,
p
). Hence c
I
(
ℳ
,
Λ
,
p
) is a linear space. Similarly, we can prove that c
0
I
(
ℳ
,
Λ
,
p
), m
I
(
ℳ
,
Λ
,
p
), and m
0
I
(
ℳ
,
Λ
,
p
) are linear spaces.Theorem 2.
Letℳ
=
(
M
k
) be a Musielak-Orlicz function. Then,
(20)
c
0
I
(
ℳ
,
Λ
,
p
)
⊂
c
I
(
ℳ
,
Λ
,
p
)
⊂
l
∞
(
ℳ
,
Λ
,
p
)
.Proof.
Letx
∈
c
I
(
ℳ
,
Λ
,
p
). Then, there exist L
∈
ℂ and ρ
>
0 such that
(21)
I
-
lim
⁡
k
M
k
(
|
Λ
k
(
x
)
-
L
|
ρ
)
p
k
=
0
.
We have(22)
M
k
(
|
Λ
k
(
x
)
|
2
ρ
)
p
k
≤
1
2
M
k
(
|
Λ
k
(
x
)
-
L
|
ρ
)
p
k
+
M
k
1
2
(
|
L
|
ρ
)
p
k
.
Taking supremum overk on both sides, we get x
∈
l
∞
(
ℳ
,
Λ
,
p
). The inclusion c
0
I
(
ℳ
,
Λ
,
p
)
⊂
c
I
(
ℳ
,
Λ
,
p
) is obvious. Thus,
(23)
c
0
I
(
ℳ
,
Λ
,
p
)
⊂
c
I
(
ℳ
,
Λ
,
p
)
⊂
l
∞
(
ℳ
,
Λ
,
p
)
.
This completes the proof of the theorem.Theorem 3.
Letℳ
=
(
M
k
) be a Musielak-Orlicz function and let p
=
(
p
k
) be a bounded sequence of positive real numbers. Then, l
∞
(
ℳ
,
Λ
,
p
) is a paranormed space with paranorm defined by
(24)
g
(
x
)
=
inf
⁡
{
ρ
>
0
:
sup
⁡
k
M
k
(
|
Λ
k
(
x
)
|
ρ
)
p
k
≤
1
}
.Proof.
It is clear thatg
(
x
)
=
g
(
-
x
). Since M
k
(
0
)
=
0, we get g
(
0
)
=
0. Let us take x
,
y
∈
l
∞
(
ℳ
,
Λ
,
p
). Let
(25)
B
(
x
)
=
{
ρ
>
0
:
sup
⁡
k
M
k
(
|
Λ
k
(
x
)
|
ρ
)
p
k
≤
1
}
,
B
(
y
)
=
{
ρ
>
0
:
sup
⁡
k
M
k
(
|
Λ
k
(
y
)
|
ρ
)
p
k
≤
1
}
.
Letρ
1
∈
B
(
x
) and ρ
2
∈
B
(
y
). If ρ
=
ρ
1
+
ρ
2, then we have
(26)
sup
⁡
k
M
k
(
|
Λ
k
(
x
+
y
)
|
ρ
)
≤
(
ρ
1
ρ
1
+
ρ
2
)
sup
⁡
k
M
k
(
|
Λ
k
(
x
)
|
ρ
1
)
+
(
ρ
2
ρ
1
+
ρ
2
)
sup
⁡
k
M
k
(
|
Λ
k
(
y
)
|
ρ
2
)
.
Thus,sup
⁡
k
M
k
(
|
Λ
(
x
+
y
)
|
/
(
ρ
1
+
ρ
2
)
)
p
k
≤
1 and
(27)
g
(
x
+
y
)
≤
inf
⁡
{
(
ρ
1
+
ρ
2
)
>
0
:
ρ
1
∈
B
(
x
)
,
ρ
2
∈
B
(
y
)
}
≤
inf
⁡
⁡
{
ρ
1
>
0
:
ρ
1
∈
B
(
x
)
}
+
inf
⁡
{
ρ
2
>
0
:
ρ
2
∈
B
(
y
)
}
=
g
(
x
)
+
g
(
y
)
.
Letσ
s
→
σ, where σ
,
σ
s
∈
ℂ and let g
(
x
s
-
x
)
→
0 as s
→
∞. We have to show that g
(
σ
s
x
s
-
σ
x
)
→
0 as s
→
∞. Let
(28)
B
(
x
s
)
=
{
ρ
s
>
0
:
sup
⁡
k
M
k
(
|
Λ
k
(
x
s
)
|
ρ
s
)
p
k
≤
1
}
,
B
(
x
s
-
x
)
=
{
ρ
s
′
>
0
:
sup
⁡
k
M
k
(
|
Λ
k
(
x
s
-
x
)
|
ρ
s
′
)
p
k
≤
1
}
.
Ifρ
s
∈
B
(
x
s
) and ρ
s
′
∈
B
(
x
s
-
x
), then we observe that
(29)
M
k
(
|
Λ
k
(
σ
s
x
s
-
σ
x
)
|
ρ
s
|
σ
s
-
σ
|
+
ρ
s
′
|
σ
|
)
≤
M
k
(
|
Λ
k
(
σ
s
x
s
-
σ
x
s
)
|
ρ
s
|
σ
s
-
σ
|
+
ρ
s
′
|
σ
|
+
|
(
σ
x
s
-
σ
x
)
|
ρ
s
|
σ
s
-
σ
|
+
ρ
s
′
|
σ
|
)
≤
|
σ
s
-
σ
|
ρ
s
ρ
s
|
σ
s
-
σ
|
+
ρ
s
′
|
σ
|
M
k
(
(
|
Λ
k
(
x
s
)
|
)
ρ
s
)
+
|
σ
|
ρ
s
′
ρ
s
|
σ
s
-
σ
|
+
ρ
s
′
|
σ
|
M
k
(
|
Λ
k
(
x
s
-
x
)
|
ρ
s
′
)
.
From the above inequality, it follows that(30)
M
k
(
|
Λ
k
(
σ
s
x
s
-
σ
x
)
|
ρ
s
|
σ
s
-
σ
|
+
ρ
s
′
|
σ
|
)
p
k
≤
1
and, consequently,
(31)
g
(
σ
s
x
s
-
σ
x
)
≤
inf
⁡
⁡
{
(
ρ
s
|
σ
s
-
σ
|
+
ρ
s
′
|
σ
|
)
>
0
:
h
h
h
h
h
h
ρ
s
∈
B
(
x
s
)
,
ρ
s
′
∈
B
(
x
s
-
x
)
}
≤
(
|
σ
s
-
σ
|
)
>
0
inf
⁡
{
ρ
>
0
:
ρ
s
∈
B
(
x
s
)
}
+
(
|
σ
|
)
>
0
inf
⁡
{
(
ρ
s
′
)
p
n
/
H
:
ρ
s
′
∈
B
(
x
s
-
x
)
}
⟶
0
as
s
⟶
∞
.
This completes the proof.Theorem 4.
Letℳ
′
=
(
M
k
′
) and ℳ
′′
=
(
M
k
′′
) be Musielak-Orlicz functions that satisfy the Δ
2-condition. Then,
(i)Z
(
ℳ
′′
,
Λ
,
p
)
⊆
Z
(
ℳ
′
∘
ℳ
′′
,
Λ
,
p
),
(ii)Z
(
ℳ
′
,
Λ
,
p
)
∩
Z
(
ℳ
′′
,
Λ
,
p
)
⊆
Z
(
ℳ
′
+
ℳ
′′
,
Λ
,
p
) for Z
=
c
I
,
c
0
I
,
m
I
,
m
0
I.Proof.
(i) Letx
∈
c
0
I
(
ℳ
′′
,
Λ
,
p
). Then, there exists ρ
>
0 such that
(32)
I
-
lim
⁡
k
ℳ
k
′′
(
|
Λ
k
(
x
)
|
ρ
)
p
k
=
0
.
Letϵ
>
0 and choose δ with 0
<
δ
<
1 such that M
k
′
(
t
)
<
ϵ for 0
≤
t
≤
δ. Write y
k
=
M
k
′′
(
|
Λ
k
(
x
)
|
/
ρ
)
p
k and consider
(33)
lim
⁡
k
∈
ℕ
0
≤
y
k
≤
δ
M
k
′
(
y
k
)
=
lim
⁡
k
∈
ℕ
y
k
≤
δ
M
k
′
(
y
k
)
+
lim
⁡
k
∈
ℕ
y
k
>
δ
M
k
′
(
y
k
)
.
Sinceℳ
=
(
M
k
) satisfies Δ
2-condition, we have
(34)
lim
⁡
k
∈
ℕ
y
k
≤
δ
M
k
′
(
y
k
)
≤
M
k
′
(
2
)
lim
⁡
k
∈
ℕ
y
k
≤
δ
(
y
k
)
.
Fory
k
>
δ, we have
(35)
y
k
<
y
k
δ
<
1
+
y
k
δ
.
Sinceℳ
'
=
(
M
k
′
) is nondecreasing and convex, it follows that
(36)
M
k
′
(
y
k
)
<
M
k
′
(
1
+
y
k
δ
)
<
1
2
M
k
′
(
2
)
+
1
2
M
k
′
(
2
y
k
)
δ
.
Sinceℳ
′
=
(
M
k
′
) satisfies Δ
2-condition, we have
(37)
M
k
′
(
y
k
)
<
1
2
K
y
k
δ
M
k
′
(
2
)
+
1
2
K
y
k
δ
M
k
′
(
2
)
=
K
y
k
δ
M
k
′
(
2
)
.
Hence,(38)
lim
⁡
k
∈
ℕ
y
k
>
δ
M
k
′
(
y
k
)
≤
max
⁡
(
1
,
K
δ
-
1
M
k
′
(
2
)
)
lim
⁡
k
∈
ℕ
y
k
≤
δ
(
y
k
)
.
From (32), (34), and (38), we have x
=
(
x
k
)
∈
c
0
I
(
ℳ
′
∘
ℳ
′′
,
Λ
,
p
). Thus, c
0
I
(
ℳ
′′
,
Λ
,
p
)
⊆
c
0
I
(
ℳ
′
∘
ℳ
′′
,
Λ
,
p
). Similarly, we can prove the other cases.
(ii) Letx
∈
c
0
I
(
ℳ
′
,
Λ
,
p
)
∩
c
0
I
(
ℳ
′′
,
Λ
,
p
). Then, there exists ρ
>
0 such that
(39)
I
-
lim
⁡
k
M
k
′
(
|
Λ
k
(
x
)
|
ρ
)
p
k
=
0
,
I
-
lim
⁡
k
M
k
′′
(
|
Λ
k
(
x
)
|
ρ
)
p
k
=
0
.
The rest of the proof follows from the following equality:(40)
lim
⁡
k
∈
ℕ
(
M
k
′
+
M
k
′′
)
(
|
Λ
k
(
x
)
|
ρ
)
p
k
=
lim
⁡
k
∈
ℕ
M
k
′
(
|
Λ
k
(
x
)
|
ρ
)
p
k
+
lim
⁡
k
∈
ℕ
M
k
′′
(
|
Λ
k
(
x
)
|
ρ
)
p
k
.Corollary 5.
Letℳ
=
(
M
k
) be a Musielak-Orlicz function which satisfies Δ
2-condition. Then, Z
(
p
,
Λ
)
⊆
Z
(
ℳ
,
Λ
,
p
) holds for  Z
=
c
I
,
c
0
I
,
m
I, and m
0
I.Proof.
The proof follows from Theorem3 by putting M
k
′′
(
x
)
=
x and M
k
′
(
x
)
=
M
k
(
x
)
∀
x
∈
[
0
,
∞
).Theorem 6.
The spacesc
0
I
(
ℳ
,
Λ
,
p
) and m
0
I
(
ℳ
,
Λ
,
p
) are solid.Proof.
We will prove for the spacec
0
I
(
ℳ
,
Λ
,
p
Λ
). Let x
∈
c
0
I
(
ℳ
,
Λ
,
p
). Then, there exists ρ
>
0 such that
(41)
I
-
lim
⁡
k
M
k
(
|
Λ
k
(
x
)
|
ρ
)
p
k
=
0
.
Let(
α
k
) be a sequence of scalars with |
α
k
|
≤
1
∀
k
∈
ℕ. Then, the result follows from the following inequality:
(42)
lim
⁡
k
M
k
(
|
Λ
k
(
α
x
)
|
ρ
)
p
k
≤
lim
⁡
k
M
k
(
|
Λ
k
(
x
)
|
ρ
)
p
k
and this completes the proof. Similarly, we can prove for the space m
0
I
(
ℳ
,
Λ
,
p
).Corollary 7.
The spacesc
0
I
(
ℳ
,
Λ
,
p
) and m
0
I
(
ℳ
,
Λ
,
p
) are monotone.Proof.
It is easy to prove, so we omit the details.Theorem 8.
The spacesc
I
(
ℳ
,
Λ
,
p
) and c
0
I
(
ℳ
,
Λ
,
p
) are sequence algebra.Proof.
Letx
,
y
∈
c
0
I
(
ℳ
,
Λ
,
p
). Then,
(43)
I
-
lim
⁡
k
M
k
(
|
Λ
k
(
x
)
|
ρ
1
)
p
k
=
0
,
for
some
ρ
1
>
0
,
I
-
lim
⁡
k
M
k
(
|
Λ
k
(
y
)
|
ρ
2
)
p
k
=
0
,
for
some
ρ
2
>
0
.
Letρ
=
ρ
1
+
ρ
2. Then, we can show that
(44)
I
-
lim
⁡
k
M
k
(
|
Λ
k
(
x
·
y
)
|
ρ
)
p
k
=
0
.
Thus,(
x
·
y
)
∈
c
0
I
(
ℳ
,
Λ
,
p
). Hence, c
0
I
(
ℳ
,
Λ
,
p
) is a sequence algebra. Similarly, we can prove that c
I
(
ℳ
,
Λ
,
p
) is a sequence algebra.



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*Source: 134534-2014-01-28.xml*