# The Ideal Convergence of Strongly ofΓ

Authors: N. Subramanian; K. Balasubramanian; K. Chandrasekhara Rao
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The aim of this paper is to introduce and study a new concept of theΓ
2 space via ideal convergence defined by modulus and also some topological properties of the resulting sequence spaces were examined.


---

## Body

## 1. Introduction

Let(
x
m
n
) be a double sequence of real or complex numbers. Then the series ∑
m
,
n
=
1
∞
‍
x
m
n is called a double series. The double series ∑
m
,
n
=
1
∞
‍
x
m
n is said to be convergent if and only if the double sequence (
S
m
n
) is convergent, where
(1)
S
m
n
=
∑
i
,
j
=
1
m
,
n
x
i
j
,
(
m
,
n
=
1,2
,
3
,
…
)
.
We denote w
2 as the class of all complex double sequences (
x
m
n
). A sequence x
=
(
x
m
n
) is said to be double analytic if
(2)
sup
⁡
m
n
|
x
m
n
|
1
/
m
+
n
<
∞
.
The vector space of all prime sense double analytic sequences is usually denoted by Λ
2. A sequence x
=
(
x
m
n
) is called double entire sequence if
(3)
(
|
x
m
n
|
)
1
/
m
+
n
⟶
0
as
m
,
n
⟶
∞
.
The vector space of all prime sense double entire sequences is usually denoted by Γ
2. The space Λ
2 is a metric space with the metric
(4)
d
(
x
,
y
)
=
sup
⁡
m
n
{
|
x
m
n
-
y
m
n
|
1
/
m
+
n
:
m
,
n
:
1,2
,
3
,
…
}
.
The space Γ
2 is a metric space with the metric
(5)
d
(
x
,
y
)
=
sup
⁡
mn
⁡
{
(
|
x
m
n
-
y
m
n
|
)
1
/
m
+
n
:
m
,
n
:
1,2
,
3
,
…
}
,
for all x
=
{
x
m
n
} and y
=
{
y
m
n
} in Γ
2.Consider a double sequencex
=
(
x
i
j
). The (
m
,
n
)
th section x
[
m
,
n
] of the sequence is defined by x
[
m
,
n
]
=
∑
‍
i
,
j
=
0
m
,
n
x
i
j
δ
i
j for all m
,
n
∈
N,
(6)
δ
m
n
=
(
0
0
⋯
0
0
⋯
0
0
⋯
0
0
⋯
⋮
0
0
⋯
1
0
⋯
0
0
⋯
0
0
⋯
)
,
with 1 in the (
m
,
n
)
th position and zero otherwise. An FK-space (or a metric space) X is said to have AK property if (
δ
m
n
) is a Schauder basis for X. Or equivalently x
[
m
,
n
]
→
x. We need the following inequality in the sequel of the paper.Lemma 1.
Fora
,
b
≥
0 and 0
<
p
<
1, one has
(7)
(
a
+
b
)
p
≤
a
p
+
b
p
.Some initial work on double sequence spaces is found in Bromwich. Later on it was investigated by Moricz [1], Moricz and Rhoades [2], Basarir and Solancan [3], Tripathy [4], Turkmenoglu [5], Subramanian and Misra [6, 7], and many others. Tripathy and Dutta [8] introduced and investigated different types of fuzzy real valued double sequence spaces. Generalizing the concept of ordinary convergence for real sequences Kostyrko et al. introduced the concept of ideal convergence which is a generalization of statistical convergence, by using the ideal I of the subsets of the set of natural numbers.The notion of different sequence spaces (for single sequences) was introduced by Kizmaz [9] as follows:
(8)
Z
(
Δ
)
=
{
x
=
(
x
k
)
∈
w
:
(
Δ
x
k
)
∈
Z
}
,
for Z
=
c
,
c
0 and l
∞, where Δ
x
k
=
x
k
-
x
k
+
1 for all k
∈
N. Here w, c, c
0, and l
∞ denote the classes of all, convergent, null, and bounded scalar valued single sequences, respectively. The above spaces are Banach spaces normed by
(9)
∥
x
∥
=
|
x
1
|
+
sup
⁡
k
≥
1
⁡
|
Δ
x
k
|
.
Later on the notion was further investigated by many others. We now introduce the following difference double sequence spaces defined by
(10)
Z
(
Δ
)
=
{
x
=
(
x
m
n
)
∈
w
2
:
(
Δ
x
m
n
)
∈
Z
}
,
where Z
=
Λ
2 and Γ
2, respectively. Δ
x
m
n
=
(
x
m
n
-
x
m
n
+
1
)
-
(
x
m
+
1
n
-
x
m
+
1
n
+
1
)
=
x
m
n
-
x
m
n
+
1
-
x
m
+
1
n
+
x
m
+
1
n
+
1 for all m
,
n
∈
N. We further generalized this notion and introduced the following notion. For m
,
n
≥
1,
(11)
Z
(
Δ
γ
μ
)
=
{
x
=
x
m
n
:
(
Δ
γ
μ
x
m
n
)
∈
Z
}
,
for
Z
=
Λ
2
,
Γ
2
.
An Orlicz function is a function f
:
[
0
,
∞
)
→
[
0
,
∞
) which is continuous, nondecreasing, and convex with f
(
0
)
=
0, f
(
x
)
>
0, for x
>
0 and f
(
x
)
→
∞ as x
→
∞. If convexity of Orlicz function f is replaced by f
(
x
+
y
)
≤
f
(
x
)
+
f
(
y
), then this function is called modulus function. A modulus function f is said to satisfy Δ
2-condition for all values u, if there exists K
>
0 such that f
(
2
u
)
≤
K
f
(
u
), u
≥
0.Remark 2.
A modulus function satisfies the inequalityf
(
λ
x
)
≤
λ
f
(
x
) for all λ with 0
<
λ
<
1.Lemma 3.
Letf be a modulus function which satisfies Δ
2-condition and let 0
<
δ
<
1. Then for each t
≥
δ, one has f
(
t
)
<
K
δ
-
1
f
(
2
) for some constant K
>
0.Spaces of strongly summable sequences were discussed by Kuttner, Maddox, and others. The class of sequences which are strongly Cesàro summable with respect to a modulus was introduced by Maddox as an extension of the definition of strongly Cesàro summable sequences. Connor further extended this definition to a definition of strongA-summability with respect to a modulus where A
=
(
a
n
,
k
) is a nonnegative regular matrix and established some connections between strong A-summability, strong A-summability with respect to a modulus, and A-statistical convergence. The notion of convergence of double sequences was presented by A. Pringsheim. Also, the four-dimensional matrix transformation (
A
x
)
k
,
l
=
∑
m
=
1
∞
‍
∑
n
=
1
∞
‍
a
k
l
m
n
x
m
n was studied extensively by Robison and Hamilton.


## 2. Definitions and Preliminaries

LetX be a nonempty set. A nonvoid class I
⊆
2
X (power set, of X) is called an ideal if I is additive (i.e., A
,
B
∈
I
⇒
A
⋃
B
∈
I) and hereditary (i.e., A
∈
I and B
⊆
A
⇒
B
∈
I). A nonempty family of sets F
⊆
2
X is said to be a filter on X if ϕ
∉
F; A
,
B
∈
F
⇒
A
⋂
B
∈
F and A
∈
F, A
⊆
B
⇒
B
∈
F. For each ideal I there is a filter F
(
I
) given by F
(
I
)
=
{
K
⊆
N
:
N
∖
K
∈
I
}. A nontrivial ideal I
⊂
2
X is called admissible if and only if {
{
x
}
:
x
∈
X
}
⊂
I.A double sequence spaceE is said to be solid or normal if (
α
m
n
x
m
n
)
∈
E, whenever (
x
m
n
)
∈
E and for all double sequences α
=
(
α
m
n
) of scalars with |
α
m
n
|
≤
1, for all m
,
n
∈
N.Letn
∈
N and let X be a real vector space of dimension w, where n
≤
w. A real valued function d
p
(
x
1
,
…
,
x
n
)
=
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p on X satisfies the following four conditions:(i)
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
=
0 if and and only if d
1
(
x
1
)
,
…
,
d
n
(
x
n
) are linearly dependent,
(ii)
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p is invariant under permutation,
(iii)
∥
(
α
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
=
|
α
|
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p, α
∈
R,
(iv)
d
p
(
(
x
1
,
y
1
)
,
(
x
2
,
y
2
)
⋯
(
x
n
,
y
n
)
)
=
(
d
X
(
x
1
,
x
2
,
…
,
x
n
)
p
+
d
Y
(
y
1
,
y
2
,
…
,
y
n
)
p
)
1
/
p for 1
≤
p
<
∞, or
(v)
d
(
(
x
1
,
y
1
)
,
(
x
2
,
y
2
)
,
…
,
(
x
n
,
y
n
)
)
≔
sup
⁡
{
d
X
(
x
1
,
x
2
,
…
,
x
n
)
,
d
Y
(
y
1
,
y
2
,
…
,
y
n
)
}, for x
1
,
x
2
,
…
,
x
n
∈
X, y
1
,
y
2
,
…
,
y
n
∈
Y, is called the p product metric of the Cartesian product of n metric spaces which is the p norm of the n-vector of the norms of the n subspaces.A trivial example ofp product metric of n metric spaces is the p norm space X
=
R equipped with the following Euclidean metric in the product space which is the p norm:
(12)
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
E
=
sup
⁡
(
|
det
⁡
(
d
m
n
(
x
m
n
)
)
|
)
=
sup
⁡
(
|
d
11
(
x
11
)
d
12
(
x
12
)
⋯
d
1
n
(
x
1
n
)
d
21
(
x
21
)
d
22
(
x
22
)
⋯
d
2
n
(
x
1
n
)
⋮
d
n
1
(
x
n
1
)
d
n
2
(
x
n
2
)
⋯
d
n
n
(
x
n
n
)
|
)
,
where x
i
=
(
x
i
1
,
…
,
x
i
n
)
∈
R
n for each i
=
1,2
,
…
,
n.


## 3. Main Results

In this section we introduce the notion of different types ofI-convergent double sequences. This generalizes and unifies different notions of convergence for Γ
2. We will denote the ideal of 2
N
×
N by I
2.LetI
2 be an ideal of 2
N
×
N, f a modulus function, η
=
(
η
m
n
) a double analytic sequence of strictly positive real numbers, and (
X
,
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
) a p-product of n metric spaces which is the p norm of the n-vector of the norms of the n subspaces. Further Γ
2
(
p
-
X
) denotes X-valued sequence space. Now, we define the following sequence spaces:
(13)
Γ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η
iii
=
x
=
(
x
m
n
)
∈
Γ
2
(
p
-
X
)
:
∀
ϵ
>
0
,
{
∑
n
=
1
s
[
f
∥
(
|
x
m
n
|
)
1
/
m
+
n
,
1
r
s
∑
m
=
1
r
(
r
,
s
)
∈
N
×
N
:
i
i
i
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
∥
(
|
x
m
n
|
)
1
/
m
+
n
,
)
∥
p
[
f
∥
(
|
x
m
n
|
)
1
/
m
+
n
,
]
η
m
n
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
∥
(
|
x
m
n
|
)
1
/
m
+
n
,
)
∥
p
[
f
∥
(
|
x
m
n
|
)
1
/
m
+
n
,
]
η
m
n
f
∥
(
|
x
m
n
|
)
1
/
m
+
n
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
∥
(
|
x
m
n
|
)
1
/
m
+
n
,
∥
p
[
f
∥
(
|
x
m
n
|
)
1
/
m
+
n
,
]
η
m
n
≥
ϵ
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
f
∥
(
|
x
m
n
|
)
1
/
m
+
n
,
}
∈
I
2
,
iiiiiiiiiiiiiiiiiiiii
i
i
i
i
ii
ii
for
every
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
∈
X
.
Λ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η
ii
i
=
x
=
(
x
m
n
)
∈
Λ
2
(
p
-
X
)
:
∃
K
>
0
,
{
∑
m
=
1
r
{
∑
m
=
1
r
(
r
,
s
)
∈
N
×
N
:
i
i
i
i
i
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
|
x
m
n
|
1
/
m
+
n
)
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
|
x
m
n
|
1
/
m
+
n
∥
p
[
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
f
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
|
x
m
n
|
1
/
m
+
n
]
η
m
n
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
|
x
m
n
|
1
/
m
+
n
)
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
|
x
m
n
|
1
/
m
+
n
∥
p
[
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
f
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
|
x
m
n
|
1
/
m
+
n
]
η
m
n
f
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
|
x
m
n
|
1
/
m
+
n
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
|
x
m
n
|
1
/
m
+
n
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
|
x
m
n
|
1
/
m
+
n
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
|
x
m
n
|
1
/
m
+
n
)
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
|
x
m
n
|
1
/
m
+
n
∥
p
[
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
f
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
|
x
m
n
|
1
/
m
+
n
]
η
m
n
i
i
i
i
i
i
≥
K
∑
m
=
1
r
}
∈
I
2
{
(
r
,
s
)
∈
N
×
N
:
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
f
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
|
x
m
n
|
1
/
m
+
n
{
(
r
,
s
)
∈
N
×
N
:
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
f
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
|
x
m
n
|
1
/
m
+
n
∑
m
=
1
r
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
}
,
for
every
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
∈
X
.
Λ
f
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η
ii
=
x
=
(
x
m
n
)
∈
Λ
2
(
p
-
X
)
:
∃
K
>
0
,
{
≤
K
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
}
,
≤
K
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
}
,
∑
m
=
1
r
(
r
,
s
)
∈
N
×
N
:
ii
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
(
|
x
m
n
|
1
/
m
+
n
∥
p
]
η
m
n
f
∥
(
|
x
m
n
|
1
/
m
+
n
,
iiiiiiiiiiiiiiiiiiiiii
i
i
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
(
|
x
m
n
|
1
/
m
+
n
)
∥
(
|
x
m
n
|
1
/
m
+
n
∥
p
]
η
m
n
i
i
≤
K
∑
m
=
1
r
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
}
,
for
every
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
∈
X
.
If η
=
η
m
n
=
1 for all m
,
n
∈
N we obtain
(14)
Γ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η
ii
=
Γ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
,
Λ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η
ii
=
Λ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
,
Λ
f
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η
i
=
Λ
f
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
.
The following well-known inequality will be used in this study: 0
≤
inf
⁡
m
n
η
m
n
=
H
0
≤
η
m
n
≤
sup
⁡
m
n
=
H
<
∞, D
=
max
⁡
(
1
,
2
H
-
1
); then
(15)
|
x
m
n
+
y
m
n
|
η
m
n
≤
D
{
|
x
m
n
|
η
m
n
+
|
y
m
n
|
η
m
n
}
,
for all m
,
n
∈
N and x
m
n
,
y
m
n
∈
C. Also |
x
m
n
|
η
m
n
/
m
+
n
≤
max
⁡
(
1
,
|
x
m
n
|
H
/
m
+
n
) for all x
m
n
∈
C.Theorem 4.
The setsΓ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η
m
n and Λ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η
m
n are linear spaces over the complex field CProof.
Now only proveΓ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η
m
n and the others can be proved similarly. Let x
,
y
∈
Γ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η
m
n and α
,
β
∈
C. Then
(16)
{
≥
ϵ
2
}
(
r
,
s
)
∈
N
×
N
:
i
i
i
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
iiiiiiiiiiiiiiiiiiiii
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
(
|
x
m
n
|
)
1
/
m
+
n
∥
p
f
∥
(
|
x
m
n
|
)
1
/
m
+
n
,
i
i
i
iiiiiiiiiiiiiiiiiiiii
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
(
|
x
m
n
|
)
1
/
m
+
n
∥
p
]
η
m
n
i
i
≥
ϵ
2
}
∈
I
2
,
{
≥
ϵ
2
}
(
r
,
s
)
∈
N
×
N
:
i
i
i
1
r
s
∑
m
=
1
r
∑
n
=
1
r
[
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
(
|
y
m
n
|
)
1
/
m
+
n
∥
p
f
∥
(
|
y
m
n
|
)
1
/
m
+
n
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
(
|
y
m
n
|
)
1
/
m
+
n
∥
p
]
η
m
n
i
i
≥
ϵ
2
}
∈
I
2
.
Since ∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p is a p-product of n metric spaces which is the p norm of the n-vector of the norms of the n subspaces and f is a modulus function, the following inequality holds:
(17)
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
f
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
∥
|
α
x
m
n
+
β
y
m
n
|
1
/
m
+
n
|
α
|
1
/
m
+
n
+
|
β
|
1
/
m
+
n
,
∥
p
∥
|
α
x
m
n
+
β
y
m
n
|
1
/
m
+
n
|
α
|
1
/
m
+
n
+
|
β
|
1
/
m
+
n
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
∥
|
α
x
m
n
+
β
y
m
n
|
1
/
m
+
n
|
α
|
1
/
m
+
n
+
|
β
|
1
/
m
+
n
,
∥
p
)
]
η
m
n
≤
D
r
s
∑
m
=
1
r
∑
n
=
1
s
[
|
α
|
1
/
m
+
n
|
α
|
1
/
m
+
n
+
|
β
|
1
/
m
+
n
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
×
f
(
|
α
|
1
/
m
+
n
|
α
|
1
/
m
+
n
+
|
β
|
1
/
m
+
n
∥
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
(
|
x
m
n
|
)
1
/
m
+
n
∥
p
|
α
|
1
/
m
+
n
|
α
|
1
/
m
+
n
+
|
β
|
1
/
m
+
n
)
]
η
m
n
+
D
r
s
∑
m
=
1
r
∑
n
=
1
s
[
|
β
|
1
/
m
+
n
|
α
|
1
/
m
+
n
+
|
β
|
1
/
m
+
n
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
ii
×
f
(
∥
|
β
|
1
/
m
+
n
|
α
|
1
/
m
+
n
+
|
β
|
1
/
m
+
n
(
|
y
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
|
β
|
1
/
m
+
n
|
α
|
1
/
m
+
n
+
|
β
|
1
/
m
+
n
∥
p
)
]
η
m
n
≤
D
r
s
∑
m
=
1
r
∑
n
=
1
s
[
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
)
]
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
)
∥
(
|
x
m
n
|
)
1
/
m
+
n
,
∥
p
)
]
η
m
n
f
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
)
∥
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
)
∥
(
|
x
m
n
|
)
1
/
m
+
n
,
∥
p
)
(
|
x
m
n
|
)
1
/
m
+
n
,
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
)
∥
(
|
x
m
n
|
)
1
/
m
+
n
,
∥
p
)
]
η
m
n
+
D
r
s
∑
m
=
1
r
∑
n
=
1
s
[
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
∥
(
|
y
m
n
|
)
1
/
m
+
n
,
∥
p
)
f
(
∥
(
|
y
m
n
|
)
1
/
m
+
n
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
∥
(
|
y
m
n
|
)
1
/
m
+
n
,
∥
p
)
]
η
m
n
.
From the above inequality we get
(18)
{
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
∥
(
(
|
α
x
m
n
+
β
y
m
n
|
)
1
/
m
+
n
|
α
|
1
/
m
+
n
+
|
β
|
1
/
m
+
n
,
)
∥
p
)
]
η
m
n
(
r
,
s
)
∈
N
×
N
:
i
i
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
∥
(
(
|
α
x
m
n
+
β
y
m
n
|
)
1
/
m
+
n
|
α
|
1
/
m
+
n
+
|
β
|
1
/
m
+
n
,
)
∥
p
)
]
η
m
n
f
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
∥
(
(
|
α
x
m
n
+
β
y
m
n
|
)
1
/
m
+
n
|
α
|
1
/
m
+
n
+
|
β
|
1
/
m
+
n
,
)
∥
p
)
∥
(
(
|
α
x
m
n
+
β
y
m
n
|
)
1
/
m
+
n
|
α
|
1
/
m
+
n
+
|
β
|
1
/
m
+
n
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
∥
(
(
|
α
x
m
n
+
β
y
m
n
|
)
1
/
m
+
n
|
α
|
1
/
m
+
n
+
|
β
|
1
/
m
+
n
,
)
∥
p
)
]
η
m
n
i
i
i
≥
ϵ
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
∥
(
(
|
α
x
m
n
+
β
y
m
n
|
)
1
/
m
+
n
|
α
|
1
/
m
+
n
+
|
β
|
1
/
m
+
n
,
)
∥
p
)
]
η
m
n
}
⊂
{
≥
ϵ
2
}
∑
m
=
1
r
(
r
,
s
)
∈
N
×
N
:
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
D
r
s
∑
m
=
1
r
∑
n
=
1
s
[
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
f
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
(
(
|
x
m
n
|
)
1
/
m
+
n
,
)
∥
p
]
η
m
n
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
≥
ϵ
2
∑
m
=
1
r
}
∈
I
2
⋃
{
∑
m
=
1
r
(
r
,
s
)
∈
N
×
N
:
‍
i
i
i
i
i
i
i
D
r
s
∑
m
=
1
r
∑
n
=
1
s
[
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
…
,
d
n
(
x
n
-
1
)
(
…
,
d
n
(
x
n
-
1
)
)
|
y
m
n
|
1
/
m
+
n
,
d
1
(
x
1
)
)
∥
…
,
d
n
(
x
n
-
1
)
)
∥
p
(
…
,
d
n
(
x
n
-
1
)
)
|
y
m
n
|
1
/
m
+
n
,
d
1
(
x
1
)
,
∥
p
[
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
f
∥
…
,
d
n
(
x
n
-
1
)
)
∥
p
(
…
,
d
n
(
x
n
-
1
)
)
|
y
m
n
|
1
/
m
+
n
,
d
1
(
x
1
)
,
]
η
m
n
f
∥
…
,
d
n
(
x
n
-
1
)
)
∥
p
(
…
,
d
n
(
x
n
-
1
)
)
|
y
m
n
|
1
/
m
+
n
,
d
1
(
x
1
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
(
…
,
d
n
(
x
n
-
1
)
)
|
y
m
n
|
1
/
m
+
n
,
d
1
(
x
1
)
)
∥
…
,
d
n
(
x
n
-
1
)
)
∥
p
(
…
,
d
n
(
x
n
-
1
)
)
|
y
m
n
|
1
/
m
+
n
,
d
1
(
x
1
)
,
∥
p
[
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
f
∥
…
,
d
n
(
x
n
-
1
)
)
∥
p
(
…
,
d
n
(
x
n
-
1
)
)
|
y
m
n
|
1
/
m
+
n
,
d
1
(
x
1
)
,
]
η
m
n
i
i
i
i
i
i
i
≥
ϵ
2
{
(
r
,
s
)
∈
D
r
s
∑
m
=
1
r
∑
m
=
1
r
}
∈
I
2
.
This completes the proof.Theorem 5.
Γ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η paranormed space with respect to the paranorm is defined by
(19)
g
r
s
(
x
)
=
inf
⁡
{
(
sup
⁡
rs
⁡
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
…
,
d
n
(
x
n
-
1
)
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
(
|
x
m
n
|
)
1
/
m
+
n
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
(
|
x
m
n
|
)
1
/
m
+
n
)
∥
p
]
η
m
n
f
∥
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
sup
⁡
rs
⁡
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
f
∥
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
(
|
x
m
n
|
)
1
/
m
+
n
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
(
|
x
m
n
|
)
1
/
m
+
n
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
(
|
x
m
n
|
)
1
/
m
+
n
)
∥
p
]
η
m
n
(
sup
⁡
rs
⁡
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
…
,
d
n
(
x
n
-
1
)
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
(
|
x
m
n
|
)
1
/
m
+
n
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
(
|
x
m
n
|
)
1
/
m
+
n
)
∥
p
]
η
m
n
f
∥
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
)
1
/
H
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
≤
1
{
(
sup
⁡
rs
⁡
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
f
∥
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
(
|
x
m
n
|
)
1
/
m
+
n
,
}
,
f
o
r
e
v
e
r
y
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
∈
X
.Proof.
g
r
s
(
θ
)
=
0 and g
r
s
(
-
x
)
=
g
r
s
(
x
) are easy to prove, so we omit them. Let us take x
,
y
∈
Γ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η
m
n. Let
(20)
g
r
s
(
x
)
=
inf
⁡
{
sup
⁡
rs
⁡
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
f
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
)
∥
p
]
η
m
n
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
≤
1
,
∀
x
∈
X
{
sup
⁡
rs
⁡
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
…
,
d
n
(
x
n
-
1
)
)
∥
]
η
m
n
f
}
,
g
r
s
(
y
)
=
inf
⁡
{
sup
⁡
rs
⁡
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
f
∥
(
(
|
y
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
(
(
|
y
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
)
∥
p
]
η
m
n
iiiiiiiiiiiiiiiiii
≤
1
,
∀
x
∈
X
{
sup
⁡
rs
⁡
1
r
s
}
.
Then we have
(21)
sup
⁡
rs
⁡
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
…
,
d
n
(
x
n
-
1
)
(
(
|
x
m
n
+
y
m
n
|
)
1
/
m
+
n
)
∥
p
]
η
m
n
f
∥
(
(
|
x
m
n
+
y
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
(
(
|
x
m
n
+
y
m
n
|
)
1
/
m
+
n
)
∥
p
]
η
m
n
≤
sup
⁡
rs
⁡
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
…
,
d
n
(
x
n
-
1
)
(
(
|
x
m
n
|
)
1
/
m
+
n
)
∥
p
]
η
m
n
f
∥
(
…
,
d
n
(
x
n
-
1
)
(
(
|
y
m
n
|
)
1
/
m
+
n
)
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
(
(
|
x
m
n
|
)
1
/
m
+
n
)
∥
p
]
η
m
n
+
sup
⁡
rs
⁡
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
…
,
d
n
(
x
n
-
1
)
(
(
|
y
m
n
|
)
1
/
m
+
n
)
∥
p
]
η
m
n
f
∥
(
…
,
d
n
(
x
n
-
1
)
(
(
|
y
m
n
|
)
1
/
m
+
n
)
(
|
y
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
(
(
|
y
m
n
|
)
1
/
m
+
n
)
∥
p
]
η
m
n
.
Thus
(22)
sup
⁡
rs
⁡
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
(
(
|
x
m
n
+
y
m
n
|
)
1
/
m
+
n
,
)
∥
p
f
∥
(
(
|
x
m
n
+
y
m
n
|
)
1
/
m
+
n
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
(
(
|
x
m
n
+
y
m
n
|
)
1
/
m
+
n
,
)
∥
p
]
η
m
n
≤
1
,
and g
r
s
(
x
+
y
)
=
g
r
s
(
x
)
+
g
r
s
(
y
).
Now, letλ
m
n
u
→
λ, where λ
m
n
u
,
λ
∈
C and g
r
s
(
x
m
n
u
-
x
m
n
)
→
0 as u
→
∞. We have to prove that g
r
s
(
λ
m
n
x
m
n
u
-
λ
x
m
n
)
→
0 as u
→
∞. Let
(23)
g
r
s
(
x
u
)
=
{
∑
m
=
1
r
sup
⁡
rs
⁡
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
…
,
d
n
(
x
n
-
1
)
∥
(
(
|
x
m
n
u
|
)
1
/
m
+
n
)
∥
p
]
η
m
n
…
,
d
n
(
x
n
-
1
)
(
…
,
d
n
(
x
n
-
1
)
∥
(
(
|
x
m
n
u
|
)
1
/
m
+
n
)
∥
p
(
|
x
m
n
u
|
)
1
/
m
+
n
∥
(
(
|
x
m
n
u
|
)
1
/
m
+
n
)
∥
p
]
η
m
n
f
∥
(
…
,
d
n
(
x
n
-
1
)
∥
(
(
|
x
m
n
u
|
)
1
/
m
+
n
)
∥
p
(
|
x
m
n
u
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
(
…
,
d
n
(
x
n
-
1
)
∥
(
(
|
x
m
n
u
|
)
1
/
m
+
n
)
∥
p
(
|
x
m
n
u
|
)
1
/
m
+
n
∥
(
(
|
x
m
n
u
|
)
1
/
m
+
n
)
∥
p
]
η
m
n
i
i
i
i
i
i
i
i
i
i
i
i
i
≤
1
,
∀
x
∈
X
∑
m
=
1
r
{
sup
⁡
rs
⁡
1
r
s
}
,
g
r
s
(
x
u
-
x
)
=
{
∑
m
=
1
r
sup
⁡
rs
⁡
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
…
,
d
n
(
x
n
-
1
)
(
(
|
x
m
n
u
-
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
)
∥
p
]
η
m
n
f
∥
(
…
,
d
n
(
x
n
-
1
)
(
(
|
x
m
n
u
-
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
)
(
|
x
m
n
u
-
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
…
,
d
n
(
x
n
-
1
)
(
(
|
x
m
n
u
-
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
)
∥
p
]
η
m
n
i
i
i
i
i
i
i
i
i
i
i
i
i
≤
1
,
∑
m
=
1
r
}
∀
x
∈
X
.
We observe that
(24)
f
(
…
,
d
n
(
x
n
-
1
)
(
|
λ
m
n
u
x
m
n
u
-
λ
x
m
n
|
)
1
/
m
+
n
|
λ
m
n
u
-
λ
|
1
/
m
+
n
+
|
λ
|
1
/
m
+
n
)
∥
p
∥
(
(
|
λ
m
n
u
x
m
n
u
-
λ
x
m
n
|
)
1
/
m
+
n
|
λ
m
n
u
-
λ
|
1
/
m
+
n
+
|
λ
|
1
/
m
+
n
,
d
1
(
x
1
)
,
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
(
|
λ
m
n
u
x
m
n
u
-
λ
x
m
n
|
)
1
/
m
+
n
|
λ
m
n
u
-
λ
|
1
/
m
+
n
+
|
λ
|
1
/
m
+
n
)
∥
p
)
≤
f
(
…
,
d
n
(
x
n
-
1
)
(
(
|
λ
m
n
u
x
m
n
u
-
λ
x
m
n
u
|
)
1
/
m
+
n
|
λ
m
n
u
-
λ
|
1
/
m
+
n
+
|
λ
|
1
/
m
+
n
,
d
1
(
x
1
)
)
∥
p
∥
(
(
|
λ
m
n
u
x
m
n
u
-
λ
x
m
n
u
|
)
1
/
m
+
n
|
λ
m
n
u
-
λ
|
1
/
m
+
n
+
|
λ
|
1
/
m
+
n
,
d
1
(
x
1
)
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
(
(
|
λ
m
n
u
x
m
n
u
-
λ
x
m
n
u
|
)
1
/
m
+
n
|
λ
m
n
u
-
λ
|
1
/
m
+
n
+
|
λ
|
1
/
m
+
n
,
d
1
(
x
1
)
)
∥
p
)
+
f
(
…
,
d
n
(
x
n
-
1
)
(
(
|
λ
x
m
n
u
-
λ
x
m
n
|
)
1
/
m
+
n
|
λ
m
n
u
-
λ
|
1
/
m
+
n
+
|
λ
|
1
/
m
+
n
,
d
1
(
x
1
)
,
)
∥
p
∥
(
(
|
λ
x
m
n
u
-
λ
x
m
n
|
)
1
/
m
+
n
|
λ
m
n
u
-
λ
|
1
/
m
+
n
+
|
λ
|
1
/
m
+
n
,
d
1
(
x
1
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
(
(
|
λ
x
m
n
u
-
λ
x
m
n
|
)
1
/
m
+
n
|
λ
m
n
u
-
λ
|
1
/
m
+
n
+
|
λ
|
1
/
m
+
n
,
d
1
(
x
1
)
,
)
∥
p
)
≤
|
λ
m
n
u
-
λ
|
|
λ
m
n
u
-
λ
|
+
|
λ
|
×
f
(
∥
(
(
|
x
m
n
u
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
)
+
|
λ
|
|
λ
m
n
u
-
λ
|
+
|
λ
|
×
f
(
…
,
d
n
(
x
n
-
1
)
(
|
x
m
n
u
-
x
m
n
|
)
1
/
m
+
n
)
∥
p
…
,
d
n
(
x
n
-
1
)
)
∥
p
)
∥
(
(
|
x
m
n
u
-
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
(
|
x
m
n
u
-
x
m
n
|
)
1
/
m
+
n
)
∥
p
)
.
From this inequality, it follows that
(25)
[
…
,
d
n
(
x
n
-
1
)
(
|
λ
m
n
u
x
m
n
u
-
λ
x
m
n
|
)
1
/
m
+
n
|
λ
m
n
u
-
λ
|
1
/
m
+
n
+
|
λ
|
1
/
m
+
n
)
∥
p
f
(
…
,
d
n
(
x
n
-
1
)
(
|
λ
m
n
u
x
m
n
u
-
λ
x
m
n
|
)
1
/
m
+
n
|
λ
m
n
u
-
λ
|
1
/
m
+
n
+
|
λ
|
1
/
m
+
n
)
∥
p
∥
(
(
|
λ
m
n
u
x
m
n
u
-
λ
x
m
n
|
)
1
/
m
+
n
|
λ
m
n
u
-
λ
|
1
/
m
+
n
+
|
λ
|
1
/
m
+
n
,
d
1
(
x
1
)
,
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
(
|
λ
m
n
u
x
m
n
u
-
λ
x
m
n
|
)
1
/
m
+
n
|
λ
m
n
u
-
λ
|
1
/
m
+
n
+
|
λ
|
1
/
m
+
n
)
∥
p
)
]
η
m
n
≤
1
,
and consequently
(26)
g
r
s
(
λ
m
n
u
x
m
n
u
-
λ
x
m
n
)
≤
(
|
λ
m
n
u
-
λ
|
)
η
m
n
/
H
inf
⁡
{
g
r
s
(
x
m
n
u
)
}
+
(
|
λ
|
)
η
m
n
/
H
inf
⁡
{
g
r
s
(
x
m
n
u
-
x
)
}
≤
max
⁡
{
|
λ
|
,
(
|
λ
|
)
η
m
n
/
H
}
g
r
s
(
x
m
n
u
-
x
m
n
)
.
Hence by our assumption the right-hand side tends to zero as u
,
m, and n
→
∞. This completes the proof.Theorem 6.
(i) If0
<
inf
⁡
m
n
η
m
n
=
H
0
≤
η
m
n
<
1, then Γ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η
⊂
Γ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
].
(ii) If1
≤
η
m
n
≤
sup
⁡
m
n
η
m
n
=
H
<
∞, then Γ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
⊂
Γ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η.
(iii) If0
<
η
m
n
<
μ
m
n
<
∞ and {
μ
m
n
/
η
m
n
} is analytic, then Γ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η
⊂
Γ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
μ.Proof.
The proof is easy. Therefore omit it.Lemma 7.
If a sequenceE is solid, then it is monotone. (See [10, page 53].)Theorem 8.
Γ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η is solid and also monotone.Proof.
Letx
∈
Γ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η and α
=
(
α
m
n
) be scalars such that |
α
m
n
|
1
/
m
+
n
≤
1 for m
,
n
∈
N. Then we have
(27)
{
∑
m
=
1
r
(
r
,
s
)
∈
N
×
N
:
i
i
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
…
,
d
n
(
x
n
-
1
)
(
|
α
m
n
x
m
n
|
)
1
/
m
+
n
)
∥
p
f
∥
(
(
|
α
m
n
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
iiiiiiiiiiiiiiiiiiiiii
i
i
i
…
,
d
n
(
x
n
-
1
)
(
|
α
m
n
x
m
n
|
)
1
/
m
+
n
)
∥
p
]
η
m
n
i
i
≤
ϵ
∑
m
=
1
r
}
⊂
{
{
…
,
d
n
(
x
n
-
1
)
[
f
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
)
∥
p
]
η
m
n
≤
ϵ
}
∈
I
2
{
{
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
≤
ϵ
}
(
r
,
s
)
∈
N
×
N
:
}
∑
m
=
1
r
(
r
,
s
)
∈
N
×
N
:
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
T
r
s
∑
m
=
1
r
∑
n
=
1
s
[
f
∥
…
,
d
n
(
x
n
-
1
)
[
f
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
)
∥
p
]
η
m
n
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
[
f
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
)
(
(
|
x
m
n
|
)
1
/
m
+
n
∥
p
]
η
m
n
≤
ϵ
∑
m
=
1
r
}
{
{
…
,
d
n
(
x
n
-
1
)
[
f
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
)
∥
p
]
η
m
n
≤
ϵ
}
∈
I
2
{
{
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
≤
ϵ
}
(
r
,
s
)
∈
N
×
N
:
}
∑
m
=
1
r
(
r
,
s
)
∈
N
×
N
:
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
∈
I
2
{
{
…
,
d
n
(
x
n
-
1
)
[
f
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
)
∥
p
]
η
m
n
≤
ϵ
}
∈
I
2
{
{
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
≤
ϵ
}
(
r
,
s
)
∈
N
×
N
:
}
∑
m
=
1
r
(
r
,
s
)
∈
N
×
N
:
}
,
where T
=
max
⁡
m
n
{
1
,
|
α
m
n
|
H
/
m
+
n
}. Hence α
x
∈
Γ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η with |
α
|
1
/
m
+
n
≤
1 for all m
,
n
∈
N whenever x
∈
Γ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η. Also by Lemma 7, it follows that Γ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η is monotone. This completes the proof.Theorem 9.
Letf, f
1, and f
2 be modulus functions. Then one has (i)
Γ
f
1
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η
⊂
Γ
f
∘
f
1
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η,
(ii)
Γ
f
1
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η
⋂
Γ
f
2
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η
Γ
f
1
+
f
2
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η.Proof.
(i) Letinf
⁡
m
n
η
m
n
=
H
0. For given ϵ
>
0, we first choose ϵ
0
>
0 such that max
⁡
{
ϵ
0
H
,
ϵ
0
H
0
}
<
ϵ. Now using the continuity of f, choose 0
<
δ
<
1 such that 0
<
t
<
δ implies f
(
t
)
<
ϵ
0. Let x
∈
Γ
f
1
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η.
We observe that(28)
A
(
δ
)
=
{
∑
m
=
1
r
(
r
,
s
)
∈
N
×
N
:
iii
i
i
i
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
f
1
∥
…
,
d
n
(
x
n
-
1
)
(
(
|
x
m
n
|
)
1
/
m
+
n
)
∥
p
]
η
m
n
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
ii
iii
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
(
(
|
x
m
n
|
)
1
/
m
+
n
)
∥
p
]
η
m
n
≥
δ
H
∑
m
=
1
r
}
∈
I
2
.
Thus if (
r
,
s
)
∉
A
(
δ
) then
(29)
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
…
,
d
n
(
x
n
-
1
)
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
)
∥
p
]
η
m
n
f
1
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
)
∥
p
]
η
m
n
<
δ
H
⟹
∑
m
=
1
r
∑
n
=
1
s
[
…
,
d
n
(
x
n
-
1
)
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
)
∥
p
]
η
m
n
f
1
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
)
∥
p
]
η
m
n
<
r
s
δ
H
,
⟹
[
…
,
d
n
(
x
n
-
1
)
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
)
∥
p
]
η
m
n
f
1
…
,
d
n
(
x
n
-
1
)
)
∥
p
]
η
m
n
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
)
∥
p
]
η
m
n
<
δ
H
,
∀
m
,
n
=
1,2
,
…
.
⟹
f
1
(
…
,
d
n
(
x
n
-
1
)
)
∥
p
)
…
,
d
n
(
x
n
-
1
)
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
)
∥
p
)
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
)
∥
p
)
<
δ
,
∀
m
,
n
=
1,2
,
…
.
Hence from above inequality and using continuity of f, we must have
(30)
f
(
f
1
(
…
,
d
n
(
x
n
-
1
)
(
|
x
m
n
|
)
1
/
m
+
n
)
∥
p
)
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
(
|
x
m
n
|
)
1
/
m
+
n
)
∥
p
)
)
<
ϵ
0
,
∀
m
,
n
=
1,2
,
…
.
∑
m
=
1
r
∑
n
=
1
s
[
f
(
…
,
d
n
(
x
n
-
1
)
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
)
∥
p
)
f
1
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
iiiiiiiiiiiiiiiiiiiiiiii
…
,
d
n
(
x
n
-
1
)
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
)
∥
p
)
]
η
m
n
<
r
s
max
⁡
{
ϵ
0
H
,
ϵ
0
H
0
}
<
r
s
ϵ
⟹
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
f
(
…
,
d
n
(
x
n
-
1
)
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
)
∥
p
f
1
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
)
∥
p
)
]
η
m
n
<
ϵ
.
Hence we have
(31)
{
∑
m
=
1
r
(
r
,
s
)
∈
N
×
N
:
i
ii
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
…
,
d
n
(
x
n
-
1
)
)
∥
p
)
]
η
m
n
f
(
…
,
d
n
(
x
n
-
1
)
)
∥
p
)
…
,
d
n
(
x
n
-
1
)
)
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
∥
p
)
]
η
m
n
f
1
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
…
,
d
n
(
x
n
-
1
)
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
)
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
∥
p
)
]
η
m
n
i
i
≥
ϵ
∑
m
=
1
r
}
⊂
A
(
δ
)
∈
I
2
.
(ii) Letx
∈
Γ
f
1
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η
⋂
Γ
f
2
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η. Then
(32)
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
…
,
d
n
(
x
n
-
1
)
)
∥
p
)
]
η
m
n
…
,
d
n
(
x
n
-
1
)
(
…
,
d
n
(
x
n
-
1
)
)
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
)
∥
p
)
]
η
m
n
(
f
1
+
f
2
)
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
×
(
…
,
d
n
(
x
n
-
1
)
(
…
,
d
n
(
x
n
-
1
)
)
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
)
∥
p
∥
…
,
d
n
(
x
n
-
1
)
)
∥
p
(
…
,
d
n
(
x
n
-
1
)
)
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
(
…
,
d
n
(
x
n
-
1
)
)
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
)
∥
p
)
]
η
m
n
≤
D
r
s
∑
m
=
1
r
∑
n
=
1
s
[
f
1
(
…
,
d
n
(
x
n
-
1
)
)
∥
p
)
∥
…
,
d
n
(
x
n
-
1
)
)
∥
p
(
…
,
d
n
(
x
n
-
1
)
)
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
…
,
d
n
(
x
n
-
1
)
)
∥
p
)
]
η
m
n
…
,
d
n
(
x
n
-
1
)
(
…
,
d
n
(
x
n
-
1
)
)
(
|
x
m
n
|
)
1
/
m
+
n
)
∥
p
)
]
η
m
n
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
(
…
,
d
n
(
x
n
-
1
)
)
(
|
x
m
n
|
)
1
/
m
+
n
)
∥
p
)
]
η
m
n
+
D
r
s
∑
m
=
1
r
∑
n
=
1
s
[
…
,
d
n
(
x
n
-
1
)
(
…
,
d
n
(
x
n
-
1
)
)
(
|
x
m
n
|
)
1
/
m
+
n
)
∥
p
)
]
η
m
n
f
2
(
…
,
d
n
(
x
n
-
1
)
(
…
,
d
n
(
x
n
-
1
)
)
(
|
x
m
n
|
)
1
/
m
+
n
)
∥
p
∥
(
(
|
x
m
n
|
)
1
/
m
+
n
,
d
1
(
x
1
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
(
…
,
d
n
(
x
n
-
1
)
)
(
|
x
m
n
|
)
1
/
m
+
n
)
∥
p
)
]
η
m
n
.
This completes the proof.Theorem 10.
Let the double sequenceη
=
(
η
m
n
) be analytic. Then
(33)
Γ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η
⊂
Λ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η
and the inclusion are strict.Theorem 11.
The class of sequenceΛ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η is sequence algebras.Proof.
Let(
x
m
n
)
,
(
y
m
n
)
∈
Λ
f
2
I
2
[
∥
(
d
1
(
x
1
)
,
…
,
d
n
(
x
n
)
)
∥
p
]
η and 0
<
ϵ
<
1. Then the result follows from the following inclusion relation:
(34)
{
{
∑
m
=
1
r
(
r
,
s
)
∈
N
×
N
:
i
i
i
i
i
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
…
,
d
n
(
x
n
-
1
)
∥
(
|
x
m
n
⊗
y
m
n
|
1
/
m
+
n
,
d
1
(
x
1
)
)
∥
p
f
∥
(
|
x
m
n
⊗
y
m
n
|
1
/
m
+
n
,
d
1
(
x
1
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
∥
(
|
x
m
n
⊗
y
m
n
|
1
/
m
+
n
,
d
1
(
x
1
)
)
∥
p
[
f
∥
(
|
x
m
n
⊗
y
m
n
|
1
/
m
+
n
,
d
1
(
x
1
)
,
]
η
m
n
i
i
i
i
<
ϵ
∑
m
=
1
r
}
∈
I
2
{
{
∑
m
=
1
r
(
r
,
s
)
∈
N
×
N
:
}
⊇
{
{
∑
m
=
1
r
(
r
,
s
)
∈
N
×
N
:
i
i
i
i
i
i
i
i
i
i
i
i
d
d
d
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
…
,
d
n
(
x
n
-
1
)
|
x
m
n
|
1
/
m
+
n
)
(
|
x
m
n
|
1
/
m
+
n
,
d
1
(
x
1
)
∥
p
]
η
m
n
…
,
d
n
(
x
n
-
1
)
)
∥
p
f
∥
(
|
x
m
n
|
1
/
m
+
n
,
d
1
(
x
1
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
d
d
d
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
…
,
d
n
(
x
n
-
1
)
|
x
m
n
|
1
/
m
+
n
)
(
|
x
m
n
|
1
/
m
+
n
,
d
1
(
x
1
)
∥
p
]
η
m
n
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
<
ϵ
∑
m
=
1
r
}
∈
I
2
{
{
∑
m
=
1
r
(
r
,
s
)
∈
N
×
N
:
}
i
⋂
{
{
∑
m
=
1
r
(
r
,
s
)
∈
N
×
N
:
‍
i
i
i
i
i
i
i
i
i
i
i
1
r
s
∑
m
=
1
r
∑
n
=
1
s
[
…
,
d
n
(
x
n
-
1
)
|
y
m
n
|
1
/
m
+
n
)
(
|
y
m
n
|
1
/
m
+
n
,
d
1
(
x
1
)
∥
p
]
η
m
n
f
∥
(
|
y
m
n
|
1
/
m
+
n
,
d
1
(
x
1
)
,
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
i
i
i
i
…
,
d
n
(
x
n
-
1
)
|
y
m
n
|
1
/
m
+
n
)
(
|
y
m
n
|
1
/
m
+
n
,
d
1
(
x
1
)
∥
p
]
η
m
n
i
i
i
i
i
i
i
i
i
i
i
<
ϵ
∑
m
=
1
r
}
∈
I
2
{
{
∑
m
=
1
r
(
r
,
s
)
∈
N
×
N
:
}
.
Similarly we can prove the result for other cases.



---

*Source: 790950-2014-05-20.xml*