In this article we introduce the sequence spaces
for a modulus function
f
, where
p
= (
p_{k}
) is a sequence of positive reals and study some of the properties of these spaces.
AMS Mathematics Subject Classification : 65H05, 65F10.
1. Introduction
The notion of IConvergence is a generalization of the concept of statistical convergence which was first introduced by H.Fast
[5]
and later on studied by various mathematicians like J.A.Fridy
[6
,
7]
, Kostyrko, Salat and Wilezynski
[19]
, Salat, Tripathy, Ziman
[29]
and Demirci
[3]
.
Also a double sequence is a double infinite array of elements
x_{kl}
∈ ℝ for all
k, l
∈ ℕ (see
[14
,
15]
). The initial works on double sequences is found in Bromwich
[1]
, Basarir and Solancan
[2]
and many others. Throughout this article a double sequence is denoted by
x
= (
x_{ij}
).
Next we discuss some preliminaries about Iconvergence (see
[12]
,
[30]
).
Let
X
be a non empty set. Then a family of sets I⊆ 2
^{X}
(power set of
X
) is said to be an ideal if I is additive i.e A,B∈I ⇒A∪ B∈I and hereditary i.e A∈I, B⊆A⇒B∈I.
A nonempty family of sets
£
(
I
) ⊆ 2
^{X}
is said to be filter on X if and only if Φ ∉
£
(I), for A,B∈
£
(I) we have A∩B∈
£
(I) and for each A ∈
£
(I)and A⊆B implies B∈
£
(I). An Ideal I⊆ 2
^{X}
is called nontrivial if I ≠ 2
^{X}
. A nontrivial ideal I⊆ 2
^{X}
is called admissible if {
x
: {
x
} ∈
X
} ⊆I.
A nontrivial ideal I is maximal if there cannot exist any nontrivial ideal J ≠ I containing I as a subset. For each ideal I, there is a filter
£
(I) corresponding to I. i.e
£
(I) = {
K
⊆
N
:
K^{c}
∈
I
},where K
^{c}
= NK.
Definition 1.1.
A double sequence (
x_{ij}
) ∈
ω
is said to be Iconvergent to a number L if for every
ϵ
> 0. {(
i, j
) ∈
×
: 
x_{ij}
−
L
 ≥
ϵ
} ∈ I. In this case we write Ilim
x_{ij}
=
L
. (see
[17]
)
The space
_{2}
c^{I}
of all Iconvergent sequences to
L
is given by
Definition 1.2.
A sequence (
x_{ij}
) ∈
ω
is said to be Inull if L = 0. In this case we write Ilim
x_{ij}
= 0.
Definition 1.3.
A sequence (
x_{ij}
) ∈
ω
is said to be Icauchy if for every
ϵ
> 0 there exists a number m = m(
ϵ
) and n = n(
ϵ
) such that
Definition 1.4.
A sequence (
x_{ij}
) ∈
ω
is said to be Ibounded if there exists M > 0 such that {(
i, j
) ∈
×
: 
x_{ij}
 >
M
}.
Definition 1.5.
Let (
x_{ij}
), (
y_{ij}
) be two sequences. We say that (
x_{ij}
) = (
y_{ij}
) for
almost all (i,j) relative to I (a.a.k.r.I),
if {(
i, j
) ∈
×
:
x_{ij}
≠
y_{ij}
} ∈
I
Definition 1.6.
For any set E of sequences the space of multipliers of E, denoted by
M
(
E
) is given by
Definition 1.7.
A map
ħ
defined on a domain
D
⊂
X
i.e
ħ
:
D
⊂
X
→
is said to satisfy Lipschitz condition if 
ħ
(
x
)−
ħ
(
y
) ≤
K

x
−
y
 where
K
is known as the Lipschitz constant.The class of KLipschitz functions defined on D is denoted by
ħ
∈ (
D,K
).
Definition 1.8.
A convergence field of Iconvergence is a set
The convergence field
F
(
I
) is a closed linear subspace of
l
_{∞}
with respect to the supremum norm,
F
(
I
) =
l
_{∞}
∩
_{2}
c^{I}
(See
[23]
).
Define a function
ħ
:
F
(
I
) →
such that
ħ
(
x
) =
I
− lim
x
, for all
x
∈
F
(
I
), then the function
ħ
:
F
(
I
) →
is a Lipschitz function (
[11
,
4
,
13]
).
Definition 1.9.
The concept of paranorm is closely related to linear metric spaces
[16]
. It is a generalization of that of absolute value.
Let
X
be a linear space. A function
g
:
X
→
R
is called paranorm, if for all
x, y, z
∈
X
,
(PI)
g
(
x
) = 0
if x
=
θ
, (P2)
g
(−
x
) =
g
(
x
), (P3)
g
(
x
+
y
) ≤
g
(
x
+
g
(
y
),
(P4) If (
λ_{n}
) is a sequence of scalars with
λ_{n}
→
λ
(
n
→ ∞) and
x_{n}
,
a
∈
X
with
x_{n}
→
a
(
n
→ ∞) , in the sense that
g
(
x_{n}
−
a
) → 0 (
n
→ ∞) , in the sense that
g
(
λ_{n}
x_{n}
−
λa
) → 0 (
n
→ ∞).
A paranorm
g
for which
g
(
x
) = 0 implies
x
=
θ
is called a total paranorm on
X
, and the pair (
X, g
) is called a totally paranormed space.(See
[23]
). The idea of modulus was structured in 1953 by Nakano.(See
[24]
). A function
f
: [0,∞)→[0,∞) is called a modulus if
(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 and (4)
f
is continuous from the right at zero.
Ruckle in
[25
,
26
,
27]
used the idea of a modulus function
f
to construct the sequence space
This space is an FK space ,and Ruckle proved that the intersection of all such
X
(
f
) spaces is
ϕ
, the space of all finite sequences.
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 proved that,for any modulus
f
Where
The space
X
(
f
) is a Banach space with respect to the norm
Spaces of the type
X
(
f
) are a special case of the spaces structured by B.Gramsch in
[10]
. From the point of view of local convexity, spaces of the type
X
(
f
) are quite interesting.
Symmetric sequence spaces, which are locally convex have been frequently studied by D.J.H Garling
[8
,
9]
, G.Köthe
[18]
.
The following subspaces of
ω
were first introduced and discussed by Maddox
[22
,
23]
.
where
p
= (
p_{k}
) is a sequence of striclty positive real numbers.
After then Lascarides
[20
,
21]
defined the following sequence spaces
where
for all
k
∈
.
We need the following lemmas in order to establish some results of this article.
Lemma 1.10.
Then the following conditions are equivalent.(See[18]).
(a) H
< ∞
and h
> 0.
(b)
c
_{0}
(
p
) =
c
_{0}
or l
_{∞}
(
p
) =
l
_{∞}
.
(c) l_{∞}
{
p
} =
l
_{∞}
(
p
).
(d) c
_{0}
{
p
} =
c
_{0}
(
p
).
(e) l
{
p
} =
l
(
p
).
Lemma 1.11.
Let K
∈
£
(
I
)
and M
⊆
N
.
If M
∉
I, then M
∩
K
∉
I
.(See
[29
,
30]
).
Lemma 1.12.
If I
⊂ 2
^{X}
and M
⊆
X
.
If M
∉
I, then M
∩
K
∉
I
.(See
[29
,
30]
).
Throughout the article
represent the bounded , Iconvergent, Inull, bounded Iconvergent and bounded Inull sequence spaces respectively. In this article we introduce the following classes of sequence spaces.
Also we write
2. Main results
Theorem 2.1.
Let
(
p_{ij}
) ∈
_{2}
l_{∞}
.
are linear spaces.
Proof
. Let (
x_{ij}
), (
y_{ij}
) ∈
_{2}
c^{I}
(
f, p
) and
α, β
be two scalars. Then for a given
ϵ
> 0.
We have
where
be such that
Then
Thus
Hence (
αx_{ij}
+
βy_{ij}
) ∈
_{2}
c^{I}
(
f, p
). Therefore
_{2}
c^{I}
(
f, p
) is a linear space. The rest of the result follows similarly.
Theorem 2.2.
Let
(
p_{ij}
) ∈
_{2}
l_{∞}
.
Then
_{2}
m^{I}
(
f, p
)
and
are paranormed spaces, paranormed by
Proof
. Let
x
= (
x_{ij}
),
y
= (
y_{ij}
) ∈
_{2}
m^{I}
(
f, p
).
(1) Clearly,
g
(
x
) = 0 if and only if
x
= 0.
(2)
g
(
x
) =
g
(
x
) is obvious.
(3) Since
using Minkowski’s inequality and the definition of
f
we have
(4) Now for any complex
λ
we have (
λ_{ij}
) such that
λ_{ij}
→
λ
, (
i, j
→ ∞).
Let
x_{ij}
∈
_{2}
m^{I}
(
f, p
) such that
f
(
x_{ij}
−
L

^{pij}
) ≥
ϵ
.
Therefore,
Hence
g
(
λ_{ij}
x_{ij}

λL
) ≤
g
(
λ_{ij}
x_{ij}
) +
g
(
λL
) =
λ_{ij}
g
(
x_{ij}
) +
λg
(
L
) as (
i, j
→ ∞).
Hence
_{2}
m^{I}
(
f, p
) is a paranormed space. The rest of the result follows similarly.
Theorem 2.3.
A sequence
x
= (
x_{ij}
) ∈
_{2}
m^{I}
(
f, p
)
Iconverges if and only if for every ϵ
> 0
there exists N_{ϵ}
∈
×
where N_{ϵ}
= (
m, n
),
m and n depending upon ϵ such that
Proof
. Suppose that
L
=
I
− lim
x
. Then
Fixing some
N_{ϵ}
∈
B_{ϵ}
, we get
which holds for all (
i, j
) ∈
B_{ϵ}
. Hence
Conversely, suppose that
That is {(
i, j
) ∈
×
: (
x_{ij}
−
x_{Nϵ}

^{pij}
) <
ϵ
} ∈
_{2}
m^{I}
(
f, p
) for all
ϵ
> 0. Then the set
Let
J_{ϵ}
= [
x_{Nϵ}
−
ϵ
,
x_{Nϵ}
+
ϵ
]. If we fix an
ϵ
> 0 then we have
C_{ϵ}
∈
_{2}
m^{I}
(
f, p
) as well as
This implies that
that is
that is
where the diam of J denotes the length of interval J. In this way, by induction we get the sequence of closed intervals
with the property that
and {(
i, j
) ∈
×
:
x_{ij}
∈
I_{k}
} ∈
_{2}
m^{I}
(
f, p
) for (k=1,2,3,4,......).
Then there exists a
ξ
∈ ∩
I_{k}
where (
i, j
) ∈
×
such that
ξ
=
I
− lim
x
. So that
f
(
ξ
) =
I
− lim
f
(
x
), that is
L
=
I
− lim
f
(
x
).
Theorem 2.4.
and I be an admissible ideal. Then the following are equivalent.
(a)
(
x_{ij}
) ∈
_{2}
c^{I}
(
f, p
);
(b) there exists
(
y_{ij}
) ∈
_{2}
c
(
f, p
)
such that x_{ij}
=
y_{ij}
, for
a.a.k.r.I
;
(c) there exists
(
y_{ij}
) ∈
_{2}
c
(
f, p
)
and
such that x_{ij}
=
y_{ij}
+
z_{ij}
for all
(
i, j
) ∈
×
and
{(
i, j
) ∈
×
:
f
(
y_{ij}
− L
^{pij}
) ≥
ϵ
} ∈
I
;
(d) there exists a subset J
×
K where J
= {
j
_{1}
,
j
_{2}
, ...}
and K
= {
k
_{1}
<
k
_{2}
....}
of
×
such that
Proof
.
(a) implies (b)
Let (
x_{ij}
) ∈
_{2}
c^{I}
(
f, p
). Then there exists
L
∈
that
Let (
m_{t}
)
and
(
n_{t}
) be increasing sequences with
m_{t}
and
n_{t}
∈
such that
Define a sequence (
y_{ij}
) as
For
m_{t}
<
k
≤
m_{t}
_{+1}
,
t
∈
.
Then (
y_{ij}
) ∈
_{2}
c
(
f, p
) and form the following inclusion
We get
x_{ij}
=
y_{ij}
, for a.a.k.r.I.
(b) implies (c)
For (
x_{ij}
) ∈
_{2}
c^{I}
(
f, p
). Then there exists (
y_{ij}
) ∈
_{2}
c
(
f, p
) such that
x_{ij}
=
y_{ij}
, for a.a.k.r.I. Let
K
= {(
i, j
) ∈
×
:
x_{ij}
≠
y_{ij}
}, then (
i, j
) ∈
I
.
Define a sequence (
z_{ij}
) as
Then
and
y_{ij}
∈
_{2}
c
(
f, p
).
(c) implies (d)
Let
P
_{1}
= {(
i, j
) ∈
×
:
f
(
x_{ij}

^{pij}
) ≥
ϵ
} ∈
I
and
Then we have
(d) implies (a)
Let
K
= {(
i
_{1}
,
j
_{1}
) < (
i
_{2}
,
j
_{2}
) < (
i
_{3}
,
j
_{3}
) < ...} ∈
£
(
I
) and
Then for an
ϵ
> 0, and Lemma 1.10, we have
Thus (
x_{ij}
) ∈
_{2}
c^{I}
(
f, p
).
Theorem 2.5.
Let
(
p_{ij}
)
and
(
q_{ij}
)
be a sequence of positive real numbers. Then
where
K^{c}
⊆
×
such that K
∈
I
.
Proof
. Let
Then there exists
β
> 0 such that
p_{ij}
>
βq_{ij}
, for all sufficiently large (
i, j
) ∈
K
.
Since
for a given
ϵ
> 0, we have
Let
G
_{0}
=
K^{c}
∪
B
_{0}
Then
G
_{0}
∈
I
. Then for all sufficiently large (
i, j
) ∈
G
_{0}
,
Therefore
Corollary 2.6.
Let
(
p_{ij}
)
and
(
q_{ij}
)
be two sequences of positive real numbers. Then
where
K^{c}
⊆
×
such that K
∈
I
.
Theorem 2.7.
Let
(
p_{ij}
)
and
(
q_{ij}
)
be two sequences of positive real numbers. Then
where
K
⊆
×
such that K^{c}
∈
I
.
Proof
. On combining Theorem 2.5 and 2.6 we get the required result.
Theorem 2.8.
Then the following results are equivalent
.
(a) H
< ∞
and h
> 0.
Proof
. Suppose that
H
< ∞ and
h
> 0,then the inequalities
min
{1,
s^{h}
} ≤
s^{pij}
≤
max
{1,
s^{H}
} hold for any
s
> 0 and for all (
i, j
) ∈
×
. Therefore the equivalence of (a) and (b) is obvious.
Theorem 2.9.
Let f be a modulus function. Then
(see [13]).
Proof
. Let (
x_{ij}
) ∈
_{2}
c^{I}
(
f, p
). Then there exists
L
∈
that
We have
Taking supremum over (
i, j
) both sides we get
and the inclusion
is obvious. Hence
and the inclusions are proper.
Theorem 2.10.
then for any modulus f, we have
where the inclusion may be proper.
Proof
. Let
This implies that
for some
K
> 0 and all (
i, j
). Therefore
x
= (
x_{ij}
) ∈
_{2}
m^{I}
(
f, p
) implies
which gives
To show that the inclusion may be proper, consider the case when
for all (
i, j
). Take
a_{ij}
= (
i
×
j
) for all (
i, j
).
Therefore
x
∈
_{2}
m^{I}
(
f, p
) implies
Thus in this case
a
= (
a_{ij}
) ∈
M
(
_{2}
m^{I}
(
f, p
)) while
Theorem 2.11.
The function ħ
:
_{2}
m^{I}
(
f, p
) →
is the Lipschitz function,where
_{2}
m^{I}
(
f, p
) =
_{2}
c^{I}
(
f, p
) ∩
_{2}
l_{∞}
(
f, p
),
and hence uniformly continuous.
Proof
. Let
x, y
∈
_{2}
m^{I}
(
f, p
),
x
≠
y
. Then the sets
Here
Thus the sets,
Hence also
B
=
B_{x}
∩
B_{y}
∈
_{2}
m^{I}
(
f, p
), so that
B
≠
ϕ
. Now taking (
i, j
) in
B
such that
Thus
ħ
is a Lipschitz function. For
the result can be proved similarly.
Theorem 2.12.
If x, y
∈
_{2}
m^{I}
(
f, p
),
then
(
x.y
) ∈
_{2}
m^{I}
(
f, p
) and
ħ
xy
) =
ħ
(
x
)
ħ
(
y
).
Proof
. For
ϵ
> 0
Now,
As
_{2}
m^{I}
(
f, p
) ⊆
l_{∞}
(
f, p
),there exists an
M
∈
such that 
x_{ij}

^{pij}
<
M
and 
ħ
(
y
)
^{pk}
<
M
. Using eqn(2) we get
For all (
i, j
) ∈
B_{x}
∩
B_{y}
∈
_{2}
m^{I}
(
f.p
). Hence (
x.y
) ∈
m^{I}
(
f, p
) and
ħ
(
xy
) =
ħ
(
x
)
ħ
(
y
). For
the result can be proved similarly.
Acknowledgements
The authors would like to record their gratitude to the reviewer for his careful reading and making some useful corrections which improved the presentation of the paper.
BIO
Vakeel A. Khan received M.Sc., M.Phil and Ph.D at Aligarh Muslim University. He is currently an Associate Professor at Aligarh Muslim University. He has published a number of research articles and some books to his name. His research interests include Functional Analysis, sequence spaces, Iconvergence, invariant meanszweir sequences and so on.
Department of Mathematics, Aligarh Muslim University, Aligarh, 200002, India.
email: vakhanmaths@gmail.com
Nazneen Khan received M.Sc. and M.Phil. from Aligarh Muslim University, and is currently a Ph.D. scholar at Aligarh Muslim University. Her research interests are Functional Analysis, sequence spaces and double sequences.
Department of Mathematics, Aligarh Muslim University, Aligarh, 200002, India.
email: nazneen4maths@gmail.com
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