Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 482392, 8 pages
doi:10.1155/2010/482392
Research Article
On Some New Sequence Spaces in
2-Normed Spaces Using Ideal Convergence and
an Orlicz Function
E. Savaş
Department of Mathematics, Istanbul Ticaret University, Üsküdar, 34672 Istanbul, Turkey
Correspondence should be addressed to E. Savaş, ekremsavas@yahoo.com
Received 25 July 2010; Accepted 17 August 2010
Academic Editor: Radu Precup
Copyright q 2010 E. Savaş. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
The purpose of this paper is to introduce certain new sequence spaces using ideal convergence and
an Orlicz function in 2-normed spaces and examine some of their properties.
1. Introduction
The notion of ideal convergence was introduced first by Kostyrko et al. 1 as a generalization
of statistical convergence which was further studied in topological spaces 2. More
applications of ideals can be seen in 3, 4.
The concept of 2-normed space was initially introduced by Gähler 5 as an interesting
nonlinear generalization of a normed linear space which was subsequently studied by many
authors see, 6, 7. Recently, a lot of activities have started to study summability, sequence
spaces and related topics in these nonlinear spaces see, 8–10.
Recall in 11 that an Orlicz function M : 0, ∞ → 0, ∞ is continuous, convex,
nondecreasing function such that M0 0 and Mx > 0 for x > 0, and Mx → ∞ as
x → ∞.
Subsequently Orlicz function was used to define sequence spaces by Parashar and
Choudhary 12 and others.
If convexity of Orlicz function, M is replaced by Mx y ≤ Mx My, then this
function is called Modulus function, which was presented and discussed by Ruckle 13 and
Maddox 14.
Note that if M is an Orlicz function then Mλx ≤ λMx for all λ with 0 < λ < 1.
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Journal of Inequalities and Applications
Let X, · be a normed space. Recall that a sequence xn n∈N of elements of X is called
to be statistically convergent to x ∈ X if the set Aε {n ∈ N : xn − x ≥ ε} has natural
density zero for each ε > 0.
A family I ⊂ 2Y of subsets a nonempty set Y is said to be an ideal in Y if i ∅ ∈ I; ii
A, B ∈ I imply A ∪ B ∈ I; iii A ∈ I, B ⊂ A imply B ∈ I, while an admissible ideal I of Y
further satisfies {x} ∈ I for each x ∈ Y , 9, 10.
Given I ⊂ 2N is a nontrivial ideal in N. The sequence xn n∈N in X is said to be Iconvergent to x ∈ X, if for each ε > 0 the set Aε {n ∈ N : xn − x ≥ ε} belongs to I,
1, 3.
Let X be a real vector space of dimension d, where 2 ≤ d < ∞. A 2-norm on X is a
function ·, · : X × X → R which satisfies i x, y 0 if and only if x and y are linearly
dependent, ii x, y y, x, iii αx, y |α|x, y, α ∈ R, and iv x, y z ≤ x, y x, z. The pair X, ·, · is then called a 2-normed space 6.
Recall that X, ·, · is a 2-Banach space if every Cauchy sequence in X is convergent
to some x in X.
Quite recently Savaş 15 defined some sequence spaces by using Orlicz function and
ideals in 2-normed spaces.
In this paper, we continue to study certain new sequence spaces by using Orlicz
function and ideals in 2-normed spaces. In this context it should be noted that though
sequence spaces have been studied before they have not been studied in nonlinear structures
like 2-normed spaces and their ideals were not used.
2. Main Results
Let Λ λn be a nondecreasing sequence of positive numbers tending to ∞ such that λn1 ≥
λn 1, λ1 0 and let I be an admissible ideal of N, let M be an Orlicz function, and let
X, ·, · be a 2-normed space. Further, let p pk be a bounded sequence of positive real
numbers. By S2 − X we denote the space of all sequences defined over X, ·, ·. Now, we
define the following sequence spaces:
W I λ, M, p, , ·, xk − L pk
1 x ∈ S2 − X : ∀ε > 0 n ∈ N :
≥ε ∈I
M ρ , z
λn k∈I
n
for some ρ > 0, L ∈ X and each z ∈ X ,
W0I λ, M, p, , ·, xk pk
1 x ∈ S2 − X : ∀ε > 0 n ∈ N :
≥ε ∈I
M ρ , z
λn k∈I
n
for some ρ > 0, and each z ∈ X ,
Journal of Inequalities and Applications
W∞ λ, M, p, , ·, 3
xk pk
1 ,
z
M ≤K
ρ n∈N λn k∈In
x ∈ S2 − X : ∃K > 0 s.t. sup
for some ρ > 0, and each z ∈ X ,
I
W∞
λ, M, p, , ·, xk pk
1 x ∈ S2 − X : ∃K > 0 s.t. n ∈ N :
≥K ∈I
M , z
λn k∈I
ρ
n
for some ρ > 0, and each z ∈ X ,
2.1
where In n − λn 1, n.
The following well-known inequality 16, page 190 will be used in the study.
If
0 ≤ pk ≤ sup pk H,
D max 1, 2H−1
2.2
then
|ak bk |pk ≤ D |ak |pk |bk |pk
2.3
for all k and ak , bk ∈ C. Also |a|pk ≤ max1, |a|H for all a ∈ C.
I
Theorem 2.1. W I λ, M, p, , ·, , W0I λ, M, p, , ·, , and W∞
λ, M, p, , ·, are linear spaces.
Proof. We will prove the assertion for W0I λ, M, p, , ·, only and the others can be proved
similarly. Assume that x, y ∈ W0I λ, M, , ·, and α, β ∈ R, so
xk pk
1 n∈N:
≥ε ∈I
M ρ , z
λn k∈I
1
for some ρ1 > 0,
n
xk pk
1 n∈N:
M , z
≥ε ∈I
λn k∈I
ρ2
r
2.4
for some ρ2 > 0.
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Journal of Inequalities and Applications
Since , ·, is a 2-norm, and M is an Orlicz function the following inequality holds:
pk
αxk βyk 1 , z
M |α|ρ1 βρ2
λn k∈I
n
pk
xk 1 |α|
M ≤D
ρ , z
λn k∈I
|α|ρ1 βρ2
1
n
pk
β
yk 1 M D
ρ , z
λn k∈I
|α|ρ1 βρ2
2
n
xk pk
yk pk
1 1 , z
≤ DF
,
z
DF
,
M M
ρ
ρ
λn k∈I
λn k∈I
1
2
n
2.5
n
where
⎡ F max⎣1, |α|
|α|ρ1 βρ2
H ,
H ⎤
β
⎦.
|α|ρ1 βρ2
2.6
From the above inequality, we get
pk
αxk βyk 1 , z
n∈N:
M |α|ρ1 βρ2
λn k∈I
n
xk pk
1 ⊆ n ∈ N : DF
M ,
z
≥
ρ
λn k∈I
1
n
≥ε
ε
2
2.7
yk pk ε
1 ∪ n ∈ N : DF
.
≥
M , z
λn k∈I
ρ2
2
n
Two sets on the right hand side belong to I and this completes the proof.
It is also easy to see that the space W∞ λ, M, p, , ·, is also a linear space and we now
have the following.
Theorem 2.2. For any fixed n ∈ N, W∞ λ, M, p, , ·, is paranormed space with respect to the
paranorm defined by
⎧
⎨
gn x inf ρpn /H : ρ > 0 s.t.
⎩
⎫
1/H
⎬
xk pk
1 ,
z
sup
≤
1,
∀z
∈
X
.
M ρ ⎭
n λn k∈I
n
Proof. That gn θ 0 and gn −x gx are easy to prove. So we omit them.
2.8
Journal of Inequalities and Applications
5
iii Let us take x xk and y yk in W∞ λ, M, p, , ·, . Let
Ax A y xk pk
1 ρ > 0 : sup
≤ 1, ∀z ∈ X ,
M , z
ρ
n λn k∈I
n
yk pk
1 ρ > 0 : sup
≤ 1, ∀z ∈ X .
M ρ , z
n λn k∈I
2.9
n
Let ρ1 ∈ Ax and ρ2 ∈ Ay, then if ρ ρ1 ρ2 , then, we have
1 M
sup
n λn n∈In
xk yk xk ρ1
1 , z ≤
sup
M , z
ρ
ρ1 ρ2 n λn k∈I
ρ1 n
yk ρ2
1 .
sup
M , z
ρ1 ρ2 n λn k∈I
ρ2 2.10
n
Thus, supn 1/λn n∈In
Mxk yk /ρ1 ρ2 , zpk ≤ 1 and
!
p /H
gn x y ≤ inf ρ1 ρ2 n : ρ1 ∈ Ax, ρ2 ∈ A y
p /H
≤ inf ρ1 n
!
!
p /H
: ρ1 ∈ Ax inf ρ2 n : ρ2 ∈ A y
2.11
gn x gn y .
iv Finally using the same technique of Theorem 2 of Savaş 15 it can be easily seen
that scalar multiplication is continuous. This completes the proof.
Corollary 2.3. It should be noted that for a fixed F ∈ I the space
W∞ F λ, M, p, , ·, xk pk
1 ,
z
≤K
M ρ n∈N−F λn k∈I
x ∈ S2 − X : ∃K > 0 s.t. sup
n
2.12
for some ρ > 0, and each z ∈ X ,
I
λ, M, p, , ·, is a paranormed space with the paranorms gn for
which is a subspace of the space W∞
F
n/
∈ F and g infn∈N−F gn .
Theorem 2.4. Let M,M1 , M2 , be Orlicz functions. Then we have
i W0I λ, M1 , p, , ·, ⊆ W0I λ, M ◦ M1 , p, , ·, provided pk is such that H0 inf pk >
0.
ii W0I λ, M1 , p, , ·, ∩ W0I λ, M2 , p, , ·, ⊆ W0I λ, M1 M2 , p, , ·, .
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Journal of Inequalities and Applications
Proof. i For given ε > 0, first choose ε0 > 0 such that max{ε0H , ε0H0 } < ε. Now using
the continuity of M choose 0 < δ < 1 such that 0 < t < δ ⇒ Mt < ε0 . Let xk ∈
W0 λ, M1 , p, , ·, . Now from the definition
Aδ xk pk
1 H
∈ I.
n∈N:
≥δ
M1 ρ , z
λn n∈In
2.13
Thus if n /
∈ Aδ then
xk pk
1 ,
z
< δH ,
M1 ρ λn n∈In
2.14
xk pk
< λn δ H ,
M1 , z
ρ n∈In
2.15
that is,
that is,
xk pk
M1 ,
z
< δH ,
ρ ∀k ∈ In ,
2.16
that is,
xk M1 ,
z
ρ < δ,
∀k ∈ In .
2.17
Hence from above using the continuity of M we must have
xk < ε0 ,
M M1 , z
ρ ∀k ∈ In ,
2.18
which consequently implies that
k∈In
pk
!
xk H H0
< λn ε,
,
z
<
λ
max
ε
,
ε
M M1 n
0
ρ 0
2.19
that is,
pk
xk 1 < ε.
M M1 , z
λn k∈I
ρ n
2.20
Journal of Inequalities and Applications
7
This shows that
pk
xk 1 n∈N:
≥ ε ⊂ Aδ
M M1 , z
λn k∈I
ρ
2.21
n
and so belongs to I. This proves the result.
ii Let xk ∈ W0I M1 , p, , ·, ∩ W0I M2 , p, , ·, , then the fact
xk pk
xk pk
xk pk
1
1
1
≤D
D
M1 M2 , z
M1 , z
M2 , z
λn
ρ
λn
ρ
λn
ρ 2.22
gives us the result.
Definition 2.5. Let X be a sequence space. Then X is called solid if αk xk ∈ X whenever
xk ∈ X for all sequences αk of scalars with |αk | ≤ 1 for all k ∈ N.
I
Theorem 2.6. The sequence spaces W0I λ, M, p, , ·, , W∞
λ, M, p, , ·, are solid.
Proof. We give the proof for W0I λ, M, p, , ·, only. Let xk ∈ W0I λ, M, p, , ·, and let αk be a sequence of scalars such that |αk | ≤ 1 for all k ∈ N. Then we have
αk xk pk
xk pk
1 C
n∈N:
≥ε ⊆ n∈N:
≥ ε ∈ I,
M M ρ , z
ρ , z
λn k∈I
λn k∈I
n
n
2.23
where C maxk {1, |αk |H }. Hence αk xk ∈ W0I λ, M, p, , ·, for all sequences of scalars αk with |αk | ≤ 1 for all k ∈ N whenever xk ∈ W0I λ, M, p, , ·, .
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