Volume 2010, Article ID 482392,8pages doi:10.1155/2010/482392
Research Article
On Some New Sequence Spaces in
2-Normed Spaces Using Ideal Convergence and an Orlicz Function
E. Savas¸
Department of Mathematics, Istanbul Ticaret University, ¨Usk ¨udar, 34672 Istanbul, Turkey
Correspondence should be addressed to E. Savas¸,[email protected] Received 25 July 2010; Accepted 17 August 2010
Academic Editor: Radu Precup
Copyrightq2010 E. Savas¸. 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.1as a generalization of statistical convergence which was further studied in topological spaces 2. More applications of ideals can be seen in3,4.
The concept of 2-normed space was initially introduced by G¨ahler5as an interesting nonlinear generalization of a normed linear space which was subsequently studied by many authorssee,6,7. Recently, a lot of activities have started to study summability, sequence spaces and related topics in these nonlinear spacessee,8–10.
Recall in11 that an Orlicz function M : 0,∞ → 0,∞is continuous, convex, nondecreasing function such thatM0 0 andMx > 0 forx > 0, andMx → ∞ as x → ∞.
Subsequently Orlicz function was used to define sequence spaces by Parashar and Choudhary12and others.
If convexity of Orlicz function,Mis replaced byMxy≤Mx My, then this function is called Modulus function, which was presented and discussed by Ruckle13and Maddox14.
Note that ifMis an Orlicz function thenMλx≤λMxfor allλwith 0< λ <1.
LetX,·be a normed space. Recall that a sequencexnn∈Nof elements ofXis called to be statistically convergent tox ∈X if the setAε {n ∈ N :xn−x ≥ ε}has natural density zero for eachε >0.
A familyI ⊂2Y of subsets a nonempty setY is said to be an ideal inYifi∅ ∈ I;ii A, B∈ IimplyA∪B∈ I;iiiA∈ I, B⊂ AimplyB ∈ I, while an admissible idealIofY further satisfies{x} ∈ Ifor eachx∈Y,9,10.
GivenI ⊂ 2N is a nontrivial ideal inN. The sequencexnn∈N inX is said to beI- convergent tox ∈ X,if for eachε > 0 the setAε {n ∈ N : xn−x ≥ ε}belongs toI, 1,3.
LetX be a real vector space of dimensiond,where 2 ≤ d < ∞.A 2-norm onX is a function·,· :X×X → Rwhich satisfiesix, y 0 if and only ifxandyare linearly dependent,iix, y y, x,iiiαx, y |α|x, y, α∈R, andivx, yz ≤ x, y x, z. The pairX,·,·is then called a 2-normed space6.
Recall thatX,·,·is a 2-Banach space if every Cauchy sequence inXis convergent to somexinX.
Quite recently Savas¸15defined 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
LetΛ λnbe a nondecreasing sequence of positive numbers tending to∞such thatλn1 ≥ λn 1, λ1 0 and letI be an admissible ideal of N, letM be an Orlicz function, and let X,·,· be a 2-normed space. Further, letp pk be a bounded sequence of positive real numbers. ByS2−X we denote the space of all sequences defined overX,·,·. Now, we define the following sequence spaces:
WI
λ, M, p,,·,
x∈S2−X:∀ε >0
n∈N: 1 λn
k∈In
M
xk−L ρ , z
pk ≥ε
∈I
for someρ >0, L∈X and eachz∈X
, W0I
λ, M, p,,·,
x∈S2−X:∀ε >0
n∈N: 1 λn
k∈In
M
xk ρ , z
pk ≥ε
∈I
for someρ >0, and eachz∈X
,
W∞
λ, M, p,,·,
x∈S2−X:∃K >0 s.t. sup
n∈N
1 λn
k∈In
M
xk ρ , z
pk ≤K for someρ >0,and eachz∈X
, W∞I
λ, M, p,,·,
x∈S2−X:∃K >0 s.t.
n∈N: 1 λn
k∈In
M
xk ρ , z
pk ≥K
∈I
for someρ >0,and eachz∈X
,
2.1
whereIn n−λn1, n.
The following well-known inequality 16, page 190will be used in the study.
If 0≤pk≤suppkH, Dmax
1,2H−1
2.2
then
|akbk|pk ≤D
|ak|pk|bk|pk
2.3
for allkandak, bk∈C. Also|a|pk≤max1,|a|Hfor alla∈C.
Theorem 2.1. WIλ, M, p,,·,, W0Iλ, M, p,,·,, andW∞I λ, M, p,,·,are linear spaces.
Proof. We will prove the assertion forW0Iλ, M, p,,·,only and the others can be proved similarly. Assume thatx, y∈W0Iλ, M,,·, andα, β∈R, so
n∈N: 1 λn
k∈In
M
xk
ρ1, z
pk ≥ε
∈I for someρ1>0,
n∈N: 1 λn
k∈Ir
M
xk
ρ2, z
pk ≥ε
∈I for someρ2>0.
2.4
Since,·, is a 2-norm, andMis an Orlicz function the following inequality holds:
1 λn
k∈In
M
αxkβyk |α|ρ1βρ2, z
pk
≤D 1 λn
k∈In
|α|
|α|ρ1βρ2
M xk
ρ1
, z pk D 1
λn
k∈In
β |α|ρ1βρ2
M yk
ρ2
, z pk
≤DF 1 λn
k∈In
M
xk
ρ1, z
pkDF 1 λn
k∈In
M
yk ρ2, z
pk,
2.5
where
F max
⎡
⎣1,
|α|
|α|ρ1βρ2 H
,
β |α|ρ1βρ2
H⎤
⎦. 2.6
From the above inequality, we get
n∈N: 1 λn
k∈In
M
αxkβyk |α|ρ1βρ2, z
pk
≥ε
⊆
n∈N:DF 1 λn
k∈In
M
xk
ρ1, z
pk≥ ε 2
∪
n∈N:DF 1 λn
k∈In
M
yk
ρ2, z
pk ≥ ε 2
.
2.7
Two sets on the right hand side belong toIand this completes the proof.
It is also easy to see that the spaceW∞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
gnx inf
⎧⎨
⎩ρpn/H:ρ >0 s.t.
sup
n
1 λn
k∈In
M
xk ρ , z
pk 1/H
≤1, ∀z∈X
⎫⎬
⎭. 2.8
Proof. Thatgnθ 0 andgn−x gxare easy to prove. So we omit them.
iiiLet us takex xkandy ykinW∞λ, M, p,,·,. Let
Ax
ρ >0 : sup
n
1 λn
k∈In
M
xk ρ , z
pk ≤1, ∀z∈X
,
A y
ρ >0 : sup
n
1 λn
k∈In
M
yk
ρ , z
pk ≤1, ∀z∈X
.
2.9
Letρ1∈Axandρ2∈Ay, then ifρρ1ρ2, then, we have
sup
n
1 λn
n∈In
M
xkyk
ρ , z
≤ ρ1 ρ1ρ2
sup
n
1 λn
k∈In
M
xk ρ1
, z ρ2
ρ1ρ2sup
n
1 λn
k∈In
M
yk ρ2, z
.
2.10
Thus, supn1/λn
n∈InMxkyk/ρ1ρ2, zpk ≤1 and gn
xy
≤inf ρ1ρ2
pn/H
:ρ1∈Ax, ρ2∈A y!
≤inf ρ1pn/H:ρ1∈Ax!
inf ρp2n/H :ρ2∈A y! gnx gn
y .
2.11
ivFinally using the same technique of Theorem 2 of Savas¸15it can be easily seen that scalar multiplication is continuous. This completes the proof.
Corollary 2.3. It should be noted that for a fixedF∈Ithe space W∞F
λ, M, p,,·,
x∈S2−X:∃K >0 s.t. sup
n∈N−F
1 λn
k∈In
M
xk ρ , z
pk ≤K for some ρ >0, and eachz∈X
,
2.12
which is a subspace of the spaceW∞Iλ, M, p,,·,is a paranormed space with the paranormsgnfor n /∈FandgF infn∈N−Fgn.
Theorem 2.4. LetM,M1, M2,be Orlicz functions. Then we have
iW0Iλ, M1, p,,·,⊆W0Iλ, M◦M1, p,,·,providedpkis such thatH0inf pk>
0.
iiW0Iλ, M1, p, ,·,∩W0Iλ, M2, p,,·,⊆W0Iλ, M1M2, p, ,·,.
Proof. i For given ε > 0, first choose ε0 > 0 such that max{ε0H, εH00} < ε. Now using the continuity of M choose 0 < δ < 1 such that 0 < t < δ ⇒ Mt < ε0. Let xk ∈ W0λ, M1, p,,·,. Now from the definition
Aδ
n∈N: 1 λn
n∈In
M1
xk
ρ, z
pk ≥δH
∈I. 2.13
Thus ifn /∈Aδthen
1 λn
n∈In
M1
xk ρ , z
pk < δH, 2.14
that is,
n∈In
M1
xk
ρ , z
pk < λnδH, 2.15
that is,
M1
xk ρ, z
pk < δH, ∀k∈In, 2.16
that is,
M1
xk
ρ , z
< δ, ∀k∈In. 2.17
Hence from above using the continuity ofMwe must have
M
M1
xk
ρ , z
< ε0, ∀k∈In, 2.18
which consequently implies that
k∈In
M
M1
xk ρ, z
pk
< λnmax εH0 , εH00!
< λnε, 2.19
that is,
1 λn
k∈In
M
M1
xk
ρ , z pk
< ε. 2.20
This shows that
n∈N: 1 λn
k∈In
M
M1
xk ρ , z
pk
≥ε
⊂Aδ 2.21
and so belongs toI. This proves the result.
iiLetxk∈W0IM1, p,,·,∩W0IM2, p,,·,, then the fact 1
λn
M1M2
xk ρ , z
pk ≤D 1 λn
M1
xk ρ, z
pkD 1 λn
M2
xk ρ , z
pk 2.22
gives us the result.
Definition 2.5. Let X be a sequence space. Then X is called solid if αkxk ∈ X whenever xk∈Xfor all sequencesαkof scalars with|αk| ≤1 for allk∈N.
Theorem 2.6. The sequence spacesW0Iλ, M, p,,·,, W∞Iλ, M, p,,·,are solid.
Proof. We give the proof forW0Iλ, M, p,,·,only. Letxk∈W0Iλ, M, p,,·,and letαk be a sequence of scalars such that|αk| ≤1 for allk∈N. Then we have
n∈N: 1 λn
k∈In
M
αkxk ρ , z
pk ≥ε
⊆
n∈N: C λn
k∈In
M
xk
ρ , z
pk ≥ε
∈I, 2.23 whereCmaxk{1,|αk|H}. Henceαkxk∈W0Iλ, M, p,,·,for all sequences of scalarsαk with|αk| ≤1 for allk∈Nwheneverxk∈W0Iλ, M, p,,·,.
References
1 P. Kostyrko, T. ˇSal´at, and W. Wilczy ´nski, “I-convergence,” Real Analysis Exchange, vol. 26, no. 2, pp.
669–686, 2000.
2 B. K. Lahiri and P. Das, “IandI∗-convergence in topological spaces,” Mathematica Bohemica, vol. 130, no. 2, pp. 153–160, 2005.
3 P. Kostyrko, M. Maˇcaj, T. ˇSal´at, and M. Sleziak, “I-convergence and extremal I-limit points,”
Mathematica Slovaca, vol. 55, no. 4, pp. 443–464, 2005.
4 P. Das and P. Malik, “On the statistical andI- variation of double sequences,” Real Analysis Exchange, vol. 33, no. 2, pp. 351–364, 2008.
5 S. G¨ahler, “2-metrische R¨aume und ihre topologische Struktur,” Mathematische Nachrichten, vol. 26, pp. 115–148, 1963.
6 H. Gunawan and Mashadi, “On finite-dimensional 2-normed spaces,” Soochow Journal of Mathematics, vol. 27, no. 3, pp. 321–329, 2001.
7 R. W. Freese and Y. J. Cho, Geometry of Linear 2-Normed Spaces, Nova Science, Hauppauge, NY, USA, 2001.
8 A. S¸ahiner, M. G ¨urdal, S. Saltan, and H. Gunawan, “Ideal convergence in 2-normed spaces,” Taiwanese Journal of Mathematics, vol. 11, no. 5, pp. 1477–1484, 2007.
9 M. G ¨urdal and S. Pehlivan, “Statistical convergence in 2-normed spaces,” Southeast Asian Bulletin of Mathematics, vol. 33, no. 2, pp. 257–264, 2009.
10 M. G ¨urdal, A. S¸ahiner, and I. Ac¸ık, “Approximation theory in 2-Banach spaces,” Nonlinear Analysis:
Theory, Methods & Applications, vol. 71, no. 5-6, pp. 1654–1661, 2009.
11 M. A. Krasnoselskii and Y. B. Rutisky, Convex Function and Orlicz Spaces, Noordhoff, Groningen, The Netherlands, 1961.
12 S. D. Parashar and B. Choudhary, “Sequence spaces defined by Orlicz functions,” Indian Journal of Pure and Applied Mathematics, vol. 25, no. 4, pp. 419–428, 1994.
13 W. H. Ruckle, “FKspaces in which the sequence of coordinate vectors is bounded,” Canadian Journal of Mathematics, vol. 25, pp. 973–978, 1973.
14 I. J. Maddox, “Sequence spaces defined by a modulus,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 100, no. 1, pp. 161–166, 1986.
15 E. Savas¸, “Δm-strongly summable sequences spaces in 2-normed spaces defined by ideal convergence and an Orlicz function,” Applied Mathematics and Computation, vol. 217, no. 1, pp. 271–276, 2010.
16 I. J. Maddox, Elements of Functional Analysis, Cambridge University Press, London, UK, 1970.
