# Generalized Differenceλ-Sequence Spaces Defined by Ideal Convergence and the Musielak-Orlicz Function

Authors: Awad A. Bakery
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We introduced the ideal convergence of generalized difference sequence spaces combining an infinite matrix of complex numbers with respect toλ-sequences and the Musielak-Orlicz function over n-normed spaces. We also studied some topological properties and inclusion relations between these spaces.


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## Body

## 1. Introduction

Throughout the paperω,ℓ∞,c,c0, andℓpdenote the classes of all, bounded, convergent, null, and p-absolutely summable sequences of complex numbers. The sets of natural numbers and real numbers will be denoted by ℕ and ℝ, respectively. Many authors studied various sequence spaces using normed or seminormed linear spaces. In this paper, using an infinite matrix of complex numbers and the notion of ideal, we aimed to introduce some new sequence spaces with respect to generalized difference operatorΔmsonλ-sequences and the Musielak-Orlicz function inn-normed linear spaces. By an ideal we mean a familyI⊂2Yof subsets of a nonempty setYsatisfying the following: (i)ϕ∈I; (ii)A,B∈IimplyA∪B∈I; (iii)A∈I, B⊂AimplyB∈I, while an admissible idealIofYfurther satisfies{x}∈Ifor eachx∈Y. The notion of ideal convergence was introduced first by Kostyrko et al. [1] as a generalization of statistical convergence. The concept of 2-normed spaces was initially introduced by Gähler [2] in the 1960s, while that ofn-normed spaces can be found in [3]; this concept has been studied by many authors; see for instance [4–7]. The notion of ideal convergence in a 2-normed space was initially introduced by Gürdal [8]. Later on, it was extended ton-normed spaces by Gürdal and Şahiner [9]. Given thatI⊂2ℕis a nontrivial ideal in ℕ, the sequence(xn)n∈ℕin a normed space(X;∥·∥)is said to beI-convergent tox∈X, if, for eachε>0,
(1)A(ε)={n∈ℕ:∥xn-x∥≥ε}∈I.A sequence(xk)in a normed space(X,∥·∥)is said to beI-bounded if there existsL>0such that
(2){k∈ℕ:∥xk∥>L}∈I.A sequence(xk)in a normed space(X,∥·∥)is said to beI-Cauchy if, for eachε>0, there exists a positive integerm=m(ε)such that
(3){k∈ℕ:∥xk-xm∥≥ε}∈I.In paper [10], the notion ofλ-convergent and bounded sequences is introduced as follows: letλ=(λj)j=1∞be a strictly increasing sequence of positive real numbers tending to infinity; that is,
(4)0<λ1<λ2<⋯,λj⟶∞asj⟶∞.We say that a sequencex=(xj)∈ω is λ-convergent to the numberl∈ℂ, called theλ-limit ofx, ifΛj(x)→lasj→∞, where
(5)Λj(x)=1λj∑r=1j‍(λr-λr-1)xr,j∈ℕ.The class of all sequences(λj)satisfying this property is denoted byΛ.In particular, we say thatx is a λ-null sequence ifΛj(x)→0asj→∞. Further, we say thatxis λ-bounded ifsupj|Λj(x)|<∞. Here and in the sequel, we will use the convention that any term with a zero subscript is equal to naught; for example,λ0=0andx0=0. Now, it is well known [10] that iflimjxj=ain the ordinary sense of convergence, then
(6)limj→∞(1λj∑r=1j‍(λr-λr-1)|xr-a|)=0.This implies that(7)limj|Λj(x)-a|=limj→∞|1λj∑r=1j‍(λr-λr-1)(xr-a)|=0,
which yields that limjΛj(x)=a and hence x is λ-convergent toa. We therefore deduce that the ordinary convergence implies theλ-convergence to the same limit.An Orlicz function is a functionM:[0,∞)→[0,∞)which is continuous, nondecreasing, and convex withM(0)=0 andM(x)>0forx>0andM(x)→∞, asx→∞. If convexity ofMis replaced byM(x+y)≤M(x)+M(y), then it is called a modulus function, introduced by Nakano [11]. Ruckle [12] and Maddox [13] used the idea of a modulus function to construct some spaces of complex sequences. An Orlicz functionMis said to satisfy the Δ2-condition for all values ofx≥0, if there exists a constantk>0, such thatM(2x)≤kM(x). TheΔ2-condition is equivalent toM(lx)≤klM(x)for all values ofxand forl>1. Lindentrauss and Tzafriri [14] used the idea of an Orlicz function to define the following sequence spaces:
(8)ℓM={x∈ω:∑k=1∞‍M(|x(k)|ρ)<∞},
which is a Banach space with the Luxemburg norm defined by
(9)∥x∥=inf{ρ>0:∑k=1∞‍M(|x(k)|ρ)≤1}.The spaceℓMis closely related to the spaceℓp, which is an Orlicz sequence space withM(x)=xpfor1≤p<∞. Recently different classes of sequences have been introduced using Orlicz functions. See [7, 9, 15–17].A sequenceM=(Mk)of Orlicz functionsMkfor allk∈ℕ is called a Musielak-Orlicz function.Kizmaz [18] defined the difference sequences ℓ∞(Δ), c(Δ), and c0(Δ) as follows.Z
(
Δ
)
=
{
x
=
(
x
k
)
:
(
Δ
x
k
)
∈
Z
}. ForZ=ℓ∞, c, and c0, whereΔx=(xk-xk+1), for allk∈ℕ. The above spaces are Banach spaces, normed by∥x∥=|x1|+supk|Δxk|. The notion of difference sequence spaces was generalized by Et and Colak [19] as follows:Z(Δs)={x=(xk):(Δsxk)∈Z}. ForZ=ℓ∞, candc0, wheres∈ℕ, (Δsxk)=(Δs-1xk-Δs-1xk+1)and so thatΔsxk=∑n=0s‍(-1)nCnsxk+n. Tripathy and Esi [20] introduced the following new type of difference sequence spaces.Z
(
Δ
m
)
=
{
x
=
(
x
k
)
:
(
Δ
m
x
k
)
∈
Z
}, Z=ℓ∞,c, andc0, whereΔmxk=(xk-xk+m), for allk∈ℕ. Tripathy et al. [21], generalized the previous notions and unified them as follows.Letm and s be nonnegative integers, then forZa given sequence space we have
(10)Z(Δms)={x=(xk):(Δmsxk)∈Z},whereΔmsxk=∑n=0s‍(-1)nCnsxk+mn(forwarddifference),Z(Δm(s))={x=(xk):(Δm(s)xk)∈Z},whereΔm(s)xk=∑n=0s‍(-1)nCnsxk-mn(backwarddifference),
wherexk=0, fork<0.


## 2. Definitions and Preliminaries

Letn∈ℕ andXbe a linear space over the fieldKof dimensiond, whered≥n≥2andKis the field of real or complex numbers. A real valued function ∥·…·∥ onXnsatisfies the following four conditions:(1)
∥x1,x2,…,xn∥=0if and only ifx1,x2,…,andxnare linearly dependent inX;(2)
∥x1,x2,…,xn∥is invariant under permutation;(3)
∥αx1,x2,…,xn∥=|α|∥x1,x2,…,xn∥for anyα∈K;(4)
∥x+x′,x2,…,xn∥≤∥x,x2,…,xn∥+∥x1,x2,…,xn∥, which is called ann-norm onXand the pair(X;∥·…·∥)is called ann-normed space over the fieldK. For example, we may takeX=ℝnbeing equipped with then-norm∥x1,x2,…,xn∥E=the volume of then-dimensional parallelepiped spanned by the vectorsx1,x2,…,andxnwhich may be given explicitly by the formula
(11)∥x1,x2,…,xn∥E=|det(xij)|=abs(|x11x12⋯x1nx21x22⋯x2n⋮⋮⋱⋮xn1xn2⋯xnn|),where xi=(xi1,xi2,…,xin)for eachi∈ℕ.Let(X,∥·…·∥)be ann-normed space of dimensiond≥n≥2and{a1,a2,a3,…,an}a linearly independent set inX. Then, the function∥·…·∥∞onXn-1defined by
(12)∥x1,x2,…,xn∥∞=max1≤i≤n∥x1,x2,…,xn-1,ai∥
defines an (n-1)-norm on X with respect to a1,a2,a3,…,andan and this is known as the derived (n-1)-norm. The standard (n)-norm on X, a real inner product space of dimension d≥n, is as follows:
(13)∥x1,x2,…,xn∥s=abs(|〈x1,x1〉〈x1,x2〉⋯〈x1,xn〉〈x2,x1〉〈x2,x2〉⋯〈x2,xn〉⋮⋮⋱⋮〈xn,x1〉〈xn,x2〉⋯〈xn,xn〉|)1/2,
where 〈·,·〉 denotes the inner product onX. If we takeX=ℝn, then
(14)∥x1,x2,…,xn∥E=∥x1,x2,…,xn∥s.Forn=1, thisn-norm is the usual norm∥x1∥=〈x1,x1〉.Definition 1.
A sequence(xk)in ann-normed space is said to be convergent tox∈Xif
(15)limk→∞∥(z1,z2,…,zn-1,xk-x)∥n=0,hhhhhhhhhhhh∀z1,z2,…,zn-1∈X.Definition 2.
A sequence(xk)in ann-normed space is called Cauchy (with respect ton-norm) if
(16)limk,j→∞∥z1,z2,…,zn-1,xk-xj∥n=0,hhhhhhhhhhhh∀z1,z2,…,zn-1∈X.If every Cauchy sequence inXconverges to anx∈X, thenXis said to be complete (with respect to then-norm). A completen-normed space is called an n-Banach space.Definition 3.
A sequence(xk)in ann-normed space (X,∥·…·∥) is said to beI-convergent tox0∈Xwith respect ton-norm, if, for each ε>0, the set
(17){k∈ℕ:∥xk-x0,z1,z2,…,zn-1∥≥ε,foreveryz1,z2,…,zn-1}∈I.Definition 4.
A sequence(xk)in ann-normed space(X,∥·…·∥)is said to beI-Cauchy if, for each ε>0, there exists a positive integerm=m(ε)such that the set
(18){k∈ℕ:∥xk-xm,z1,z2,…,zn-1∥≥ε,foreveryz1,z2,…,zn-1}∈I.
Letx=(xk)be a sequence; thenS(x)denotes the set of all permutations of the elements of(xk); that is,S(x)=(xπ(n)):πis a permutation of ℕ.Definition 5.
A sequence spaceEis said to be symmetric ifS(x)⊂Efor allx∈E.Definition 6.
A sequence spaceEis said to be normal (or solid) if(αkxk)∈E, whenever(xk)∈Eand for all sequences (αk)of scalars with|αk|≤1for allk∈ℕ.Definition 7.
A sequence spaceEis said to be a sequence algebra ifx,y∈E; thenx·y=(xkyk)∈E.Lemma 8.
Everyn-normed space is an(n-r)-normed space for allr=1,2,3,…,n-1. In particular, everyn-normed space is a normed space.Lemma 9.
On a standardn-normed spaceX, the derived(n-1)-norm∥·…·∥∞defined with respect to the orthogonal set{e1,e2,…,en}is equivalent to the standard(n-1)-norm∥·…·∥s. To be precise, one has
(19)∥x1,x2,…,xn-1∥∞≤∥·…·∥s≤n∥x1,x2,…,xn-1∥∞,
for allx1,x2,…,xn-1∈X, where∥x1,x2,…,xn-1∥∞=max1≤i≤n{∥x1,x2,…,xn-1,ei∥S}.
For any bounded sequence(pn)of positive numbers, one has the following well known inequality: if0≤pk≤supkpk=GandD=max(1,2G-1), then|an+bn|pn≤D(|an|pn+|bn|pn), for allkandak,bk∈ℂ.


## 3. Main Results

In this section, we define some new ideal convergent sequence spaces and investigate their linear topological structures. We find out some relations related to these sequence spaces. LetI be an admissible ideal of ℕ, ℳ=(Mj) a Musielak-Orlicz function, and Δms the forward generalized difference operator on the class of all sequences(λj)satisfying the propertyΛand ann-normed space(X,∥·…·∥). Further, letp=(pk)be any bounded sequence of positive real numbers; we will define the following sequence spaces:
(20)W[A,ℳ,Δms,Λ,p,∥·…·∥]I={z1,z2,…,zn-1Δms(Λj(x))-lρ,∥)]pj≥εx∈ω(n-X):∀ε>0×{z1,z2,…,zn-1Δms(Λj(x))-lρ,∥)]pj≥εk∈ℕ:∑j=1∞‍akj×[Mj(∥Δms(Λj(x))-lρ,z1,z2,…,zn-1Δms(Λj(x))-lρ,∥)]pj≥ε}∈I,forsomeρ>0,l∈Xandeachz1,z2,…,zn-1∈Xz1,z2,…,zn-1Δms(Λj(x))-lρ,∥)]pj≥ε},W[A,ℳ,Δms,Λ,p,∥·…·∥]0I={z1,z2,…,zn-1Δms(Λj(x))ρ,∥)]pjx∈ω(n-X):∀ε>0×{z1,z2,…,zn-1Δms(Λj(x))ρ,∥)]pjk∈ℕ:∑j=1∞‍akj×[Mj(∥Δms(Λj(x))ρ,z1,z2,…,zn-1Δms(Λj(x))ρ,∥)]pj≥ε}∈I,forsomeρ>0,andeachz1,z2,…,zn-1∈Xz1,z2,…,zn-1Δms(Λj(x))ρ,∥)]pj},W[A,ℳ,Δms,Λ,p,∥·…·∥]∞={z1,z2,…,zn-1Δms(Λj(x))ρ∥)]pjx∈ω(n-X):∃K>0st.supk∑j=1∞‍akjkkkk×[Mj(∥Δms(Λj(x))ρ,z1,z2,…,zn-1Δms(Λj(x))ρ∥)]pj<∞,forsomeρ>0andeachz1,z2,…,zn-1∈Xz1,z2,…,zn-1Δms(Λj(x))ρ∥)]pj},W[A,ℳ,Δms,Λ,p,∥·…·∥]∞I={{z1,z2,…,zn-1Δms(Λj(x))ρ∥)]pj∥Δms(Λj(x))ρ,k∈ℕ:∑j=1∞‍akjx∈ω(n-X):∃K>0,s.t.{∥Δms(Λj(x))ρz1,z2,…,zn-1Δms(Λj(x))ρ∥)]pj,k∈ℕ:∑j=1∞‍akjssssssssssssssssssssssss×[Mj(∥Δms(Λj(x))ρ,z1,z2,…,zn-1∥Δms(Λj(x))ρ,∥)]pj≥K∥Δms(Λj(x))ρ,}∈I,HHHHforsomeρ>0,HHHHandeachz1,z2,…,zn-1∈X{z1,z2,…,zn-1Δms(Λj(x))ρ∥)]pj∥Δms(Λj(x))ρ,k∈ℕ:∑j=1∞‍akj}.Let us consider a few special cases of the aforementioned sets.(1) IfMk(x)=M(x), for allk∈ℕ then the previous classes of sequences are denoted byW[A,M,Δms,Λ,p,∥·…·∥]I, W[A,M,Δms,Λ,p,∥·…·∥]0I, W[A,M,Δms,Λ,p,∥·…·∥]∞, and W[A,M,Δms,Λ,p,∥·…·∥]∞I, respectively.(2) Ifpk=1for allk∈ℕ then the previous classes of sequences are denoted by W[A,ℳ,Δms,Λ,∥·…·∥]I, W[A,ℳ,Δms,Λ,∥·…·∥]0I, W[A,ℳ,Δms,Λ,∥·…·∥]∞, and W[A,ℳ,Δms,Λ,∥·…·∥]∞I, respectively.(3) IfMk(x)=x, for allk∈ℕ andx∈[0,∞[, then the previous classes of sequences are denoted by W[A,Δms,Λ,p,∥·…·∥]I, W[A,Δms,Λ,p,∥·…·∥]0I, W[A,Δms,Λ,p,∥·…·∥]∞, and W[A,Δms,Λ,p,∥·…·∥]∞I, respectively.(4) If we takeMk(x)=M(x), for allk∈ℕandA=(akj)as
(21)akj={1k,k≥j,0,otherwise,
then we denote the previous classes of sequences byW[C,M,Δms,Λ,p,∥·…·∥]I, W[C,M,Δms,Λ,p,∥·…·∥]0I, W[C,M,Δms,p,∥·…·∥]∞, andW[C,M,Δms,Λ,p,∥·…·∥]∞I, respectively.(5) If we takeMk(x)=M(x)andA=(akj)as
(22)akj={1ϕk,j∈Ik=[k-ϕk+1,k],0,otherwise,
where(ϕk)is a nondecreasing sequence of positive numbers tending to ∞,ϕ1=1, and ϕk+1≤ϕk+1, then we denote the previous classes of sequences byW[Φ,M,Δms,Λ,p,∥·…·∥]I,W[Φ,M,Δms,Λ,p,∥·…·∥]0I, W[Φ,M,Δms,Λ,p,∥·…·∥]∞, and W[Φ,M,Δms,Λ,p,∥·…·∥]∞I.(6) IfA=(akj)as in (22), then we denote the previous classes of sequences byW[Φ,ℳ,Δms,Λ,p,∥·…·∥]I, W[Φ,ℳ,Δms,Λ,p,∥·…·∥]0I, W[Φ,ℳ,Δms,Λ,p,∥·…·∥]∞, andW[Φ,ℳ,Δms,Λ,p,∥·…·∥]∞I.And ifλj=jfor allj∈ℕ, then the previous classes of sequences are denoted byW[Φ,ℳ,Δms,C,p,∥·…·∥]I, W[Φ,ℳ,Δms,C,p,∥·…·∥]0I, W[Φ,ℳ,Δms,C,p,∥·…·∥]∞, andW[Φ,ℳ,Δms,C,p,∥·…·∥]∞Iand they are a generalization of the sequence spaces defined by Bakery et al. [22].(7) By a lacunaryθ=(jr), r=0,1,2,…, wherej0=0, we will mean an increasing sequence of nonnegative integers withjr-jr-1→∞asr→∞. The interval determined byθwill be denoted byIr=]jr-1,jr]andhr=jr-jr-1and letA=(akj)as
(23)akj={1hr,j∈Ir=]jr-1,jr],0,otherwise.Then we denote the previous classes of sequences byW[θ,M,Δms,Λ,p,∥·…·∥]I, W[θ,M,Δms,Λ,p,∥·…·∥]0I, W[θ,M,Δms,Λ,p,∥·…·∥]∞, andW[θ,M,Δms,Λ,p,∥·…·∥]∞I, respectively.(8) IfMk(x)=M(x), for allk∈ℕ, A=I, andλj=j, then the previous classes of sequences are denoted byW[M,Δms,C,p,∥·…·∥]I, W[M,Δms,C,p,∥·…·∥]0I, W[M,Δms,C,p,∥·…·∥]∞, andW[M,Δms,C,p,∥·…·∥]∞I.(9) Ifs=1, then the previous classes of sequences are denoted byW[A,ℳ,Δm,Λ,p,∥·…·∥]I, W[M,Δm,C,p,∥·…·∥]0I, W[M,Δm,C,p,∥·…·∥]∞, and W[M,Δm,C,p,∥·…·∥]∞I.(10) Ifm=1, then the previous classes of sequences are denoted by[A,ℳ,Δs,Λ,p,∥·…·∥]I, W[M,Δs,C,p,∥·…·∥]0I, W[M,Δs,C,p,∥·…·∥]∞, andW[M,Δs,C,p,∥·…·∥]∞I.Theorem 10.
The spacesW[A,ℳ,Δms,Λ,p,∥·…·∥]I, W[A,ℳ,Δms,Λ,p,∥·…·∥]0I and W[A,ℳ,Δms,Λ,p,∥·…·∥]∞I are linear spaces.Proof.
We will prove the assertion forW[A,ℳ,Δms,Λ,p,∥·…·∥]0I; the others can be proved similarly. Assume thatx=(xk), y=(yk)∈W[A,ℳ,Δms,Λ,p,∥·…·∥]0I, andα,β∈ℂ. Then, there existρ1andρ2such that the sets
(24){Δms(Λj(x))ρ1z1,z2,…,zn-1Δms(Λj(x))ρ1∥)]pjk∈ℕ:∑j=1∞‍akj×[Mj(∥Δms(Λj(x))ρ1,z1,z2,…,zn-1Δms(Λj(x))ρ1∥)]pj≥ε2∑j=1∞‍z1,z2,…,zn-1Δms(Λj(x))ρ1∥)]pj}∈I,(25){∑j=1∞‍k∈ℕ:∑j=1∞akj[Mj(∥Δms(Λj(y)ρ2,z1,z2,…,zn-1∥)]pj≥ε2∑j=1∞‍}∈I.
Since(X,∥·…·∥)is ann-norm,ΔmsandΛjare linear, and the Orlicz functionMjis convex for allj∈ℕ, the following inequality holds:
(26)∑j=1∞‍akj[Mj(∥Δms(Λj(αx+βy))|α|ρ1F+|β|ρ2,z1,z2,…,zn-1Δms(Λj(αx+βy))|α|ρ1F+|β|ρ2∥)]pj≤D∑j=1∞‍akj|α|ρ1|α|ρ1+|β|ρ2×[Mj(∥Δms(Λj(x))ρ1,z1,z2,…,zn-1∥)]pj+D∑j=1∞‍akj|β|ρ2|α|ρ1+|β|ρ2×[Mj(∥Δms(Λj(y))ρ2,z1,z2,…,zn-1∥)]pj≤DL∑j=1∞‍akj×[Mj(∥Δms(Λj(x))ρ1,z1,z2,…,zn-1∥)]pj+DL∑j=1∞‍akj×[Mj(∥Δms(Λj(y))ρ2,z1,z2,…,zn-1Δms(Λj(y))ρ2∥)Δms(Λj(y))ρ2]pj,
whereL=max{|α|ρ1/(|α|ρ1+|β|ρ2),|β|ρ2/(|α|ρ1+|β|ρ2)}. On the other hand from the above inequality we get
(27){z1,z2,…,zn-1∥Δms(Λj(αx+βy))|α|ρ1+|β|ρ2∥)∥Δms(Λj(αx+βy))|α|ρ1+|β|ρ2]pjk∈ℕ:∑j=1∞‍akj[Mj(∥Δms(Λj(αx+βy))|α|ρ1+|β|ρ2,z1,z2,…,zn-1∥Δms(Λj(αx+βy))|α|ρ1+|β|ρ2∥)∥Δms(Λj(αx+βy))|α|ρ1+|β|ρ2]pj≥ε∑j=1∞‍}⊆{Δms(Λj(x))ρ1z1,z2,…,zn-1Δms(Λj(x))ρ1∥)]pjk∈ℕ:DL×∑j=1∞‍akj[Mj(∥Δms(Λj(x))ρ1,z1,z2,…,zn-1Δms(Λj(x))ρ1∥)]pj≥ε2}∪{Δms(Λj(x))ρ1×∑j=1∞‍akj[Mj(∥Δms(Λj(y))ρ2,z1,z2,…,zn-1∥)]pjk∈ℕ:DL×∑j=1∞‍akj[Mj(∥Δms(Λj(y))ρ2,z1,z2,…,zn-1∥)]pjhhhhhhh≥ε2z1,z2,×∑j=1∞‍akj[Mj(∥Δms(Λj(y))ρ2,z1,z2,…,zn-1∥)]pj…,zn-1Δms(Λj(x))ρ1∥)]pj}.
Since the two sets on the right hand side belong toI, this completes the proof.Theorem 11.
The spacesW[A,ℳ,Δms,Λ,p,∥·…·∥]I, W[A,ℳ,Δms,Λ,p,∥·…·∥]0I, and W[A,ℳ,Δms,Λ,p,∥·…·∥]∞I are paranormed spaces (not totally paranormed) with respect to the paranorm gΔ defined by
(28)gΔ(x)=∑j=1ms∥xj,z1,z2,…,zn-1∥‍+inf{[Mjz1,z2,…,zn-1Δms(Λj(x))ρ1∥)]pj]1/H(∥Δms(Λj(x))ρ1,ρpk/H:supk[∑j=1∞‍akj[Mj(∥Δms(Λj(x))ρ1,z1,z2,…,zn-1Δms(Λj(x))ρ1∥)]pj]1/HhhhhhhhhhJJJJh≤1,forsomeρ>0,hhhhhhhhhJJJJhandeachz1,z2,…,zn-1∈X∑j=1∞‍akj[(∥Δms(Λj(x))ρ1z1,z2,…,zn-1Δms(Λj(x))ρ1∥)]pj]1/H,z1,z2,…,zn-1∥)Mj},
whereH=max{1,supkpk}.Proof.
ClearlygΔ(-x)=gΔ(x)andgΔ(θ)=0. Letx=(xk)andy=(yk)∈W[A,ℳ,Δms,Λ,p,∥·…·∥]0I. Then, forρ>0we set
(29)A1={[∑j=1∞‍akjz1,z2,…,zn-1∥Δms(Λj(x))ρ,∥)]pj]1/H[Mj(∥Δms(Λj(x))ρ,z1,z2,…,zn-1∥)]pj]ρ:supk[∑j=1∞‍akj×[Mj(∥Δms(Λj(x))ρ,z1,z2,…,zn-1∥Δms(Λj(x))ρ,∥)]pj]1/H≤1,foreachz1,z2,…,zn-1∈Xz1,z2,…,zn-1∥Δms(Λj(x))ρ,∥)]pj]1/H},A2={z1,z2,…,zn-1∑j=1∞‍∥)]pj]1/Hρ:supk[∑j=1∞‍akj×[∑j=1∞‍Mj(∥Δms(Λj(y))ρ,HHHHHHHHHHHHHz1,z2,…,zn-1∑j=1∞‍∥)]pj]1/H≤1,foreachz1,z2,…,zn-1∈X{ρ:supkz1,z2,…,zn-1∑j=1∞‍∥)]pj]1/H[∑j=1∞‍akj}.
Letρ1∈A1, ρ2∈A2, andρ=ρ1+ρ2; then we have
(30)supk[∑j=1∞‍akj×[Mj(∥Δms(Λj(x+y))ρ,×1111111z1,z2,…,zn-1(∥Δms(Λj(x+y))ρ,∥)]pj]1/H≤ρ1ρ1+ρ2×supk[∑j=1∞‍akj[Mj(∥Δms(Λj(x))ρ1,hhhhhhhhhhhhhhhhhhhhhz1,z2,…,zn-1Δms(Λj(x))ρ1∥)]pj]1/Hhhh+ρ2ρ1+ρ2×supk[∑j=1∞‍akj[Mj(∥Δms(Λj(y))ρ2,hhhhhhhhhhhhhhhhhhhhz1,z2,…,zn-1Δms(Λj(x))ρ1∥)]pj]1/H≤1,gΔ(x+y)=∑j=1ms‍∥xj+yj,z1,z2,…,zn-1∥+inf{(ρ1+ρ2)pk/H:ρ1∈A1,ρ2∈A2}≤∑j=1ms‍∥xj,z1,z2,…,zn-1∥+inf{(ρ1)pk/H:ρ1∈A1}+∑j=1ms‍∥yj,z1,z2,…,zn-1∥+inf{(ρ2)pk/H:ρ2∈A2}=gΔ(x)+gΔ(y).
Letλt→λwhereλt,λ∈ℂ, and let gΔ(xt-x)→0 as t→∞. We have to show that gΔ(λtxt-λx)→0as t→∞. We set
(31)A3={z1,z2,…,zn-1Δms(Λj(y))ρt1∥)]pj]1/Hρt:supk[∑j=1∞‍akj×[Mj(∥Δms(Λj(x))ρt,z1,z2,…,zn-1∥Δms(Λj(x))ρt,∥)]pj]1/H≤1,ffforeachz1,z2,…,zn-1∈Xz1,z2,…,zn-1∥Δms(Λj(x))ρt,∥)]pj]1/H},A4={∑j=1∞‍z1,z2,…,zn-1Δms(Λj(y))ρt1∥)]pj]1/Hρt1:supk[∑j=1∞‍akjA4={∑j=1∞‍ρt1:supk×[[(∥Δms(Λj(y))ρt1,z1,z2,…,zn-1∥)]Mj(∥Δms(Λj(y))ρt1,0000000000z1,z2,…,zn-1Δms(Λj(y))ρt1∥)]pj]1/H≤1,ddforeachz1,z2,…,zn-1∈Xz1,z2,…,zn-1Δms(Λj(y))ρt1∥)]pj]1/H∑j=1∞‍ρt1}.
Ifρt∈A3 and ρt1∈A4, then by using non-decreasing and convexity of the Orlicz functionMjfor all j∈ℕ we get
(32)supk[∑j=1∞‍akj×[Mj(∥Δms(λtxjt-λxj)|λt-λ|ρt+|λ|ρt1,z1,z2,…,zn-1∥(Δmsλtxjt-λxjt)|λt-λ|ρt+|λ|ρt1,∥)]pj]1/H≤supk[∑j=1∞‍akj×[Mj(∥(Δmsλtxjt-λxjt)|λt-λ|ρt+|λ|ρt1,z1,z2,…,zn-1∥(Δmsλtxjt-λxjt)|λt-λ|ρt+|λ|ρt1,∥)]pj]1/H+supk[∑j=1∞‍akj×[Mj(∥Δms(λxjt-λxj)|λt-λ|ρt+|λ|ρt1,z1,z2,…,zn-1∥(Δmsλtxjt-λxjt)|λt-λ|ρt+|λ|ρt1,∥)]pj]1/H≤|λt-λ|ρt|λt-λ|ρt+|λ|ρt1×supk[∑j=1∞‍akj[Mj(∥Δmsxjtρt,z1,z2,…,zn-1∥)]pj]1/H+|λ|ρt1|λt-λ|ρt+|λ|ρt1×supk[∑j=1∞‍akj[Mj(∥Δms(xjt-xj)ρt1,z1,z2,…,zn-1∥(Δmsλtxjt-λxjt)|λt-λ|ρt+|λ|ρt1,∥)]pj]1/H.
From the previous inequality, it follows that(33)supk[∑j=1∞‍akj×[∥Δms(λtxjt-λxj)|λt-λ|ρt+|λ|ρt1,z1,z2,…,zn-1∥Mj(∥Δms(λtxjt-λxj)|λt-λ|ρt+|λ|ρt1,z1,z2,…,zn-1Δms(λtxjt-λxj)|λt-λ|ρt+|λ|ρt1∥)]pj]1/H≤1,
and consequently
(34)gΔ(λtxt-λx)=∑j=1ms‍∥λtxjt-λxj,z1,z2,…,zn-1∥+inf{(|λt-λ|ρt+|λ|ρt1)pk/H:ρt∈A3,ρt1∈A4}≤|λt-λ|∑j=1ms‍∥xjt,z1,z2,…,zn-1∥+|λt-λ|pk/Hinf{(ρt)pk/H:ρt∈A3}+|λ|∑j=1ms‍∥λtxjt-λxj,z1,z2,…,zn-1∥+|λ|pk/Hinf{(ρt1)pk/H:ρt1∈A4}≤max{|λt-λ|,|λt-λ|pk/H}gΔ(xt)+max{|λ|,|λ|pk/H}gΔ(xt-x).
Note thatgΔ(xt)≤gΔ(x)+gΔ(xt-x), for allt∈ℕ. Hence, by our assumption, the right hand of (34) tends to 0 ast→∞, and the result follows. This completes the proof of the theorem.Theorem 12.
Letℳ=(Mj), ℳ′=(Mj′), and ℳ′′=(Mj′′) be the Musielak-Orlicz functions. Then, the following hold:(a)
W[A,ℳ′,Δms,Λ,p,∥·…·∥]0I⊆W[A,ℳ.ℳ′,Δms,Λ,p,∥·…·∥]0I, providedp=(pk) such thatG0=infpk>0,(b)
W[A,ℳ′,Δms,Λ,p,∥·…·∥]0I⊆W[A,ℳ′+ℳ′′,Δms,Λ,p,∥·…·∥]0I.Proof.
(a) Letε>0be given. Chooseε1>0such thatsupk(∑j=1∞‍akj)max{ε1G,ε1G0}<ε. Using the continuity of the Orlicz functionM, choose0<δ<1such that0<t<δimplies thatM(t)<ε1.
Letx=(xk) be any element in W[A,ℳ′,Δms,Λ,p,∥·…·∥]0I and put
(35)Aδ={Δms(Λj(x))ρ1∑j=1∞‍akj[Mj′(∥Δms(Λj(x))ρ1,z1,z2,…,zn-1∥)]pjk∈ℕ:∑j=1∞‍akj[Mj′(∥Δms(Λj(x))ρ1,z1,z2,…,zn-1∥)]pj≥δG∑j=1∞‍akj[Mj′(∥Δms(Λj(x))ρ1,z1,z2,…,zn-1∥)]pj∥Δms(Λj(x))ρ1,z1,z2,…,zn-1∥}.
Then, by the definition of ideal convergent, we have the setAδ∈I. Ifn∉Aδ, then we have
(36)∑j=1∞‍akj[Mj′(∥Δms(Λj(x))ρ1,z1,z2,…,zn-1∥)]pj<δG[Mj′(∥Δms(Λj(x))ρ1,z1,z2,…,zn-1∥)]pj<δG⟹Mj′(∥Δms(Λj(x))ρ1,z1,z2,…,zn-1∥)<δ.
Using the continuity of the Orlicz functionMjfor alljand the relation (36), we have
(37)Mj[Mj′(∥Δms(Λj(x))ρ1,z1,z2,…,zn-1∥)]<ε1.
Consequently, we get(38)∑j=1∞‍akj[MjMj′(∥Δms(Λj(x))ρ1,z1,z2,…,zn-1∥)]pj<supk(∑j=1∞‍akj)max{ε1G,ε1G0}<ε⟹∑j=1∞‍akj[MjMj′(∥Δms(Λj(x))ρ1,z1,z2,…,zn-1∥)]pj<ε.
This shows that(39){∥Δms(Λj(x))ρ1z1,z2,…,zn-1Δms(Λj(x))ρ1∥)]pj,z1,z2,…,zn-1∥k∈ℕ:∑j=1∞‍akj[MjMj′(∥Δms(Λj(x))ρ1,z1,z2,…,zn-1Δms(Λj(x))ρ1∥)]pj≥ε[MjMj′z1,z2,…,zn-1Δms(Λj(x))ρ1∥)]pj(∥Δms(Λj(x))ρ1,z1,z2,…,zn-1∥)]pj}⊆Aδ∈I.
This proves the assertion.
(b) Letx=(xk)be any element in W[A,ℳ′,Λ,p,∥·…·∥]0I. Then, by the following inequality, the results follow:
(40)∑j=1∞‍akj[(Mj′+Mj′′)(∥Δms(Λj(x))ρ1,z1,z2,…,zn-1∥)]pj≤D∑j=1∞‍akj[Mj′(∥Δms(Λj(x))ρ1,z1,z2,…,zn-1∥)]pj+D∑j=1∞‍akj[Mj′′(∥Δms(Λj(x))ρ1,z1,z2,…,zn-1∥)]pj.Theorem 13.
The inclusionsZ[A,ℳ,Δms-1,Λ,∥·…·∥]⊆Z[A,ℳ,Δms,Λ,∥·…·∥] are strict for s,m≥1 in general whereZ=WI,W0I, andW∞I.Proof.
We will give the proof forW[A,ℳ,Δms-1,Λ,∥·…·∥]0I⊆W[A,ℳ,Δms,Λ,∥·…·∥]0I only. The others can be proved by similar arguments. Let x=(xk)∈W[A,ℳ,Δms-1,Λ,∥·…·∥]0I. Then let ε>0 be given; there exist ρ>0 such that
(41){k∈ℕ:∑j=1∞‍akjMj(∥Δms-1(Λj(x))ρ,z1,z2,…,zn-1∥)ssssssss≥ε2∑j=1∞‍(∥Δms-1(Λj(x))ρ,z1,z2,…,zn-1∥)}∈I.
SinceMj for all j∈ℕ is non-decreasing and convex, it follows that
(42)∑j=1∞‍akjMj(∥Δms(Λj(x))2ρ,z1,z2,…,zn-1∥)=∑j=1∞‍akj×Mj(∥Δms-1Λj+1(x)-Δms-1(Λj(x))2ρ,0000000000z1,z2,…,zn-1Δms-1Λj+1(x)-Δms-1(Λj(x))2ρ∥)≤12∑j=1∞‍akjMj(∥Δms-1(Λj+1(x))ρ,z1,z2,…,zn-1∥)+12∑j=1∞‍akjMj(∥Δms-1(Λj(x))ρ,z1,z2,…,zn-1∥),
and then we have
(43){Mj(∥Δms(Λj(x))2ρ,k∈ℕ:∑j=1∞‍akj×Mj(∥Δms(Λj(x))2ρ,z1,z2,…,zn-1Mj(∥Δms(Λj(x))2ρ,∥)≥ε}⊆{×∑j=1∞‍akjMj(∥Δms-1xj+1ρ,k∈ℕ:12⊆×∑j=1∞‍akjMj(∥Δms-1xj+1ρ,⊆z1,z2,…,zn-1∥Δms-1xj+1ρ,∥)≥ε2×∑j=1∞‍akjMj(∥Δms-1xj+1ρ,}∪{×∑j=1∞‍(∥Δms-1(Λj(x))ρ,k∈ℕ:12×∑j=1∞‍akjMj(∥Δms-1(Λj(x))ρ,z1,z2,…,zn-1∥Δms-1xj+1ρ,∥)≥ε2(×∑j=1∞‍∥Δms-1(Λj(x))ρ,}.
LetMk(x)=M(x)=x for allx∈[0,∞[, k∈ℕ and λk=k for all k∈ℕ. Consider a sequence x=(xk)=(ks). Then, x∈W[A,ℳ,Δms,Λ,∥·…·∥]0Ibut does not belong to W[A,ℳ,Δms-1,Λ,∥·…·∥]0I, for s=m=1. This shows that the inclusion is strict.Theorem 14.
Let0<pk≤qk for all k∈ℕ; then
(44)W[A,ℳ,Δms,Λ,p,∥·…·∥]∞⊆W[A,ℳ,Δms,Λ,q,∥·…·∥]∞.Proof.
Letx=(xj)∈W[A,ℳ,Δms,Λ,p,∥·…·∥]∞; then there exists some ρ>0 such that
(45)supk∑j=1∞‍akj[Mj(∥Δms(Λj(x))ρ,z1,z2,…,zn-1∥)]pj<∞.
This implies that(46)Mj(∥Δms(Λj(x))ρ,z1,z2,…,zn-1∥)<1,
for a sufficiently large value of j. Since Mj for all j∈ℕ is non-decreasing, we get
(47)supk∑j=1∞‍akj[Mj(∥Δms(Λj(x))ρ,z1,z2,…,zn-1∥)]qj≤supk∑j=1∞‍akj[Mj(∥Δms(Λj(x))ρ,z1,z2,…,zn-1∥)]pj<∞.
Thus,x∈W[A,ℳ,Δms,Λ,q,∥·…·∥]∞. This completes the proof of the theorem.Theorem 15.
(i) If0<infpk≤pk<1, then
(48)W[A,ℳ,Δms,Λ,p,∥·…·∥]∞⊆W[A,ℳ,Δms,Λ,∥·…·∥]∞.
(ii) If1<pk≤supkpk<∞, then W[A,ℳ,Δms,Λ,∥·…·∥]∞⊆W[A,ℳ,Δms,Λ,p,∥·…·∥]∞.Proof.
(i) Letx=(xj)∈W[A,ℳ,Δms,Λ,p,∥·…·∥]∞; since 0<infkpk≤pk<1, then we have
(49)supk∑j=1∞‍akj[Mj(∥Δms(Λj(x))ρ,z1,z2,…,zn-1∥)]≤supk∑j=1∞‍akj[Mj(∥Δms(Λj(x))ρ,z1,z2,…,zn-1∥)]pj<∞,
and hence x∈W[A,ℳ,Δms,Λ,∥·…·∥]∞.
(ii) Let1<pk≤supkpk<∞ and x=(xj)∈W[A,ℳ,Δms,Λ,∥·…·∥]∞. Then for each 0<ε<1 there exists a positive integer j0 such that
(50)supk∑j=1∞‍akj[Mj(∥Δms(Λj(x))ρ,z1,z2,…,zn-1∥)]≤ε<1,
for all j≥j0. This implies that
(51)supk∑j=1∞‍akj[Mj(∥Δms(Λj(x))ρ,z1,z2,…,zn-1∥)]pj≤supk∑j=1∞‍akj[Mj(∥Δms(Λj(x))ρ,z1,z2,…,zn-1∥)]<∞.
Thus x∈W[A,ℳ,Δms,Λ,p,∥·…·∥]∞ and this completes the proof.Theorem 16.
For any sequence of the Orlicz functionsℳ=(Mj) which satisfies the Δ2-condition, we have W[A,Δms,Λ,p,∥·…·∥]I⊂W[A,ℳ,Δms,Λ,p,∥·…·∥]I.Proof.
Letx=(xj)∈W[A,Δms,Λ,p,∥·…·∥]I and ε>0be given. Then, there existρ>0such that the set
(52){[∥∑j=1∞‍Δms(Λj(x))-lρ,k∈ℕ:∑j=1∞‍akj[∥Δms(Λj(x))-lρ,z1,z2,…,zn-1Δms(Λj(x))-lρ∥]pj≥ε∑j=1∞‍}∈I,forsomel.
By takingyj=∥(Δms(Λj(x))-l)/ρ,z1,z2,…,zn-1∥, let ε>0 and choose δ wit 0<δ<1 such that Mj(t)<ε for all j∈ℕ; for 0≤t≤δ, consider that
(53)∑j=1∞‍[Mj(yj)]pj=∑j=1,yj≤δ∞‍[Mj(yj)]pj+∑j=1,yj>δ∞‍[Mj(yj)]pj,
sinceMjis continuous for alln∈ℕ.
∑
j
∈
I
k
,
y
j
≤
δ
‍
[
M
j
(
y
j
)
]
p
j
<
ε and foryj>δ, we use the fact thatyj<yj/δ<1+yj/δ. Since ℳ=(Mj)is non-decreasing and convex, it follows that
(54)Mj(yj)<Mj(1+yjδ)<12Mj(2)+12Mj(2yjδ).
Sinceℳ=(Mj)satisfies the Δ2-condition, then
(55)Mj(yj)<yj2δLMj(2)+yj2δLMj(2)=yjδLMj(2).
Hence(56)∑j=1,yj>δ∞‍[Mj(yj)]pj<max{1,supj(Lδ-1Mj(2))pj}×∑j=1,yj>δ∞‍(yj)pj,
and then we have
(57)∑j=1∞‍[Mj(yj)]pj<ε+max{1,supj(Lδ-1Mj(2))pj}∑j=1,yj>δ∞‍(yj)pj.
This proves thatW[A,Δms,Λ,p,∥·…·∥]I⊂W[A,ℳ,Δms,Λ,p,∥·…·∥]I.Theorem 17.
Let0<pn≤qn<1and(qn/pn)be bounded; then
(58)W[A,ℳ,Δms,Λ,q,∥·…·∥]I⊆W[A,ℳ,Δms,Λ,p,∥·…·∥]I.Proof.
Letx=(xj)∈W[A,ℳ,Λ,q,∥·…·∥]∞and we put
(59)yj=[Mj(∥Δms(Λj(x))-lρ,z1,z2,…,zn-1∥)]qj,βj=pjqj∀j∈ℕ.
Then0<βj≤1, for allj∈ℕ. Let it be such that0<β≤βjfor allj∈ℕ. Define the sequences(aj)and(bj)as follows: foryj≥1, letaj=yjandbj=0; foryj<1, letaj=0andbj=yj. Then clearly, for allj∈ℕ we haveyj=aj+bj,yjβj=ajβj+bjβj, ajβj≤aj≤yj, andbjβj≤bjβ. Therefore, we have
(60)∑j=1∞‍akjyjβj≤∑j=1∞‍akjyj≤[∑j=1∞‍akjyj]β.
Hencex∈W[A,ℳ,Δms,Λ,p,∥·…·∥]∞.Theorem 18.
For any two sequencesp=(pk)andq=(qk)of positive real numbers and for any two n-norms∥·…·∥1and∥·…·∥2onX, the following holds:
(61)Z[A,ℳ,Δms,Λ,p,∥·…·∥1]∩Z[A,ℳ,Δms,Λ,q,∥·…·∥2]≠ϕ,
whereZ=WI,W0I,W∞I, andW∞.Proof.
The proof of the theorem is obvious, because the zero element belongs to each of the sequence spaces involved in the intersection.Theorem 19.
The sequence spacesW[A,ℳ,Δms,Λ,p,∥·…·∥]I, W[A,ℳ,Δms,Λ,p,∥·…·∥]0I, W[A,ℳ,Δms,Λ,p,∥·…·∥]∞, and W[A,ℳ,Δms,Λ,p,∥·…·∥]∞I are neither solid nor symmetric nor sequence algebras fors,m≥1in general.Proof.
The proof is obtained by using the same techniques of Et [23] and Theorems 15, 17, and 18.Note 1.
It is clear from definitions that(62)W[A,ℳ,Δms,Λ,p,∥·…·∥]0I⊆W[A,ℳ,Δms,Λ,p,∥·…·∥]I⊆W[A,ℳ,Δms,Λ,p,∥·…·∥]∞I.Theorem 20.
The spacesZ[A,ℳ,Δms,Λ,p,∥·…·∥] and Z[A,ℳ,p,∥·…·∥] are equivalent as topological spaces, where Z=WI,W0I,W∞I, and W∞.Proof.
Consider the mapping(63)T:Z[A,ℳ,Δms,Λ,p,∥·…·∥]⟶Z[A,ℳ,p,∥·…·∥],
defined by T(x)=(Δms(Λjx)) for each x∈Z[A,ℳ,Δms,Λ,p,∥·…·∥]. Then, clearly T is a linear homeomorphism and the proof follows.Remark 21.
If we replace the difference operatorΔms by Δm(s), then for each ε>0 we get the following sequence spaces:
(64)W[A,ℳ,Δm(s),Λ,p,∥·…·∥]I={z1,z2,…,∑j∈Ik‍zn-1∥Δm(s)xj-lρ,∥∥)]pjx∈ω(n-X):{∑j∈Ik‍z1,z2,…,zn-1∥Δm(s)xj-lρ,∥∥)]pjk∈ℕ:λk-1∑j∈Ik‍[Mj(∥Δm(s)xj-lρ,W[A,ℳ,Δm(s),Λ,p,∥·…·∥]Iz1,z2,…,zn-1∥Δm(s)xj-lρ,∥∥)]pj≥ε∑j∈Ik‍z1,z2,…,zn-1∥Δm(s)xj-lρ,∥∥)]pj}∈I,forsomeρ>0,l∈Xandeachhhhhhhhhhhhhhhhhhhhhz1,z2,…,zn-1∈Xz1,z2,…,zn-1∥Δm(s)xj-lρ,∑j∈Ik‍∥∥)]pj∥},W[A,ℳ,Δm(s),Λ,p,∥·…·∥]0I={∥Δm(s)xjρ,z1,z2,…,zn-1∥z1,z2,…,zn-1Δm(s)xjρ,∥)]pjx∈ω(n-X):{z1,z2,…,zn-1Δm(s)xjρ,∥)]pjk∈ℕ:λk-1∑j∈Ik[Mj(∥Δm(s)xjρ,‍λk-1∑j∈Ik[Mj(∥Δm(s)xjρ,‍[A,ℳ,Δm(s),Λ,p,∥·…·∥]0I,∥·…·∥z1,z2,…,zn-1Δm(s)xjρ,∥)]pj≥ε∑j∈Ikz1,z2,…,zn-1Δm(s)xjρ,∥)]pj[Mj(∥Δm(s)xjρ,‍}∈I,forsomeρ>0andeachz1,z2,…,zn-1∈X∑j∈Ik‍z1,z2,…,zn-1Δm(s)xjρ,∥)]pj},W[A,ℳ,Δm(s),Λ,p,∥·…·∥]∞={∥Δm(s)xjρ,z1,z2,…z1,z2,…,zn-1Δm(s)xjρ,∥)]pj,zn-1∥x∈ω(n-X):supkλk-1∑j∈Ik‍[Mj(∥Δm(s)xjρ,z1,z2,…,zn-1∥)]pj<∞,forsomeρ>0andeachz1,z2,…,zn-1∈X∥Δm(s)xjρ,z1,z2,…,zn-1Δm(s)xjρ,∥)]pjz1,z2,…,zn-1∥},W[A,ℳ,Δm(s),Λ,p,∥·…·∥]∞I={∑j∈Ik‍z1,z2,…,zn-1∑j∈Ik‍∥)]pjx∈ω(n-X):∃K>0,s.t.{z1,z2,…,zn-1∑j∈Ik‍∥)]pjk∈ℕ:λk-1∑j∈Ik‍[Mj(∥Δm(s)xjρ,hhhhhhhhhhhhhhhhhhhhhhhhhhz1,z2,…,zn-1∑j∈Ik‍∥)]pjssssss≥K∥Δm(s)xjρz1,z2,…,zn-1∑j∈Ik‍∥)]pj,}∈I,ssforsomeρ>0andeachz1,z2,…,zn-1∈Xz1,z2,…,zn-1∑j∈Ik‍∥)]pj}.Corollary 22.
The sequence spacesZ[A,ℳ,Δms,Λ,p,∥·…·∥], whereZ=WI,W0I,W∞I, andW∞, are paranormed spaces (not totally paranormed) with respect to the paranormhΔdefined by
(65)hΔ(x)=∑j=1ms‍∥xj,z1,z2,…,zn-1∥+inf{∑j=1∞‍akjz1,z2,…,zn-1∥Δm(s)xjρ,∥)]pj]1/Hρpk/H:hvhhhhsupk[∑j=1∞‍akjhΔ(x)=hΔ(x)=×[∥Δm(s)(Λj(x))ρ1,Mj(∥Δm(s)(Λj(x))ρ1,hhhhhhhhhhz1,z2,…,zn-1∥Δm(s)xjρ,∥)]pj]1/H00000000≤1,forsomeρ>0,sssssandeachz1,z2,…,zn-1∈Xz1,z2,…,zn-1∥Δm(s)xjρ,∥)]pj]1/HΔm(s)(Λj(x))ρ1},
whereH=max{1,supkpk} andZ=WI,W0I,W∞I, andW∞. Also it is clear that the paranormsgΔandhΔare equivalent.We state the following theorem in view of Lemma9.Theorem 23.
LetX be a standard n-normed space and {e1,e2,…,en} an orthogonal set in X. Then, the following hold:(a)
W[A,ℳ,Δm(s),Λ,p,∥·…·∥∞]I=W[A,ℳ,Δm(s),Λ,p,∥·…·∥n-1]I,(b)
W[A,ℳ,Δm(s),Λ,p,∥·…·∥∞]0I=W[A,ℳ,Δm(s),Λ,p,∥·…·∥n-1]0I,(c)
W[A,ℳ,Δm(s),Λ,p,∥·…·∥∞]∞=W[A,ℳ,Δm(s),Λ,p,∥·…·∥n-1]∞,(d)
W[A,ℳ,Δm(s),Λ,p,∥·…·∥∞]∞I=W[A,ℳ,Δm(s),Λ,p,∥·…·∥n-1]∞I,where ∥·…·∥∞is the derived(n-1)-norm defined with respect to the set{e1,e2,…,en}and∥·…·∥n-1is the standard(n-1)-norm onX.



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*Source: 123798-2013-12-22.xml*