# Some Properties of the Sequence SpaceEnq(ℳ,p,s,||·,…,·||)

Authors: Kuldip Raj; Sunil K. Sharma
Journal: Journal of Operators
 (2013)

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## Abstract

We introduce the sequence spaceEnq(ℳ,p,s,||·,…,·||) defined by a Musielak-Orlicz function ℳ=(Mk). We also study some topological properties and prove some inclusion relations involving this space.


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## Body

## 1. Introduction and Preliminaries

The concept of 2-normed spaces was initially developed by Gähler [1] in the mid-1960s, while one can see that of n-normed spaces in Misiak [2]. Since then, many others have studied this concept and obtained various results; see Gunawan [3, 4] and Gunawan and Mashadi [5]. Let n∈ℕ and X be a linear space over the field 𝕂, where 𝕂 is the field of real or complex numbers of dimension d, where d≥n≥2. A real valued function ∥·,…,·∥ on Xn satisfying the following four conditions:  (1)
∥x1,x2,…,xn∥=0 if and only if x1,x2,…,xn are linearly dependent in X;(2)
∥x1,x2,…,xn∥ is invariant under permutation;(3)
∥αx1,x2,…,xn∥=|α|∥x1,x2,…,xn∥ for any α∈𝕂; and(4)
∥x+x′,x2,…,xn∥≤∥x,x2,…,xn∥+∥x′,x2,…,xn∥is called an n-norm on X, and the pair (X,∥·,…,·∥) is called an n-normed space over the field 𝕂.For example, we may takeX=ℝn being equipped with the n-norm ∥x1,x2,…,xn∥E = the volume of the n-dimensional parallelopiped spanned by the vectors x1,x2,…,xn which may be given explicitly by the formula
(1)∥x1,x2,…,xn∥E=|det(xij)|,
where xi=(xi1,xi2,…,xin)∈ℝn for each i=1,2,…,n. Let (X,∥·,…,·∥) be an n-normed space of dimension d≥n≥2 and {a1,a2,…,an} be linearly independent set in X. Then the following function ∥·,…,·∥∞ on Xn-1 defined by
(2)∥x1,x2,…,xn∥∞=max{∥x1,x2,…,xn-1,ai∥:i=1,2,…,n}
defines an (n-1)-norm on X with respect to {a1,a2,…,an}.A sequence(xk) in an n-normed space (X,∥·,…,·∥) is said to converge to some L∈X if
(3)limk→∞∥xk-L,z1,…,zn-1∥=0foreveryz1,…,zn-1∈X.A sequence(xk) in an n-normed space (X,∥·,…,·∥) is said to be Cauchy if
(4)limk→∞p→∞∥xk-xp,z1,…,zn-1∥=0foreveryz1,…,zn-1∈X.
If every Cauchy sequence in X converges to some L∈X, then X is said to be complete with respect to the n-norm. Any complete n-normed space is said to be n-Banach space.An Orlicz functionM is a function which is continuous, nondecreasing, and convex with M(0)=0, M(x)>0 for x>0 and M(x)→∞ as x→∞.Lindenstrauss and Tzafriri [6] used the idea of Orlicz function to define the following sequence space. Let w be the space of all real or complex sequences x=(xk); then
(5)ℓM={x∈w:∑k=1∞‍M(|xk|ρ)<∞},
which is called an Orlicz sequence space. The space ℓM is a Banach space with the norm
(6)∥x∥=inf{ρ>0:∑k=1∞‍M(|xk|ρ)≤1}.It is shown in [6] that every Orlicz sequence space ℓM contains a subspace isomorphic to ℓp(p≥1). The Δ2-condition is equivalent to M(Lx)≤kLM(x) for all values of x≥0 and for L>1. A sequence ℳ=(Mk) of Orlicz function is called a Musielak-Orlicz function; see [7, 8]. A sequence 𝒩=(Nk) defined by
(7)Nk(v)=sup{|v|u-(Mk):u≥0},k=1,2,…,
is called the complementary function of a Musielak-Orlicz function ℳ. For a given Musielak-Orlicz function ℳ, the Musielak-Orlicz sequence space tℳ and its subspace hℳ are defined as follows:
(8)tℳ={x∈w:Iℳ(cx)<∞forsomec>0},hℳ={x∈w:Iℳ(cx)<∞∀c>0},
where Iℳ is a convex modular defined by
(9)Iℳ(x)=∑k=1∞‍Mk(xk),x=(xk)∈tℳ.We considertℳ equipped with the Luxemburg norm
(10)∥x∥=inf{k>0:Iℳ(xk)≤1}
or equipped with the Orlicz norm
(11)∥x∥0=inf{1k(1+Iℳ(kx)):k>0}.LetX be a linear metric space. A function p: X→ℝ is called paranorm if  (1)
p(x)≥0 for all x∈X,(2)
p(-x)=p(x) for all x∈X,(3)
p(x+y)≤p(x)+p(y) for all x,y∈X,(4)
if(λn) is a sequence of scalars with λn→λ as n→∞, and (xn) is a sequence of vectors with p(xn-x)→0 as n→∞; then p(λnxn-λx)→0 as n→∞.A paranormp for which p(x)=0 implies x=0 is called total paranorm, and the pair (X,p) is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see [9, Theorem 10.4.2, p. 183]). For more details about sequence spaces, see [10–23] and the references therein.Let∑k=0∞‍ak be an infinite series with sequence of partial sums (sk) and q be any positive real number. The Euler transform (E,q)s of the sequence s=(sn) is defined by
(12)Enq(s)=1(1+q)n∑v=0n‍(nv)qn-vsv.The series∑n=0∞‍an is said to be summable (E,q) to the number s if
(13)Enq(s)=1(1+q)n∑v=0n‍(nv)qn-vsv⟶sasn⟶∞
and is said to be absolutely summable (E,q) or summable |E,q| if
(14)∑k‍|Ekq(s)-Ek-1q(s)|<∞.Letx=(xn) be a sequence of scalars; we denote Nn(x)=Enq(x)-En-1q(x), where Enq(x) is defined by (12). After applications of Abel's transform, we have
(15)Nn(x)=-1(1+q)n-1∑k=0n-2‍xk+1Ak+sn-1An-1(1+q)n-1+sn(1+q)n-qn-1(1+q)ns0,
where
(16)Ak=∑i=0k‍[q1+q(ni)-(n-1i)]qn-i-1.Note that, for any sequencesx=(xn), y=(yn) and scalar λ, we haveNn(x+y)=Nn(x)+Nn(y) and Nn(λx)=λNn(x).Letℳ=(Mk) be a Musielak-Orlicz function and p=(pk) be a bounded sequence of positive real numbers. We define the following sequence space in the present paper:
(17)Enq(ℳ,p,s,∥·,…,·∥)={x=(xk):∑k=1∞‍1ks[Mk(∥Nk(x)ρ,z1,…,zn-1∥)]pk<∞,s≥0,forsomeρ>0∑k=1∞‍}.
If we take p=(pk)=1 for all k∈ℕ, we have
(18)Enq(ℳ,s,∥·,…,·∥)={x=(xk):∑k=1∞‍1ks[Mk(∥Nk(x)ρ,z1,…,zn-1∥)]<∞,s≥0,forsomeρ>0∑k=1∞‍}.
If we take s=0, we get
(19)Enq(ℳ,p,∥·,…,·∥)={x=(xk):∑k=1∞‍[Mk(∥Nk(x)ρ,z1,…,zn-1∥)]pk<∞,s≥0,forsomeρ>0∑k=1∞‍}.The following inequality will be used throughout the paper. If0≤pk≤suppk=H, K=max(1,2H-1), then
(20)|ak+bk|pk≤K{|ak|pk+|bk|pk}
for all k and ak,bk∈ℂ. Also, |a|pk≤max(1,|a|H) for all a∈ℂ.In this paper, we study some topological properties and inclusion relations between the previously defined sequence spaces.


## 2. Main Results

Theorem 1.
Letℳ=(Mk) be a Musielak-Orlicz function and p=(pk) be a bounded sequence of positive real numbers; the space Enq(ℳ,p,s,∥·,…,·∥) is linear over the complex field ℂ.Proof.
Letx=(xk),y=yk∈Enq(ℳ,p,s,∥·,…,·∥), and α,β∈ℂ. Then, there exist positive integers ρ1 and ρ2 such that
(21)∑k=1∞‍1ks[Mk(∥Nk(x)ρ1,z1,…,zn-1∥)]pk<∞,∑k=1∞1ks[Mk(∥Nk(y)ρ2,z1,…,zn-1∥)]pk<∞.
Let ρ3=max(2|α|ρ1,2|β|ρ2). Since (Mk) is nondecreasing, convex function and by using inequality (20), we have
(22)∑k=1∞‍1ks[Mk(∥Nk(αx+βy)ρ3,z1,…,zn-1∥)]pk≤∑k=1∞‍1ks[Mk(∥Nk(αx)ρ3,z1,…,zn-1∥)+(∥Nk(βy)ρ3,z1,…,zn-1∥)]pk≤K∑k=1∞‍1ks[Mk(∥Nk(x)ρ1,z1,…,zn-1∥)]pk+K∑k=1∞‍1ks[Mk(∥Nk(y)ρ2,z1,…,zn-1∥)]pk<∞.
Therefore, αx+βy∈Enq(ℳ,p,s,∥·,…,·∥). Hence, Enq(ℳ,p,s,∥·,…,·∥) is a linear space.Theorem 2.
Letℳ=(Mk) be a Musielak-Orlicz function and p=(pk) be a bounded sequence of positive real numbers. Then, the space Enq(ℳ,p,s,∥·,…,·∥) is a paranormed space with
(23)g(x)=inf{ρpn/H:(∑k=1∞‍1ks[Mk(∥Nk(x)ρ,z1,…,zn-1∥)]pk)1/H≤1,n=1,2,3,…(∑k=1∞‍1ks[Mk(∥Nk(x)ρ,z1,…,zn-1∥)]pk)(1/H)},
where H=max(1,supkpk).Proof.
Clearly,g(x)=g(-x) and g(x+y)≤g(x)+g(y). Since Mk(0)=0, we get inf{ρpn/H}=0 for x=0. Finally, we prove that multiplication is continuous. Let λ be any number such that
(24)g(λx)=inf{ρpn/H:∑k=1∞‍1ks[Mk(∥λNk(x)ρ,z1,…,zn-1∥)]pk∑k=1∞‍≤1,n=1,2,3,…∑k=1∞‍[Mk(∥λNk(x)ρ,z1,…,zn-1∥)]pk}.
Thus, we have
(25)g(λx)=inf{(λs)pn/H:∑k=1∞‍1ks[Mk(∥Nk(x)s,z1,…,zn-1∥)]pk≤1,n=1,2,3,…∑k=1∞‍[Mk(∥Nk(x)s,z1,…,zn-1∥)]pk},
where s=ρ/|λ|. Since |λ|pk≤max(1,|λ|H), then |λ|pk/H≤(max(1,|λ|H))1/H. Hence,
(26)g(λx)≤(max(1,|λ|H))1/Hinf{(s)pn/H:(∑k=1∞‍1ks[Mk(∥Nk(x)ρ,z1,…,zn-1∥)]pk)1/H≤1,n=1,2,…(∑k=1∞‍1ks[Mk(∥Nk(x)ρ,z1,…,zn-1∥)]pk)(1/H)},
and therefore g(λx) converges to zero when g(x) converges to zero in Enq(ℳ,p,s,∥·,…,·∥). Now, suppose that λn→0 as n→∞ and x in Enq(ℳ,p,s,∥·,…,·∥). For arbitrary ϵ>0, let n0 be a positive integer such that
(27)∑k=n0+1∞‍1ks[Mk(∥Nk(x)ρ,z1,…,zn-1∥)]pk<ϵ2
for some ρ>0. This implies that
(28)(∑k=n0+1∞‍1ks[Mk(∥Nk(x)ρ,z1,…,zn-1∥)]pk)1/H≤ϵ2.
Let0<|λ|<1; then, using convexity of (Mk), we get
(29)∑k=n0+1∞‍1ks[Mk(∥λNk(x)ρ,z1,…,zn-1∥)]pk<|λ|∑k=n0+1∞‍1ks[Mk(∥Nk(x)ρ,z1,…,zn-1∥)]pk<(ϵ2)H.
Since(Mk) is continuous everywhere in [0,∞), then
(30)h(t)=∑k=1n0‍1ks[Mk(∥tNk(x)ρ,z1,…,zn-1∥)]pk
is continuous at 0. So there is 0<δ<1 such that |h(t)|<ϵ/2 for 0<t<δ. Let K be such that |λn|<δ for n>K; we have
(31)(∑k=1n0‍1ks[Mk(∥λnNk(x)ρ,z1,…,zn-1∥)]pk)1/H<ϵ2.
Thus,(32)(∑k=1∞‍1ks[Mk(∥λnNk(x)ρ,z1,…,zn-1∥)]pk)1/H<ϵ,forn>K.
Hence,g(λx)→0 as λ→0. This completes the proof of the theorem.Theorem 3.
Ifℳ′=(Mk′) and ℳ′′=(Mk′′) are two Musielak-Orlicz functions and s,s1,ands2 are nonnegative real numbers, then one has(i)
Enq(ℳ′,p,s,∥·,…,·∥)∩Enq(ℳ′′,p,s,∥·,…,·∥)⊆Enq(ℳ′+ℳ′′,p,s,∥·,…,·∥),(ii)
Ifs1≤s2, then Enq(ℳ′,p,s1,∥·,…,·∥)⊆Enq(ℳ′,p,s2,∥·,…,·∥),(iii)
Ifℳ′ and ℳ′′ are equivalent, then  Enq(ℳ′,p,s,∥·,…,·∥)=Enq(ℳ′′p,s,∥·,…,·∥).Proof.
It is obvious, so we omit the details.Theorem 4.
Suppose that0<rk≤pk<∞ for each k. Then Enq(ℳ,r,s,∥·,…,·∥)⊆Enq(ℳ,p,s,∥·,…,·∥).Proof.
Letx∈Enq(ℳ,r,s,∥·,…,·∥). Then there exists some ρ>0 such that
(33)∑k=1∞‍1ks[Mk(∥Nk(x)ρ,z1,…,zn-1∥)]rk<∞.
This implies thatMk(∥Nk(x)/ρ,z1,…,zn-1∥)≤1 for sufficiently large value of k, say k≥k0, for some fixed k0∈ℕ. Since (Mk) is nondecreasing, we get
(34)∑k≥k0∞‍1ks[Mk(∥Nk(x)ρ,z1,…,zn-1∥)]pk≤∑k≥k0∞‍1ks[Mk(∥Nk(x)ρ,z1,…,zn-1∥)]rk<∞.
Hence,x∈Enq(ℳ,p,s,∥·,…,·∥).Theorem 5.
(i) If0<pk≤1 for each k, then Enq(ℳ,p,s,∥·,…,·∥)⊆Enq(ℳ,s,∥·,…,·∥).
(ii) Ifpk≥1 for all k, then Enq(ℳ,s,∥·,…,·∥)⊆Enq(ℳ,p,s,∥·,…,·∥).Proof.
It is easy to prove by using Theorem4; so we omit the details.


## 3. Statistical Convergence

The notion of statistical convergence was introduced by Fast [24] and Schoenberg [25] independently. Over the years and under different names, statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory, and number theory. Later on, it was further investigated from the sequence space point of view and linked with summability theory by Fridy [26], Connor [27], Šalát [28], Mursaleen [29], Işik [30], Savaş [31], Malkowsky and Savas [32], Kolk [33], Maddox [34], Mohiuddine and Aiyub [35], and many others. In the recent years, generalizations of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on locally compact spaces. Statistical convergence and its generalizations are also connected with subsets of the Stone-Cech compactification of natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability.Letθ=(kr) be the sequence of positive integers such that k0=0, 0<kr<kr+1 and hr=kr-kr-1→∞ as r→∞. Then θ is called a lacunary sequence. The intervals determined by θ will be denoted by Ir=(kr-1,kr], and the ratio kr/(kr-1) will be denoted by qr (see [36]).Letθ=(kr) be a lacunary sequence; then the sequence x=(xk) is said to be Nk(u)-lacunary statistically convergent to the number l provided that, for every ϵ>0,
(35)limr1hr|{k∈Ir:∥ukNk(x)-lρ,z1,…,zn-1∥≥ϵ}|=0.
In this case, we write [Nk(u),S]θ-limx=l or xk→l([Nk(u),S]θ).Letθ=(kr) be a lacunary sequence, ℳ=(Mk) be a Musielak-Orlicz function, u=(uk) be a sequence of strictly positive real numbers, and p=(pk) be a bounded sequence of positive real numbers. A sequence x=(xk) is said to be strongly Nk(u)-lacunary convergent to the number l with respect to the Musielak-Orlicz function ℳ provided that
(36)limr1hr∑k∈Ir‍[Mk(∥ukNk(x)-lρ,z1,…,zn-1∥)]pk=0.The set of all stronglyNk(u)-lacunary convergent sequences to the number l with respect to the Musielak-Orlicz function ℳ is denoted by [Nk,u,ℳ,p,∥·,…,·∥]θ. In this case, we write xk→l([Nk,u,ℳ,p,∥·,…,·∥]θ). In the special case ℳ(x)=x, pk=p0 for all k∈ℕ, we shall write [Nk,u,∥·,…,·∥]θ instead of [Nk,u,ℳ,p,∥·,…,·∥]θ.In this section, we give some results aboutNk(u)-lacunary statistical convergence and give some relations between the set of Nk(u)-lacunary statistical convergence sequences and other spaces which are defined with respect to Musielak-Orlicz function ℳ.Theorem 6.
Letθ=(kr) be a lacunary sequence.(i)
If a sequencex=(xk) is strongly Nk(u)-lacunary convergent to l, then it is Nk(u)-lacunary statistically convergent to l.(ii)
If a bounded sequencex=(xk) is Nk(u)-lacunary statistically convergent to l, then it is strongly Nk(u)-lacunary convergent to l.Proof.
(i) Letϵ>0 and xk→l([Nk,u,∥·,…,·∥]θ). Then, we can write
(37)∑k∈Ir‍∥ukNk(x)-lρ,z1,…,zn-1∥p0≥∑k∈Ir∥(ukNk(x)-l)/ρ,z1,…,zn-1∥≥ϵ∥ukNk(x)-lρ,z1,…,zn-1∥p0≥ϵp0|{k∈Ir:∥ukNk(x)-lρ,z1,…,zn-1∥≥ϵ}|.
Hence, xk→l([Nk,u,S]θ).
(ii) Suppose thatxk→l([Nk,u,S]θ), and let x∈l∞. Let ϵ>0 be given and take Nϵ such that
(38)1hr|{k∈Ir:∥ukNk(x)-lρ,z1,…,zn-1∥≥(ϵ2)(1/p0)}|≤ϵ2Kp0
for all r>Nϵ and set Tr={k∈Ir:∥(ukNk(x)-l)/ρ,z1,…,zn-1∥≥(ϵ/2)1/p0}, where K=supk|xk|<∞. Now, for all r>Nϵ, we have
(39)1hr∑k∈Ir‍∥ukNk(x)-lρ,z1,…,zn-1∥p0=1hr∑k∈Irk∈Tr∥ukNk(x)-lρ,z1,…,zn-1∥p0+1hr∑k∈Irk∉Tr∥ukNk(x)-lρ,z1,…,zn-1∥p0≤1hr(hrϵ2Kp0)Kp0+ϵ2hrhr=ϵ.
Thus,(xk)∈[Nk,u,∥·,…,·∥]θ. This completes the proof of the theorem.Theorem 7.
For any lacunary sequenceθ, if limr→∞infqr>1, then [Nk,u,S]⊂[Nk,u,S]θ.Proof.
Iflimr→∞infqr>1, then there exists a δ>0 such that 1+δ≤qr for sufficiently large r. Since hr=kr-kr-1, we have kr/hr≤(1+δ)/δ. Let xk→l([Nk,u,S]). Then, for every ϵ>0, we have
(40)1kr|{k≤kr:∥ukNk(x)-lρ,z1,…,zn-1∥≥ϵ}|≥1kr|{k∈Ir:∥ukNk(x)-lρ,z1,…,zn-1∥≥ϵ}|≥δ1+δ1kr|{k∈Ir:∥ukNk(x)-lρ,z1,…,zn-1∥≥ϵ}|.
Hence,[Nk,u,S]⊂[Nk,u,S]θ.Theorem 8.
Letθ=(kr) be a lacunary sequence, ℳ=(Mk) be a Musielak-Orlicz function, and 0<h=infkpk≤pk≤supkpk=H. Then, [Nk,u,ℳ,∥·,…,·∥]θ⊂[Nk,u,S]θ.Proof.
Letx∈[Nk,u,ℳ,∥·,…,·∥]θ. Then there exists a number ρ>0 such that
(41)1hr∑k∈Ir‍[Mk(∥ukNk(x)-lρ,z1,…,zn-1∥)]pk⟶0,asr→∞.
Then, givenϵ>0, we have
(42)1hr∑k∈Ir‍[Mk(∥ukNk(x)-lρ,z1,…,zn-1∥)]pk≥1hr∑∥(ukNk(x)-l)/ρ,z1,…,zn-1∥≥ϵk∈Ir‍[Mk(∥ukNk(x)-lρ,z1,…,zn-1ukNk(x)-lρ∥)Mk(∥ukNk(x)-lρ]pk≥1hr∑∥(ukNk(x)-l)/ρ,z1,…,zn-1∥≥ϵk∈Ir‍[Mk(ϵ1)]pk,whereϵ/ρ=ϵ1≥1hr∑‍min{[Mk(ϵ1)]h,[Mk(ϵ1)]H}≥1hr|{k∈Ir:∥ukNk(x)-lρ,z1,…,zn-1∥≥ϵ}|·min{[Mk(ϵ1)]h,[Mk(ϵ1)]H}.
Hence,x∈[Nk,u,S]θ. This completes the proof of the theorem.Theorem 9.
Letθ=(kr) be a lacunary sequence, ℳ=(Mk) be a Musielak-Orlicz function, and p=(pk) be a bounded sequence; then [Nk,u,S]θ⊂[Nk,u,ℳ,∥·,…,·∥]θ.Proof.
Letx=(xk)∈l∞ with xk→l([Nk,u,S]θ). Since x∈l∞, there is a constant T>0 such that ∥(ukNk(x)-l)/ρ,z1,…,zn-1∥≤T, and given ϵ>0, we have
(43)1hr∑k∈Ir‍[Mk(∥ukNk(x)-lρ,z1,…,zn-1∥)]pk=1hr∑∥(ukNk(x)-l)/ρ,z1,⋯,zn-1∥≥ϵk∈Ir‍[Mk(∥ukNk(x)-lρ,z1,…,zn-1ukNk(x)-lρ∥∥ukNk(x)-lρ,)Mk(∥ukNk(x)-lρ,]pk+1hr∑∥(ukNk(x)-l)/ρ,z1,…,zn-1∥<ϵk∈Ir‍[Mk(∥ukNk(x)-lρ,z1,…,zn-1ukNk(x)-lρ∥∥ukNk(x)-lρ,)Mk(∥ukNk(x)-lρ,]pk≤1hr∑∥(ukNk(x)-l)/ρ,z1,…,zn-1∥≥ϵk∈Ir‍max{[Mk(Tρ)]h,[Mk(Tρ)]H}+1hr∑∥(ukNk(x)-l/ρ),z1,…,zn-1∥<ϵk∈Ir‍[Mk(ϵρ)]pk≤max{[Mk(K)]h,[Mk(K)]H}×1hr|{k∈Ir:∥ukNk(x)-lρ,z1,…,zn-1∥≥ϵ}|+max{[Mk(ϵ1)]h,[Mk(ϵ1)]H},Tρ=K,ϵρ=ϵ1.
Hence,x∈[Nk,u,ℳ,∥·,…,·∥]θ.



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