# On Some Geometric Properties of a New Paranormed Sequence Space

Authors: Vatan Karakaya; Fatma Altun
Journal: Journal of Function Spaces
 (2014)

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## Abstract

We introduce a new sequence space which is defined by the operatorW=(wnk) on the sequence space ℓ(p). We define a modular functional on this space and investigate structure of this space equipped with Luxemburg norm. Also we study some geometric properties which are called Kadec-Klee, k-NUC, and uniform Opial properties and prove that this new space possesses these properties.


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## Body

## 1. Introduction

In literature, there are many papers about geometric properties and their applications on different sequence spaces. Some of them are as follows.In [1], Opial defined the Opial property with his name mentioned and he proved that ℓp(1<p<∞) satisfies this property but the space Lp[0,2π](p≠2,1<p<∞) does not.Franchetti [2] has shown that any infinite dimensional Banach space has an equivalent norm satisfying the Opial property. Later, Prus [3] has introduced and investigated uniform Opial property for Banach spaces.In [4], the notion of nearly uniform convexity for Banach spaces was introduced by Huff. Also Huff proved that every nearly uniformly convex space is reflexive and it has uniform Kadec-Klee property. However, Kutzarova [5] defined k-nearly uniformly convex Banach spaces.Shue [6] first defined Cesaro sequence spaces with norm. In [7], it is shown that the Cesaro sequence spaces cesp(1≤p<∞) have Kadec-Klee and local uniform rotundity properties.In [8], it was shown that Banach-Saks of type-p property holds in these spaces.Later, Sanhan and Suantai [9] generalized the normed sequence spaces to the paranormed sequence spaces. He showed that the sequence spaces ces(p) equipped Luxemburg norm are rotund and have Kadec-Klee property.Petrot and Suantai [10] studied the uniform Opial property of these spaces. In [9], Sanhan and Suantai have showed that the Cesaro sequence space ces(p), where the sum runs over 2r≤k≤2r+1, equipped with Luxemburg norm has property (H) but it is not rotund.Karakaya [11] introduced a new sequence space involving lacunary sequences connected with Cesaro sequence space and examined some geometric properties of this space equipped with Luxemburg norm. In [12], Karakaş et al. defined and studied a new difference sequence space involving lacunary sequences by using difference operator.In [13], Khan and Rahman introduced sequence spaces ces[(pn),(qn)]. Afterwards, Mursaleen and Khan [14] generalized this space to the vector-valued sequence space. In the space ces[(pn),(qn)], if we specialize qn=1 for all n∈ℕ, then we get ces[(pn),(qn)]=ces(p) defined in [9].In [15], Şimşek and Karakaya generalized sequence space ces[(pn),(qn)] to vector-valued space ces(X,pn,qn) and investigated some topological and geometrical properties as Kadec-Klee and rotund according to Luxemburg norm of this space.In [16], Savaş et al. introduced an ℓp-type new sequence space and examined some geometrical properties of this space concerning Banach-Saks of type-p and Gurarii’s modulus of convexity. Also, in [17], Şimşek et al. investigated the k-nearly uniform convexity (k-NUC) property and some fixed point results in modular space Vρ(λ;p); Şimşek and Karakaya [18] introduced modular sequence space ℓρ(u,v,p) obtained from paranormed ones by generalized weighted means on Köthe sequence spaces and investigated Kadec-Klee property of this space.


## 2. Preliminaries and Notation

Let(X,∥·∥) (for the brevity X=(X,∥·∥)) be a normed linear space and let B(X) (resp. S(X)) be the closed unit ball (resp. unit sphere) of X. The space of all real sequences is denoted by w. For any sequence {xn} in X, we denote by conv({xn}) the convex hull of the elements of {xn}.A Banach spaceX is calleduniformly convex (UC) if for each ɛ>0, there is δ>0 such that, for x,y∈S(X), the inequality ∥x-y∥>ɛ implies that
(1)∥12(x+y)∥<1-δ.
Recall that for a number ɛ>0 a sequence {xn} is said to be an ɛ-seperated sequence if
(2)sep({xn})=inf⁡{∥xn-xm∥,n≠m}>ɛ.
A Banach space X is said to have theKadec-Klee property (H property) if every weakly convergent sequence on the unit sphere is convergent in norm.A Banach spaceX is said to have theuniform Kadec-Klee property (UKK) if for every ɛ>0 there exists δ>0 such that if x is the weak limit of a normalized ɛ-separated sequence, then ∥x∥<1-δ (see [4]). We have that every (UKK) Banach space has the Kadec-Klee property.A Banach spaceX is said to be thenearly uniformly convex (NUC) if for every ɛ>0 there exists δ>0 such that, for every sequence {xn}⊂B(X) with sep({xn})>ɛ, we have
(3)conv({xn})∩(1-δ)B(X)≠∅.
Let k≥2 be an integer. A Banach space X is said to be k-nearly uniformly convex (k-NUC) if for any ɛ>0 there exists δ>0 such that, for every sequence {xn}⊂B(X) with sep({xn})>ɛ, there are n1,n2,…,nk∈ℕ such that
(4)∥xn1+xn2+⋯+xnkk∥<1-δ.
Of course a Banach space X is (NUC) whenever it is (k-NUC) for some integer k≥2. Clearly, (k-NUC) Banach spaces are (NUC) but the opposite implication does not hold in general (see [5]).A Banach spaceX is said to have theOpial property if every sequence {xn} that is weakly convergent to x0 satisfies
(5)lim⁡n→∞inf⁡∥xn-x0∥<lim⁡n→∞inf⁡∥xn-x∥,
for every x∈X and x≠x0 (see [1]).A Banach spaceX is said to have theuniform Opial property if every ɛ>0 there exists δ>0 such that, for each weakly null sequence {xn}⊂S(X) and x∈X with ∥x∥≥ɛ, we have (see [3])
(6)1+τ≤lim⁡n→∞inf⁡∥xn+x∥.
A point x∈S(X) is called anextreme point if for any y,z∈B(X) the equality 2x=y+z implies that y=z. A Banach space X is said to berotund (abbreviated as (R)) if every point of S(X) is an extreme point. A Banach space X is said to befullyk-rotund (written as kR) (see [19]) if for every sequence {xn}⊂B(X)(7)∥xn1+xn2+⋯+xnk∥⟶kasn1,n2,…,nk⟶∞
implies that {xn} is convergent.It is well known that (UC) implies(kR) and (kR) implies ((k+1)R), and (kR) spaces are reflexive and rotund, and it is easy to see that (k-NUC) implies (kR).For a real vector spaceX, a function ρ:X→[0,∞] is called a modular if it satisfies the following conditions:(i)
ρ(x)=0⇔x=0,(ii)
ρ(αx)=ρ(x) for all α∈F with |α|=1,(iii)
ρ(αx+βy)≤ρ(x)+ρ(y) for all x,y∈X and all α,β≥0 with α+β=1. Further, the modular ρ is called convex if(iv)
ρ(αx+βy)≤αρ(x)+βρ(y) holds for all x,y∈X and all α,β≥0 with α+β=1.For any modularρ on X, the space
(8)Xρ={x∈X:ρ(λx)⟶0asλ⟶0+}
is called the modular space.A sequence(xn) of elements of Xρ is called modular convergent to x∈Xρ if there exists a λ>0 such that ρ(λ(xn-x))→0 as n→∞. If ρ is a convex modular, then the following formula defines a norm on Xρ which is called the Luxemburg norm:
(9)∥x∥L=inf⁡{λ>0:ρ(xλ)≤1}.A modularρ is said to satisfy the Δ2-condition (ρ∈Δ2) if for any ɛ>0 there exist constants K≥2 and a>0 such that
(10)ρ(2x)≤Kρ(x)+ɛ,
for all x∈Xρ with ρ(x)≤a.Ifρ satisfies the Δ2-condition for all a>0 with K≥2 dependent on a, we say that ρ satisfies the strong Δ2-condition (ρ∈Δ2s).Lemma 1.
Ifρ∈Δ2s, then, for any L>0 and ɛ>0, there exists δ>0 such that
(11)|ρ(u+v)-ρ(u)|<ɛ
whenever u,v∈Xρ with ρ(u)≤L and ρ(v)≤δ.Proof.
See [20].Lemma 2.
Ifρ∈Δ2s, then, for any ɛ>0, there exists δ>0 such that ∥x∥≤1-δ whenever ρ(x)≤1-ɛ.See [21].Lemma 3.
Ifρ∈Δ2s, then for any (xn)∈Xρ(12)∥xn∥⟶0⟺ρ(xn)⟶0.
See [20].Lemma 4.
Ifρ∈Δ2s, then for any ɛ>0 there exists δ>0 such that ∥x∥≥1-δ whenever ρ(x)≥1-ɛ.See [20].In this paper, we will need the following inequalities in the sequel:(13)|ak+bk|p≤2p-1(|ak|p+|bk|p),
for p≥1.In [22], Polat et al. defined the matrix V=(vnk) by
(14)vnk={un(vk-vk+1),k<nunvn,k=n0,k>n.
Here, for all n∈ℕ, un≠0, vn≠0, and (un) depend on n; (vk) depend on k.Using this matrix, we defineℓΔ(u,v,p) sequence space as follows:
(15)ℓΔ(u,v,p)={(∑k=1n-1un(vk-vk+1)xk+unvnxn)x=(xn)∈w:(yn)=(∑k=1n-1un(vk-vk+1)xk+unvnxn)∈ℓ(p),n∈ℕ.(∑k=1n-1un(vk-vk+1)xk+unvnxn)},
where Δxk=xk-xk-1 is back difference, and Δvk=vk-vk+1 is forward difference. Throughout this study, p=(pk) is a bounded sequence of positive real numbers; H=sup⁡pr and M=max⁡{1,H}. We denote (Vx)n=(∑k=1n-1un(vk-vk+1)xk+unvnxn)n for short in proof.Theorem 5.
The sequence spaceℓΔ(u,v,p) is a complete metric space of nonabsolute type with respect to the paranorm defined by
(16)g(x)=(∑‍n|∑k=1n-1un(vk-vk+1)xk+unvnxn|pk)1/M.Proof.
The linearity ofℓΔ(u,v,p) with respect to the coordinatewise and scalar multiplication follows from the following inequalities which are satisfied for x,y∈ℓΔ(u,v,p):
(17)(∑‍n|∑k=1n-1un(vk-vk+1)(xk+yk)+unvn(xn+yn)|pk)1/M≤(∑‍n|∑k=1n-1un(vk-vk+1)xk+unvnxn|pk)1/M+(∑‍n|∑k=1n-1un(vk-vk+1)yk+unvnyn|pk)1/M,
and for any α∈ℝ(18)|α|pk≤max⁡{1,|α|M}.
It is clear that g(θ)=0 and g(x)=g(-x) for all x∈ℓΔ(u,v,p). From (17), it can be seen the subadditivity of g and g(αx)≤max⁡{1,|α|M}g(x).
Let(xm) be any sequence in ℓΔ(u,v,p) such that g(xm-x)→0 and (αm) are any sequence of scalars such that αm→α. Then, since the inequality
(19)g(xm)≤g(x)+g(xm-x)
holds, the subadditivity of g, (g(xm)), is bounded and thus we have
(20)g(αmxm-αx)=(∑‍n|∑k=1n-1un(vk-vk+1)(αmxkm-αxk)ghhhh+unvn(αmxnm-αxn)∑k=1n-1un(vk-vk+1)xk|pk)1/M≤|αm-α|pk/Mg(xm)+|α|pk/Mg(xm-x),
which tends to zero as n→∞. Therefore, the scalar multiplication is continuous. Hence, g is a paranorm on the space ℓΔ(u,v,p). It remains to prove the completeness of the space ℓΔ(u,v,p). Let (xj) be any Cauchy sequence in the space ℓΔ(u,v,p). Then, for a given ɛ>0, there exists a positive integer m0(ɛ)∈ℕ such that g(xi-xj)<ɛ/2 for all i,j≥m0(ɛ). Using definition of g, we obtain for each fixed n∈ℕ that
(21)|(Vxi)n-(Vxj)n|pk/M≤(∑‍n|(Vxi)n-(Vxj)n|pk)1/M<ɛ2,
for every i,j≥m0(ɛ) which leads us to the fact that {(Vx0)n,(Vx1)n,…} is a Cauchy sequence of real numbers for every fixed n∈ℕ. Since ℝ is complete, (Vxi)n→V(x) as i→∞. Using these infinitely many limits (Vx)0,(Vx)1,(Vx)2,…, we may write the sequence {(Vx)0,(Vx)1,…}.
For allj≥m0(ɛ) and every fixed n∈ℕ(22)|(Vxj)n-(Vx)n|<ɛ2.
Now, we have to show x∈ℓΔ(u,v,p). To do this, we have
(23)|(Vx)n|pk=|(Vx)n-(Vxi)n+(Vxi)n|pk(∑‍n|(Vx)n|pn)1/M≤(∑‍n|(Vx)n-(Vxi)n|pn)1/M+(∑‍n|(Vxi)n|pn)1/M.
Hence, we get x∈ℓΔ(u,v,p). As a result ℓΔ(u,v,p) is a complete metric space.We introduce a modular sequence spaceℓρΔ(u,v,p) by
(24)ℓρΔ(u,v,p)={x=(xn)∈w:∑k=1∞(∑i=1k-1ukΔvi|xi|+ukvk|xk|)pk<∞.(∑i=1k-1ukΔvi|xi|+ukvk|xk|)pk}.
The Luxemburg norm on the sequence space ℓρΔ(u,v,p) is defined as follows:
(25)∥x∥L=inf⁡{λ>0:ρ(xλ)≤1}hhhforeveryx∈ℓρΔ(u,v,p).
Here, the modular defined by
(26)ρ(x)=∑k=1∞(∑i=1k-1ukΔvi|xi|+ukvk|xk|)pk
is a convex modular on ℓρΔ(u,v,p).


## 3. Main Results

In this section, we will give some basic properties of the modularρ on the space ℓρΔ(u,v,p). Also, we will investigate some relationships between the modular ρ and the Luxemburg norm on ℓρΔ(u,v,p). Finally, we study some geometric properties on this space.Let us start with some lemmas which will be used in the proof of the theorems about geometric properties of this space.Lemma 6.
The functionalρ is a convex modular on ℓρΔ(u,v,p).Proof.
Letx,y∈ℓρΔ(u,v,p). It is obvious that(i)
ρ(x)=0⇔x=0.(ii)
ρ(λx)=ρ(x) for all scalar λ with |λ|=1(27)ρ(λx)=∑k=1∞(∑i=1k-1ukΔvi|αxi|+ukvk|αxk|)pk=∑k=1∞|α|pk(∑i=1k-1ukΔvi|xi|+ukvk|xk|)pk=ρ(x).(iii)
Forα,β≥0 with α+β=1, by the convexity of t→|t|pk for every k∈ℕ, we have
(28)ρ(λx+βy)=∑k=1∞(∑i=1k-1ukΔvi|αxi+βyi|+ukvk|αxk+βyk|)pk≤∑k=1∞(∑i=1k-1ukΔvi|αxi|+ukvk|αxk|ghhhh+∑i=1k-1ukΔvi|βyi|+ukvk|βyk|)pk≤∑k=1∞|α|pk(∑i=1k-1ukΔvi|xi|+ukvk|xk|)pk+∑k=1∞|β|pk(∑i=1k-1ukΔvi|yi|+ukvk|yk|)pk≤αρ(x)+βρ(y).For x∈ℓρΔ(u,v,p), the modular ρ on ℓρΔ(u,v,p) satisfies the following properties:(i)
if0<a<1, then aMρ(x/a)≤ρ(x) and ρ(ax)≤aρ(x),
(ii)
ifa≥1, then ρ(x)≤aMρ(x/a),
(iii)
ifa≥1, then ρ(x)≤aρ(x)≤ρ(ax).Proof.
It can be proved with standard techniques in a similar way as in [23].Lemma 7.
For anyx∈ℓρΔ(u,v,p),(i)
if∥x∥<1, then ρ(x)≤∥x∥,(ii)
if∥x∥>1, then ρ(x)≥∥x∥,(iii)
∥x∥=1⇔ρ(x)=1,(iv)
if∥x∥<1, then ρ(x)<1,(v)
if∥x∥>1, then ρ(x)>1.Proof.
It can be proved with standard techniques in a similar way as in [23].Lemma 8.
Let{xn} be a sequence in ℓρΔ(u,v,p):(i)
iflim⁡n→∞⁡∥xn∥=1, then lim⁡n→∞⁡ρ(xn)=1,(ii)
iflim⁡n→∞⁡ρ(xn)=0, then lim⁡n→∞⁡∥xn∥=0.Proof.
It can be proved with standard techniques in a similar way as in [23].Lemma 9.
For anyL>0 and ɛ>0, there exists δ>0 such that
(29)|ρ(u+v)-ρ(u)|<ɛ,
whenever u,v∈ℓρΔ(u,v,p) with ρ(u)≤L and ρ(v)≤δ.Proof.
Sincep=(pk) is bounded, it is easy to see that ρ∈Δ2s. Hence, the lemma is obtained directly from Lemma 1.Lemma 10.
For any sequence(xn)∈ℓρΔ(u,v,p),
(30)∥xn∥⟶0⟺ρ(xn)⟶0.Proof.
Sinceρ∈Δ2s, the lemma is obtained directly from Lemma 3.Lemma 11.
For anyx∈ℓρΔ(u,v,p) and ɛ∈(0,1), there exists δ∈(0,1) such that ρ(x)≤1-ɛ implies ∥x∥≤1-δ.Proof.
Sinceρ∈Δ2s, the lemma is obtained directly from Lemma 2.Now we will show that theℓρΔ(u,v,p) is a Banach space with respect to the Luxemburg normTheorem 12.
The spaceℓρΔ(u,v,p) is a Banach space with respect to the Luxemburg norm defined by
(31)∥x∥=inf⁡{λ>0:ρ(xλ)≤1}.Proof.
We will show that every Cauchy sequence inℓρΔ(u,v,p) is convergent according to the Luxemburg norm. Let (xkn) be a Cauchy sequence ℓρΔ(u,v,p) and ɛ∈(0,1). Thus, there exists n0(ɛ) such that ∥xn-xm∥<ɛ for all m,n≥n0. By the Lemma 8(i), we obtain
(32)ρ(xn-xm)<∥xn-xm∥<ɛ,
for all m,n≥n0(ɛ); that is,
(33)∑k=1∞(∑i=1k-1ukΔvi|xn(i)-xm(i)|+ukvk|xn(k)-xm(k)|)pk<ɛ.
For fixed k we get that
(34)|xn(i)-xm(i)|<ɛ.
Hence, we obtain that the sequence (xn(i)) is a Cauchy sequence in ℝ. Since the real number ℝ is complete, xm(i)→x(i) as m→∞. Therefore, for fixed k and
(35)(∑i=1k-1ukΔvi|xn(i)-x(i)|+ukvk|xn(k)-x(k)|)pk<ɛ∀n≥n0(ɛ).
So, we obtain that for all n≥n0(ɛ) and as m→∞(36)ρ(xn-xm)⟶ρ(xn-x).
So, for all n≥n0(ɛ) from Lemma 8(i),
(37)ρ(xn-x)<∥xn-x∥<ɛ.
It can be seen that, for all n≥n0, xn→x and (xn-x)∈ℓρΔ(u,v,p).
From the linearity of the sequence spaceℓρΔ(u,v,p), we can write that
(38)x=xn-(xn-x)∈ℓρΔ(u,v,p).
Hence, the sequence space ℓρΔ(u,v,p) is a Banach space with respect to the Luxemburg norm. This completes the proof of the theorem.Lemma 13.
Letx∈ℓρΔ(u,v,p) and {xn}⊆ℓρΔ(u,v,p). If ρ(xn)→ρ(x) as n→∞ and xn(i)→x(i) as n→∞ for all i∈ℕ, then xn→x as n→∞.Now, we shall give the main theorems of this paper involving the geometric properties of the spaceℓρΔ(u,v,p).Theorem 14.
The spaceℓρΔ(u,v,p) has the Kadec-Klee property.Proof.
Letx∈S(ℓρΔ(u,v,p)) and xn(i)⊆ℓρΔ(u,v,p) such that ∥xn(i)∥→1 and xn(i)→wx(i) as n→∞. From Lemma 8(iii), we get ρ(x)=1. So, from Lemma 9(i), it follows that ρ(xn)→ρ(x) as n→∞. Since mapping πi:ℓρΔ(u,v,p)→ℝ defined by πi(y)=yi is a continuous linear functional on ℓρΔ(u,v,p). It follows that xn(i)→x(i) as n→∞ for all i∈ℕ. So from Lemma 13,   xn→x as n→∞.Theorem 15.
The spaceℓρΔ(u,v,p) is k-NUC for any integer k≥2 where 1<p<∞.Proof.
Letɛ>0 and {xn}⊆B(ℓρΔ(u,v,p)) with sep(xn)≥ɛ. For each m∈ℕ let
(39)xnm=(0,0,…,0︸m-1,xn(m),xn(m+1),…),
since for each i∈ℕ(xn(i))n=1∞ is bounded; by using the diagonal method, we have that, for each m∈ℕ, we can find that a subsequence (xnj) of (xn) such that (xnj) converges for each i∈ℕ, 1≤i≤m. Therefore, there exists an increasing sequence of positive integer (tm) such that sep((xnjm)j>tm)≥ɛ. Hence, there is a sequence of positive integer (rm)m=1∞ with r1<r2<r3<⋯ such that ∥xrmm∥≥ɛ/2 for all m∈ℕ. Then, by Lemma 11, we may assume that there exists μ>0 such that
(40)ρ(xrmm)≥μ∀m∈ℕ.
Let α>0 and 1<α<lim⁡n→∞⁡inf⁡pn. For fixed integer k≥2, let ε1=((kα-1-1)/(k-1)kα)(μ/2). Then, by Lemma 10, there is a δ>0 such that
(41)|ρ(u+v)-ρ(u)|<ε1,
whenever ρ(u)≤1 and ρ(v)≤δ.
Since, by Lemma8(i), ρ(xn)≤1 for all n∈ℕ, there exist positive integers mi(i=1,2,…,k-1) with m1<m2<⋯<mk-1 such that ρ(ximi)≤δ and α≤pj for all j≥mk-1. Define mk=mk-1+1. From (40), we have ρ(xrmkmk)≥μ. Let si=i for 1≤i≤k-1 and sk=rmk. Then, in virtue of (40), (41), and convexity of function fi(u)=|u|pi(i∈ℕ), we have
(42)ρ(xs1+xs2+⋯+xskk)=∑n=1∞(∑i=1n-1unΔvi|xs1(i)+xs2(i)+⋯+xsk(i)k|hh+unvn|xs1(n)+xs2(n)+⋯+xsk(n)k|∑i=1∞unΔvi|xs1(i)+xs2(i)+⋯+xsk(i)k|)pn=∑n=1m1(∑i=1n-1unΔvi|xs1(i)+xs2(i)+⋯+xsk(i)k|hh+unvn|xs1(n)+xs2(n)+⋯+xsk(n)k|∑i=1n-1unΔvi|xs1(i)+xs2(i)+⋯+xsk(i)k|)pn+∑n=m1+1∞(∑i=1n-1unΔvi|xs1(i)+xs2(i)+⋯+xsk(i)k|ghhhhhhhhh+unvn|xs1(n)+xs2(n)+⋯+xsk(n)k|∑i=1n-1unΔvi|xs1(i)+xs2(i)+⋯+xsk(i)k|)pn≤∑n=1m11k∑j=1k(∑i=1n-1unΔvi|xsj(i)|+unvn|xsj(n)|)pn+∑n=m1+1∞(∑i=1n-1unΔvi|xs2(i)+xs3(i)+⋯+xsk(i)k|ghhghhh+unvn|xs2(n)+xs3(n)+⋯+xsk(n)k|∑i=1n-1unΔvi|xs2(i)+xs3(i)+⋯+xsk(i)k|)pn+ε1=∑n=1m11k∑j=1k(∑i=1n-1unΔvi|xsj(i)|+unvn|xsj(n)|)pn+∑n=m1+1m21k∑j=2k(∑i=1n-1unΔvi|xsj(i)|+unvn|xsj(n)|)pn+∑n=m2+1∞(∑i=1n-1unΔvi|xs3(i)+xs4(i)+⋯+xsk(i)k|ghghhhhhh+unvn|xs3(n)+xs4(n)+⋯+xsk(n)k|∑i=1n-1unΔvi|xs3(i)+xs4(i)+⋯+xsk(i)k|)pn+2ε1≤∑n=1m11k∑j=1k(∑i=1n-1unΔvi|xsj(i)|+unvn|xsj(n)|)pn+∑n=m1+1m21k∑j=2k(∑i=1n-1unΔvi|xsj(i)|+unvn|xsj(n)|)pn+∑n=m2+1m31k∑j=3k(∑i=1n-1unΔvi|xsj(i)|+unvn|xsj(n)|)pn+⋯+∑n=mk-2+1mk-11k∑j=k-1k(∑i=1n-1unΔvi|xsj(i)|+unvn|xsj(n)|)pn+∑n=mk-1+1∞(∑i=1n-1unΔvi|xsk(i)k|+unvn|xsk(n)n|)pn+(k-1)ε1≤ρ(xs1)+ρ(xs2)+⋯+ρ(xsk)k+1k∑n=1mk(∑i=1n-1unΔvi|xsj(i)|+unvn|xsj(n)|)pn+∑n=mk-1+1∞(∑i=1n-1unΔvi|xsk(i)k|+unvn|xsk(n)n|)pn+(k-1)ε1≤k-1k+1k∑n=1mk(∑i=1n-1unΔvi|xsj(i)|+unvn|xsj(n)|)pn+1kα∑n=mk+1∞(∑i=1n-1unΔvi|xsj(i)|+unvn|xsj(n)|)pn+(k-1)ε1≤1-1k+1k×[1-∑n=mk+1∞(∑i=1n-1unΔvi|xsj(i)|+unvn|xsj(n)|)pn]+1kα∑n=mk+1∞(∑i=1n-1unΔvi|xsj(i)|+unvn|xsj(n)|)pn+(k-1)ε1=1-kα-1kα∑n=mk+1∞(∑i=1n-1unΔvi|xsj(i)|+unvn|xsj(n)|)pn+(k-1)ε1≤1+(k-1)ε1-kα-1-1kαμ=1-kα-1-1kαμ2.
By Lemma 13, there exists γ>0 such that
(43)∥xs1+xs2+⋯+xskk∥<1-γ.
Therefore, ℓρΔ(u,v,p) is k-NUC.Theorem 16.
For any1<p<∞, the space ℓρΔ(u,v,p) has the uniform Opial property.Proof.
Take anyɛ>0 and x∈ℓρΔ(u,v,p) with ∥x∥≥ɛ. Let (xn) be weakly null sequence in S(ℓρΔ(u,v,p)). By lim⁡n→∞⁡sup⁡pr<∞, that is, ρ∈δ2s by Lemma 13, there exists δ∈(0,1) independent of x such that ρ(x)>δ. Also by ρ∈δ2s and Lemma 10 asserts that there exists δ1∈(δ,1) such that
(44)|ρ(y+z)-ρ(y)|<δ4,
whenever ρ(y)≤1 and ρ(z)≤δ1.
Choosek0∈ℕ such that
(45)∑k=k0+1∞(∑i=1k-1ukΔvi|xi|+ukvk|xk|)pk<δ14
so, we have
(46)δ<∑k=1k0(∑i=1k-1ukΔvi|xi|+ukvk|xk|)pk+∑k=k0+1∞(∑i=1k-1ukΔvi|xi|+ukvk|xk|)pk≤∑k=1k0(∑i=1k-1ukΔvi|xi|+ukvk|xk|)pk+δ14
which implies that
(47)∑k=1k0(∑i=1k-1ukΔvi|xi|+ukvk|xk|)pk>δ-δ14>δ-δ4=3δ4.
Since xn→w0, then there exists n0∈ℕ such that
(48)3δ4≤∑k=1k0(∑i=1k-1ukΔvi|xn(i)+x(i)|+ukvk|xn(k)+x(k)|)pk,
for all n>n0, since weak convergence implies coordinatewise convergence. We denote
(49)xn|k0=(xn(1),xn(2),…,xn(k0),0,0…)xn|ℕ-k0=(0,0…,0,xn(k0+1),xn(k0+2),…).
Againxn→w0, and then there exists n1∈ℕ such that
(50)∥xn|k0∥<1-(1-δ4)M,
for all n>n1.
Hence, by the triangle inequality of the norm, we get(51)∥xn|ℕ-k0∥>(1-δ4)M.
It follows by the definition of norm that we have
(52)1<ρ(xn|ℕ-k0(1-δ/4)M)=∑k=k0+1∞(∑i=1k-1ukΔvi|xn(i)(1-δ/4)M|+ukvk|xn(k)(1-δ/4)M|)pk≤1(1-δ/4)M∑k=k0+1∞(∑i=1k-1ukΔvi|xn(i)|+ukvk|xn(k)|)pk,
which implies that
(53)∑k=k0+1∞(∑i=1k-1ukΔvi|xn(i)|+ukvk|xn(k)|)pk>(1-δ4)M2,
for all n>n1.
By inequalities (44), (45), and (48), (53) yields for any n>n1 that
(54)ρ(xn+x)=∑k=1k0(∑i=1k-1ukΔvi|xn(i)+x(i)|+ukvk|xn(k)+x(k)|)pk+∑k=k0+1∞(∑i=1k-1ukΔvi|xn(i)+x(i)|+ukvk|xn(k)+x(k)|)pk≥3δ4+∑k=k0+1∞(∑i=1k-1ukΔvi|xn(i)|+ukvk|xn(k)|)pk≥3δ4+(1-δ4)M2-δ4.
Since ρ∈δ2s and by Lemma 4, there exists τ depending on δ only such that ∥xn+x∥≥1+τ, which implies that
(55)lim⁡n→∞inf⁡∥xn+x∥≥1+τ.
Therefore, ℓρΔ(u,v,p) has the uniform Opial property.



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*Source: 685382-2014-04-24.xml*