# Some geometric properties of the metric space $V[λ,p]$

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## Abstract

In this study, we consider the space $V[λ,p]$ with an invariant metric. Then, we examine some geometric properties of the linear metric space $V[λ,p]$ such as property (β), property (H) and k-NUC property.

MSC:40A05, 46A45, 46B20.

## 1 Introduction

Let X be a vector space over the scalar field of real numbers and d be an invariant metric on X. We denote $B d (X)$ and $S d (X)$ as follows: Let $(X,d)$ be a linear metric space and $B d (X)$ (resp., $S d (X)$) be a closed unit ball (resp., the unit sphere) of X. A linear metric space $(X,d)$ has property (β) if and only if for each $r>0$ and $ε>0$, there exists $δ>0$ such that for each element $x∈ B d (0,r)$ and each sequence $( x n )$ in $B d (0,r)$ with $sep( x n )≥ε$, there is an index k for which $d( x + x k 2 ,0)≤1−δ$, where $sep( x n )=inf{d( x n , x m ):n≠m}>ε$ . If for each $x∈ S d (0,r)$ and $( x n )⊂ S d (0,r)$, $x n → w x$ implies $x n →x$, a linear metric space $(X,d)$ is said to have property (H). Let $k≥2$ be an integer. A linear metric space $(X,d)$ is said to be k-nearly uniform convex (k-NUC) if for every $ε>0$ and $r>0$, there exists $δ>0$ such that for any sequence $( x n )⊂ B d (0,r)$ with $sep( x n )≥ε$, there are $s 1 , s 2 ,…, s k$ such that $d( x s 1 + x s 2 + ⋯ + x s k k ,0)≤r−δ$ . These properties have been studied by Mongkolkeha and Pumam , Sanhan and Suantai , Cui et al.  and Cui and Hudzik .

Ahuja et al.  introduced the notions of strict convexity and U.C.I (uniform convexity) in linear metric spaces which are generalizations of the corresponding concepts in linear normed spaces. Later, Sastry and Naidu  introduced the notions of U.C.II and U.C.III in linear metric spaces and showed that these three forms are not always equivalent. Further, Junde et al. [9, 10] showed that if a linear metric space is complete and U.C.I, then it is reflexive.

In summability theory, de la Vallée-Poussin mean was first used to define the $(V,λ)$-summability by Leindler . $(V,λ)$-summable sequences have been studied by many authors including Et et al. [12, 13], Savas , Savas and Malkowsky  and Şimsek et al. [20, 21]. Let $Λ=( λ k )$ be a nondecreasing sequence of positive real numbers tending to infinity and let $λ 1 =1$ and $λ k + 1 ≤ λ k +1$. The generalized de la Vallée-Poussin mean is defined by $t n (x)= 1 λ n ∑ k ∈ I n x k$, where $I n =[n− λ n +1,n]$ for $n=1,2,…$ . A sequence $x=( x k )$ is said to be $(V,λ)$-summable to a number if $t n (x)→ℓ$ as $n→∞$. If $λ n =n$, then $(V,λ)$-summability is reduced to Cesàro summability.

Let w be the space of all real sequences. Let $p=( p k )$ be a bounded sequence of positive real numbers. Şimşek et al.  defined the space $V[λ,p]$ as follows:

$V[λ,p]= { x = ( x k ) ∈ ω : ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | x j | ) p k < ∞ } .$

If $λ k =k$, then $V[λ,p]=ces(p)$ . If $λ k =k$ and $p k =p$ for all $k∈N$, then $V[λ,p]= ces p$ . Paranorm on $V[λ,p]$ is given by

$h(x)= ( ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | x j | ) p k ) 1 M ,$

where $M=max{1,H}$ and $H=sup p k$. If $p k =p$ for all $k∈N$, the notation $V p (λ)$ is used in place of $V[λ,p]$ and the norm on $V p (λ)$ is as follows:

$∥ x ∥ V p ( λ ) = ( ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | x j | ) p ) 1 p .$

$ρ: V ρ [λ,p]→[0,∞]$, $ρ(x)=( ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | x j | ) p k )$ is a modular on $V ρ [λ,p]$ and the Luxemburg norm on $V ρ [λ,p]$ is defined by $∥ x ∥ L =inf{σ>0:ρ( x σ )≤1}$ for all $x∈ V ρ [λ,p]$. The Amemiya norm on the space $V ρ [λ,p]$ can be similarly introduced as follows:

## 2 Main results

In this part of the paper, our main purpose is to define a metric on $V[λ,p]$ and show that $V[λ,p]$ possesses property (β), property (H) and k-NUC property. Let $p=( p k )$ be a bounded sequence of real numbers with $p k >1$ for all $k∈N$. The mapping $d(x,y)= ( ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | x ( j ) − y ( j ) | ) p k ) 1 / H$ is a metric on the space $V[λ,p]$, where $M=max(1,H=sup p k )$ and $m=inf p k$ since the function $| t | p$ is convex for $p>1$. First, we will show that the space $V[λ,p]$ has property (β) under the above metric. To do this, we need the following two lemmas. To prove these lemmas, we use the technique given in Sanhan and Mongkolkeha .

Lemma 2.1 Let $y,z∈(V[λ,p],d)$. If $β∈(0,1)$, then

$( d ( y + z , 0 ) ) M ≤ ( d ( y , 0 ) ) M + 2 M β ( d ( y , 0 ) ) M + 2 M β M − 1 ( d ( z , 0 ) ) M .$

Proof Let $y,z∈(V[λ,p],d)$ and $0<β<1$. Then

$( d ( y + z , 0 ) ) M = ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | y ( j ) + z ( j ) | ) p k ≤ ∑ k = 1 ∞ ( ( 1 − β ) 1 λ k ∑ j ∈ I k | y ( j ) | + β 1 λ k ∑ j ∈ I k | y ( j ) + z ( j ) β | ) p k ≤ ( 1 − β ) ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | y ( j ) | ) p k + β ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | y ( j ) + z ( j ) β | ) p k ≤ ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | y ( j ) | ) p k + 2 M β ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | y ( j ) | ) p k + 2 M ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | z ( j ) β | ) p k ≤ ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | y ( j ) | ) p k + 2 M β ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | y ( j ) | ) p k + 2 M β M − 1 ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | z ( j ) | ) p k = ( d ( y , 0 ) ) M + 2 M β ( d ( y , 0 ) ) M + 2 M β M − 1 ( d ( z , 0 ) ) M .$

□

Lemma 2.2 Let $y,z∈(V[λ,p],d)$. Then for any $ε>0$ and $L>0$, there exists $δ>0$ such that

$| ( d ( y + z , 0 ) ) M − ( d ( y , 0 ) ) M |<ε,$

where $( d ( y , 0 ) ) M ≤L$ and $( d ( z , 0 ) ) M ≤δ$.

Proof Let $ε>0$ and $L>0$. For $β= ε 2 M + 1 ( L + ε )$, we take $δ= ε β M − 1 2 M + 1$. From Lemma 2.1, we have

$( d ( y + z , 0 ) ) M ≤ ( d ( y , 0 ) ) M + 2 M β ( d ( y , 0 ) ) M + 2 M β M − 1 ( d ( z , 0 ) ) M ≤ ( d ( y , 0 ) ) M + 2 M β L + 2 M β M − 1 δ ≤ ( d ( y , 0 ) ) M + 2 M ε 2 M + 1 L L + ε + 2 M β M − 1 ε β M − 1 2 M + 1 ≤ ( d ( y , 0 ) ) M + ε 2 + ε 2 ≤ ( d ( y , 0 ) ) M + ε$
(2.1)

and

$( d ( y , 0 ) ) M ≤ ( d ( y + z , 0 ) ) M + 2 M β ( d ( y + z , 0 ) ) M + 2 M β M − 1 ( d ( − z , 0 ) ) M ≤ ( d ( y + z , 0 ) ) M + 2 M β ( ( d ( y , 0 ) ) M + ε ) + 2 M β M − 1 δ ≤ ( d ( y + z , 0 ) ) M + 2 M β ( L + ε ) + 2 M β M − 1 ε β M − 1 2 M + 1 = ( d ( y + z , 0 ) ) M + 2 M ε 2 M + 1 ( L + ε ) ( L + ε ) + ε 2 = ( d ( y + z , 0 ) ) M + ε 2 + ε 2 = ( d ( y + z , 0 ) ) M + ε .$
(2.2)

From (2.1) and (2.2), we obtain that $| ( d ( y + z , 0 ) ) M − ( d ( y , 0 ) ) M |<ε$. □

Theorem 2.3 The space $(V[λ,p],d)$ has property (β).

Proof Let $ε>0$ and $( x n )⊂B(V[λ,p],d)$ such that $sep( x n )≥ε$ and $x∈B(V[λ,p],d)$. We take $y N =(0,0,…,0, ∑ k = 1 N y(k),y(N+1),y(N+2),…)$. By using the diagonal method, we can find a subsequence $( x n r )$ of $( x n )$ for each $N∈N$ such that $( x n r (k))$ converges for each $k∈N$ with $1≤k≤N$, since $( x n ( k ) ) k = 1 ∞$ is bounded for each $k∈N$. Therefore, there is $t N ∈N$ for each $N∈N$ such that $sep( ( x n N ) r > t N )≥ε$. So, there is a sequence of positive integers $( t N ) N = 1 ∞$ with $t 1 < t 2 < t 3 ⋯$ such that $d( x t N N ,0)≥ ε 2$ for all $N∈N$. Then there exists $κ>0$ such that for all $N∈N$,

$∑ k = N ∞ ( 1 λ k ∑ j ∈ I k | x t N | ) p k ≥κ.$
(2.3)

By Lemma 2.2, there exists $δ 0$ such that

$| ( d ( y + z , 0 ) ) M − ( d ( y , 0 ) ) M |< κ 2 m ,$
(2.4)

where $( d ( y , 0 ) ) M < j M$ and $( d ( z , 0 ) ) M ≤ δ 0$. There exists $N 1 ∈N$ such that $( d ( x N 1 , 0 ) ) M ≤ δ 0$ if $x∈B(V[λ,p])$ and $( d ( x , 0 ) ) M ≤ δ 0$. Let us take $y= x t N 1 N 1$ and $z= x N 1$. Hence, we have

$∑ k = N 1 ∞ ( 1 λ k ∑ j ∈ I k | x ( j ) + x t N 1 ( j ) 2 | ) p k ≤ ∑ k = N 1 ∞ ( 1 λ k ∑ j ∈ I k | x t N 1 ( j ) 2 | ) p k + κ 2 m .$
(2.5)

From (2.3), (2.4), (2.5) and by using the convexity of the function $f(t)= | t | p k$ for all $k∈N$, we obtain that

$( d ( y + z 2 , 0 ) ) M = ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | x ( j ) + x t N 1 ( j ) 2 | ) p k = ∑ k = 1 N 1 − 1 ( 1 λ k ∑ j ∈ I k | x ( j ) + x t N 1 ( j ) 2 | ) p k + ∑ k = N 1 ∞ ( 1 λ k ∑ j ∈ I k | x ( j ) + x t N 1 ( j ) 2 | ) p k ≤ ∑ k = 1 N 1 − 1 ( 1 λ k ∑ j ∈ I k | x ( j ) + x t N 1 ( j ) 2 | ) p k + ∑ k = N 1 ∞ ( 1 λ k ∑ j ∈ I k | x t N 1 ( k ) 2 | ) p k + κ 2 m ≤ 1 2 ∑ k = 1 N 1 − 1 ( 1 λ k ∑ j ∈ I k | x ( j ) | ) p k + 1 2 ∑ k = 1 N 1 − 1 ( 1 λ k ∑ j ∈ I k | x t N 1 ( j ) | ) p k + 1 2 m ∑ k = N 1 ∞ ( 1 λ k ∑ j ∈ I k | x t N 1 ( j ) | ) p k + κ 2 m ≤ 1 2 ∑ k = 1 N 1 − 1 ( 1 λ k ∑ j ∈ I k | x ( j ) | ) p k + 1 2 ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | x t N 1 ( j ) | ) p k − 2 m − 2 2 m + 1 ∑ k = N 1 ∞ ( 1 λ k ∑ j ∈ I k | x t N 1 ( j ) | ) p k + κ 2 m < j M 2 + j M 2 − 2 m − 2 2 m + 1 κ + κ 2 m = j M − κ 2 .$

Therefore, we have $d( y + z 2 ,0)< ( j M − κ 2 ) 1 / M whenever $δ∈(0,j− ( j M − κ 2 ) 1 / M )$. Consequently, the space $(V[λ,p],d)$ possesses property (β). □

Now, we will show that the space $(V[λ,p],d)$ has k-NUC property.

Theorem 2.4 The space $V[λ,p]$ is k-NUC for any integer $k≥2$.

Proof Let $ε>0$ and $( x n )⊂ B d (V[λ,p])$ with $sep( x n )≥ε$. For each $m∈N$, let

$x n m = ( 0 , 0 , … , x n ( m ) , x n ( m + 1 ) , … ) .$
(2.6)

Since the sequence $( x n ( i ) ) i = 1 ∞$ is bounded for each $i∈N$, by using the diagonal method, we can find a subsequence $( x n l )$ of $( x n )$ such that $( x n l (k))$ converges for each $k∈N$. Therefore, there is an increasing sequence $t m$ with $sep( ( x n l m ) l > t m )≥ε$. Hence, there exists a sequence of positive integers $( r m ) m = 1 ∞$ with $r 1 < r 2 < r 3 <⋯$ such that $d( x r m m ,0)≥ ε 2$ for all $m∈N$. Then there is $ζ>0$ such that

$∑ k = m ∞ ( 1 λ k ∑ j ∈ I k | x r m | ) p k ≥ζ.$
(2.7)

Let $α>0$ such that $1<α< lim k → ∞ inf p k$. Let $ε 1 = n α − 1 − 1 ( n − 1 ) n α ζ 2$ for $k≥2$. From Lemma 2.2, there is a $δ>0$ such that

$| ( d ( y + z , 0 ) ) M − ( d ( y , 0 ) ) M |< ε 1 ,$
(2.8)

where $( d ( y , 0 ) ) M < r M$ and $( d ( z , 0 ) ) M ≤δ$. Then there exist positive integers $m i$ ($i=1,2,…,n−1$) with $m 1 < m 2 <⋯< m n − 1$ such that $d( x i m i ,0)≤δ$. Now, define $m n = m n − 1 +1$. Then we have $d( x r m n m n ,0)≥ζ$ for all $m∈N$. For $1≤i≤n−1$, let $s i =i$ and $s n = r m n$. By using (2.6), (2.7), (2.8) and the convexity of the function $f i (u)= | u | p i$ ($i∈N$), we obtain Thus, we have $d( x s 1 ( j ) + x s 2 ( j ) + ⋯ + x s n ( j ) n ,0)< ( r M − ( n α − 1 − 1 n α ) ζ 2 ) 1 / M for $δ∈(0,r− ( r M − ( n α − 1 − 1 n α ) ζ 2 ) 1 / M )$. Hence, $(V[λ,p],d)$ is k-NUC. □

Since k-NUC implies NUC and NUC implies property (H), by using the previous theorem, we can give the following result.

Corollary 2.5 The space $(V[λ,p],d)$ has property (H).

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## Author information

Correspondence to Murat Karakaş.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

MC, MK and ME have contributed to all parts of the article. All authors read and approved the final manuscript.

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Çinar, M., Karakaş, M. & Et, M. Some geometric properties of the metric space $V[λ,p]$. J Inequal Appl 2013, 28 (2013) doi:10.1186/1029-242X-2013-28

• #### DOI

https://doi.org/10.1186/1029-242X-2013-28

### Keywords

• linear metric space
• Luxemburg norm
• de la Vallée-Poussin mean
• k-NUC property
• property (H) 