Some geometric properties of the metric space $V[\lambda ,p]$

Abstract

In this study, we consider the space $V[\lambda ,p]$ with an invariant metric. Then, we examine some geometric properties of the linear metric space $V[\lambda ,p]$ 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 $\epsilon >0$, there exists $\delta >0$ such that for each element $x\in B_d(0,r)$ and each sequence $(x_n)$ in $B_d(0,r)$ with $sep(x_n)\ge \epsilon$, there is an index k for which $d(\frac{x+x_k}{2},\mathbf{0})\le 1-\delta$, where $sep(x_n)=inf\{d(x_n,x_m):n\ne m\}>\epsilon$ [1]. If for each $x\in S_d(0,r)$ and $(x_n)\subset S_d(0,r)$, $x_n\stackrel{w}{\to }x$ implies $x_n\to x$, a linear metric space $(X,d)$ is said to have property (H). Let $k\ge 2$ be an integer. A linear metric space $(X,d)$ is said to be k-nearly uniform convex (k-NUC) if for every $\epsilon >0$ and $r>0$, there exists $\delta >0$ such that for any sequence $(x_n)\subset B_d(0,r)$ with $sep(x_n)\ge \epsilon$, there are $s_1,s_2,\dots ,s_k$ such that $d(\frac{x_{s_1}+x_{s_2}+\cdots +x_{s_k}}{k},\mathbf{0})\le r-\delta$ [2]. These properties have been studied by Mongkolkeha and Pumam [3], Sanhan and Suantai [4], Cui et al. [5] and Cui and Hudzik [6].

Ahuja et al. [7] 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 [8] 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 $(V,\lambda )$-summability by Leindler [11]. $(V,\lambda )$-summable sequences have been studied by many authors including Et et al. [12, 13], Savas [14–18], Savas and Malkowsky [19] and Şimsek et al. [20, 21]. Let $\mathrm{\Lambda }=({\lambda }_k)$ be a nondecreasing sequence of positive real numbers tending to infinity and let ${\lambda }_1=1$ and ${\lambda }_{k+1}\le {\lambda }_k+1$. The generalized de la Vallée-Poussin mean is defined by $t_n(x)=\frac{1}{{\lambda }_n}{\sum }_{k\in I_n}{x}_k$, where $I_n=[n-{\lambda }_n+1,n]$ for $n=1,2,\dots$ . A sequence $x=(x_k)$ is said to be $(V,\lambda )$-summable to a number ℓ if $t_n(x)\to \ell$ as $n\to \mathrm{\infty }$. If ${\lambda }_n=n$, then $(V,\lambda )$-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. [20] defined the space $V[\lambda ,p]$ as follows:

$$V[\lambda ,p]=\{x=(x_k)\in \omega :\sum _{k=1}^{\mathrm{\infty }}{(\frac{1}{{\lambda }_k}\sum _{j\in I_k}|x_j|)}^{p_k}<\mathrm{\infty }\}.$$

If ${\lambda }_k=k$, then $V[\lambda ,p]=ces(p)$ [22]. If ${\lambda }_k=k$ and $p_k=p$ for all $k\in \mathrm{N}$, then $V[\lambda ,p]={ces}_p$ [23]. Paranorm on $V[\lambda ,p]$ is given by

$$h(x)={\left(\sum _{k=1}^{\mathrm{\infty }}{(\frac{1}{{\lambda }_k}\sum _{j\in I_k}|x_j|)}^{p_k}\right)}^{\frac{1}{M}},$$

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

$${\parallel x\parallel }_{V_p(\lambda )}={\left(\sum _{k=1}^{\mathrm{\infty }}{(\frac{1}{{\lambda }_k}\sum _{j\in I_k}|x_j|)}^p\right)}^{\frac{1}{p}}.$$

$\rho :V_{\rho }[\lambda ,p]\to [0,\mathrm{\infty }]$, $\rho (x)=({\sum }_{k=1}^{\mathrm{\infty }}{(\frac{1}{{\lambda }_k}{\sum }_{j\in I_k}|x_j|)}^{p_k})$ is a modular on $V_{\rho }[\lambda ,p]$ and the Luxemburg norm on $V_{\rho }[\lambda ,p]$ is defined by ${\parallel x\parallel }_L=inf\{\sigma >0:\rho (\frac{x}{\sigma })\le 1\}$ for all $x\in V_{\rho }[\lambda ,p]$. The Amemiya norm on the space $V_{\rho }[\lambda ,p]$ can be similarly introduced as follows:

$${\parallel x\parallel }_A=\underset{\sigma >0}{inf}\frac{1}{\sigma }(1+\rho (\sigma x))\text{for all }x\in V_{\rho }[\lambda ,p].$$

2 Main results

In this part of the paper, our main purpose is to define a metric on $V[\lambda ,p]$ and show that $V[\lambda ,p]$ 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 $k\in \mathrm{N}$. The mapping $d(x,y)={({\sum }_{k=1}^{\mathrm{\infty }}{(\frac{1}{{\lambda }_k}{\sum }_{j\in I_k}|x(j)-y(j)|)}^{p_k})}^{1/H}$ is a metric on the space $V[\lambda ,p]$, 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 $V[\lambda ,p]$ 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 [1].

Lemma 2.1 Let $y,z\in (V[\lambda ,p],d)$. If $\beta \in (0,1)$, then

$${(d(y+z,\mathbf{0}))}^M\le {(d(y,\mathbf{0}))}^M+2^M\beta {(d(y,\mathbf{0}))}^M+\frac{2^M}{{\beta }^{M-1}}{(d(z,\mathbf{0}))}^M.$$

Proof Let $y,z\in (V[\lambda ,p],d)$ and $0<\beta <1$. Then

$$\begin{array}{rcl}{(d(y+z,\mathbf{0}))}^M& =& \sum _{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_k}\sum _{j\in I_k}\right|y(j)+z(j)\left|\right)}^{p_k}\\ \le & \sum _{k=1}^{\mathrm{\infty }}{((1-\beta )\frac{1}{{\lambda }_k}\sum _{j\in I_k}|y(j)|+\beta \frac{1}{{\lambda }_k}\sum _{j\in I_k}|y(j)+\frac{z(j)}{\beta }\left|\right)}^{p_k}\\ \le & (1-\beta )\sum _{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_k}\sum _{j\in I_k}\right|y(j)\left|\right)}^{p_k}+\beta \sum _{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_k}\sum _{j\in I_k}\right|y(j)+\frac{z(j)}{\beta }\left|\right)}^{p_k}\\ \le & \sum _{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_k}\sum _{j\in I_k}\right|y(j)\left|\right)}^{p_k}+2^M\beta \sum _{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_k}\sum _{j\in I_k}\right|y(j)\left|\right)}^{p_k}\\ +2^M\sum _{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_k}\sum _{j\in I_k}\right|\frac{z(j)}{\beta }\left|\right)}^{p_k}\\ \le & \sum _{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_k}\sum _{j\in I_k}\right|y(j)\left|\right)}^{p_k}+2^M\beta \sum _{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_k}\sum _{j\in I_k}\right|y(j)\left|\right)}^{p_k}\\ +\frac{2^M}{{\beta }^{M-1}}\sum _{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_k}\sum _{j\in I_k}\right|z(j)\left|\right)}^{p_k}\\ =& {(d(y,\mathbf{0}))}^M+2^M\beta {(d(y,\mathbf{0}))}^M+\frac{2^M}{{\beta }^{M-1}}{(d(z,\mathbf{0}))}^M.\end{array}$$

 □

Lemma 2.2 Let $y,z\in (V[\lambda ,p],d)$. Then for any $\epsilon >0$ and $L>0$, there exists $\delta >0$ such that

$$|{(d(y+z,\mathbf{0}))}^M-{(d(y,\mathbf{0}))}^M|<\epsilon ,$$

where ${(d(y,\mathbf{0}))}^M\le L$ and ${(d(z,\mathbf{0}))}^M\le \delta$.

Proof Let $\epsilon >0$ and $L>0$. For $\beta =\frac{\epsilon }{2^{M+1}(L+\epsilon )}$, we take $\delta =\frac{\epsilon {\beta }^{M-1}}{2^{M+1}}$. From Lemma 2.1, we have

$$\begin{array}{rcl}{(d(y+z,\mathbf{0}))}^M& \le & {(d(y,\mathbf{0}))}^M+2^M\beta {(d(y,\mathbf{0}))}^M+\frac{2^M}{{\beta }^{M-1}}{(d(z,\mathbf{0}))}^M\\ \le & {(d(y,\mathbf{0}))}^M+2^M\beta L+\frac{2^M}{{\beta }^{M-1}}\delta \\ \le & {(d(y,\mathbf{0}))}^M+2^M\frac{\epsilon }{2^{M+1}}\frac{L}{L+\epsilon }+\frac{2^M}{{\beta }^{M-1}}\frac{\epsilon {\beta }^{M-1}}{2^{M+1}}\\ \le & {(d(y,\mathbf{0}))}^M+\frac{\epsilon }{2}+\frac{\epsilon }{2}\\ \le & {(d(y,\mathbf{0}))}^M+\epsilon \end{array}$$

(2.1)

and

$$\begin{array}{rcl}{(d(y,\mathbf{0}))}^M& \le & {(d(y+z,\mathbf{0}))}^M+2^M\beta {(d(y+z,\mathbf{0}))}^M+\frac{2^M}{{\beta }^{M-1}}{(d(-z,\mathbf{0}))}^M\\ \le & {(d(y+z,\mathbf{0}))}^M+2^M\beta ({(d(y,\mathbf{0}))}^M+\epsilon )+\frac{2^M}{{\beta }^{M-1}}\delta \\ \le & {(d(y+z,\mathbf{0}))}^M+2^M\beta (L+\epsilon )+\frac{2^M}{{\beta }^{M-1}}\frac{\epsilon {\beta }^{M-1}}{2^{M+1}}\\ =& {(d(y+z,\mathbf{0}))}^M+2^M\frac{\epsilon }{2^{M+1}(L+\epsilon )}(L+\epsilon )+\frac{\epsilon }{2}\\ =& {(d(y+z,\mathbf{0}))}^M+\frac{\epsilon }{2}+\frac{\epsilon }{2}\\ =& {(d(y+z,\mathbf{0}))}^M+\epsilon .\end{array}$$

(2.2)

From (2.1) and (2.2), we obtain that $|{(d(y+z,\mathbf{0}))}^M-{(d(y,\mathbf{0}))}^M|<\epsilon$. □

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

Proof Let $\epsilon >0$ and $(x_n)\subset B(V[\lambda ,p],d)$ such that $sep(x_n)\ge \epsilon$ and $x\in B(V[\lambda ,p],d)$. We take $y^N=(0,0,\dots ,0,{\sum }_{k=1}^{N}y(k),y(N+1),y(N+2),\dots )$. By using the diagonal method, we can find a subsequence $(x_{n_r})$ of $(x_n)$ for each $N\in \mathrm{N}$ such that $(x_{n_r}(k))$ converges for each $k\in \mathrm{N}$ with $1\le k\le N$, since ${(x_n(k))}_{k=1}^{\mathrm{\infty }}$ is bounded for each $k\in \mathrm{N}$. Therefore, there is $t_N\in \mathrm{N}$ for each $N\in \mathrm{N}$ such that $sep({(x_n^N)}_{r>t_N})\ge \epsilon$. So, there is a sequence of positive integers ${(t_N)}_{N=1}^{\mathrm{\infty }}$ with $t_1<t_2<t_3\cdots$ such that $d(x_{t_N}^N,\mathbf{0})\ge \frac{\epsilon }{2}$ for all $N\in \mathrm{N}$. Then there exists $\kappa >0$ such that for all $N\in \mathrm{N}$,

$$\sum _{k=N}^{\mathrm{\infty }}{(\frac{1}{{\lambda }_k}\sum _{j\in I_k}|x_{t_N}|)}^{p_k}\ge \kappa .$$

(2.3)

By Lemma 2.2, there exists ${\delta }_0$ such that

$$|{(d(y+z,\mathbf{0}))}^M-{(d(y,\mathbf{0}))}^M|<\frac{\kappa }{2^m},$$

(2.4)

where ${(d(y,\mathbf{0}))}^M<j^M$ and ${(d(z,\mathbf{0}))}^M\le {\delta }_0$. There exists $N_1\in \mathrm{N}$ such that ${(d(x^{N_1},\mathbf{0}))}^M\le {\delta }_0$ if $x\in B(V[\lambda ,p])$ and ${(d(x,\mathbf{0}))}^M\le {\delta }_0$. Let us take $y=x_{t_{N_1}}^{N_1}$ and $z=x^{N_1}$. Hence, we have

$$\sum _{k=N_1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_k}\sum _{j\in I_k}\right|\frac{x(j)+x_{t_{N_1}}(j)}{2}\left|\right)}^{p_k}\le \sum _{k=N_1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_k}\sum _{j\in I_k}\right|\frac{x_{t_{N_1}}(j)}{2}\left|\right)}^{p_k}+\frac{\kappa }{2^m}.$$

(2.5)

From (2.3), (2.4), (2.5) and by using the convexity of the function $f(t)={|t|}^{p_k}$ for all $k\in \mathrm{N}$, we obtain that

$$\begin{array}{rcl}{\left(d(\frac{y+z}{2},\mathbf{0})\right)}^M& =& \sum _{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_k}\sum _{j\in I_k}\right|\frac{x(j)+x_{t_{N_1}}(j)}{2}\left|\right)}^{p_k}\\ =& \sum _{k=1}^{N_1-1}{\left(\frac{1}{{\lambda }_k}\sum _{j\in I_k}\right|\frac{x(j)+x_{t_{N_1}}(j)}{2}\left|\right)}^{p_k}+\sum _{k=N_1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_k}\sum _{j\in I_k}\right|\frac{x(j)+x_{t_{N_1}}(j)}{2}\left|\right)}^{p_k}\\ \le & \sum _{k=1}^{N_1-1}{\left(\frac{1}{{\lambda }_k}\sum _{j\in I_k}\right|\frac{x(j)+x_{t_{N_1}}(j)}{2}\left|\right)}^{p_k}+\sum _{k=N_1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_k}\sum _{j\in I_k}\right|\frac{x_{t_{N_1}}(k)}{2}\left|\right)}^{p_k}+\frac{\kappa }{2^m}\\ \le & \frac{1}{2}\sum _{k=1}^{N_1-1}{\left(\frac{1}{{\lambda }_k}\sum _{j\in I_k}\right|x(j)\left|\right)}^{p_k}+\frac{1}{2}\sum _{k=1}^{N_1-1}{\left(\frac{1}{{\lambda }_k}\sum _{j\in I_k}\right|x_{t_{N_1}}(j)\left|\right)}^{p_k}\\ +\frac{1}{2^m}\sum _{k=N_1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_k}\sum _{j\in I_k}\right|x_{t_{N_1}}(j)\left|\right)}^{p_k}+\frac{\kappa }{2^m}\\ \le & \frac{1}{2}\sum _{k=1}^{N_1-1}{\left(\frac{1}{{\lambda }_k}\sum _{j\in I_k}\right|x(j)\left|\right)}^{p_k}+\frac{1}{2}\sum _{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_k}\sum _{j\in I_k}\right|x_{t_{N_1}}(j)\left|\right)}^{p_k}\\ -\frac{2^m-2}{2^{m+1}}\sum _{k=N_1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_k}\sum _{j\in I_k}\right|x_{t_{N_1}}(j)\left|\right)}^{p_k}+\frac{\kappa }{2^m}\\ <& \frac{j^M}{2}+\frac{j^M}{2}-\frac{2^m-2}{2^{m+1}}\kappa +\frac{\kappa }{2^m}\\ =& j^M-\frac{\kappa }{2}.\end{array}$$

Therefore, we have $d(\frac{y+z}{2},\mathbf{0})<{(j^M-\frac{\kappa }{2})}^{1/M}<j-\delta$ whenever $\delta \in (0,j-{(j^M-\frac{\kappa }{2})}^{1/M})$. Consequently, the space $(V[\lambda ,p],d)$ possesses property (β). □

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

Theorem 2.4 The space $V[\lambda ,p]$ is k-NUC for any integer $k\ge 2$.

Proof Let $\epsilon >0$ and $(x_n)\subset B_d(V[\lambda ,p])$ with $sep(x_n)\ge \epsilon$. For each $m\in \mathrm{N}$, let

$$x_n^m=(0,0,\dots ,x_n(m),x_n(m+1),\dots ).$$

(2.6)

Since the sequence ${(x_n(i))}_{i=1}^{\mathrm{\infty }}$ is bounded for each $i\in \mathrm{N}$, 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 $k\in \mathrm{N}$. Therefore, there is an increasing sequence $t_m$ with $sep({(x_{n_l}^m)}_{l>t_m})\ge \epsilon$. Hence, there exists a sequence of positive integers ${(r_m)}_{m=1}^{\mathrm{\infty }}$ with $r_1<r_2<r_3<\cdots$ such that $d(x_{r_m}^m,\mathbf{0})\ge \frac{\epsilon }{2}$ for all $m\in \mathrm{N}$. Then there is $\zeta >0$ such that

$$\sum _{k=m}^{\mathrm{\infty }}{(\frac{1}{{\lambda }_k}\sum _{j\in I_k}|x_{r_m}|)}^{p_k}\ge \zeta .$$

(2.7)

Let $\alpha >0$ such that $1<\alpha <{lim}_{k\to \mathrm{\infty }}inf{p}_k$. Let ${\epsilon }_1=\frac{n^{\alpha -1}-1}{(n-1)n^{\alpha }}\frac{\zeta }{2}$ for $k\ge 2$. From Lemma 2.2, there is a $\delta >0$ such that

$$|{(d(y+z,\mathbf{0}))}^M-{(d(y,\mathbf{0}))}^M|<{\epsilon }_1,$$

(2.8)

where ${(d(y,\mathbf{0}))}^M<r^M$ and ${(d(z,\mathbf{0}))}^M\le \delta$. Then there exist positive integers $m_i$ ($i=1,2,\dots ,n-1$) with $m_1<m_2<\cdots <m_{n-1}$ such that $d(x_i^{m_i},\mathbf{0})\le \delta$. Now, define $m_n=m_{n-1}+1$. Then we have $d(x_{r_{m_n}}^{m_n},\mathbf{0})\ge \zeta$ for all $m\in \mathrm{N}$. For $1\le i\le n-1$, let $s_i=i$ and $s_n=r_{m_n}$. By using (2.6), (2.7), (2.8) and the convexity of the function $f_i(u)={|u|}^{p_i}$ ($i\in \mathrm{N}$), we obtain

Thus, we have $d(\frac{x_{s_1}(j)+x_{s_2}(j)+\cdots +x_{s_n}(j)}{n},\mathbf{0})<{(r^M-(\frac{n^{\alpha -1}-1}{n^{\alpha }})\frac{\zeta }{2})}^{1/M}<r-\delta$ for $\delta \in (0,r-{(r^M-(\frac{n^{\alpha -1}-1}{n^{\alpha }})\frac{\zeta }{2})}^{1/M})$. Hence, $(V[\lambda ,p],d)$ 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 $(V[\lambda ,p],d)$ has property (H).