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# Mappings of type generalized de La Vallée Poussin’s mean

Journal of Inequalities and Applications20132013:518

https://doi.org/10.1186/1029-242X-2013-518

• Received: 21 April 2013
• Accepted: 9 September 2013
• Published:

## Abstract

In the present paper, we study the operator ideals generated by the approximationnumbers and generalized de La Vallée Poussin’s mean defined in(Şimşek et al. in J. Comput. Anal. Appl. 12(4):768-779, 2010). Ourresults coincide with those in (Faried and Bakery in J. Inequal. Appl. 2013,doi:10.1186/1029-242X-2013-186) for the generalized Cesáro sequence space.

## Keywords

• approximation numbers
• operator ideal
• generalized de La Vallée Poussin’s mean sequence space

## 1 Introduction

By $L\left(X,Y\right)$ we denote the space of all bounded linear operators from anormed space X into a normed space Y. The set of nonnegative integersis denoted by $\mathbb{N}=\left\{0,1,2,\dots \right\}$ and the real numbers by . By ω wedenote the space of all real sequences. A map which assigns to every operator$T\in L\left(X,Y\right)$ a unique sequence ${\left({s}_{n}\left(T\right)\right)}_{n=0}^{\mathrm{\infty }}$ is called an s-function and the number${s}_{n}\left(T\right)$ is called the n th s-numbers ofT if the following conditions are satisfied:
1. (a)

$\parallel T\parallel ={s}_{0}\left(T\right)\ge {s}_{1}\left(T\right)\ge \cdots \ge 0$ for all $T\in L\left(X,Y\right)$.

2. (b)

${s}_{n}\left({T}_{1}+{T}_{2}\right)\le {s}_{n}\left({T}_{1}\right)+\parallel {T}_{2}\parallel$ for all ${T}_{1},{T}_{2}\in L\left(X,Y\right)$.

3. (c)

${s}_{n}\left(RST\right)\le \parallel R\parallel {s}_{n}\left(S\right)\parallel T\parallel$ for all $T\in L\left({X}_{0},X\right)$, $S\in L\left(X,Y\right)$ and $R\in L\left(Y,{Y}_{0}\right)$.

4. (d)

${s}_{n}\left(\lambda T\right)=|\lambda |{s}_{n}\left(T\right)$ for all $T\in L\left(X,Y\right)$, $\lambda \in \mathbb{R}$.

5. (e)

$rank\left(T\right)\le n$ if ${s}_{n}\left(T\right)=0$ for all $T\in L\left(X,Y\right)$.

6. (f)

where ${I}_{n}$ is the identity operator on the Euclidean space${\ell }_{2}^{n}$.

As examples of s-numbers, we mention approximation numbers${\alpha }_{n}\left(T\right)$, Gelfand numbers ${c}_{n}\left(T\right)$, Kolmogorov numbers ${d}_{n}\left(T\right)$ and Tichomirov numbers ${d}_{n}^{\ast }\left(T\right)$ defined by:
1. (I)

.

2. (II)

${c}_{n}\left(T\right)={a}_{n}\left({J}_{Y}T\right)$, where ${J}_{Y}$ is a metric injection (a metric injection is a one-to-one operator with closed range and with norm equal to one) from the space Y into a higher space ${\ell }^{\mathrm{\infty }}\left(\mathrm{\Lambda }\right)$ for a suitable index set Λ.

3. (III)

${d}_{n}\left(T\right)={inf}_{dimY\le n}{sup}_{\parallel x\parallel \le 1}{inf}_{y\in Y}\parallel Tx-y\parallel$.

4. (IV)

${d}_{n}^{\ast }\left(T\right)={d}_{n}\left({J}_{Y}T\right)$.

All these numbers satisfy the following condition:
1. (g)

${s}_{n+m}\left({T}_{1}+{T}_{2}\right)\le {s}_{n}\left({T}_{1}\right)+{s}_{m}\left({T}_{2}\right)$ for all ${T}_{1},{T}_{2}\in L\left(X,Y\right)$.

An operator ideal U is a subclass of such that its components satisfy the following conditions:
1. (i)

${I}_{K}\in U$, where K denotes the 1-dimensional Banach space, where $U\subset L$.

2. (ii)

If ${T}_{1},{T}_{2}\in U\left(X,Y\right)$, then ${\lambda }_{1}{T}_{1}+{\lambda }_{2}{T}_{2}\in U\left(X,Y\right)$ for any scalars ${\lambda }_{1}$, ${\lambda }_{2}$.

3. (iii)

If $V\in L\left({X}_{0},X\right)$, $T\in U\left(X,Y\right)$ and $R\in L\left(Y,{Y}_{0}\right)$, then $RTV\in U\left({X}_{0},{Y}_{0}\right)$. See [1, 2] and .

For a sequence $p=\left({p}_{n}\right)$ of positive real numbers with ${p}_{n}\ge 1$, for all $n\in \mathbb{N}$, the generalized Cesáro sequence space is defined by

The space $Ces\left(p\right)$ is a Banach space with the norm $\parallel x\parallel =inf\left\{\lambda >0:\rho \left(\frac{x}{\lambda }\right)\le 1\right\}$.

If $p=\left({p}_{n}\right)$ is bounded, we can simply write $Ces\left(p\right)=\left\{x\in \omega :{\sum }_{n=0}^{\mathrm{\infty }}{\left(\frac{1}{n+1}{\sum }_{k=0}^{n}|{x}_{k}|\right)}^{{p}_{n}}<\mathrm{\infty }\right\}$. Also, some geometric properties of$Ces\left(p\right)$ are studied in  and .

Let $\mathrm{\Lambda }=\left({\lambda }_{n}\right)$ be a nondecreasing sequence of positive real numberstending to infinity, and let ${\lambda }_{0}=1$ and ${\lambda }_{n+1}\le {\lambda }_{n}+1$.

De La Vallée Poussin’s means of a sequence $x=\left({x}_{k}\right)$ are defined as follows:
The generalized de La Vallée Poussin’s mean sequence space was defined in .
The space $V\left(\lambda ,p\right)$ is a Banach space with the norm
$\parallel x\parallel =inf\left\{\lambda >0:\rho \left(\frac{x}{\lambda }\right)\le 1\right\}.$
If $p=\left({p}_{n}\right)$ is bounded, we can simply write
$V\left(\lambda ,p\right)=\left\{x\in \omega :\sum _{n=0}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}|{x}_{k}|\right)}^{{p}_{n}}<\mathrm{\infty }\right\}.$

Also, some geometric properties of $V\left(\lambda ,p\right)$ are studied in [9, 10] and .

Throughout this paper, the sequence $\left({p}_{n}\right)$ is a bounded sequence of positive real numbers with

(b1) the sequence $\left({p}_{n}\right)$ of positive real numbers is increasing and bounded with$lim sup{p}_{n}<\mathrm{\infty }$ and $liminf{p}_{n}>1$,

(b2) the sequence $\left({\lambda }_{n}\right)$ is a nondecreasing sequence of positive real numberstending to infinity, ${\lambda }_{0}=1$ and ${\lambda }_{n+1}\le {\lambda }_{n}+1$ with ${\sum }_{n=0}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\right)}^{{p}_{n}}<\mathrm{\infty }$.

Also we define ${e}_{i}=\left(0,0,\dots ,1,0,0,\dots \right)$, where 1 appears at the i th place for all$i\in \mathbb{N}$.

Different classes of paranormed sequence spaces have been introduced and their differentproperties have been investigated. See  and .

For any bounded sequence of positive numbers $\left({p}_{n}\right)$, we have the following well-known inequality${|{a}_{n}+{b}_{n}|}^{{p}_{n}}\le {2}^{h-1}\left({|{a}_{n}|}^{{p}_{n}}+{|{b}_{n}|}^{{p}_{n}}\right)$, $h={sup}_{n}{p}_{n}$, and ${p}_{n}\ge 1$ for all $n\in \mathbb{N}$. See .

## 2 Preliminary and notation

Definition 2.1 A class of linear sequence spaces E is called a specialspace of sequences (sss) having the following conditions:
1. (1)

E is a linear space and ${e}_{n}\in E$ for each $n\in \mathbb{N}$.

2. (2)

If $x\in \omega$, $y\in E$ and $|{x}_{n}|\le |{y}_{n}|$ for all $n\in \mathbb{N}$, then $x\in E$i.e., E is solid’.

3. (3)

If ${\left({x}_{n}\right)}_{n=0}^{\mathrm{\infty }}\in E$, then ${\left({x}_{\left[\frac{n}{2}\right]}\right)}_{n=0}^{\mathrm{\infty }}=\left({x}_{0},{x}_{0},{x}_{1},{x}_{1},{x}_{2},{x}_{2},\dots \right)\in E$, where $\left[\frac{n}{2}\right]$ denotes the integral part of $\frac{n}{2}$.

Example 2.2${\ell }_{p}$ is a special space of sequences for$0.

Example 2.3${ces}_{p}$ is a special space of sequences for$1.

Definition 2.4, where ${U}_{E}^{\mathrm{app}}\left(X,Y\right):=\left\{T\in L\left(X,Y\right):{\left({\alpha }_{n}\left(T\right)\right)}_{n=0}^{\mathrm{\infty }}\in E\right\}$.

Theorem 2.5${U}_{E}^{\mathrm{app}}$is an operator ideal if E is a specialspace of sequences (sss).

Proof See . □

We give here the sufficient conditions on the generalized de La ValléePoussin’s mean such that the class of all bounded linear operators between anyarbitrary Banach spaces with n th approximation numbers of the bounded linearoperators in the generalized de La Vallée Poussin’s mean form an operatorideal.

## 3 Main results

Theorem 3.1${U}_{V\left(\lambda ,p\right)}^{\mathrm{app}}$is an operator ideal, if conditions (b1)and (b2) are satisfied.

Proof (1-i) Let $x,y\in V\left(\lambda ,p\right)$ since
$\sum _{n=0}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}|{x}_{k}+{y}_{k}|\right)}^{{p}_{n}}\le {2}^{h-1}\left(\sum _{n=0}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}|{x}_{k}|\right)}^{{p}_{n}}+\sum _{n=0}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}|{y}_{k}|\right)}^{{p}_{n}}\right),$

$h={sup}_{n}{p}_{n}$, then $x+y\in V\left(\lambda ,p\right)$.

(1-ii) Let $\lambda \in \mathbb{R}$, $x\in V\left(\lambda ,p\right)$, then
$\sum _{n=0}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}|\lambda {x}_{k}|\right)}^{{p}_{n}}\le \underset{n}{sup}|\lambda {|}^{{p}_{n}}\sum _{n=0}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}|{x}_{k}|\right)}^{{p}_{n}}<\mathrm{\infty },$

we get $\lambda x\in V\left(\lambda ,p\right)$, from (1-i) and (1-ii), $V\left(\lambda ,p\right)$ is a linear space.

To show that ${e}_{m}\in V\left(\lambda ,p\right)$ for each $m\in \mathbb{N}$, since ${\sum }_{n=0}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\right)}^{{p}_{n}}<\mathrm{\infty }$. Thus we get
$\rho \left({e}_{m}\right)=\sum _{n=m}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}|{e}_{m}\left(k\right)|\right)}^{{p}_{n}}=\sum _{n=m}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\right)}^{{p}_{n}}<\mathrm{\infty }.$
Hence ${e}_{m}\in V\left(\lambda ,p\right)$.
1. (2)

Let $|{x}_{n}|\le |{y}_{n}|$ for each $n\in \mathbb{N}$, then ${\sum }_{n=0}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}{\sum }_{k\in {I}_{n}}|{x}_{k}|\right)}^{{p}_{n}}\le {\sum }_{n=0}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}{\sum }_{k\in {I}_{n}}|{y}_{k}|\right)}^{{p}_{n}}$ since $y\in V\left(\lambda ,p\right)$. Thus $x\in V\left(\lambda ,p\right)$.

2. (3)
Let $\left({x}_{n}\right)\in V\left(\lambda ,p\right)$, then we have
$\begin{array}{c}\sum _{n=0}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}|{x}_{\left[\frac{k}{2}\right]}|\right)}^{{p}_{n}}\hfill \\ \phantom{\rule{1em}{0ex}}=\sum _{n=0}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{2n}}\sum _{k\in {I}_{2n}}|{x}_{\left[\frac{k}{2}\right]}|\right)}^{{p}_{2n}}+\sum _{n=0}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{2n+1}}\sum _{k\in {I}_{2n+1}}|{x}_{\left[\frac{k}{2}\right]}|\right)}^{{p}_{2n+1}}\hfill \\ \phantom{\rule{1em}{0ex}}=\sum _{n=0}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{2n}}\left(\left(\sum _{k\in {I}_{n}}2|{x}_{k}|\right)+|{x}_{n}|\right)\right)}^{{p}_{n}}+\sum _{n=0}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{2n+1}}\left(\sum _{k\in {I}_{n}}2|{x}_{k}|\right)\right)}^{{p}_{n}}\hfill \\ \phantom{\rule{1em}{0ex}}\le {2}^{h-1}\left(\sum _{n=0}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\left(2\sum _{k\in {I}_{n}}|{x}_{k}|\right)\right)}^{{p}_{n}}+\sum _{n=0}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}|{x}_{k}|\right)}^{{p}_{n}}\right)\hfill \\ \phantom{\rule{2em}{0ex}}+{2}^{h}\sum _{n=0}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}|{x}_{k}|\right)}^{{p}_{n}}\hfill \\ \phantom{\rule{1em}{0ex}}\le {2}^{h-1}\left({2}^{h}+1\right)\sum _{n=0}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}|{x}_{k}|\right)}^{{p}_{n}}+{2}^{h}\sum _{n=0}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}|{x}_{k}|\right)}^{{p}_{n}}\hfill \\ \phantom{\rule{1em}{0ex}}\le \left({2}^{2h-1}+{2}^{h-1}+{2}^{h}\right)\sum _{n=0}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}|{x}_{k}|\right)}^{{p}_{n}}<\mathrm{\infty }.\hfill \end{array}$

Hence ${\left({x}_{\left[\frac{n}{2}\right]}\right)}_{n=0}^{\mathrm{\infty }}\in V\left(\lambda ,p\right)$. Hence from Theorem 2.5 it follows that${U}_{V\left(\lambda ,p\right)}^{\mathrm{app}}$ is an operator ideal. □

Corollary 3.2${U}_{ces\left(p\right)}^{\mathrm{app}}$is an operator ideal if$\left({p}_{n}\right)$is an increasing sequence of positive realnumbers, ${lim}_{n\to \mathrm{\infty }}sup{p}_{n}<\mathrm{\infty }$and${lim}_{n\to \mathrm{\infty }}inf{p}_{n}>1$.

Corollary 3.3${U}_{{ces}_{p}}^{\mathrm{app}}$is an operator ideal if$1.

Theorem 3.4 The linear space$F\left(X,Y\right)$is dense in${U}_{V\left(\lambda ,p\right)}^{\mathrm{app}}\left(X,Y\right)$if conditions (b1) and (b2) aresatisfied.

Proof First we prove that every finite mapping $T\in F\left(X,Y\right)$ belongs to ${U}_{V\left(\lambda ,p\right)}^{\mathrm{app}}\left(X,Y\right)$. Since ${e}_{m}\in V\left(\lambda ,p\right)$ for each $m\in \mathbb{N}$ and $V\left(\lambda ,p\right)$ is a linear space, then for every finite mapping$T\in F\left(X,Y\right)$, i.e., the sequence ${\left({\alpha }_{n}\left(T\right)\right)}_{n=0}^{\mathrm{\infty }}$ contains only finitely many numbers different from zero.Now we prove that ${U}_{V\left(\lambda ,p\right)}^{\mathrm{app}}\left(X,Y\right)\subseteq \overline{F\left(X,Y\right)}$. Since letting $T\in {U}_{V\left(\lambda ,p\right)}^{\mathrm{app}}\left(X,Y\right)$ we get ${\left({\alpha }_{n}\left(T\right)\right)}_{n=0}^{\mathrm{\infty }}\in V\left(\lambda ,p\right)$, and since $\rho \left({\left({\alpha }_{n}\left(T\right)\right)}_{n=0}^{\mathrm{\infty }}\right)<\mathrm{\infty }$, let $\epsilon \in \phantom{\rule{0.2em}{0ex}}\right]0,1\left[$, then there exists a natural number$s>0$ such that $\rho \left({\left({\alpha }_{n}\left(T\right)\right)}_{n=s}^{\mathrm{\infty }}\right)<\frac{\epsilon }{{2}^{h+2}\delta c}$ for some $c\ge 1$, where $\delta =max\left\{1,{\sum }_{n=s}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\right)}^{{p}_{n}}\right\}$. Since ${\alpha }_{n}\left(T\right)$ is decreasing for each $n\in \mathbb{N}$, we get
$\begin{array}{rl}\sum _{n=s+1}^{2s}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}{\alpha }_{2s}\left(T\right)\right)}^{{p}_{n}}& \le \sum _{n=s+1}^{2s}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}{\alpha }_{n}\left(T\right)\right)}^{{p}_{n}}\\ \le \sum _{n=s}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}{\alpha }_{k}\left(T\right)\right)}^{{p}_{n}}<\frac{\epsilon }{{2}^{h+2}\delta c},\end{array}$
(1)
then there exists $A\in {F}_{2s}\left(X,Y\right)$, $rank\left(A\right)\le 2s$ with
$\sum _{n=2s+1}^{3s}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}\parallel T-A\parallel \right)}^{{p}_{n}}\le \sum _{n=s+1}^{2s}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}\parallel T-A\parallel \right)}^{{p}_{n}}<\frac{\epsilon }{{2}^{h+2}\delta c},$
(2)
and since $\left({p}_{n}\right)$ is a bounded sequence of positive real numbers, so we cantake
$\underset{n=s}{\overset{\mathrm{\infty }}{sup}}{\left(\sum _{k\in {I}_{s}}\parallel T-A\parallel \right)}^{{p}_{n}}<\frac{\epsilon }{{2}^{h}\delta },$
(3)
also . Then there exists a natural number$N>0$, ${A}_{N}$ with $rank\left({A}_{N}\right)\le N$ and $\parallel T-{A}_{N}\parallel \le 2{\alpha }_{N}\left(T\right)$. Since ${\alpha }_{n}\left(T\right)\stackrel{n\to \mathrm{\infty }}{⟶}0$, then
(4)
Since $\left({p}_{n}\right)$ is an increasing sequence, by using (1), (2), (3) and (4),we get
$\begin{array}{rl}d\left(T,A\right)=& \rho {\left({\alpha }_{n}\left(T-A\right)\right)}_{n=0}^{\mathrm{\infty }}\\ =& \sum _{n=0}^{3s-1}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}{\alpha }_{k}\left(T-A\right)\right)}^{{p}_{n}}+\sum _{n=3s}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}{\alpha }_{k}\left(T-A\right)\right)}^{{p}_{n}}\\ \le & \sum _{n=0}^{3s}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}\parallel T-A\parallel \right)}^{{p}_{n}}+\sum _{n=s}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n+2s}}{\alpha }_{k}\left(T-A\right)\right)}^{{p}_{n+2s}}\\ \le & 3\sum _{n=0}^{s}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}\parallel T-A\parallel \right)}^{{p}_{n}}\\ +\sum _{n=s}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{2s-1}}{\alpha }_{k}\left(T-A\right)+\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n+2s}\mathrm{\setminus }{I}_{2s-1}}{\alpha }_{k}\left(T-A\right)\right)}^{{p}_{n}}\\ \le & 3\sum _{n=0}^{s}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}\parallel T-A\parallel \right)}^{{p}_{n}}\\ +{2}^{h-1}\left(\sum _{n=s}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{2s-1}}{\alpha }_{k}\left(T-A\right)\right)}^{{p}_{n}}+\sum _{n=s}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n+2s}\mathrm{\setminus }{I}_{2s-1}}{\alpha }_{k}\left(T-A\right)\right)}^{{p}_{n}}\right)\\ \le & 3\sum _{n=0}^{s}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}\parallel T-A\parallel \right)}^{{p}_{n}}\\ +{2}^{h-1}\left(\sum _{n=s}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{s}}\parallel T-A\parallel \right)}^{{p}_{n}}+\sum _{n=s}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}{\alpha }_{k+2s}\left(T-A\right)\right)}^{{p}_{n}}\right)\\ \le & 3\sum _{n=0}^{s}{\left(\frac{1}{{\lambda }_{n}}\sum _{k=0}^{n}\parallel T-A\parallel \right)}^{{p}_{n}}\\ +{2}^{h-1}\underset{n=s}{\overset{\mathrm{\infty }}{sup}}{\left(\sum _{k\in {I}_{s}}\parallel T-A\parallel \right)}^{{p}_{n}}\sum _{n=s}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\right)}^{{p}_{n}}+{2}^{h-1}\sum _{n=s}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}\sum _{k\in {I}_{n}}{\alpha }_{k}\left(T\right)\right)}^{{p}_{n}}<\epsilon .\end{array}$

□

Definition 3.5 A class of special space of sequences (sss)${E}_{\rho }$ is called a pre-modular special space of sequences ifthere exists a function $\rho :E\to \left[0,\mathrm{\infty }\left[$ satisfying the following conditions:
1. (i)

$\rho \left(x\right)\ge 0$$\mathrm{\forall }x\in {E}_{\rho }$ and $\rho \left(x\right)=0⇔x=\theta$, where θ is the zero element of E,

2. (ii)

there exists a constant $l\ge 1$ such that $\rho \left(\lambda x\right)\le l|\lambda |\rho \left(x\right)$ for all values of $x\in E$ and for any scalar λ,

3. (iii)

for some numbers $k\ge 1$, we have the inequality $\rho \left(x+y\right)\le k\left(\rho \left(x\right)+\rho \left(y\right)\right)$ for all $x,y\in E$,

4. (iv)

if $|{x}_{n}|\le |{y}_{n}|$ for all $n\in \mathbb{N}$, then $\rho \left(\left({x}_{n}\right)\right)\le \rho \left(\left({y}_{n}\right)\right)$,

5. (v)

for some numbers ${k}_{0}\ge 1$, we have the inequality $\rho \left(\left({x}_{n}\right)\right)\le \rho \left(\left({x}_{\left[\frac{n}{2}\right]}\right)\right)\le {k}_{0}\rho \left(\left({x}_{n}\right)\right)$,

6. (vi)

for each $x={\left(x\left(i\right)\right)}_{i=0}^{\mathrm{\infty }}\in E$, there exists $s\in \mathbb{N}$ such that $\rho {\left(x\left(i\right)\right)}_{i=s}^{\mathrm{\infty }}<\mathrm{\infty }$. This means the set of all finite sequences is ρ-dense in E,

7. (vii)

for any $\lambda >0$, there exists a constant $\zeta >0$ such that $\rho \left(\lambda ,0,0,0,\dots \right)\ge \zeta \lambda \rho \left(1,0,0,0,\dots \right)$.

It is clear from condition (ii) that ρ is continuous at θ.The function ρ defines a metrizable topology in E endowed withthis topology which is denoted by ${E}_{\rho }$.

Example 3.6${\ell }_{p}$ is a pre-modular special space of sequences for$0, with $\rho \left(x\right)={\sum }_{n=0}^{\mathrm{\infty }}|{x}_{n}{|}^{p}$.

Example 3.7${ces}_{p}$ is a pre-modular special space of sequences for$1, with $\rho \left(x\right)={\sum }_{n=0}^{\mathrm{\infty }}{\left(\frac{1}{n+1}{\sum }_{k=0}^{n}|{x}_{n}|\right)}^{p}$.

Theorem 3.8$V\left(\lambda ,p\right)$with$\rho \left(x\right)={\sum }_{n=0}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}{\sum }_{k\in {I}_{n}}|{x}_{n}|\right)}^{{p}_{n}}$is a pre-modular special space of sequences ifconditions (b1) and (b2) are satisfied.

Proof (i) Clearly, $\rho \left(x\right)\ge 0$ and $\rho \left(x\right)=0⇔x=\theta$.
1. (ii)

Since $\left({p}_{n}\right)$ is bounded, then there exists a constant $l\ge 1$ such that $\rho \left(\lambda x\right)\le l|\lambda |\rho \left(x\right)$ for all values of $x\in E$ and for any scalar λ.

2. (iii)

For some numbers $k=max\left(1,{2}^{h-1}\right)\ge 1$, we have the inequality $\rho \left(x+y\right)\le k\left(\rho \left(x\right)+\rho \left(y\right)\right)$ for all $x,y\in V\left(\lambda ,p\right)$.

3. (iv)

Let $|{x}_{n}|\le |{y}_{n}|$ for all $n\in \mathbb{N}$, then ${\sum }_{n=0}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}{\sum }_{k\in {I}_{n}}|{x}_{n}|\right)}^{{p}_{n}}\le {\sum }_{n=0}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}{\sum }_{k\in {I}_{n}}|{y}_{n}|\right)}^{{p}_{n}}$.

4. (v)

There exist some numbers ${k}_{0}={2}^{h-1}\left({2}^{h}+1\right)+{2}^{h}\ge 1$; by using (iv) we have the inequality $\rho \left(\left({x}_{n}\right)\right)\le \rho \left(\left({x}_{\left[\frac{n}{2}\right]}\right)\right)\le {k}_{0}\rho \left(\left({x}_{n}\right)\right)$.

5. (vi)

It is clear that the set of all finite sequences is ρ-dense in $V\left(\lambda ,p\right)$.

6. (vii)

For any $\lambda >0$, there exists a constant $0<\zeta <{\lambda }^{{p}_{0}-1}$ such that $\rho \left(\lambda ,0,0,0,\dots \right)\ge \zeta \lambda \rho \left(1,0,0,0,\dots \right)$. □

Theorem 3.9 Let X be a normed space, Y be aBanach space, and let conditions (b1) and (b2) besatisfied, then${U}_{{V}_{\rho }\left(\lambda ,p\right)}^{\mathrm{app}}\left(X,Y\right)$is complete.

Proof Let $\left({T}_{m}\right)$ be a Cauchy sequence in ${U}_{{V}_{\rho }\left(\lambda ,p\right)}^{\mathrm{app}}\left(X,Y\right)$. Since $V\left(\lambda ,p\right)$ with $\rho \left(x\right)={\sum }_{n=0}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{n}}{\sum }_{k\in {I}_{n}}|{x}_{n}|\right)}^{{p}_{n}}$ is a pre-modular special space of sequences, then, byusing condition (vii) and since ${U}_{{V}_{\rho }\left(\lambda ,p\right)}^{\mathrm{app}}\left(X,Y\right)\subseteq L\left(X,Y\right)$, we have $\rho \left({\left({\alpha }_{n}\left({T}_{i}-{T}_{j}\right)\right)}_{n=0}^{\mathrm{\infty }}\right)\ge \rho \left({\alpha }_{0}\left({T}_{i}-{T}_{j}\right),0,0,0,\dots \right)=\rho \left(\parallel {T}_{i}-{T}_{j}\parallel ,0,0,0,\dots \right)\ge \zeta \parallel {T}_{i}-{T}_{j}\parallel \rho \left(1,0,0,0,\dots \right)$, then $\left({T}_{m}\right)$ is also a Cauchy sequence in $L\left(X,Y\right)$. Since the space $L\left(X,Y\right)$ is a Banach space, then there exists$T\in L\left(X,Y\right)$ such that $\parallel {T}_{m}-T\parallel \stackrel{m\to \mathrm{\infty }}{⟶}0$ and since ${\left({\alpha }_{n}\left({T}_{m}\right)\right)}_{n=0}^{\mathrm{\infty }}\in E$ for all $m\in \mathbb{N}$, ρ is continuous at θ andusing (iii), we have

Hence ${\left({\alpha }_{n}\left(T\right)\right)}_{n=0}^{\mathrm{\infty }}\in {V}_{\rho }\left(\lambda ,p\right)$ as such $T\in {U}_{{V}_{\rho }\left(\lambda ,p\right)}^{\mathrm{app}}\left(X,Y\right)$. □

Corollary 3.10 Let X be a normed space, Y bea Banach space and$\left({p}_{n}\right)$be an increasing sequence of positive real numberswith$lim sup{p}_{n}<\mathrm{\infty }$and$lim inf{p}_{n}>1$, then${U}_{{ces}_{\left(p\right)}}^{\mathrm{app}}\left(X,Y\right)$is complete.

Corollary 3.11 Let X be a normed space, Y bea Banach space and$\left({p}_{n}\right)$be an increasing sequence of positive real numberswith$1, then${U}_{{ces}_{p}}^{\mathrm{app}}\left(X,Y\right)$is complete.

## Declarations

### Acknowledgements

The author is most grateful to the editor and anonymous referee for careful readingof the paper and valuable suggestions which helped in improving an earlier version ofthis paper.

## Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science and Arts, King Abdulaziz University (KAU), P.O. Box 80200, Khulais, Code 21589, Saudi Arabia
(2)
Department of Mathematics, Faculty of Science, Ain Shams University, Abbassia, Cairo, P.O. Box 1156, 11566, Egypt

## References 