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# Almost convergence and generalized weighted mean II

- Murat Kirişci
^{1}Email author

**2014**:93

https://doi.org/10.1186/1029-242X-2014-93

© Kirişci; licensee Springer. 2014

**Received:**8 January 2013**Accepted:**28 January 2014**Published:**25 February 2014

## Abstract

In this paper, we investigate some new sequence spaces, which naturally emerge from the concepts of almost convergence and generalized weighted mean. The object of this paper is to introduce the new sequence spaces obtained as the matrix domain of generalized weighted mean in the spaces of almost null and almost convergent sequences. Furthermore, the beta and gamma dual spaces of the new spaces are determined and some classes of matrix transformations are characterized.

**MSC:**47A15, 46A32, 47D20.

## Keywords

- almost convergence
- matrix domain
- sequence spaces
- generalized weighted mean
*β*- and*γ*-duals- matrix transformations

## 1 Introduction

By a *sequence space*, we understand a linear subspace of the space ${\mathbb{C}}^{\mathbb{N}}$ of all complex sequences which contains *ϕ*, the set of all finitely non-zero sequences, where ℂ denotes the complex field and $\mathbb{N}=\{0,1,2,\dots \}$. We write ${\ell}_{\mathrm{\infty}}$, *c*, and ${c}_{0}$ for the classical spaces of all bounded, convergent and null sequences, respectively. Also by *bs*, *cs*, ${\ell}_{1}$, and ${\ell}_{p}$, we denote the spaces of all bounded, convergent, absolutely, and *p*-absolutely convergent series, respectively.

A sequence spaces *μ* with a linear topology is called a *K*-space if each of the maps ${p}_{i}:\mu \to \mathbb{C}$ defined by ${p}_{i}(x)={x}_{i}$ is continuous for all $i\in \mathbb{N}$. A *K*-space is called an *FK*-space if *μ* is a complete linear metric space; a *BK*-space is a normed *FK*-space.

*μ*is called a

*Schauder basis*if for every $x\in \mu $, there is a unique sequence ${({\beta}_{n})}_{n=0}^{\mathrm{\infty}}$ of scalars such that

The sequence $({e}^{(n)})$ is the Schauder basis for ${\ell}_{p}$ and ${c}_{0}$, and $\{e,{e}^{(n)}\}$ is the Schauder basis for the space *c*, while the space ${\ell}_{\mathrm{\infty}}$ has no Schauder basis, where ${e}^{(n)}$ and *e* denote the sequences whose only non-zero entry is a 1 in the *n* th place for each $n\in \mathbb{N}$ and $e=(1,1,1,\dots )$.

A subset *M* of a metric space $(X,d)$ is said to be *dense* in *X* if $\overline{M}=X$. A metric space $(X,d)$ is said to be *separable* if it contains a countable subset which is dense in *X*. Note that a nonseparable space has no Schauder basis.

*λ*and

*μ*be two sequence spaces, and $A=({a}_{nk})$ be an infinite matrix of complex numbers ${a}_{nk}$, where $k,n\in \mathbb{N}$. Then we say that

*A*defines a matrix mapping from

*λ*into

*μ*, and we denote it by writing $A:\lambda \to \mu $ if for every sequence $x=({x}_{k})\in \lambda $ the sequence $Ax=\{{(Ax)}_{n}\}$, the

*A*-transform of

*x*, is in

*μ*; here

*A*such that $A:\lambda \to \mu $. Thus, $A\in (\lambda :\mu )$ if and only if the series on the right side of (1.1) converges for each $n\in \mathbb{N}$ and each $x\in \lambda $ and we have $Ax={\{{(Ax)}_{n}\}}_{n\in \mathbb{N}}\in \mu $ for all $x\in \lambda $. A sequence

*x*is said to be

*A*-summable to

*α*if

*Ax*converges to

*α*, which is called the

*A*-limit of

*x*. Also by $(\lambda :\mu ;p)$, we denote the subset of $(\lambda :\mu )$ for which limits or sums are preserved whenever there is a limit or sum on the spaces

*λ*and

*μ*. The matrix domain ${\lambda}_{A}$ of an infinite matrix

*A*in a sequence space

*λ*is defined by

*λ*are linearly isomorphic,

*i.e.*, ${\lambda}_{A}\cong \lambda $ [1]. We write

*U*for the set of all sequences $u=({u}_{k})$ such that $({u}_{k})\ne 0$ for all $k\in \mathbb{N}$. For $u\in U$, let $1/u=(1/{u}_{k})$. Let $u,v\in U$ and define the

*generalized weighted mean*or

*factorable matrix*$G(u,v)=({g}_{nk})$ by

for all $k,n\in \mathbb{N}$; here ${u}_{n}$ depends only on *n* and ${v}_{k}$ only on *k*.

for all $k,m,n\in \mathbb{N}$.

The main purpose of present paper is to introduce the sequence spaces ${f}_{0}(G)$ and $f(G)$ derived as the domain of the generalized weighted mean in the spaces ${f}_{0}$ and *f* of almost null and almost convergent sequences, and to determine the *β*- and *γ*-duals of these spaces. Furthermore, some classes of matrix mappings on/in the space $f(G)$ are characterized.

## 2 Spaces of almost null and almost convergent sequences

*P*is defined on

*ω*by ${(Px)}_{m}={x}_{m+1}$ for all $n\in \mathbb{N}$. A Banach limit

*L*is defined on ${\ell}_{\mathrm{\infty}}$, as a non-negative linear functional, such that $L(Px)=L(x)$ and $L(e)=1$. A sequence $x=({x}_{k})\in {\ell}_{\mathrm{\infty}}$ is said to be almost convergent to the generalized limit

*α*if all Banach limits of

*x*are

*α*[2] and is denoted by $f\phantom{\rule{0.2em}{0ex}}\text{-}\text{lim}\phantom{\rule{0.2em}{0ex}}{x}_{k}=\alpha $. Let ${P}^{i}$ be the composition of

*P*with itself

*i*times and let us write for a sequence $x=({x}_{k})$

Lorentz [2] proved that $f\phantom{\rule{0.2em}{0ex}}\text{-}\text{lim}\phantom{\rule{0.2em}{0ex}}{x}_{k}=\alpha $ if and only if ${lim}_{n\to \mathrm{\infty}}{t}_{mn}(x)=\alpha $ uniformly in *m*. It is well known that a convergent sequence is almost convergent such that its ordinary and generalized limits are equal.

*f*of almost null and almost convergent sequences are defined as follows:

where ${t}_{mn}(x)$ is defined by (2.1). Also, by *fs*, we denote the space of all almost convergent series.

## 3 New sequence spaces and their duals

In this section, we introduce the sequence spaces ${f}_{0}(G)$ and $f(G)$ and give some results concerning them, and we determine their beta and gamma duals.

where $\lambda \in \{{\ell}_{\mathrm{\infty}},c,{c}_{0}\}$.

Şimşek *et al.* [6] have introduced a modular structure of the sequence spaces defined by Altay and Başar [5] and studied Kadec-Klee and uniform Opial properties of this sequence space on Köthe sequence spaces.

*f*, that is,

**Theorem 3.1**

*The sequence spaces*$f(G)$

*and*${f}_{0}(G)$

*are*

*BK*-

*spaces with the same norm given by*

*where*

*Proof* ${f}_{0}$ and *f* endowed with the norm ${\parallel \cdot \parallel}_{\mathrm{\infty}}$ are *BK*-spaces (Boos [[7], Example 7.3.2(b)]) and $G(u,v)$ is a triangle matrix. Theorem 4.3.2 of Wilansky [[8], p. 61] gives the fact that $f(G)$ and ${f}_{0}(G)$ are *BK*-spaces with the norm ${\parallel \cdot \parallel}_{f(G)}$. □

**Theorem 3.2** *The sequence spaces* ${f}_{0}(G)$ *and* $f(G)$ *strictly include the spaces* ${f}_{0}$ *and* *f*, *respectively*.

*Proof* By the definition on the sequence spaces ${f}_{0}$ and *f*, it is immediate that $f\subset f(G)$ and ${f}_{0}\subset {f}_{0}(G)$.

The sequence is not almost convergent but *Gt* is almost convergent to $1/2$. This step completes the proof. □

**Theorem 3.3** *The inclusions* $c\subset f(G)$ *and* $f(G)\subset {\ell}_{\mathrm{\infty}}$ *strictly hold*.

*Proof* It is clear that $c\subset f(G)$ and $f(G)\subset {\ell}_{\mathrm{\infty}}$ because of Theorem 3.2 and $c\subset f\subset {\ell}_{\mathrm{\infty}}$. Further, we show that these inclusions are strict.

Consider the sequence $x={G}^{-1}(u,v)y$ with the sequence *y* in the set ${\ell}_{\mathrm{\infty}}\setminus f$ given by Miller and Orhan [9] as $y=(0,\dots ,0,1,\dots ,1,0,\dots ,0,1,\dots ,1,\dots )$, where the blocks of 0’s are increasing by factors of 100 and blocks of 1’s are increasing by factors of 10. Then the sequence *x* is not in $f(G)$ but in the space ${\ell}_{\mathrm{\infty}}$. This shows that the inclusion $f(G)\subset {\ell}_{\mathrm{\infty}}$ is strict.

Since the inclusion $f\subset f(G)$ strictly holds by Theorem 3.2, by combining this fact with the well-known strict inclusion $c\subset f$, one can easily see that the inclusion $c\subset f(G)$ also strictly holds, as was desired. □

It is known from Corollary 3.3 of Başar and Kirişçi [10] that the Banach space *f* has no Schauder basis. It is also known from Theorem 2.3 of Jarrah and Malkowsky [11] that the domain ${\mu}_{A}$ of a matrix *A* in a normed sequence space *μ* has a basis if and only if *μ* has a basis whenever $A=({a}_{nk})$ is a triangle. Combining these two facts one can immediately conclude that neither the space ${f}_{0}(G)$ nor the space $f(G)$ have a Schauder basis.

**Lemma 3.4** [[12], Theorem 2.1]

*Let*

*λ*,

*μ*

*be the BK*-

*spaces and*${B}_{\mu}^{U}=({b}_{nk})$

*be defined via the sequence*$\alpha =({\alpha}_{k})\in \mu $

*and triangle matrix*$U=({u}_{nk})$

*by*

*for all* $k,n\in \mathbb{N}$. *Then the inclusion* $\mu {\lambda}_{U}\subset {\lambda}_{U}$ *holds if and only if the matrix* ${B}_{\mu}^{U}=U{D}_{\alpha}{U}^{-1}$ *is in the classes* $(\lambda :\lambda )$, *where* ${D}_{\alpha}$ *is the diagonal matrix defined by* ${[{D}_{\alpha}]}_{nn}={\alpha}_{n}$ *for all* $n\in \mathbb{N}$.

**Lemma 3.5** [[12], Theorem 3.1]

*be defined via a sequence*$a=({a}_{k})\in \omega $

*and inverse of the triangle matrix*$U=({u}_{nk})$

*by*

*for all*$k,n\in \mathbb{N}$.

*Then*

*and*

From Lemma 3.4 and Lemma 3.5, we may give the theorem determining the *β*- and *γ*-duals of the sequence space $f(G)$.

**Theorem 3.6**

*Let*$u,v\in U$

*and*$z=({z}_{k})\in \omega $.

*Define the matrix*$E=({e}_{nk})$

*by*

*for all*$k,n\in \mathbb{N}$.

*Then*

*and*

*Proof*Consider the equality

where $E=({e}_{nk})$ is defined by (3.4). We therefore observe by (3.5) that $zx=({z}_{k}{x}_{k})\in cs$ or *bs* whenever $x=({x}_{k})\in f(G)$ if and only if $Ey\in c$ or ${\ell}_{\mathrm{\infty}}$ whenever $y=({y}_{k})\in f$. We obtain from Lemma 3.4 and Lemma 3.5 the result that $z=({z}_{k})\in {\{f(G)\}}^{\beta}$ or $z=({z}_{k})\in {\{f(G)\}}^{\gamma}$ if and only if $E\in (f:c)$ or $E\in (f:{\ell}_{\mathrm{\infty}})$, which is what we wished to prove. □

As a direct consequence of Theorem 3.6, we have the following.

**Corollary 3.7**

*Let*$u,v\in U$

*for all*$k\in \mathbb{N}$.

*Then*

*and*

## 4 Some matrix mappings related to the space $f(G)$

## 5 Conclusion

As an essential work on the algebraic and topological properties of the spaces ${f}_{0}$ and *f*, Başar and Kirişçi [10] have recently introduced the sequence spaces ${\stackrel{\u02c6}{f}}_{0}$ and $\stackrel{\u02c6}{f}$ derived by the domain of the generalized difference matrix $B(r,s)$ in the sequence spaces ${f}_{0}$ and *f*, respectively. Following Başar and Kirişçi [10], Kayaduman and Şengönül have studied the domain ${\tilde{f}}_{0}$ and $\tilde{f}$ of the Cesàro mean of order one in the spaces ${f}_{0}$ and *f*, in [20]. They have determined the *β*- and *γ*-duals of the new spaces ${\tilde{f}}_{0}$ and $\tilde{f}$, and they characterize some classes of matrix transformations on/in the new sequence spaces. They complete the paper by a nice section including some core theorems related to the matrix classes on/in the new sequence space $\tilde{f}$. Quite recently, in [26], Sönmez has introduced the domain $f(B)$ of the triple band matrix $B(r,s,t)$ in the sequence space *f*. In this paper, the *β*- and *γ*-duals of the space $f(B)$ are determined. Furthermore, the classes $(f(B):\mu )$ and $(\mu :f(B))$ of infinite matrices are characterized together with some other classes, where *μ* is any given sequence space. Finally, in [27] Candan has studied the sequence spaces ${f}_{0}(\tilde{B})$ and $f(\tilde{B})$ as the domain of the double sequential band matrix $\tilde{B}(\tilde{r},\tilde{s})$ in the sequence spaces ${f}_{0}$ and *f*.

Since Kirişçi and Başar [28], Başar and Kirişçi [10], Kayaduman and Şengönül [20], Sönmez [26, 29], and Candan [27, 30] are recent works on the domain of certain triangle matrices in the spaces ${f}_{0}$, *f*, and in the classical sequence spaces, the present paper is their natural continuation. Also these spaces are special cases of the notion of *A*-almost convergence and ${\mathcal{F}}_{B}$-convergence ([31, 32]) as well as analogous to the definition introduced in [33].

Finally, we should note that the investigation of the domain of some particular limitation matrices, namely Cesàro means of order *m*, Nörlund means, *etc.*, in the spaces ${f}_{0}$ and *f* will lead to new results which are not comparable with the present results.

## Author’s contributions

MK defined the new almost sequence spaces derived by generalized weighted mean and studied some properties. MK computed the duals of new spaces and characterized the matrix classes. In last section, it was summarized to studies in manuscripts and given some open problems by MK. The author read and approved the final manuscript.

## Article’s information

Some of the results of this study presented in First International Conference on Analysis and Applied Mathematics (ICAAM 2012, Gumushane University, Turkey) [34].

## Declarations

### Acknowledgements

I would like to express thanks to Professor Feyzi Başar, Department of Mathematics, Fatih University, Buyukcekmece, Istanbul-34500, Turkey, for his valuable help on some results and the useful comments which improved the presentation paper. I have benefited much from the constructive reports of the anonymous referees, and I am grateful for their valuable comments on the first draft on this paper, which improved the presentation and readability. This work was supported by Scientific Projects Coordination Unit of Istanbul University. Project number 27008.

## Authors’ Affiliations

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