# Weakly unconditionally Cauchy series and Fibonacci sequence spaces

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## Abstract

We study new sequence spaces associated to sequences in normed spaces and the band matrix *F̂* defined by the Fibonacci sequence. We give some characterizations of continuous linear operators and weakly unconditionally Cauchy series by means of completeness of the new sequence spaces. Also, we characterize the barreledness of a normed space via weakly^{∗} unconditionally Cauchy series in \(X^{*}\).

## Keywords

Fibonacci sequence spaces weakly unconditionally Cauchy series completeness barreledness## MSC

46B15 40A05 46B45## 1 Introduction

By *w*, we denote the space of all real sequences \(x=(x_{k})\). Any vector subspace of *w* is called a *sequence space*. We have \(\ell_{\infty }\), *c* and \(c_{0}\) for the spaces of all bounded, convergent and null sequences \(x=(x_{k})\), respectively, normed by \(\Vert x \Vert _{\infty }=\sup_{k}\vert x_{k} \vert \), where \(k\in \mathbb{N}\), the set of positive integers.

A sequence space *λ* with a linear topology is called a *K-space* provided each of the maps \(p_{i}:\lambda \to \mathbb{R}\) defined by \(p_{i}(x)=x_{i}\) is continuous for all \(i\in \mathbb{N}\). A K-space *λ* is called an *FK-space* provided *λ* is a complete linear metric space. We say that an FK space \(\lambda \supset c_{00}\) has AD if \(c_{00}\) is dense in *λ*, where \(c_{00}=\operatorname{\operatorname{span}}\{e^{n}:n\in \mathbb{N}\}\), the set of all finitely non-zero sequences.

*A*-transform of \(x\in w\), if \((Ax)_{n}=\sum_{k} a_{nk}x_{k}\) converges for each \(n\in \mathbb{N}\). For a sequence space

*λ*, the matrix domain \(\lambda_{A}\) of an infinite matrix

*A*is defined by

A series \(\sum_{k}x_{k}\) in a real Banach space *X* is called weakly unconditionally Cauchy series (wuCs) if \(\sum_{k}\vert f(x_{k}) \vert <\infty \) for every \(f\in X^{*}\) (the dual space of *X*), and is called unconditionally convergent (ucs) if \(\sum_{k}x_{\pi (k)}\) is convergent for every permutation *π* of \(\mathbb{N}\). By \(\operatorname{ucs}(X)\), \(\ell_{1}(X)\), \(\operatorname{cs}(X)\), \(\operatorname{wcs}(X)\) and \(\operatorname{wuCs}(X)\), we denote the *X*-valued sequence spaces of unconditionally convergent, absolutely convergent, convergent, weakly convergent and weakly unconditionally Cauchy series, respectively.

*X*has a copy of \(c_{0}\) if and only if \(\operatorname{wuCs}(X)\setminus \operatorname{ucs}(X)\neq \emptyset \), and if

*X*is a normed space then \(x=(x_{k})\in \operatorname{wuCs}(X)\) if and only if the set

In [15], for a sequence \(x=(x_{k})\) in a normed space *X* the spaces \(S(x)\) and \(S_{w}(x)\) were defined by the set of all sequences \(a=(a_{i})\in \ell_{\infty }\) such that \((a_{i}x_{i})\in \operatorname{cs}(X)\) and \((a_{i}x_{i})\in \operatorname{wcs}(X)\), respectively and several types of convergence of a series in a normed space have been characterized via these spaces. The completeness and barreledness of a normed space can also be characterized by means of the sequence spaces obtained by series in a normed space in [16] and [17, 18]. The characterizations of wucs are studied on locally convex spaces in [19].

In this paper, we introduce the sets \(S\widehat{F}(x)\), \(S\widehat{F} _{w}(x)\) and \(S\widehat{F}_{w}^{*}(g)\) by means of sequences in normed spaces and the Fibonacci matrix \(\widehat{F}= (\widehat{f}_{nk})\). We will characterize wucs by means of completeness of the spaces \(S\widehat{F}(x)\) and \(S\widehat{F}_{w}(x)\), and we will obtain necessary and sufficient conditions for the operator \(T:S\widehat{F}(x) (\textrm{ and } S\widehat{F}_{w}(x))\rightarrow X\) to be continuous. Finally, we will give a characterization of the barreledness of a normed space through w^{∗}ucs in \(X^{*}\).

## 2 Main results

*X*and \(X^{*}\), respectively. We introduce the subspaces of \(\ell_{\infty }( \widehat{F})\) which are defined by

^{∗}topology, respectively.

In the following theorem we obtain a sufficient condition for the space \(S\widehat{F}(x)\) to be a Banach space.

### Theorem 2.1

*Let*

*X*

*be a normed space and*\(x=(x_{k})\)

*be a sequence in X*.

*If*

- (i)
*X**is a Banach space*, - (ii)
\(x\in \operatorname{wuCs}(X)\),

*then*\(S\widehat{F}(x)\)

*is a Banach space*.

### Proof

Since \(x\in \operatorname{wuCs}(X)\), the set *E* given in (1.1), is bounded. Therefore, there exists \(M>0\) such that \(\Vert E \Vert \leq M\). Let \((a^{m})\) be a Cauchy sequence in \(S\widehat{F}(x)\). Since \(\ell_{\infty }( \widehat{F})\) is a Banach space, there exists \(a=(a_{k}^{0})\in \ell_{\infty }(\widehat{F})\) such that \(\lim_{m} a^{m}=a^{0}\) in \(\ell_{\infty }(\widehat{F})\). We will show that \(a^{0}\in S \widehat{F}(x)\).

*X*. Then for \(\epsilon >0\) there exists \(y_{0}\in X\) such that for \(m>m_{1}\)

### Remark 2.2

The theorem that follows gives us a characterization of wucs.

### Theorem 2.3

*Let* *X* *is a normed space and* \(x=(x_{k})\) *be a sequence in X*. *If* *X* *is a Banach space*, *then* \(x\in \operatorname{wuCs}(X)\) *if and only if* \(S \widehat{F}(x)\) *is a Banach space*.

### Proof

### Remark 2.4

*X*is not Banach space, then the above theorem is not satisfied. Actually, If

*X*is not Banach space then there exists a sequence \(x=(x_{k})\in \ell_{1}(X)\setminus cs(X)\) such that for every \(k\in \mathbb{N}\) and \(x^{**}\in X^{**}\setminus X\)

### Theorem 2.5

*Let*

*X*

*be a normed space and*\(x=(x_{k})\)

*be a sequence in X*.

*We also define the linear operator*

*Then*

*T*

*is continuous if and only if*\(x=(x_{k})\in \operatorname{wuCs}(X)\).

### Proof

If the operator *T* is continuous, then we prove that \(x=(x_{k}) \in \operatorname{wuCs}(X)\). Since *T* is continuous, there exists \(K>0\) such that \(\Vert T(a_{k}) \Vert \leq K\Vert (a_{k}) \Vert \) for \(a=(a_{k})\in S\widehat{F}(x)\).

*E*, defined in (1.1), is bounded and hence \(x=(x_{k})\in \operatorname{wuCs}(X)\).

*E*is bounded, there exists \(K>0\) such that \(\Vert e \Vert < K\) for every \(e\in E\). If we take \((a_{k})\in S\widehat{F}(x)\), then

*T*is continuous. □

Now, we will extend some of the above results to weak topology. First, let us start with the following result.

### Theorem 2.6

*Let* *X* *be a Banach space and* \(x=(x_{k})\) *be a sequence in X*. *If* \(x\in \operatorname{wuCs}(X)\), *then* \(S\widehat{F}_{w}(x)\) *is a Banach space*.

### Proof

In the first place, as in Theorem 2.1, since \(x\in \operatorname{wuCs}(X)\), we suppose that \(\Vert e \Vert \leq M\) for every \(e\in E\) and \((a^{m})\) be a Cauchy sequence in \(S\widehat{F}_{w}(x)\) such that \(a^{m}\rightarrow a^{0} \in \ell_{\infty }(\widehat{F})\), as \(m\rightarrow \infty \).

*f*in \(X^{*}\) such that

*X*. Let us suppose that \(y_{0}\in X\) such that for \(m>m_{1}\)

### Theorem 2.7

*Let* *X* *be a normed space and* \(x=(x_{k})\) *be a sequence in X*. *If* *X* *is a Banach space*, *then* \(x\in \operatorname{wuCs}(X)\) *if and only if* \(S \widehat{F}_{w}(x)\) *is a Banach space*.

### Proof

### Theorem 2.8

*Let*

*X*

*be a normed space and*\(x=(x_{k})\)

*be a sequence in X*.

*We also define the linear operator*

*Then*

*T*

*is continuous if and only if*\(x=(x_{k})\in \operatorname{wuCs}(X)\).

### Proof

The proof is similar to that of Theorem 2.5. □

*X*and a sequence \(g=(g_{i})\) in \(X^{*}\), the set \(S\widehat{F}_{w^{*}}(g)\) was defined by

*X*is barreled, then weakly unconditionally Cauchy series and weakly

^{∗}unconditionally Cauchy series in \(X^{*}\) are equivalent.

### Theorem 2.9

*Let*

*X*

*be a normed space and*\(g=(g_{i})\)

*be a sequence in*\(X^{*}\).

*Consider the following statements*:

- (i)
\(g\in \operatorname{wuCs}(X^{*})\).

- (ii)
\(S\widehat{F}_{w^{*}}(g)=\ell_{\infty }(\widehat{F})\).

- (iii)
\(g\in w^{*}\operatorname{ucs}(X^{*})\);

*that is*, \(\sum_{k}\vert g_{k}(x) \vert < \infty \)*for every*\(x\in X\).

*We have*(i) ⇒ (ii) ⇒ (iii).

*Furthermore*

*X*

*is a barreled normed space if and only if the three conditions are equivalent*.

### Proof

(i) ⇒ (ii). Since \(S\widehat{F}_{w^{*}}(g)\subset \ell_{\infty }(\widehat{F})\), we will show that \(\ell_{\infty }( \widehat{F})\subset S\widehat{F}_{w^{*}}(g)\). If \(a=(a_{k})\in \ell_{\infty }(\widehat{F})\), then \((\widehat{F}(a_{k})g_{k})\in \operatorname{wuCs}(X ^{*})\). Thus \(\sum_{k=1}^{\infty }\widehat{F}(a_{k})g_{k}\) is weak^{∗} convergent in \(X^{*}\) and hence \(a=(a_{k})\in S\widehat{F}_{w ^{*}}(g)\).

(ii) ⇒ (iii). It is obvious.

*X*is a barreled space then we will show that (iii) ⇒ (i). We define the set \(E'\) by

*X*is barreled, \(E'\) is bounded for the norm topology of \(X^{*}\). Therefore \((g_{k})\in \operatorname{wuCs}(X^{*})\).

Conversely, if (iii) ⇒ (i) are equivalent, then we will prove that *X* is a barreled space. Let us suppose that *X* is not a barreled space. Then there exists a weak^{∗}-bounded set \(N\subseteq X^{*}\) which is not bounded. Let \((g_{k})\in N\) such that \(\Vert g_{k} \Vert >2^{k}.2^{k}\) for \(k\in \mathbb{N}\). If we take \(h_{k}= \frac{1}{2^{k}}g_{k}\) for \(k\in \mathbb{N}\) then it is clear that \((h_{k}(x))\in \ell_{1}\) for every \(x\in X\). Since \(\Vert h_{k} \Vert >2^{k}\) for every \(k\in \mathbb{N}\), the series \(\sum_{k=1}^{\infty } \frac{1}{2^{k}}h_{k}\) does not convergence. Hence \((h_{k})\notin \operatorname{wuCs}(X ^{*})\). □

## 3 Conclusion

In this paper, we introduced and studied the sets \(S\widehat{F}(x)\), \(S\widehat{F}_{w}(x)\) and \(S\widehat{F}_{w}^{*}(g)\) via sequences in normed spaces and the Fibonacci matrix \(\widehat{F}= (\widehat{f}_{nk})\). We obtained the characterizations of continuous linear operator and weakly unconditionally Cauchy series by means of completeness of the space \(S\widehat{F}(x)\), and we extended the obtained results to weak topology. Also, we gave necessary and sufficient conditions for a normed space *X* to be barreled space. Furthermore, one can obtain more general conclusion corresponding to the results of this paper by taking more general matrices instead of the Fibonacci matrix.

## Notes

### Acknowledgements

We would like to express our thanks to the anonymous reviewers for their valuable comments.

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