- Research
- Open Access
Weakly unconditionally Cauchy series and Fibonacci sequence spaces
- Ramazan Kama^{1}Email author and
- Bilal Altay^{2}
https://doi.org/10.1186/s13660-017-1407-y
© The Author(s) 2017
- Received: 29 December 2016
- Accepted: 25 May 2017
- Published: 8 June 2017
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 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.
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
In the following theorem we obtain a sufficient condition for the space \(S\widehat{F}(x)\) to be a Banach space.
Theorem 2.1
- (i)
X is a Banach space,
- (ii)
\(x\in \operatorname{wuCs}(X)\),
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 \(a^{0}\in S \widehat{F}(x)\).
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
Theorem 2.5
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 \(a=(a_{k})\in S\widehat{F}(x)\).
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 \(m\rightarrow \infty \).
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
Proof
The proof is similar to that of Theorem 2.5. □
Theorem 2.9
- (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 \(x\in X\).
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.
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.
Declarations
Acknowledgements
We would like to express our thanks to the anonymous reviewers for their valuable comments.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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