The binomial sequence spaces of nonabsolute type
- Mustafa Cemil Bişgin^{1}Email author
https://doi.org/10.1186/s13660-016-1256-0
© The Author(s) 2016
Received: 13 August 2016
Accepted: 22 November 2016
Published: 29 November 2016
Abstract
In this paper, we introduce the binomial sequence spaces \(b^{r,s}_{0}\) and \(b^{r,s}_{c}\) of nonabsolute type which include the spaces \(c_{0}\) and c, respectively. Also, we prove that the spaces \(b^{r,s}_{0}\) and \(b^{r,s}_{c}\) are linearly isomorphic to the spaces \(c_{0}\) and c, in turn, and we investigate some inclusion relations. Moreover, we obtain the Schauder bases of those spaces and determine their α-, β-, and γ-duals. Finally, we characterize some matrix classes related to those spaces.
Keywords
MSC
1 The basic definitions and notations
Let w be the set of all real (or complex) valued sequences. Then w becomes a vector space under point-wise addition and scalar multiplication. A sequence space is a vector subspace of w. We use the notations of \(\ell_{\infty}, c_{0}, c\), and \(\ell_{p}\) for the spaces of all bounded, null, convergent, and absolutely p-summable sequences, respectively, where \(1\leq p <\infty\).
The theory of matrix transformations is of great importance in the summability which was obtained by Cesàro, Borel, Riesz and others. Therefore, many authors have defined new sequence spaces by using this theory. For example, \((\ell_{\infty})_{N_{q}}\) and \(c_{N_{q}}\) in [3], \(X_{p}\) and \(X_{\infty}\) in [4], c̃ and \(\tilde{c}_{0}\) in [5], \(a_{0}^{r}\) and \(a_{c}^{r}\) in [6]. Moreover, many authors have constructed new sequence spaces by using especially the Euler matrix. For instance, \(e_{0}^{r}\) and \(e_{c}^{r}\) in [7], \(e_{p}^{r}\) and \(e_{\infty}^{r}\) in [8] and [9], \(e_{0}^{r}(\Delta), e_{c}^{r}(\Delta)\), and \(e_{\infty}^{r}(\Delta)\) in [10], \(e_{0}^{r}(\Delta^{(m)}), e_{c}^{r}(\Delta^{(m)})\) and \(e_{\infty}^{r}(\Delta^{(m)})\) in [11], \(e_{0}^{r}(B^{(m)}), e_{c}^{r}(B^{(m)})\), and \(e_{\infty}^{r}(B^{(m)})\) in [12], \(e_{0}^{r}(\Delta, p), e_{c}^{r}(\Delta, p)\), and \(e_{\infty }^{r}(\Delta, p)\) in [13], \(e_{0}^{r}(u , p)\) and \(e_{c}^{r}(u , p)\) in [14].
In this paper, we introduce the binomial sequence spaces \(b^{r,s}_{0}\) and \(b^{r,s}_{c}\) of nonabsolute type which include the spaces \(c_{0}\) and c, respectively. Also, we prove that the spaces \(b^{r,s}_{0}\) and \(b^{r,s}_{c}\) are linearly isomorphic to the spaces \(c_{0}\) and c, in turn and investigate some inclusion relations. Moreover, we obtain the Schauder basis of those spaces and determine their α-, β-, and γ-duals. Finally, we characterize some matrix classes related to those spaces.
2 The binomial sequence spaces of nonabsolute type
In this chapter, we introduce the binomial sequence spaces \(b^{r,s}_{0}\) and \(b^{r,s}_{c}\) of nonabsolute type and prove that the spaces \(b^{r,s}_{0}\) and \(b^{r,s}_{c}\) are linearly isomorphic to the spaces \(c_{0}\) and c, respectively. Moreover, we deal with an inclusion relation related to those spaces.
- (i)
\(\sup_{n \in\mathbb{N}}\sum_{k=0}^{n} \vert \frac {1}{(s+r)^{n}}\binom{n}{k}s^{n-k}r^{k} \vert <\infty\),
- (ii)
\(\lim_{n \rightarrow\infty}\frac{1}{(s+r)^{n}}\binom {n}{k}s^{n-k}r^{k}=0\),
- (iii)
\(\lim_{n\rightarrow\infty}\sum_{k=0}^{n}\frac {1}{(s+r)^{n}}\binom{n}{k}s^{n-k}r^{k}=1\).
Here, we would like to emphasize that if we take \(r+s=1\), we obtain the Euler matrix \(E^{r}=(e_{nk}^{r})\). So the binomial matrix \(B^{r,s}=(b^{r,s}_{nk})\) generalizes the Euler matrix \(E^{r}=(e_{nk}^{r})\).
Now, we want to start with the following theorem related to the theory of BK-spaces, which is of great importance in the characterization of matrix transformations between sequence spaces.
Theorem 2.1
Proof
The sequence spaces \(c_{0}\) and c are BK-spaces according to their \(sup\)-\(norms\). Moreover, the binomial matrix \(B^{r,s}=(b^{r,s}_{nk})\) is a triangle matrix and (2.1) holds. By combining these three facts and Theorem 4.3.12 of Wilansky [2], we deduce that the binomial sequence spaces \(b^{r,s}_{0}\) and \(b^{r,s}_{c}\) are BK-spaces. This completes the proof. □
Let \(\vert x\vert =(\vert x_{k}\vert )\) for all \(k \in\mathbb{N}\). Because of \(\Vert x\Vert _{b^{r,s}_{0}} \neq \Vert \vert x\vert \Vert _{b^{r,s}_{0}}\) and \(\Vert x\Vert _{b^{r,s}_{c}} \neq \Vert \vert x\vert \Vert _{b^{r,s}_{c}}\) for at least one sequence in the binomial sequence spaces \(b^{r,s}_{0}\) and \(b^{r,s}_{c}\), \(b^{r,s}_{0}\) and \(b^{r,s}_{c}\) are sequence spaces of nonabsolute type.
Theorem 2.2
The binomial sequence spaces \(b^{r,s}_{0}\) and \(b^{r,s}_{c}\) are linearly isomorphic to the sequence spaces \(c_{0}\) and c, respectively.
Proof
Because a repetition of similar statements is redundant, the proof of the theorem is given for only the space \(b^{r,s}_{c}\). For this purpose, we should show the existence of a linear bijection between the spaces \(b^{r,s}_{c}\) and c. Let us consider the transformation L defined by \(L:b^{r,s}_{c}\longrightarrow c ,L(x)=B^{r,s}x\). Then it is obvious that, for all \(x \in b^{r,s}_{c}, L(x)=B^{r,s}x \in c\). Moreover, it is clear that L is a linear transformation and \(x=0\) whenever \(L(x)=0\). Because of this, L is injective.
Theorem 2.3
The inclusions \(e_{0}^{r} \subset b_{0}^{r,s}\) and \(e_{c}^{r} \subset b_{c}^{r,s}\) strictly hold, where \(e_{0}^{r}\) and \(e_{c}^{r}\) are Euler sequence spaces of nonabsolute type.
Proof
If \(r+s=1\), we obtain \(E^{r}=B^{r,s}\). So, the inclusion \(e_{0}^{r} \subset b_{0}^{r,s}\) holds. Assume that \(0< r<1\) and \(s=4\). Now, we define a sequence \(x=(x_{k})\) such that \(x_{k}= (-\frac{3}{r} )^{k}\) for all \(k \in\mathbb{N}\). Then it is obvious that \(x= ( (-\frac{3}{r} )^{k} ) \notin c_{0}\) and \(E^{r}x= ( (-2-r )^{k} ) \notin c_{0}\). On the other hand, \(B^{r,s}x= ( (\frac {1}{4+r} )^{k} )\in c_{0}\). As a consequence, \(x=(x_{k}) \in b^{r,s}_{0}\setminus e_{0}^{r}\).
This shows that the inclusion \(e_{0}^{r} \subset b_{0}^{r,s}\) strictly holds. Another part of the theorem can be proved in a similar way. This completes the proof. □
Theorem 2.4
The inclusions \(c_{0} \subset b_{0}^{r,s}\) and \(c \subset b_{c}^{r,s}\) strictly hold. But the sequence spaces \(b_{0}^{r,s}\) and \(\ell_{\infty }\) do not include each other.
Proof
If we consider regularity of the binomial matrix \(B^{r,s}\), we can easily conclude that \(B^{r,s}x\in c_{0}\) whenever \(x \in c_{0}\). This means that \(x \in b_{0}^{r,s}\) for all \(x \in c_{0}\), namely \(c_{0} \subset b_{0}^{r,s}\). Now we define a sequence \(u=(u_{k})\) such that \(u_{k}=(-1)^{k}\) for all \(k \in\mathbb{N}\). Then we obtain \(B^{r,s}x= ( (\frac{s-r}{s+r} )^{k} ) \in c_{0}\). As a consequence, u is in \(b_{0}^{r,s}\) but not in \(c_{0}\). So, the inclusion \(c_{0} \subset b_{0}^{r,s}\) is strict. By using a similar way, one can show that the inclusion \(c \subset b_{c}^{r,s}\) is strict.
To prove the second part of the theorem, we consider the sequences \(e=(1,1,1,\ldots)\) and \(v=(v_{k})\) defined by \(v_{k}= (-\frac{s}{r} )^{k}\) for all \(k \in\mathbb{N}\), where \(\vert \frac{s}{r} \vert >1\). Then we obtain \(B^{r,s}e=e\) and \(B^{r,s}v=(1,0,0,\ldots)\). Hence, e is in \(\ell_{\infty}\) but not in \(b_{0}^{r,s}\) and v is in \(b_{0}^{r,s}\) but not in \(\ell_{\infty}\). This shows that the sequence spaces \(b_{0}^{r,s}\) and \(\ell_{\infty}\) overlap but these spaces do not include each other. This completes the proof. □
Definition 2.1
see [2]
An infinite matrix \(A=(a_{nk})\) is called coregular, if \(A=(a_{nk})\) is conservative and \(\chi(A)=\lim_{n\rightarrow\infty}\sum_{k}a_{nk}-\sum_{k}\lim_{n\rightarrow\infty}a_{nk}\neq0\).
By taking into account the regularity of the binomial matrix \(B^{r,s}=(b_{nk}^{r,s})\), we obtain \(\chi(B^{r,s})=1\neq0\). So, the binomial matrix \(B^{r,s}=(b_{nk}^{r,s})\) is coregular.
Definition 2.2
see [15]
Let \(A=(a_{nk})\) be an infinite matrix with bounded columns. Then A is defined to be of type M if \(tA=0\) implies \(t=0\) for every \(t \in \ell\).
Definition 2.3
see [2]
For a conservative triangle \(A=(a_{nk})\), \(c\subset c_{A}\). Its closure c̄ in \(c_{A}\) is called the perfect part of \(c_{A}\). If c is dense, \(A=(a_{nk})\) is called perfect.
Now we give the following two theorems, which are needed.
Theorem 2.5
see [15]
Theorem 2.7
Each regular binomial matrix \(B^{r,s}=(b^{r,s}_{nk})\) is perfect.
Proof
We know that the regular binomial matrix \(B^{r,s}=(b^{r,s}_{nk})\) is coregular. So, for the proof, we should show that \(B^{r,s}=(b^{r,s}_{nk})\) is of type M.
3 The Schauder basis and α-, β-, γ- and continuous duals
In this chapter, we construct the Schauder basis and designate the α-, β-, γ-, and continuous duals of the binomial sequence spaces \(b^{r,s}_{0}\) and \(b^{r,s}_{c}\).
Theorem 3.1
- (a)The sequence \(\{g^{(k)}(r,s) \}_{k \in\mathbb {N}}\) is a Schauder basis for the binomial sequence space \(b_{0}^{r,s}\), and every \(x \in b_{0}^{r,s}\) has a unique representation of the form$$ x=\sum_{k}\mu_{k}g^{(k)}(r,s). $$
- (b)The set \(\{e, g^{(0)}(r,s), g^{(1)}(r,s),\ldots \}\) is a Schauder basis for the binomial sequence space \(b_{c}^{r,s}\), and any \(x \in b_{c}^{r,s}\) has a unique representation of the form$$ x=le+\sum_{k} [\mu_{k}-l ]g^{(k)}(r,s). $$
Proof
(b) We know that \(\{g^{(k)}(r,s) \}\subset b_{0}^{r,s}\) and \(B^{r,s}e=e \in c\). So, the inclusion \(\{e, g^{(k)}(r,s) \} \subset b_{c}^{r,s}\) trivially holds.
For a given arbitrary sequence \(x=(x_{k}) \in b_{c}^{r,s}\), we define a sequence \(y=(y_{k})\) such that \(y=x-le\), where \(l=\lim_{k \rightarrow \infty}\mu_{k}\). Then it is obvious that \(y=(y_{k}) \in b_{0}^{r,s}\). By considering the part (a), one can say that \(y=(y_{k})\) has a unique representation. This implies that \(x=(x_{k})\) has a unique representation, as desired in part (b). This completes the proof. □
By taking into account the results of Theorems 2.1 and 3.1, we give the following result.
Corollary 3.2
The binomial sequence spaces \(b_{0}^{r,s}\) and \(b_{c}^{r,s}\) are separable.
By \(X^{*}\), we denote the space of all bounded linear functionals on X. \(X^{*}\) is called the continuous dual of a normed space X.
Theorem 3.4
Proof
Theorem 3.5
- (I)
\(\{b_{0}^{r,s} \}^{\beta}=v_{2}^{r,s}\cap v_{3}^{r,s}\),
- (II)
\(\{b_{c}^{r,s} \}^{\beta}=v_{2}^{r,s}\cap v_{3}^{r,s}\cap v_{4}^{r,s}\),
- (III)
\(\{b_{0}^{r,s} \}^{\gamma}= \{ b_{c}^{r,s} \}^{\gamma}=v_{2}^{r,s}\).
Proof
These results show that \(\{b_{0}^{r,s} \}^{\beta }=v_{2}^{r,s}\cap v_{3}^{r,s}\).
(III) \(ax=(a_{k}x_{k}) \in bs\) whenever \(x=(x_{k}) \in b_{0}^{r,s}\) or \(b_{c}^{r,s}\) if and only if \(H^{r,s}y \in\ell_{\infty}\) whenever \(y=(y_{k}) \in c_{0}\) or c. This means that \(a=(a_{k}) \in \{ b_{0}^{r,s} \}^{\gamma}= \{b_{c}^{r,s} \}^{\gamma}\) if and only if \(H^{r,s} \in(c_{0}:\ell_{\infty})=(c:\ell_{\infty})\). By combining this and Lemma 3.3(iv), we deduce that (3.5) holds. Hence, \(\{b_{0}^{r,s} \}^{\gamma}= \{ b_{c}^{r,s} \}^{\gamma}=v_{2}^{r,s}\). This completes the proof. □
Theorem 3.6
\(\{b_{0}^{r,s} \}^{*}\) and \(\{b_{c}^{r,s} \}^{*}\) are equivalent to \(\ell_{1}\).
Proof
To avoid a repetition of similar statements, the proof of the theorem is given for only the binomial sequence space \(b_{c}^{r,s}\). For the proof, the existence of a linear surjective norm preserving mapping \(L : \{b_{c}^{r,s} \}^{*}\longrightarrow\ell_{1}\) should be shown.
4 Some matrix classes related to the binomial sequence spaces
In this chapter, we characterize some matrix classes related to the binomial sequence spaces \(b_{0}^{r,s}\) and \(b_{c}^{r,s}\). Now, we start with two lemmas which are required in the proof of the *theorems.
Lemma 4.2
see [17]
Let \(X, Y\) be any two sequence spaces, A be an infinite matrix and B be a triangle matrix. Then \(A\in(X:Y_{B})\) if and only if \(BA \in(X:Y)\).
Theorem 4.3
- (I)\(A \in(b_{c}^{r,s}:\ell_{p})\) if and only if$$\begin{aligned} &\sup_{K\in\mathcal{F}}\sum_{n} \biggl\vert \sum_{k\in K}t_{nk}^{r,s} \biggr\vert ^{p} < \infty, \end{aligned}$$(4.1)$$\begin{aligned} &t_{nk}^{r,s} \quad\textit{exists for all } k,n\in\mathbb{N} , \end{aligned}$$(4.2)$$\begin{aligned} &\sum_{k}t_{nk}^{r,s} \quad\textit{converges for all }n\in\mathbb{N} , \end{aligned}$$(4.3)where \(1\leq p <\infty\).$$\begin{aligned} & \sup_{m \in\mathbb{N}}\sum_{k=0}^{m} \Biggl\vert \sum_{j=k}^{m} \binom {j}{k}(-s)^{j-k}r^{-j}(r+s)^{k}a_{nj} \Biggr\vert < \infty,\quad n\in\mathbb{N} , \end{aligned}$$(4.4)
- (II)
Proof
Given a sequence \(x=(x_{k}) \in b_{c}^{r,s}\), we suppose that the conditions (4.1)-(4.4) hold. Then, by taking into account Theorem 3.5(II), we conclude that \(\{a_{nk} \} _{k\in\mathbb{N}} \in \{b_{c}^{r,s} \}^{\beta}\) for all \(n \in\mathbb{N}\). Thus, the A-transform of x exists. Let us consider a matrix \(U^{r,s}=(u_{nk}^{r,s})\) defined by \(u_{nk}^{r,s}=t_{nk}^{r,s}\) for all \(n,k \in\mathbb{N}\). Since \(U^{r,s}=(u_{nk}^{r,s})\) satisfies Lemma 3.3(v), we deduce that \(U^{r,s}=(u_{nk}^{r,s}) \in(c:\ell_{p})\).
According to the assumption, A can be applied to the binomial sequence space \(b_{c}^{r,s}\). So, it is trivial that the conditions (4.2)-(4.4) hold. This completes the proof of part (I).
If we take Lemma 3.3(iv) instead of Lemma 3.3(v), then part (II) can be proved in a similar way. □
Theorem 4.4
Proof
Suppose that A satisfies the conditions (4.2), (4.4), (4.5), (4.9), and (4.10). Given an arbitrary sequence \(x=(x_{k}) \in b_{c}^{r,s}\) with \(\lim_{k\rightarrow\infty}x_{k}=l\), then Ax exists. Since \(B^{r,s}=(b_{nk}^{r,s})\) is regular and \(y=(y_{k})\) is connected with the sequence \(x=(x_{k})\) by equation (2.2), we obtain \(y=(y_{k})\in c\) such that \(\lim_{k\rightarrow \infty}y_{k}=l\).
On the contrary, suppose that \(A \in(b_{c}^{r,s}:c)\). It is well known that every convergent sequence is also bounded, namely \(c\subset\ell _{\infty}\). By combining this fact and Theorem 4.3(II), we deduce that the necessity of the conditions (4.2), (4.4), and (4.5) holds. Since Ax exists and belongs to c for all \(x \in b_{c}^{r,s}\), if we take \(g^{(k)}(r,s)= \{ g^{(k)}_{n}(r,s) \}_{n\in\mathbb{N}}\) instead of an arbitrary sequence \(x=(x_{k})\), we deduce that \(Ag^{(k)}(r,s)= \{ t^{r,s}_{nk} \} _{n\in\mathbb{N}}\in c\) for all \(k \in\mathbb{N}\). This shows us that the necessity of (4.9) holds.
Moreover, if we take \(x=e\) in (4.7), we obtain \(Ax= \{ \sum_{k}t^{r,s}_{nk} \}_{n\in\mathbb{N}}\in c\). The last result is the necessity of (4.10). This completes the proof. □
Corollary 4.5
Let \(A=(a_{nk})\) be an infinite matrix with complex entries. Then \(A \in(b_{c}^{r,s}:c)_{m}\) if and only if the conditions (4.2), (4.4), and (4.5) hold, and the conditions (4.9) and (4.10) hold with \(\alpha_{k}=0\) for all \(k \in\mathbb{N}\) and \(\alpha=m\), in turn.
Lemma 4.6
see [18]
Theorem 4.7
\((b_{c}^{r,s}:c)_{m}\cap(b_{\infty}^{r,s}:c)=\emptyset\).
Proof
Now, by using Lemma 4.2, we can give some more results.
Corollary 4.8
Corollary 4.9
Corollary 4.10
Corollary 4.11
Corollary 4.12
5 Conclusion
By considering the definition of the binomial matrix \(B^{r,s}=(b_{nk}^{r,s})\), we deduce that \(B^{r,s}=(b_{nk}^{r,s})\) reduces in the case \(r+s=1\) to the \(E^{r}=(e_{nk}^{r})\), which is called the method of Euler means of order r. So, our results obtained from the matrix domain of the binomial matrix \(B^{r,s}=(b_{nk}^{r,s})\) are more general and more extensive than the results on the matrix domain of the Euler means of order r. Moreover, the binomial matrix \(B^{r,s}=(b_{nk}^{r,s})\) is not a special case of the weighed mean matrices. So, this paper filled up a gap in the existent literature.
Declarations
Acknowledgements
We would like to express our thanks to the anonymous reviewers for their valuable comments.
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Authors’ Affiliations
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