On some properties of new paranormed sequence space defined by -convergent sequences
© Braha; licensee Springer. 2014
Received: 26 March 2014
Accepted: 16 June 2014
Published: 23 July 2014
In this paper, we introduce the new sequence space and we will show some topological properties like completeness, isomorphism, and some inclusion relations between this sequence spaces and some of the other sequence spaces. In addition we will compute the α-, β-, and γ-duals of these spaces. At the end of the article we will show some matrix transformations between the space and the other spaces.
By w we denote the space of all complex sequences. If , then we simply write instead of . Also, we shall use the conventions that and is the sequence whose only non-zero term is 1 in the n th place for each , where . Any vector subspace of w is called a sequence space. We shall write , c, and for the sequence spaces of all bounded, convergent, and null sequences, respectively. Further, by (), we denote the sequence space of all p-absolutely convergent series, that is, for . Moreover, we write bs, cs, and for the sequence spaces of all bounded, convergent, and null series, respectively. A sequence space X is called an FK space if it is a complete linear metric space with continuous coordinates (), where ℂ denotes the complex field and for all and every . A normed FK space is called a BK space, that is, a BK space is a Banach sequence space with continuous coordinates.
provided the series on (1) is convergent for each .
which is a sequence space. The approach constructing a new sequence space by means of the matrix domain of a particular limitation method has recently been employed by several authors; see for instance [1–18]. In this paper, we introduce the new sequence space and we will show some topological properties as completeness, isomorphism, and some inclusion relations between this sequence spaces and some of the other sequence spaces. In addition we will compute the α-, β-, and γ-duals of these spaces.
2 Notion of the -convergent sequences
Let be a nondecreasing sequence of positive numbers tending to ∞, as , and , for each . From this relation it follows that . The first difference is defined as follows: , where , and the second difference is defined as .
This generalizes the concept of Λ-strong convergence in .
3 The sequence space
Hence, we get . Therefore, the space , is complete. □
Theorem 2 The sequence space is a BK space.
Now the proof of the theorem follows from Theorem 4.3.12 given in . □
Theorem 3 The sequence space is linearly isomorphic to the space , where .
As a consequence of Theorem 2 and Theorem 3 we get the following result.
The sequence is a Schauder basis for the space and every has a unique representation: .
The sequence is a Schauder basis for the space and every has a unique representation: , where .
Proof Since and for every , the proof of the theorem follows from Corollary 2.3 given in . □
Theorem 4 The inclusion holds. The inclusion is strict.
From the last relation we get , as , respectively, . To prove that the inclusion is strict we will show the following.
Hence . On the other hand , for , . With which we have proved the theorem. □
Theorem 5 The inclusions strictly hold.
Proof It is clear that the inclusion holds. Further, since is strict, from Lemmas 1 and 2 from  it follows that is also strict. In what follows we will show that the last inclusion is strict, too. For this reason we will show the following.
From the last relation it follows that . □
Theorem 6 The inclusion holds if and only if for every sequence , where . Here and .
Proof The proof of the theorem is similar to Theorem 3.3 given in . □
If for all , then the inclusion holds.
If for all , then the inclusion holds.
- (2)Let us suppose that . Then , , such that
for all . Hence . □
4 Duals of the space
In this section we will give the theorems in which the α-, β-, and γ-duals are determined of the sequence space . In proving the theorems we apply the technique used in . Also we will give some matrix transformations from into by using the matrix given in .
where is defined by (18). From (21) it follows that or whenever if and only if or whenever . This means that or if and only if or . □
As a direct consequence of Theorem 8, we get the following.
for some constant M.
In what follows we will characterize the β- and γ-dual of the sequence space .
From (24) it follows that or bs if and only if or . This means that or . With which the theorem is proved. □
As an immediate result of the above theorem, we get the following.
5 Some matrix transformations related to sequence space
From the above conditions we get the following.
if and only if (27), (28), (29), and (40) hold,
if and only if (30), (31), (31), and (40) hold,
if and only if (33), (34), (35), (36), (37), and (41) hold,
if and only if (38), (39), and (42) hold.
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