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.
MSC:46A45.
Hinweise
Competing interests
The author declares to have no competing interests.
1 Introduction
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.
The sequence spaces , c, and are BK spaces with the usual sup-norm given by . Also, the space is a BK space with the usual -norm defined by
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where . A sequence in a normed space X is called a Schauder basis for X if for every there is a unique sequence of scalars such that , i.e., . The α-, β-, and γ-duals of a sequence space X are, respectively, defined by
and
If A is an infinite matrix with complex entries (), then we write instead of . Also, we write for the sequence in the n th row of A, that is, for every . Further, if then we define the A-transform of x as the sequence , where
(1)
provided the series on (1) is convergent for each .
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Furthermore, the sequence x is said to be A-summable to if Ax converges to a which is called the A-limit of x. In addition, let X and Y be sequence spaces. Then we say that A defines a matrix mapping from X into Y if for every sequence the A-transform of x exists and is in Y. Moreover, we write for the class of all infinite matrices that map X into Y. Thus if and only if for all and for all . For an arbitrary sequence space X, the matrix domain of an infinite matrix A in X is defined by
(2)
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 .
Let be a sequence of complex numbers, such that . In [19] is given the concept of -convergent sequence as follows: Let be any given sequence of complex numbers, we will say that it converges -strongly to number x if
This generalizes the concept of Λ-strong convergence in [20].
Let us denote
(3)
for all . Assume here and below that , are bounded sequences of strictly positive real numbers with and , for for all . The linear space as defined by Madoxx [3] is as follows:
(4)
which are complete spaces paranormed by
(5)
3 The sequence space
In this section we will define the sequence space and prove that this sequence space according to its paranorm is a complete linear space. We have
(6)
and in case where , for every we get
(7)
Let be any sequence; we will define the -transform of the sequence as follows:
(8)
Theorem 1The sequence spaceis the complete linear metric space with respect to the paranorm defined by
(9)
Proof The linearity of follows from Minkowski’s inequality. In what follows we will prove that defines a paranorm. In fact, for any we get
(10)
and for any
(11)
It is clear that , for all . From inequalities (10) and (11) we find the subadditivity of and hence . Let be any sequence of points such that and also any sequence of scalars such that . Then, since the inequality
(12)
holds by the subadditivity of g, we find that is bounded and we thus have
(13)
which tends to zero as . Therefore, the scalar multiplication is continuous. Hence g is a paranorm on the space . It remains to prove the completeness of the space . Let be any Cauchy sequence in the space , where . Then, for a given , there exists a positive integer such that for all . Using the definition of g, we obtain for each fixed
(14)
for every , which leads to the fact that is a Cauchy sequence of real numbers for every fixed . Since ℝ is complete, it converges, say as . Using these infinitely many limits, we may write the sequence . From (14) as , we have
(15)
for every fixed . By using (14) and boundedness of the Cauchy sequence, we have
(16)
Hence, we get . Therefore, the space , is complete. □
Theorem 2The sequence spaceis aBKspace.
Proof Let us denote by the following matrix:
for all . Then the -transform of a sequence is the sequence , where is given by (8) for every . Thus
Now the proof of the theorem follows from Theorem 4.3.12 given in [21]. □
Theorem 3The sequence spaceis linearly isomorphic to the space , where .
Proof Let be an operator defined by , where is given by (8). The operator T is linear and injective, from it follows that . In what follows we will prove that T is surjective. Let be any element; we define by
(17)
then we get
□
As a consequence of Theorem 2 and Theorem 3 we get the following result.
Corollary 1Define the sequencefor every fixedby
where . Then we have the following.
(1)
The sequenceis a Schauder basis for the spaceand everyhas a unique representation: .
(2)
The sequenceis a Schauder basis for the spaceand everyhas a unique representation: , where .
Proof Since and for every , the proof of the theorem follows from Corollary 2.3 given in [22]. □
Theorem 4The inclusionholds. The inclusion is strict.
Proof Let ; then it follows that , from which follows that
From the last relation we get , as , respectively, . To prove that the inclusion is strict we will show the following.
Example 1 Let , , , and . Then it follows that
Hence . On the other hand , for , . With which we have proved the theorem. □
Theorem 5The inclusionsstrictly hold.
Proof It is clear that the inclusion holds. Further, since is strict, from Lemmas 1 and 2 from [19] 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.
Example 2 Let
then it follows that
From the last relation it follows that . □
Theorem 6The inclusionholds if and only iffor every sequence , where . Hereand .
Proof The proof of the theorem is similar to Theorem 3.3 given in [6]. □
Theorem 7
(1)
Iffor all , then the inclusionholds.
(2)
Iffor all , then the inclusionholds.
Proof (1) Let for all and . Then it follows that . Hence
we find such that , for every , respectively,
From the last relation we get .
(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 [1]. Also we will give some matrix transformations from into by using the matrix given in [4].
Theorem 8Letand . Define the matrixby
(18)
Then
(19)
(20)
Proof Let , , and . Then we have
(21)
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.
Corollary 2Letfor every , whereFis the collection of all finite subsets of ℕ. Then
(1)
,
(2)
,
for some constantM.
In what follows we will characterize the β- and γ-dual of the sequence space .
Theorem 9Let , . Define the sequence , , and the matrixby
where , . Then
(22)
(23)
Proof Let . Then we obtain
(24)
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.
Corollary 3Letbe a conjugate sequence of numbers , it means thatandfor all . Then
(1)
,
(2)
.
5 Some matrix transformations related to sequence space
In this section we will show some matrix transformations between the sequence space and sequence spaces , , , and where is a sequence of nondecreasing, bounded positive real numbers. Let be connected by the relation . For an infinite matrix , taking into consideration Theorem 3, we get
(25)
where
(26)
Let N, K be a finite subsets of the natural numbers ℕ and L, T be a natural numbers. and and also let be a conjugate sequence of numbers . Prior to giving the theorems, let us suppose that is a nondecreasing bounded sequence of positive real numbers and consider the following conditions:
(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
(36)
(37)
(38)
(39)
(40)
(41)
(42)
From the above conditions we get the following.
Theorem 10
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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Competing interests
The author declares to have no competing interests.