A Note on Some Strongly Sequence Spaces
We introduce and study new sequence spaces which arise from the notions of generalized de la Vallée-Poussin means, invariant means, and modulus functions.
Let be the set of all real or complex sequences and let , , and be the Banach spaces of bounded, convergent, and null sequences , respectively, with the usual norm .
A sequence is said to be almost convergent if its Banach limit coincides. Let denote the space of all almost convergent sequences. Lorentz  proved that where
The space of strongly almost convergent sequences was introduced by Maddox  as where .
Let be a one-to-one mapping from the set of positive integers into itself such that , , where denotes the th iterate of the mapping in , see . A continuous linear functional on is said to be an invariant mean or a -mean, if and only if, (i), when the sequence is such that for all ,(ii), where ,(iii), for all .
For a certain kind of mapping , every invariant mean extends the functional limit on the space , in the sense that for all . Consequently, , where is the set of bounded sequences with equal -means. Schaefer  proved that where Thus we say that a bounded sequence is -convergent, if and only if, such that for all , . Note that similarly as the concept of almost convergence leads naturally to the concept of strong almost convergence, the -convergence leads naturally to the concept of strong -convergence.
A sequence is said to be strongly -convergent (see, Mursaleen ), if there exists a number such that as uniformly in . We write to denote the set of all strong -convergent sequences and when (1.6) holds, we write . Taking , we obtain . Then the strong -convergence generalizes the concept of strong almost convergence. We also note that It is also well known that the concept of paranorm is closely related to linear metric spaces. In fact, it is a generalization of absolute value. Let be a linear space. A function is called a paranorm, if (P:1), (P:2), for all , (P:3), for all , (P:4), for all (triangle inequality), (P:5) if is a sequence of scalars, with , and is a sequence of vectors with , then (continuity of multiplication by scalars).
A complete linear metric space is said to be a Fréchet space. A Fréchet sequence space is said to be an space, if its metric is stronger than the metric of on , that is, convergence in the sequence space implies coordinatewise convergence (the letters and stand for Fréchet and Koordinate, the German word for coordinate).
Note that, by Ruckle in , a modulus function is a function from to such that (i), if and only if, , (ii), for all , (iii) increasing, (iv) is continuous from the right at zero.
Since , it follows from condition that is continuous on . Furthermore, from condition (ii), we have for all , and thus hence In , Ruckle used the idea of a modulus function in order to construct a class of spaces From the definition, we can easily see that the space is closely related to the space , if we consider for all real numbers . Several authors study these types of spaces. For example, Maddox introduced and examined some properties of the sequence spaces , and , defined by using a modulus , which generalized the well-known spaces , and of strongly summable sequences, see . Similarly, Savaş in  generalized the concept of strong almost convergence by using a modulus and examined some further properties of the corresponding new sequence spaces.
The generalized de la Vallé-Poussin mean is defined by where for . Then a sequence is said to be - summable to a number (see ), if as , and we write for the sets of sequences that are, respectively, strongly summable to zero, strongly summable, and strongly bounded by the de la Vallé-Poussin method. In the special case where , for , the sets , , and reduce to the sets , , and , which were introduced and studied by Maddox, see .
We also note that the sets of sequence spaces such as strongly -summable to zero, strongly -summable, and strongly -bounded with respect to the modulus function were defined by Nuray and Savaş in .
2. Main Results
Let be a sequence of real numbers such that for all , and . This assumption is made throughout the rest of this paper. Then we now write In particular, if we take for all , we have Similarly, when , then , and are reduced to In particular, when for all , then we have the spaces which were introduced and studied by Malkowsky and Savaş in . Further, when , for , the sets and are reduced to and respectively, see . Now, if we consider , then one can easily obtain If for all , then we can obtain the spaces , , and . Throughout this paper, we use the notation instead of .
If , then it is clear that , , and are linear spaces over the complex field .
Lemma 2.1. Let be any modulus. Then
Proof. Let . Then there is a constant such that for all , and so . Let . Then there is a constant such that for all and , and so for all and . Thus . This completes the proof.
If , with as uniformly in , then we write .
The following well-known inequality (, page 190) will be used later.
If and , then for all and .
In the following theorem, we prove implies and we also prove the uniqueness of the limit . To prove the theorem, we need the following lemma.
Lemma 2.2 (see ). Let , . Then , if and only if, , where .
Note that no other relation between and is needed in Lemma 2.2.
Theorem 2.3. Let . Then implies . Let . If , then is unique.
Proof. Let . By the definition of modulus, we have . Since , it follows from the above lemma that and consequently, .
Let . Suppose that , and . Now, from (2.9) and the definition of modulus, we have Hence, Further, as and, therefore, From (2.11) and (2.12), it follows that and by the definition of modulus, we have . Hence and this completes the proof.
(i) Let . Then,
(ii) Let . Then,
Proof. (i) Let . Since , we get
and hence .
(ii) Let for each , and . Let . Then, for each , , there exists a positive integer such that for all . This implies that Therefore, . This completes the proof.
Finally, we conclude this paper by stating the following theorem. We omit the proof, since it involves routine verification and can be obtained by using standard techniques.
Theorem 2.5. and are complete linear topological spaces, with paranorm , where is defined by where .
The authors express their sincere thanks to the referee(s) for careful reading of the paper and several helpful suggestions.
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