A Note on Some Strongly Sequence Spaces

and Applied Analysis 3 P:4 p x y ≤ p x p y , for all x, y ∈ X triangle inequality , P:5 if λn is a sequence of scalars, with λn → λ n → ∞ , and xn is a sequence of vectors with p xn −x → 0 n → ∞ , then p λnxn −λx → 0 n → ∞ continuity of multiplication by scalars . A complete linear metric space is said to be a Fréchet space. A Fréchet sequence space X is said to be an FK space, if its metric is stronger than the metric of w on X, that is, convergence in the sequence space X implies coordinatewise convergence the letters F and K stand for Fréchet and Koordinate, the German word for coordinate . Note that, by Ruckle in 5 , a modulus function f is a function from 0,∞ to 0,∞ such that i f x 0, if and only if, x 0, ii f x y ≤ f x f y , for all x, y ≥ 0, iii f increasing, iv f is continuous from the right at zero. Since |f x − f y | ≤ f |x − y| , it follows from condition iv that f is continuous on 0,∞ . Furthermore, from condition ii , we have f nx ≤ nf x for all n ∈ N, and thus f x f ( nx 1 n ) ≤ nf ( x n ) , 1.8 hence 1 n f x ≤ f ( x n ) , ∀n ∈ N. 1.9 In 5 , Ruckle used the idea of a modulus function f in order to construct a class of FK spaces L ( f ) {


Introduction
Let w be the set of all real or complex sequences and let l ∞ , c, and c 0 be the Banach spaces of bounded, convergent, and null sequences x x k , respectively, with the usual norm x sup n |x n |.
A sequence x x k ∈ l ∞ is said to be almost convergent if its Banach limit coincides.Let c denote the space of all almost convergent sequences.Lorentz 1 proved that c x ∈ l ∞ : lim m t mn x exist uniformly in n , 1.1 where The space c of strongly almost convergent sequences was introduced by Maddox 2 as c x ∈ l ∞ : lim m t mn |x − e| exist uniformly in n for some ∈ C , 1.3 where e 1, 1, . . . .

Abstract and Applied Analysis
Let σ be a one-to-one mapping from the set of positive integers into itself such that σ m n σ m−1 σ n , m 1, 2, 3, . .., where σ m n denotes the mth iterate of the mapping σ in n, see 3 .A continuous linear functional ϕ on l ∞ is said to be an invariant mean or a σ-mean, if and only if, i ϕ x ≥ 0, when the sequence x x n is such that x n ≥ 0 for all n, ii ϕ e 1, where e 1, 1, . . ., iii ϕ x σ n ϕ x , for all x ∈ l ∞ .
For a certain kind of mapping σ, every invariant mean ϕ extends the functional limit on the space c, in the sense that ϕ x lim x for all x ∈ c.Consequently, c ⊂ V σ , where V σ is the set of bounded sequences with equal σ-means.Schaefer 3 proved that where Thus we say that a bounded sequence x x k is σ-convergent, if and only if, x ∈ V σ such that σ k n / n for all n ≥ 0, k ≥ 1.Note that similarly as the concept of almost convergence leads naturally to the concept of strong almost convergence, the σ-convergence leads naturally to the concept of strong σ-convergence.
A sequence x x k is said to be strongly σ-convergent see, Mursaleen 4 , if there exists a number such that as k → ∞ uniformly in m.We write V σ to denote the set of all strong σ-convergent sequences and when 1.6 holds, we write V σ − lim x .Taking σ m m 1, we obtain V σ c .Then the strong σ-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 X be a linear space.A function p : X → R is called a paranorm, if P:1 p 0 ≥ 0, P:2 p x ≥ 0, for all x ∈ X, P:3 p −x p x , for all x ∈ X, P:4 p x y ≤ p x p y , for all x, y ∈ X triangle inequality , P:5 if λ n is a sequence of scalars, with λ n → λ n → ∞ , and x n is a sequence of vectors with p x n − x → 0 n → ∞ , then p λ n x n − λx → 0 n → ∞ continuity of multiplication by scalars .
A complete linear metric space is said to be a Fréchet space.A Fréchet sequence space X is said to be an FK space, if its metric is stronger than the metric of w on X, that is, convergence in the sequence space X implies coordinatewise convergence the letters F and K stand for Fréchet and Koordinate, the German word for coordinate .
Note that, by Ruckle in 5 , a modulus function follows from condition iv that f is continuous on 0, ∞ .Furthermore, from condition ii , we have f nx ≤ nf x for all n ∈ N, and thus In 5 , Ruckle used the idea of a modulus function f in order to construct a class of FK spaces From the definition, we can easily see that the space L f is closely related to the space l 1 , if we consider f x x for all real numbers x ≥ 0. Several authors study these types of spaces.For example, Maddox introduced and examined some properties of the sequence spaces w 0 f , w f and w ∞ f , defined by using a modulus f, which generalized the well-known spaces w 0 , w and w ∞ of strongly summable sequences, see 6 .Similarly, Savas ¸in 7 generalized the concept of strong almost convergence by using a modulus f and examined some further properties of the corresponding new sequence spaces.
The generalized de la Vallé-Poussin mean is defined by where I n n − λ n 1, n for n 1, 2, . ... Then a sequence x x k is said to be V, λsummable to a number L see 8 , if t n x → L as n → ∞, 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 λ n n, for n 1, 2, 3, . .., the sets V, λ 0 , V, λ , and V, λ ∞ reduce to the sets w 0 , w, and w ∞ , which were introduced and studied by Maddox, see 6 .
We also note that the sets of sequence spaces such as strongly σ-summable to zero, strongly σ-summable, and strongly σ-bounded with respect to the modulus function were defined by Nuray and Savas ¸in 9 .

Main Results
Let p p k be a sequence of real numbers such that p k > 0 for all k, and sup k p k < ∞.This assumption is made throughout the rest of this paper.Then we now write

2.1
In particular, if we take p k 1 for all k, we have

2.3
In particular, when p k p for all k, then we have the spaces which were introduced and studied by Malkowsky and Savas ¸in 10 .Further, when λ n n, for n 1, 2, 3, . .., the sets V , λ, f 0 and V , λ, f are reduced to c f and c 0 f respectively, see 7 .Now, if we consider f x x, then one can easily obtain

2.5
If p k 1 for all k, then we can obtain the spaces V σ , λ 0 , V σ , λ , and V σ , λ ∞ .Throughout this paper, we use the notation , and V, σ, λ, f, p ∞ are linear spaces over the complex field C. Lemma 2.1.Let f be any modulus.Then for all m, and so for all k and m, and so for all m and n.Thus x ∈ V σ , λ, f ∞ .This completes the proof.
The following well-known inequality 11 , page 190 will be used later.
for all k and a k , b k ∈ C.
In the following theorem, we prove x k → implies x k → ∈ V σ , λ, f, p and we also prove the uniqueness of the limit .To prove the theorem, we need the following lemma.
Note that no other relation between p k and q k is needed in Lemma 2.2.
. By the definition of modulus, we have a > 0. Now, from 2.9 and the definition of modulus, we have

2.12
From 2.11 and 2.12 , it follows that f a 0 and by the definition of modulus, we have a 0. Hence 1 2 and 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.V σ , λ, f, p 0 and V σ , λ, f, p are complete linear topological spaces, with paranorm g, where g is defined by where M max 1, {sup k p k } .