Generalized Cesàro Summable Difference Sequence Spaces

Et (M. Et, On some generalized Cesàro difference sequence spaces, İstanbul Üniv. Fen Fak. Mat. Derg. 55/56 (1996/97), 221–229) introduced Cesàro difference sequence spaces Cp(∆) (1 ≤ p < ∞) ,C∞(∆) and studied some topological properties. In this paper we continue to examine others relations with related the sequence spaces C1(∆) and C∞(∆).


Introduction
Let w be the set of all sequences of real or complex numbers and ∞ , c and c 0 be respectively the Banach spaces of bounded, convergent and null sequences x = (x k ) with the usual norm x ∞ = sup |x k | , where k ∈ N = {1, 2, . ..} , the set of positive integers.Also by bs, cs, 1 and p ; we denote the spaces of all bounded, convergent, absolutely summable and p−absolutely summable series, respectively.
A sequence space X with a linear topology is called a K−space provided each of the maps p i : X → C defined by p i (x) = x i is continuous for each i ∈ N, where C denotes the complex field.A K−space X is called an FK−space provided X is a complete linear metric space.An FK−space whose topology is normable is called a BK−space.We say that an FK−space X has AK (or has the AK property), if (e k ) ( the sequence of unit vectors) is a Schauder bases for X.
Study of difference sequence spaces is a recent development in the summability theory.The approach of using differences of terms does not help for all the problems involving sequences.However, for many sequences, the use of differences can be very profitable and, moreover, the differences themselves prove extremely valuable in many other contexts.Sometimes a situation may arise that we have a sequence at hand and we are interested in sequences formed by its successive differences and in the structure of these new sequences.
The notion of difference sequence spaces was introduced by Kızmaz [16] and the notion was generalized by Et and Çolak [11].Later on Et and Nuray [12] generalized these sequence spaces to the following sequence spaces.Let X be any sequence space and let m be a non-negative integer.Then, The sequence spaces ∆ m (X) are Banach spaces normed by e-mail: mikailet68@gmail.com Let X be any sequence space, if x ∈X (∆ m ) then there exists one and only one y = (y k ) ∈ X such that for sufficiently large k, for instance k > 2m.Recently, a large amount of work has been carried out by many mathematicians regarding various generalizations of difference sequence spaces.For a detailed account of difference sequence spaces one may refer to ([1-12], [14][15]).
The Cesàro sequence spaces Ces p and Ces ∞ have been introduced by Shiue [19].Jagers [13] has determined the Köthe duals of the sequence space Ces p (1 < p < ∞).It can be shown that the inclusion p ⊂ Ces p is strict for 1 < p < ∞.Later on the Cesàro sequence spaces X p and X ∞ of non-absolute type are defined by Ng and Lee ( [17], [18]).

Topological Properties of
In this section we prove some results involving the sequence spaces C 1 (∆ m ) and C ∞ (∆ m ).
Definition 1 Let m be a non-negative integer.We define the following sequence spaces: The above sequence spaces contain some unbounded sequences for m ≥ 1, for example let Proof.Prof follows from Theorem 2.2 of Et and Nuray [12].
) and the inclusion is strict.
Proof.Let x ∈ ∞ (∆ m−1 ), then there exists M > 0 such that ∆ m−1 x k ≤ M, for all k ∈ N. On the other hand, we can write Theorem 4 The sequence spaces C 1 (∆ m−1 ) and c(∆ m ) overlap without containing each other, but c(∆ m ) ⊂ C 1 (∆ m ) and the inclusion is strict.
Proof.Proof is similar to that of Theorem 3.
The proof of the following result is easy, so we state without proof.
Proof.Proof follows from the fact that C 1 (∆ m ) is a proper and complete subspace of C ∞ (∆ m ).
Theorem 7 C ∞ (∆ m ) is not separable, in general.
Proof.Proof follows from the fact that if a normed space has a Schauder basis, then it is separable.