Abstract

We introduce a new sequence space which is defined by the operator on the sequence space . We define a modular functional on this space and investigate structure of this space equipped with Luxemburg norm. Also we study some geometric properties which are called Kadec-Klee, k-NUC, and uniform Opial properties and prove that this new space possesses these properties.

1. Introduction

In literature, there are many papers about geometric properties and their applications on different sequence spaces. Some of them are as follows.

In [1], Opial defined the Opial property with his name mentioned and he proved that satisfies this property but the space does not.

Franchetti [2] has shown that any infinite dimensional Banach space has an equivalent norm satisfying the Opial property. Later, Prus [3] has introduced and investigated uniform Opial property for Banach spaces.

In [4], the notion of nearly uniform convexity for Banach spaces was introduced by Huff. Also Huff proved that every nearly uniformly convex space is reflexive and it has uniform Kadec-Klee property. However, Kutzarova [5] defined -nearly uniformly convex Banach spaces.

Shue [6] first defined Cesaro sequence spaces with norm. In [7], it is shown that the Cesaro sequence spaces have Kadec-Klee and local uniform rotundity properties.

In [8], it was shown that Banach-Saks of type- property holds in these spaces.

Later, Sanhan and Suantai [9] generalized the normed sequence spaces to the paranormed sequence spaces. He showed that the sequence spaces equipped Luxemburg norm are rotund and have Kadec-Klee property.

Petrot and Suantai [10] studied the uniform Opial property of these spaces. In [9], Sanhan and Suantai have showed that the Cesaro sequence space , where the sum runs over , equipped with Luxemburg norm has property but it is not rotund.

Karakaya [11] introduced a new sequence space involving lacunary sequences connected with Cesaro sequence space and examined some geometric properties of this space equipped with Luxemburg norm. In [12], Karaka et al. defined and studied a new difference sequence space involving lacunary sequences by using difference operator.

In [13], Khan and Rahman introduced sequence spaces . Afterwards, Mursaleen and Khan [14] generalized this space to the vector-valued sequence space. In the space , if we specialize for all , then we get defined in [9].

In [15], imek and Karakaya generalized sequence space to vector-valued space and investigated some topological and geometrical properties as Kadec-Klee and rotund according to Luxemburg norm of this space.

In [16], Sava et al. introduced an -type new sequence space and examined some geometrical properties of this space concerning Banach-Saks of type- and Gurarii’s modulus of convexity. Also, in [17], imek et al. investigated the k-nearly uniform convexity (k-NUC) property and some fixed point results in modular space ; imek and Karakaya [18] introduced modular sequence space obtained from paranormed ones by generalized weighted means on Köthe sequence spaces and investigated Kadec-Klee property of this space.

2. Preliminaries and Notation

Let (for the brevity ) be a normed linear space and let (resp. ) be the closed unit ball (resp. unit sphere) of . The space of all real sequences is denoted by . For any sequence in , we denote by the convex hull of the elements of .

A Banach space is called uniformly convex (UC) if for each , there is such that, for , the inequality implies that Recall that for a number a sequence is said to be an -seperated sequence if A Banach space is said to have the Kadec-Klee property (H property) if every weakly convergent sequence on the unit sphere is convergent in norm.

A Banach space is said to have the uniform Kadec-Klee property (UKK) if for every there exists such that if is the weak limit of a normalized -separated sequence, then (see [4]). We have that every (UKK) Banach space has the Kadec-Klee property.

A Banach space is said to be the nearly uniformly convex (NUC) if for every there exists such that, for every sequence with , we have Let be an integer. A Banach space is said to be -nearly uniformly convex (-NUC) if for any there exists such that, for every sequence with , there are such that Of course a Banach space is (NUC) whenever it is (-NUC) for some integer . Clearly, (-NUC) Banach spaces are (NUC) but the opposite implication does not hold in general (see [5]).

A Banach space is said to have the Opial property if every sequence that is weakly convergent to satisfies for every and (see [1]).

A Banach space is said to have the uniform Opial property if every there exists such that, for each weakly null sequence and with , we have (see [3]) A point is called an extreme point if for any the equality implies that . A Banach space is said to be rotund (abbreviated as ) if every point of is an extreme point. A Banach space is said to be fully -rotund (written as ) (see [19]) if for every sequence implies that is convergent.

It is well known that (UC) implies and implies , and spaces are reflexive and rotund, and it is easy to see that (-NUC) implies .

For a real vector space , a function is called a modular if it satisfies the following conditions:(i),(ii) for all with ,(iii) for all and all with . Further, the modular is called convex if(iv) holds for all and all with .

For any modular on , the space is called the modular space.

A sequence of elements of is called modular convergent to if there exists a such that as . If is a convex modular, then the following formula defines a norm on which is called the Luxemburg norm:

A modular is said to satisfy the -condition if for any there exist constants and such that for all with .

If satisfies the -condition for all with dependent on , we say that satisfies the strong -condition .

Lemma 1. If , then, for any and , there exists such that whenever with and .

Proof. See [20].

Lemma 2. If , then, for any , there exists such that whenever .

See [21].

Lemma 3. If , then for any See [20].

Lemma 4. If , then for any there exists such that whenever .

See [20].

In this paper, we will need the following inequalities in the sequel: for .

In [22], Polat et al. defined the matrix by Here, for all , , , and depend on ; depend on .

Using this matrix, we define sequence space as follows: where is back difference, and is forward difference. Throughout this study, is a bounded sequence of positive real numbers; and . We denote for short in proof.

Theorem 5. The sequence space is a complete metric space of nonabsolute type with respect to the paranorm defined by

Proof. The linearity of with respect to the coordinatewise and scalar multiplication follows from the following inequalities which are satisfied for : and for any It is clear that and for all . From (17), it can be seen the subadditivity of and .
Let be any sequence in such that and are any sequence of scalars such that . Then, since the inequality holds, the subadditivity of , , is bounded and thus we have which tends to zero as . Therefore, the scalar multiplication is continuous. Hence, is a paranorm on the space . It remains to prove the completeness of the space . Let be any Cauchy sequence in the space . Then, for a given , there exists a positive integer such that for all . Using definition of , we obtain for each fixed that for every which leads us to the fact that is a Cauchy sequence of real numbers for every fixed . Since is complete, as . Using these infinitely many limits , we may write the sequence .
For all and every fixed Now, we have to show . To do this, we have Hence, we get . As a result is a complete metric space.

We introduce a modular sequence space by The Luxemburg norm on the sequence space is defined as follows: Here, the modular defined by is a convex modular on .

3. Main Results

In this section, we will give some basic properties of the modular on the space . Also, we will investigate some relationships between the modular and the Luxemburg norm on . Finally, we study some geometric properties on this space.

Let us start with some lemmas which will be used in the proof of the theorems about geometric properties of this space.

Lemma 6. The functional is a convex modular on .

Proof. Let . It is obvious that(i).(ii) for all scalar with (iii)For with , by the convexity of for every , we have For , the modular on satisfies the following properties:

(i)if , then and ,(ii)if , then ,(iii)if , then .

Proof. It can be proved with standard techniques in a similar way as in [23].

Lemma 7. For any ,(i)if , then ,(ii)if , then ,(iii),(iv)if , then ,(v)if , then .

Proof. It can be proved with standard techniques in a similar way as in [23].

Lemma 8. Let be a sequence in :(i)if , then ,(ii)if , then .

Proof. It can be proved with standard techniques in a similar way as in [23].

Lemma 9. For any and , there exists such that whenever with and .

Proof. Since is bounded, it is easy to see that . Hence, the lemma is obtained directly from Lemma 1.

Lemma 10. For any sequence ,

Proof. Since , the lemma is obtained directly from Lemma 3.

Lemma 11. For any and , there exists such that implies .

Proof. Since , the lemma is obtained directly from Lemma 2.

Now we will show that the is a Banach space with respect to the Luxemburg norm

Theorem 12. The space is a Banach space with respect to the Luxemburg norm defined by

Proof. We will show that every Cauchy sequence in is convergent according to the Luxemburg norm. Let be a Cauchy sequence and . Thus, there exists such that for all . By the Lemma 8, we obtain for all ; that is, For fixed we get that Hence, we obtain that the sequence is a Cauchy sequence in . Since the real number is complete, as . Therefore, for fixed and So, we obtain that for all and as So, for all from Lemma 8, It can be seen that, for all , and .
From the linearity of the sequence space , we can write that Hence, the sequence space is a Banach space with respect to the Luxemburg norm. This completes the proof of the theorem.