Generalized Vector-Valued Sequence Spaces Defined by Modulus Functions

Abstract

We introduce the vector-valued sequence spaces , , and , and , using a sequence of modulus functions and the multiplier sequence of nonzero complex numbers. We give some relations related to these sequence spaces. It is also shown that if a sequence is strongly -Cesàro summable with respect to the modulus function then it is -statistically convergent.

1. Introduction

Let be the set of all sequences real or complex numbers and , , and be, respectively, the Banach spaces of bounded, convergent, and null sequences with the usual norm , where , the set of positive integers.

The studies on vector-valued sequence spaces are done by Das and Choudhury [1], Et [2], Et et al. [3], Leonard [4], Rath and Srivastava [5], J. K. Srivastava and B. K. Srivastava [6], Tripathy et al. [7, 8], and many others.

Let be a sequence of seminormed spaces such that for each . We define

(1.1)

It is easy to verify that is a linear space under usual coordinatewise operations defined by and , where .

Let be a sequences of nonzero scalar. Then for a sequence space , the multiplier sequence space , associated with the multiplier sequence , is defined as .

The notion of a modulus was introduced by Nakano [9]. We recall that a modulus is a function from to such that

(i) if and only if ,

(ii) for ,

(iii) is increasing,

(iv) is continuous from the right at 0.

It follows that must be continuous everywhere on . Maddox [10] and Ruckle [11] used a modulus function to construct some sequence spaces.

After then some sequence spaces defined by a modulus function were introduced and studied by Bilgin [12], Pehlivan and Fisher [13], Waszak [14], Bhardwaj [15], Altın [16], and many others.

The notion of difference sequence spaces was introduced by Kızmaz [17] and it was generalized by Et and Çolak [18]. Let be a fixed positive integer. Then we write

(1.2)

for , and , where , , and so we have

(1.3)

2. Main Results

In this section, we prove some results involving the sequence spaces , , and .

Definition 2.1.

Let be a sequence of seminormed spaces such that for each , a sequence of strictly positive real numbers, a sequence of seminorms, a sequence of modulus functions, and any fixed sequence of nonzero complex numbers . We define the following sequence spaces:

(2.1)

Throughout the paper will denote any one of the notation 0,1 or .

If and for all , we will write instead of .

If and for all , we will write instead of .

If , we say that is strongly -Cesàro summable with respect to the modulus function and we will write and will be called -limit of with respect to the modulus .

The proofs of the following theorems are obtained by using the known standard techniques; therefore, we give them without proofs.

Theorem 2.2.

Let the sequence be bounded. Then the spaces are linear spaces.

Theorem 2.3.

Let be a modulus function and the sequence be bounded; then

(2.2)

and the inclusions are strict.

Theorem 2.4.

is a paranormed (need not total paranorm) space with

(2.3)

where .

Theorem 2.5.

Let and be any two sequences of modulus functions. For any bounded sequences and of strictly positive real numbers and for any two sequences of seminorms and , we have

(i);

(ii);

(iii);

(iv)If is stronger than for each , then ;

(v)If equivalent to for each , then ;

(vi).

Proof.

• (i) We will only prove (i) for and the other cases can be proved by using similar arguments. Let and choose with such that for and for all . Write and consider

  

  (2.4)

where the first summation is over and second summation is over . Since is continuous, we have

(2.5)

By the definition of , we have for ,

(2.6)

Hence

(2.7)

From (2.5) and (2.7), we obtain .

The following result is a consequence of Theorem 2.5(i).

Corollary 2.6.

Let be a modulus function. Then .

Theorem 2.7.

Let and be bounded; then .

Proof.

If we take for all and using the same technique of Theorem  5 of Maddox [19], it is easy to prove the theorem.

Theorem 2.8.

Let be a modulus function; if , then .

Proof.

Omitted.

3. -Statistical Convergence

The notion of statistical convergence were introduced by Fast [20] and Schoenberg [21], independently. Over the years and under different names, statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory, and number theory. Later on it was further investigated from the sequence space point of view and linked with summability theory by Šalát [22], Fridy [23], Connor [24], Mursaleen [25], Işik [26], Malkowsky and Sava [27], and many others. In recent years, generalizations of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on locally compact spaces. Statistical convergence and its generalizations are also connected with subsets of the Stone-Čech compactification of the natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability. The notion depends on the density of subsets of the set of natural numbers.

A subset of is said to have density positive integers which is defined by if

(3.1)

where is the characteristic function of . It is clear that any finite subset of have zero natural density and .

In this section, we introduce -statistically convergent sequences and give some inclusion relations between -statistically convergent sequences and -summable sequences.

Definition 3.1.

A sequence is said to be -statistically convergent to if for every ,

(3.2)

In this case, we write . The set of all -statistically convergent sequences is denoted by . In the case , we will write instead of .

Theorem 3.2.

Let be a modulus function; then

(i)If , then ;

(ii)If and , then ;

(iii),

where .

Proof.

Omitted.

In the following theorems, we will assume that the sequence is bounded and .

Theorem 3.3.

Let be a modulus function. Then .

Proof.

Let and let be given. Let and denote the sums over with and , respectively. Then

(3.3)

Hence .

Theorem 3.4.

Let be bounded; then .

Proof.

Suppose that is bounded. Let and and be denoted in previous theorem. Since is bounded, there exists an integer such that , for all . Then

(3.4)

Hence .

Theorem 3.5.

if and only if is bounded.

Proof.

Let be bounded. By Theorems 3.3 and 3.4, we have .

Conversely suppose that is unbounded. Then there exists a sequence of positive numbers with , for . If we choose

(3.5)

then we have

(3.6)

for all and so , but for , and for all . This contradicts to .