International Journal of Analysis

Volume 2014 (2014), Article ID 684512, 4 pages

http://dx.doi.org/10.1155/2014/684512

Research Article

A New Double Sequence Space Defined by a Double Sequence of Modulus Functions

Birsen Sağır, Cenap Duyar, and Oğuz Oğur

Department of Mathematics, Art and Science Faculty, Ondokuz Mayıs University, Kurupelit Campus, 55139 Samsun, Turkey

Received 1 November 2013; Revised 8 January 2014; Accepted 9 January 2014; Published 11 February 2014

Academic Editor: Shamsul Qamar

Copyright © 2014 Birsen Sağır et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this work we introduce new spaces of double sequences defined by a double sequence of modulus functions, and we study some properties of this space.

1. Introduction

In this work, by and , we denote the spaces of single complex sequences and double complex sequences, respectively. and denote the set of positive integers and complex numbers, respectively. If, for all , there is such that where and , then a double sequence is said to be converge (in terms of Pringsheim) to . A real double sequence is nondecreasing, if for . A double series is infinity sum and its convergence implies the convergence by of partial sums sequence , where (see [1–3]).

For , denote the space of sequences such that

(see [4]).

A double sequence is said to be bounded if and only if . The space of all bounded double sequences is denoted by . It is known that is Banach space (see [5, 6]).

A double sequence space is said to be normal if whenever for all and .

The double sequence spaces in the various forms were introduced and studies by Khan and Tabassum in [7–14], by Khan in [15], and by Khan et al. in [16, 17].

Now let be a family of subsets having most elements in . Also denote the class of subsets in such that the elements of and are most and , respectively. Besides is taken as a nondecreasing double sequence of the positive real numbers such that

(see [18]).

Let be a double sequence. A set is defined by

A double sequence space is said to be symmetric if and whenever and .

A BK-space is a Banach sequence space in which the coordinate maps are continuous.

A function is said to be a modulus function if it satisfies the following:

(1) if and only if ;(2) for all ;(3) is increasing;(4) is continuous from right at .

It follows that is continuous on . The modulus function may be bounded or unbounded. For example, if we take , then is bounded. But, for , is not bounded.

The BK-spaces , introduced by Sargent in [19], is in the form

Sargent studied some properties of this space and examined relationship between this space and -space.

The space was extended to by Tripathy and Sen [20] as follows:

Recently, Raj et al. [21] introduced and studied the following sequence space .

Let be a sequence of modulus functions. Then the space is defined by

In this work we introduce sequence spaces defined by

where is a double sequence of modulus functions.

2. Main Results

The result stated in the first theorem is not hard. So, we give it without proof.

Theorem 1. The sequence space is a linear space.

Theorem 2. The sequence spaces are complete.

Proof. Let be a double Cauchy sequence in such that for all . Then

for some and for all . For each , there exists a positive integer such that

for all . Hence

for some and for all . This implies that

for all and for each fixed . Hence is a Cauchy sequence in .

Then, there exists such that as and let us define . From (10), for each fixed ,

for some , for all and .

Letting , we get

for some , for all , and . Thus we obtain

for some and for all . This implies that for all .

Hence . By (14),

for all . This means that as . Hence is a Banach space.

Theorem 3. The space is a BK-space.

Proof. Suppose that with as . For each there exists such that

for all . Thus

for some and for all . Hence we obtain

for all and for all . This implies as . This completes the proof.

Corollary 4. The space is a symmetric space.

Proof. Let and let . Then we can write . Thus we obtain

Corollary 5. The space is a normal space.

Proof. It is obvious.

Theorem 6. Consider

Proof. Suppose that . Then we have

Thus for each fixed and for ,

for some . Hence

for some . This implies that . Hence .

Theorem 7. if and only if .

Proof. Let . Then for all . If , then

for some . Thus

for some . Hence . This shows that . Conversely, let . We define . Let . Then there exists a subsequence of such that as . Then for we have

for some . This is a contradiction as and this completes the proof.

Proposition 8. Consider

Proof. Clearly, , where for when and by nondecreasing . Then, by Theorem 7, first inclusion is obtained. Suppose . Then

for some . Hence we obtain

for all . Thus and proof is completed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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