ABSTRACT

In 2000, Kostyrko, Salat, and Wilczynski introduced and studied the concept of I-convergence of sequences in metric spaces where I is an ideal. The concept of I-convergence has a wide application in the field of Number Theory, trigonometric series, summability theory, probability theory, optimization and approximation theory. In this article we introduce the double sequence spaces and for a modulus function f and study some of the properties of these spaces.

1. Introduction

The notion of I-Convergence is a generalization of the concept statistical convergence which was first introduced by H. Fast [1] and later on studied by J. A. Fridy [2,3] from the sequence space point of view and linked it with the summability theory. At the initial stage I-Convergence was studied by Kostyrko, Salat and Wilezynski [4]. Further it was studied by Salat, Tripathy, Ziman [5] and Demirci [6]. Throughout a double sequence is denoted by Also a double sequence is a double infinite array of elements for all The inital works on double sequences is found in Bromwich [7], Basarir and Solancan [8] and many others.

2. Definitions and Preliminaries

Throughout the article and denotes the set of natural, real, complex numbers and the class of all sequences respectively.

Let X be a non empty set. A set (denoting the power set of X) is said to be an ideal if I is additive i.e and hereditary i.e..

A non-empty family of sets is said to be filter on X if and only if, for we have and for each and implies.

An Ideal is called non-trivial if.

A non-trivial ideal is called admissible if.

A non-trivial ideal I is maximal if there cannot exist any non-trivial ideal containing I as a subset.

For each ideal I, there is a filter corresponding to I.

i.e., where.

The idea of modulus was structured in 1953 by Nakano (See [9]).

A function is called a modulus if

(1) if and only if(2) for all(3) is nondecreasing, and

(4) is continuous from the right at zero.

Ruckle [10] used the idea of a modulus function to construct the sequence space

This space is an FK space , and Ruckle[10] proved that the intersection of all such spaces is, the space of all finite sequences.

The space X(f) is closely related to the space which is an X(f) space with for all real. Thus Ruckle [11] proved that, for any modulus.

where

The space is a Banach space with respect to the norm

(See [10]).

Spaces of the type are a special case of the spaces structured by B. Gramsch in [12]. From the point of view of local convexity, spaces of the type are quite pathological. Therefore symmetric sequence spaces, which are locally convex have been frequently studied by D. J. H. Garling [13,14], G. Kothe [15] and W. H. Ruckle [10,16].

Definition 2.1. A sequence space E is said to be solid or normal if implies for all sequence of scalars with for all

(see [17])

Definition 2.2. Let

and E be a double sequence space. A -step space of is a sequence space

Definition 2.3. A cannonical preimage of a sequence is a sequence defined as follows

(see [18]).

Definition 2.4. A sequence space E is said to be monotone if it contains the cannonical preimages of all its stepspaces (see [19]).

Definition 2.5. A sequence space E is said to be convergence free if, whenever and implies.

Definition 2.6. A sequence space E is said to be a sequence algebra if whenever

.

Definition 2.7. A sequence space E is said to be symmetric if whenever where

and is a permutation on N.

Definition 2.8. A sequence is said to be I-convergent to a number L if for every.. In this case we write I-lim.

The space of all I-convergent sequences to is given by

Definition 2.9. A sequence is said to be I-null if. In this case we write I-lim.

Definition 2.10. A sequence is said to be I-cauchy if for every there exists a number and such that

.

Definition 2.11. A sequence is said to be I-bounded if there exists such that

Definition 2.12. A modulus function is said to satisfy condition if for all values of u there exists a constant such that for all values of.

Definition 2.13. Take for I the class of all finite subsets of. Then is a non-trivial admissible ideal and convergence coincides with the usual convergence with respect to the metric in X (see [4]).

Definition 2.14. For and with respectively. is a non-trivial admissible ideal, -convergence is said to be logarithmic statistical convergence (see [4]).

Definition 2.15. A map defined on a domain i.e. is said to satisfy Lipschitz condition if where K is known as the Lipschitz constant. The class of K-Lipschitz functions defined on D is denoted by (see [20]).

Definition 2.16. A convergence field of I-convergence is a set

The convergence field is a closed linear subspace of with respect to the supremum norm, (See [5]).

Define a function such that

, for all, then the function is a Lipschitz function (see [20]).

(c.f [18,20-30])

Throughout the article and represent the bounded, I-convergent, I-null, bounded I-convergent and bounded I-null sequence spaces respectively.

In this article we introduce the following classes of sequence spaces.

We also denote by

and

The following Lemmas will be used for establishing some results of this article.

Lemma (1) Let E be a sequence space. If E is solid then E is monotone.

Lemma (2) Let and. If, then

Lemma (3) If and. If, then.

3. Main Results

Theorem 3.1. For any modulus function f, the classes of sequences and are linear spaces.

Proof: We shall prove the result for the space.

The proof for the other spaces will follow similarly.

Let and let be scalars. Then

That is for a given, we have

(1)

(2)

Since f is a modulus function, we have

Now, by (1) and (2),

Therefore

Hence is a linear space.

Theorem 3.2. A sequence is I-convergent if and only if for every there exists such that

(3)

Proof: Suppose that. Then

Fix an. Then we have

which holds for all.

Hence

Conversely, suppose that

That is

for all. Then the set

Let. If we fix an then we have as well as

Hence This implies that

that is

that is

where the diam of N denotes the length of interval N.

In this way, by induction we get the sequence of closed intervals

with the property that for

and

for

.

Then there exists a where such that. So that, that is.

Theorem 3.3. Let and be modulus functions that satisfy the -condition.If is any of the spaces and etc, then the following assertions hold.

(i),

(ii).

Proof: (i) Let. Then

(4)

Let and choose with such that for.

Write and consider

We have

(5)

For, we have. Since f is non-decreasing,it follows that

Since satisfies the -condition, we have

Hence

(6)

From (4), (5) and (6), we have.

Thus. The other cases can be proved similarly.

(ii) Let. Then

and

Therefore

which implies that is

Corollary 3.4. for and

Proof: The result can be easily proved using for.

Theorem 3.5. The spaces and are solid and monotone.

Proof: We shall prove the result for. Let. Then

(7)

Let be a sequence of scalars with for all. Then we have

which implies that.

Therefore the space is solid. The space

is monotone follows from Lemma (1). For

the result can be proved similarly.

Theorem 3.6. The spaces and are neither solid nor monotone in general.

Proof: Here we give a counter example.

Let and for all Consider the K-step space of X defined as follows, Let and let be such that

Consider the sequence defined by for all.

Then but its K-stepspace preimage does not belong to Thus is not monotone. Hence is not solid.

Theorem 3.7. The spaces and are sequence algebras.

Proof: We prove that is a sequence algebra.

Let. Then

and

Then we have

Thus is a sequence algebra.

For the space, the result can be proved similarly.

Theorem 3.8. The spaces and are not convergence free in general.

Proof: Here we give a counter example.

Let and for all. Consider the sequence and defined by

Then and, but

and.

Hence the spaces and are not convergence free.

Theorem 3.9. If I is not maximal and, then the spaces and are not symmetric.

Proof: Let be infinite and for all

If

Then by Lemma (3) we have.

Let be such that and.

Let and be bijections, then the map defined by

is a permutation on, but and.

Hence and are not symmetric.

Theorem 3.10. Let f be a modulus function. Then and the inclusions are proper.

Proof: The inclusion is obvious.

Let Then there exists such that

We have

Taking the supremum over on both sides we get.

Next we show that the inclusion is proper.

(i)

Let then for some, which implies Hence the inclusion is proper.

(ii) Let then

Therefore, and hence the inclusion is proper.

Theorem 3.11. The function is the Lipschitz function, where

, and hence uniformly continuous.

Proof: Let. Then the sets

Thus the sets,

Hence also, so that.

Now taking in,

Thus is a Lipschitz function. For the result can be proved similarly.

Theorem 3.12. If, then

and.

Proof: For

Now,

(8)

As, there exists an such that and.

Using Equation (8) we get

For all. Hence

and.

For the result can be proved similarly.

4. Acknowledgements

The authors would like to record their gratitude to the reviewer for his careful reading and making some useful corrections which improved the presentation of the paper.

REFERENCES