On Some I-Convergent Double Sequence Spaces Defined by a Modulus Function

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     2 0 2 , I I c f c f and   2 I l f  for a modulus function f and study some of the properties of these spaces.


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   .ij x x  Also a double sequence is a double infinite array of elements kl x   for all , k l .  The inital works on double sequences is found in Bromwich [7], Basarir and Solancan [8] and many others.

Definitions and Preliminaries
Throughout the article and , , IN IR   denotes the set of natural, real, complex numbers and the class of all sequences respectively.
Let X be a non empty set.A set ( 2 denoting the power set of X) is said to be an ideal if I is additive i.e    .A non-trivial ideal I is maximal if there cannot exist any non-trivial ideal J I   containing I as a subset.
For each ideal I, there is a filter   £ I corresponding to I.
i.e.     The idea of modulus was structured in 1953 by Nakano (See [9]).
A function ) f is continuous from the right at zero.Ruckle [10] used the idea of a modulus function f to construct the sequence space


This space is an FK space , and Ruckle [10] proved that the intersection of all such   X f 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

 
X f is a Banach space with respect to the norm  (See [10]).

 
X f 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   X f 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 , : , ; and and E be a double sequence space.A -step space of is a sequence space ,for 0, otherwise.

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 Definition 2.7.A sequence space E is said to be symmetric if I is a non-trivial admissible ideal and f I convergence coincides with the usual convergence with respect to the metric in X (see [4]).

 
A  0  respectively.I  is a non-trivial admissible ideal, I  -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]).
We also denote by 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.

Main Results
Theorem 3.1.For any modulus function f, the classes of sequences and , , Proof: We shall prove the result for the space .
 

I c f
The proof for the other spaces will follow similarly.
and let ,   be scalars.Then lim 0, for some ; lim 0, for some ; That is for a given , we have Since f is a modulus function, we have Now, by ( 1) and ( 2), .

is I-convergent if and only if for every there exists
, :  .Then we have 2 2 , : Conversely, suppose that , : where the diam of N denotes the length of interval N.
In this way, by induction we get the sequence of closed intervals 0 1 .
with the property that Then there ex a ists f and g be modulus functions that satisfy the 2  -condition.If X is any of the spaces Let and choose 0 and consider We have For ij y   , we have 1 From ( 4), ( 5) and ( 6), we have     Th an be proved si

 
X X f  for Let   ij  be a sequence of scalars with solid.The space

 
I f is monotone follows from Lemma (1).For ither solid nor monotone in .Proof: Here we give a counter exampl general t solid. is no and Theorem 3. 7 paces

   
2 0 2 Taking th h e supremum over on bot sides we ge , and hence uniformly con- , : .

4.
The authors would like to record their gratit e to the reviewer for his careful reading and making some useful co s which improved the tion of the paper.lt can be proved si A non-empty family of sets is said to be filter on X if and only if sent the bounded, I-convergent, I-null, bounded I-convergent and bounded I-null sequence spaces respectively.
no example.tconvergence free in generaProof: Here we give a counter l. ij