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.