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 ![](https://www.scirp.org/html/6-8101912\8c9ccfeb-e6ba-4af9-9a0f-0cc0ddfab6a7.jpg)
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
![](https://www.scirp.org/html/6-8101912\ee55230c-3cfb-4898-ad8e-1dfbe3bb11a7.jpg)
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
.
![](https://www.scirp.org/html/6-8101912\606802dc-0200-4771-a25d-240fb37ac566.jpg)
where
![](https://www.scirp.org/html/6-8101912\849a48e4-180b-493d-b1b0-2739665cae0a.jpg)
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 ![](https://www.scirp.org/html/6-8101912\ff34a2d3-26dd-4fef-af4f-b1eb3ed9a57f.jpg)
(see [17])
Definition 2.2. Let
![](https://www.scirp.org/html/6-8101912\bd107daa-4a4a-411d-ae65-80a66cda8deb.jpg)
and E be a double sequence space. A
-step space of
is a sequence space
![](https://www.scirp.org/html/6-8101912\71a30bc9-39ad-4c1f-af93-22534fdc6705.jpg)
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
![](https://www.scirp.org/html/6-8101912\8de52d49-1777-4dd8-bae5-01dc2e1a0f37.jpg)
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
![](https://www.scirp.org/html/6-8101912\ef8facf9-1ca9-4a41-8c0a-1052b60a2172.jpg)
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
![](https://www.scirp.org/html/6-8101912\d5a1d1fc-8bb4-48ce-b84e-7ef5503220cb.jpg)
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.
![](https://www.scirp.org/html/6-8101912\e0b1b5c4-61aa-4b44-a8e8-7196c79b21ad.jpg)
We also denote by
![](https://www.scirp.org/html/6-8101912\c3db70f2-f012-4ee5-808c-2c32068ea278.jpg)
and
![](https://www.scirp.org/html/6-8101912\0704d441-adb3-48da-9086-38a8a30703bc.jpg)
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 ![](https://www.scirp.org/html/6-8101912\be901bc3-3c3b-4683-8069-74c63f543a0d.jpg)
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
![](https://www.scirp.org/html/6-8101912\ab3ee4fb-8e78-40da-9350-21e66c128b06.jpg)
That is for a given
, we have
(1)
(2)
Since f is a modulus function, we have
![](https://www.scirp.org/html/6-8101912\1895f0ac-4fe1-4a96-aa93-3a34a415e5f8.jpg)
Now, by (1) and (2),
![](https://www.scirp.org/html/6-8101912\b1384d2e-0507-43e0-8b77-10e997fd54ff.jpg)
Therefore ![](https://www.scirp.org/html/6-8101912\ac57f873-7155-40ea-94af-450be3541856.jpg)
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
![](https://www.scirp.org/html/6-8101912\f5d7a369-4a95-4ffe-b7af-1d02bf154f07.jpg)
Fix an
. Then we have
![](https://www.scirp.org/html/6-8101912\b0c4b080-0b5e-4ab8-942f-3e38a2d5ce9a.jpg)
which holds for all
.
Hence ![](https://www.scirp.org/html/6-8101912\6252709d-6352-4515-900c-37eec1e37109.jpg)
Conversely, suppose that
![](https://www.scirp.org/html/6-8101912\6f0b368d-176a-4c06-85bb-af88d11c96fa.jpg)
That is ![](https://www.scirp.org/html/6-8101912\5e1f701b-3202-445e-98f5-389b276e440b.jpg)
for all
. Then the set
![](https://www.scirp.org/html/6-8101912\ad729329-0967-42e1-90ba-af89c9a4e612.jpg)
Let
. If we fix an
then we have
as well as ![](https://www.scirp.org/html/6-8101912\077fb779-04aa-41c5-9e30-9502b01d9875.jpg)
Hence
This implies that
![](https://www.scirp.org/html/6-8101912\59629f9a-e9f9-4345-8df8-808dc44c5c56.jpg)
that is
![](https://www.scirp.org/html/6-8101912\7655a8eb-bee7-403e-937f-b2d24bd661a3.jpg)
that is
![](https://www.scirp.org/html/6-8101912\a49f67fd-f96d-4469-a0a6-5e3793ad536b.jpg)
where the diam of N denotes the length of interval N.
In this way, by induction we get the sequence of closed intervals
![](https://www.scirp.org/html/6-8101912\7446935a-de02-4252-8625-6db4f48dc6b4.jpg)
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
![](https://www.scirp.org/html/6-8101912\b1b49448-9f09-4bd8-8fdb-1988aa9ff97c.jpg)
We have
(5)
For
, we have
. Since f is non-decreasing,it follows that
![](https://www.scirp.org/html/6-8101912\d1001082-b0df-45ea-8e21-f3d900e426e5.jpg)
Since
satisfies the
-condition, we have
![](https://www.scirp.org/html/6-8101912\722fe3a2-896f-4c9f-8e1a-a7c3622c1016.jpg)
Hence
(6)
From (4), (5) and (6), we have
.
Thus
. The other cases can be proved similarly.
(ii) Let
. Then
and ![](https://www.scirp.org/html/6-8101912\869bafe5-bf6c-45d9-a4f2-37d1728855b3.jpg)
![](https://www.scirp.org/html/6-8101912\9f0f01e8-5ccd-4af3-b25b-b0a7b28bd436.jpg)
Therefore
![](https://www.scirp.org/html/6-8101912\d49478e2-5c17-48af-9cea-f2d60bdbcdc4.jpg)
which implies
that is
![](https://www.scirp.org/html/6-8101912\9b8a6451-4df5-48d1-bcc6-96de2f74714f.jpg)
Corollary 3.4.
for
and ![](https://www.scirp.org/html/6-8101912\deaffdff-f2cc-4478-bc73-20fc7895d1a0.jpg)
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
![](https://www.scirp.org/html/6-8101912\2ea474d4-4ebf-4fde-87a1-1b4d3b9d7568.jpg)
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
![](https://www.scirp.org/html/6-8101912\42f8cc73-6c47-4bab-b3ee-7bd0fe070810.jpg)
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
![](https://www.scirp.org/html/6-8101912\eed75aa2-c56c-4fd0-bf35-b2ef896d4473.jpg)
and
![](https://www.scirp.org/html/6-8101912\e78d8b35-d724-4e55-8598-8dad2f84a08b.jpg)
Then we have
![](https://www.scirp.org/html/6-8101912\876a26f0-c5ca-4b19-8b36-2d4c3ef6f0be.jpg)
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
![](https://www.scirp.org/html/6-8101912\7b4e7a3c-fa5f-4f09-b53f-120aac8dc4ee.jpg)
Then
and
, but ![](https://www.scirp.org/html/6-8101912\2e36b4a1-6d70-46e9-92a9-b6e72e76f0de.jpg)
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 ![](https://www.scirp.org/html/6-8101912\4ad9a72d-dc42-4a96-aff3-abf73ad14b65.jpg)
If
![](https://www.scirp.org/html/6-8101912\8a118cb1-a474-484e-bd4e-44d966b9226c.jpg)
Then by Lemma (3) we have
.
Let
be such that
and
.
Let
and
be bijections, then the map
defined by
![](https://www.scirp.org/html/6-8101912\e5727c1e-7d65-4973-b248-c27fcfacccf5.jpg)
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
![](https://www.scirp.org/html/6-8101912\361639ed-3d80-48e3-9403-4b6c688b76da.jpg)
We have ![](https://www.scirp.org/html/6-8101912\61b8b571-d361-4475-9e25-629b0b763161.jpg)
Taking the supremum over
on both sides we get
.
Next we show that the inclusion is proper.
(i) ![](https://www.scirp.org/html/6-8101912\87358886-5449-4518-828e-31df1a857866.jpg)
Let
then
for some
, which implies
Hence the inclusion is proper.
(ii)
Let
then
![](https://www.scirp.org/html/6-8101912\a36e25ec-dc68-4bd1-9bcf-1a873e73f240.jpg)
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
![](https://www.scirp.org/html/6-8101912\1724a789-eddd-4d73-8a42-94da320d8692.jpg)
Thus the sets,
![](https://www.scirp.org/html/6-8101912\7525d226-2348-4e5d-9e47-59172b1a645a.jpg)
Hence also
, so that
.
Now taking
in
,
![](https://www.scirp.org/html/6-8101912\1bc72998-8be9-4ab1-9645-360e55b20cd7.jpg)
Thus
is a Lipschitz function. For
the result can be proved similarly.
Theorem 3.12. If
, then
and
.
Proof: For ![](https://www.scirp.org/html/6-8101912\217b9276-4a20-411c-8c8a-81e57110cd72.jpg)
![](https://www.scirp.org/html/6-8101912\005b5cb6-d620-4cae-85b5-ff3cb86319f9.jpg)
Now,
(8)
As
, there exists an
such that
and
.
Using Equation (8) we get
![](https://www.scirp.org/html/6-8101912\2d17d61b-c769-4e55-ae10-5501cc87068f.jpg)
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.