ON SOME GENERALIZED I-CONVERGENT DOUBLE SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION†
ON SOME GENERALIZED I-CONVERGENT DOUBLE SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION†
Journal of Applied Mathematics & Informatics. 2014. Sep, 32(3_4): 331-341
• Received : December 17, 2013
• Accepted : February 19, 2014
• Published : September 28, 2014
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VAKEEL A. KHAN
NAZNEEN KHAN

Abstract
In this article we introduce the sequence spaces for a modulus function f , where p = ( pk ) is a sequence of positive reals and study some of the properties of these spaces. AMS Mathematics Subject Classification : 65H05, 65F10.
Keywords
1. Introduction
The notion of I-Convergence is a generalization of the concept of statistical convergence which was first introduced by H.Fast [5] and later on studied by various mathematicians like J.A.Fridy [6 , 7] , Kostyrko, Salat and Wilezynski [19] , Salat, Tripathy, Ziman [29] and Demirci [3] .
Also a double sequence is a double infinite array of elements xkl ∈ ℝ for all k, l ∈ ℕ (see [14 , 15] ). The initial works on double sequences is found in Bromwich [1] , Basarir and Solancan [2] and many others. Throughout this article a double sequence is denoted by x = ( xij ).
Next we discuss some preliminaries about I-convergence (see [12] , [30] ).
Let X be a non empty set. Then a family of sets I⊆ 2 X (power set of X ) is said to be an ideal if I is additive i.e A,B∈I ⇒A∪ B∈I and hereditary i.e A∈I, B⊆A⇒B∈I.
A non-empty family of sets £ ( I ) ⊆ 2 X is said to be filter on X if and only if Φ ∉ £ (I), for A,B∈ £ (I) we have A∩B∈ £ (I) and for each A ∈ £ (I)and A⊆B implies B∈ £ (I). An Ideal I⊆ 2 X is called non-trivial if I ≠ 2 X . A non-trivial ideal I⊆ 2 X is called admissible if { x : { x } ∈ X } ⊆I.
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 £ (I) = { K N : Kc I },where K c = N-K.
Definition 1.1. A double sequence ( xij ) ∈ ω is said to be I-convergent to a number L if for every ϵ > 0. {( i, j ) ∈
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×
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: | xij L | ≥ ϵ } ∈ I. In this case we write I-lim xij = L . (see [17] )
The space 2 cI of all I-convergent sequences to L is given by
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Definition 1.2. A sequence ( xij ) ∈ ω is said to be I-null if L = 0. In this case we write I-lim xij = 0.
Definition 1.3. A sequence ( xij ) ∈ ω is said to be I-cauchy if for every ϵ > 0 there exists a number m = m( ϵ ) and n = n( ϵ ) such that
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Definition 1.4. A sequence ( xij ) ∈ ω is said to be I-bounded if there exists M > 0 such that {( i, j ) ∈
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×
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: | xij | > M }.
Definition 1.5. Let ( xij ), ( yij ) be two sequences. We say that ( xij ) = ( yij ) for almost all (i,j) relative to I (a.a.k.r.I), if {( i, j ) ∈
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×
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: xij yij } ∈ I
Definition 1.6. For any set E of sequences the space of multipliers of E, denoted by M ( E ) is given by
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Definition 1.7. A map ħ defined on a domain D X i.e ħ : D X
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is said to satisfy Lipschitz condition if | ħ ( x )− ħ ( y )| ≤ K | x y | where K is known as the Lipschitz constant.The class of K-Lipschitz functions defined on D is denoted by ħ ∈ ( D,K ).
Definition 1.8. A convergence field of I-convergence is a set
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The convergence field F ( I ) is a closed linear subspace of l with respect to the supremum norm, F ( I ) = l 2 cI (See [23] ).
Define a function ħ : F ( I ) →
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such that ħ ( x ) = I − lim x , for all x F ( I ), then the function ħ : F ( I ) →
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is a Lipschitz function ( [11 , 4 , 13] ).
Definition 1.9. The concept of paranorm is closely related to linear metric spaces [16] . It is a generalization of that of absolute value.
Let X be a linear space. A function g : X R is called paranorm, if for all x, y, z X ,
(PI) g ( x ) = 0 if x = θ , (P2) g (− x ) = g ( x ), (P3) g ( x + y ) ≤ g ( x + g ( y ),
(P4) If ( λn ) is a sequence of scalars with λn λ ( n → ∞) and xn , a X with xn a ( n → ∞) , in the sense that g ( xn a ) → 0 ( n → ∞) , in the sense that g ( λn xn λa ) → 0 ( n → ∞).
A paranorm g for which g ( x ) = 0 implies x = θ is called a total paranorm on X , and the pair ( X, g ) is called a totally paranormed space.(See [23] ). The idea of modulus was structured in 1953 by Nakano.(See [24] ). A function f : [0,∞)→[0,∞) is called a modulus if
(1) f (t) = 0 if and only if t = 0, (2) f (t+u)≤ f (t)+ f (u) for all t,u≥0,
(3) f is increasing and (4) f is continuous from the right at zero.
Ruckle in [25 , 26 , 27] used the idea of a modulus function f to construct the sequence space
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This space is an FK space ,and Ruckle proved that the intersection of all such X ( f ) spaces is ϕ , the space of all finite sequences.
The space X ( f ) is closely related to the space l 1 which is an X ( f ) space with f ( x ) = x for all real x ≥ 0. Thus Ruckle proved that,for any modulus f
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Where
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The space X ( f ) is a Banach space with respect to the norm
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Spaces of the type X ( f ) are a special case of the spaces structured by B.Gramsch in [10] . From the point of view of local convexity, spaces of the type X ( f ) are quite interesting.
Symmetric sequence spaces, which are locally convex have been frequently studied by D.J.H Garling [8 , 9] , G.Köthe [18] .
The following subspaces of ω were first introduced and discussed by Maddox [22 , 23] .
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where p = ( pk ) is a sequence of striclty positive real numbers.
After then Lascarides [20 , 21] defined the following sequence spaces
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where
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for all k
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.
We need the following lemmas in order to establish some results of this article.
Lemma 1.10.
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Then the following conditions are equivalent.(See[18]).
(a) H < ∞ and h > 0. (b) c 0 ( p ) = c 0 or l ( p ) = l . (c) l { p } = l ( p ). (d) c 0 { p } = c 0 ( p ). (e) l { p } = l ( p ).
Lemma 1.11. Let K £ ( I ) and M N . If M I, then M K I .(See [29 , 30] ).
Lemma 1.12. If I ⊂ 2 X and M X . If M I, then M K I .(See [29 , 30] ).
Throughout the article
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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.
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Also we write
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2. Main results
Theorem 2.1. Let ( pij ) ∈ 2 l .
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are linear spaces.
Proof . Let ( xij ), ( yij ) ∈ 2 cI ( f, p ) and α, β be two scalars. Then for a given ϵ > 0.
We have
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where
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be such that
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Then
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Thus
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Hence ( αxij + βyij ) ∈ 2 cI ( f, p ). Therefore 2 cI ( f, p ) is a linear space. The rest of the result follows similarly.
Theorem 2.2. Let ( pij ) ∈ 2 l . Then 2 mI ( f, p ) and
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are paranormed spaces, paranormed by
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Proof . Let x = ( xij ), y = ( yij ) ∈ 2 mI ( f, p ).
(1) Clearly, g ( x ) = 0 if and only if x = 0.
(2) g ( x ) = g (- x ) is obvious.
(3) Since
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using Minkowski’s inequality and the definition of f we have
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(4) Now for any complex λ we have ( λij ) such that λij λ , ( i, j → ∞).
Let xij 2 mI ( f, p ) such that f (| xij L | pij ) ≥ ϵ .
Therefore,
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Hence g ( λij xij - λL ) ≤ g ( λij xij ) + g ( λL ) = λij g ( xij ) + λg ( L ) as ( i, j → ∞).
Hence 2 mI ( f, p ) is a paranormed space. The rest of the result follows similarly.
Theorem 2.3. A sequence x = ( xij ) ∈ 2 mI ( f, p ) I-converges if and only if for every ϵ > 0 there exists Nϵ
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×
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where Nϵ = ( m, n ), m and n depending upon ϵ such that
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Proof . Suppose that L = I − lim x . Then
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Fixing some Nϵ Bϵ , we get
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which holds for all ( i, j ) ∈ Bϵ . Hence
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Conversely, suppose that
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That is {( i, j ) ∈
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×
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: (| xij xNϵ | pij ) < ϵ } ∈ 2 mI ( f, p ) for all ϵ > 0. Then the set
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Let Jϵ = [ xNϵ ϵ , xNϵ + ϵ ]. If we fix an ϵ > 0 then we have Cϵ 2 mI ( f, p ) as well as
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This implies that
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that is
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that is
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where the diam of J denotes the length of interval J. In this way, by induction we get the sequence of closed intervals
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with the property that
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and {( i, j ) ∈
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×
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: xij Ik } ∈ 2 mI ( f, p ) for (k=1,2,3,4,......).
Then there exists a ξ ∈ ∩ Ik where ( i, j ) ∈
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×
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such that ξ = I − lim x . So that f ( ξ ) = I − lim f ( x ), that is L = I − lim f ( x ).
Theorem 2.4.
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and I be an admissible ideal. Then the following are equivalent.
(a) ( xij ) ∈ 2 cI ( f, p );
(b) there exists ( yij ) ∈ 2 c ( f, p ) such that xij = yij , for a.a.k.r.I ;
(c) there exists ( yij ) ∈ 2 c ( f, p ) and
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such that xij = yij + zij for all ( i, j ) ∈
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×
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and {( i, j ) ∈
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×
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: f (| yij − L| pij ) ≥ ϵ } ∈ I ;
(d) there exists a subset J × K where J = { j 1 , j 2 , ...} and K = { k 1 < k 2 ....} of
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×
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such that
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Proof . (a) implies (b)
Let ( xij ) ∈ 2 cI ( f, p ). Then there exists L
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that
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Let ( mt ) and ( nt ) be increasing sequences with mt and nt
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such that
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Define a sequence ( yij ) as
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For mt < k mt +1 , t
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.
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Then ( yij ) ∈ 2 c ( f, p ) and form the following inclusion
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We get xij = yij , for a.a.k.r.I.
(b) implies (c)
For ( xij ) ∈ 2 cI ( f, p ). Then there exists ( yij ) ∈ 2 c ( f, p ) such that xij = yij , for a.a.k.r.I. Let K = {( i, j ) ∈
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×
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: xij yij }, then ( i, j ) ∈ I .
Define a sequence ( zij ) as
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Then
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and yij 2 c ( f, p ).
(c) implies (d)
Let P 1 = {( i, j ) ∈
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×
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: f (| xij | pij ) ≥ ϵ } ∈ I and
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Then we have
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(d) implies (a)
Let K = {( i 1 , j 1 ) < ( i 2 , j 2 ) < ( i 3 , j 3 ) < ...} ∈ £ ( I ) and
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Then for an ϵ > 0, and Lemma 1.10, we have
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Thus ( xij ) ∈ 2 cI ( f, p ).
Theorem 2.5. Let ( pij ) and ( qij ) be a sequence of positive real numbers. Then
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where Kc
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×
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such that K I .
Proof . Let
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Then there exists β > 0 such that pij > βqij , for all sufficiently large ( i, j ) ∈ K .
Since
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for a given ϵ > 0, we have
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Let G 0 = Kc B 0 Then G 0 I . Then for all sufficiently large ( i, j ) ∈ G 0 ,
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Therefore
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Corollary 2.6. Let ( pij ) and ( qij ) be two sequences of positive real numbers. Then
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where Kc
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×
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such that K I .
Theorem 2.7. Let ( pij ) and ( qij ) be two sequences of positive real numbers. Then
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where K
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×
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such that Kc I .
Proof . On combining Theorem 2.5 and 2.6 we get the required result.
Theorem 2.8.
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Then the following results are equivalent . (a) H < ∞ and h > 0.
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Proof . Suppose that H < ∞ and h > 0,then the inequalities min {1, sh } ≤ spij max {1, sH } hold for any s > 0 and for all ( i, j ) ∈
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×
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. Therefore the equivalence of (a) and (b) is obvious.
Theorem 2.9. Let f be a modulus function. Then
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(see [13]).
Proof . Let ( xij ) ∈ 2 cI ( f, p ). Then there exists L
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that
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We have
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Taking supremum over ( i, j ) both sides we get
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and the inclusion
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is obvious. Hence
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and the inclusions are proper.
Theorem 2.10.
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then for any modulus f, we have
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where the inclusion may be proper.
Proof . Let
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This implies that
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for some K > 0 and all ( i, j ). Therefore x = ( xij ) ∈ 2 mI ( f, p ) implies
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which gives
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To show that the inclusion may be proper, consider the case when
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for all ( i, j ). Take aij = ( i × j ) for all ( i, j ).
Therefore x 2 mI ( f, p ) implies
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Thus in this case a = ( aij ) ∈ M ( 2 mI ( f, p )) while
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Theorem 2.11. The function ħ : 2 mI ( f, p ) →
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is the Lipschitz function,where 2 mI ( f, p ) = 2 cI ( f, p ) ∩ 2 l ( f, p ), and hence uniformly continuous.
Proof . Let x, y 2 mI ( f, p ), x y . Then the sets
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Here
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Thus the sets,
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Hence also B = Bx By 2 mI ( f, p ), so that B ϕ . Now taking ( i, j ) in B such that
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Thus ħ is a Lipschitz function. For
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the result can be proved similarly.
Theorem 2.12. If x, y 2 mI ( f, p ), then ( x.y ) ∈ 2 mI ( f, p ) and ħ xy ) = ħ ( x ) ħ ( y ).
Proof . For ϵ > 0
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Now,
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As 2 mI ( f, p ) ⊆ l ( f, p ),there exists an M
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such that | xij | pij < M and | ħ ( y )| pk < M . Using eqn(2) we get
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For all ( i, j ) ∈ Bx By 2 mI ( f.p ). Hence ( x.y ) ∈ mI ( f, p ) and ħ ( xy ) = ħ ( x ) ħ ( y ). For
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the result can be proved similarly.
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.
BIO
Vakeel A. Khan received M.Sc., M.Phil and Ph.D at Aligarh Muslim University. He is currently an Associate Professor at Aligarh Muslim University. He has published a number of research articles and some books to his name. His research interests include Functional Analysis, sequence spaces, I-convergence, invariant meanszweir sequences and so on.
Department of Mathematics, Aligarh Muslim University, Aligarh, 200-002, India.
e-mail: vakhanmaths@gmail.com
Nazneen Khan received M.Sc. and M.Phil. from Aligarh Muslim University, and is cur-rently a Ph.D. scholar at Aligarh Muslim University. Her research interests are Functional Analysis, sequence spaces and double sequences.
Department of Mathematics, Aligarh Muslim University, Aligarh, 200-002, India.
e-mail: nazneen4maths@gmail.com
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