On Fuzzy Real Valued I- Convergent Double Sequence Spaces
Bipan Hazarika
Department of Mathematics and Computing; Rajiv Gandhi University;
Itanagar-791 112, Arunachal Pradesh, India.
E-mail: [email protected].
Abstract: In this article we introduce the different types of fuzzy real-valued
I-convergent double sequence spaces. Also study their topological and algebraic
properties like solidness, symmetricity etc.
Key Words: Fuzzy number; ideal; I-convergent; solid space; symmetric space;
convergence free; sequence algebra.
2010 AMS Subject Classification No: 40A05; 40A99; 40G15; 46A45.
1. Introduction
Among various developments of the theory of fuzzy sets a progressive development
has been made to find the fuzzy analogues of the classical set theory. In fact the fuzzy set
theory has become an area of active research for the last 40 years. The notion of fuzziness
are using by many persons for Cybernetics, Artificial Intelligence, Expert System and
Fuzzy control, Pattern recognition, Operation research, Decision making, Image analysis,
Projectiles, Probability theory, Agriculture, Weather forecasting. The fuzzy set theory has
been used widely in many engineering applications, such as, in bifurcation of non-linear
dynamical systems, in the control of chaos, in the non-linear operator, in population
dynamics. The fuzziness of all the subjects of mathematical sciences has been
investigated. It attracted many workers on sequence spaces and summability theory to
introduce different types of sequence spaces and study their different properties.
Throughout 2 wF , 2 c F , ( 2 c0 ) F and ( 2   ) F denote the classes of all, convergent, null
and bounded fuzzy real-valued double sequence spaces respectively. Also N and R denote
the set of positive integers and set of real numbers respectively.
A fuzzy real number X is a fuzzy set on R i.e. a mapping X: RJ(=[0, 1]) associating
each real number t with its grade of membership X(t).
The -cut of a fuzzy real number X is denoted by X , 0< 1, where
X = {tR: X(t)}.
A fuzzy real number X is called convex, if X(t) X(s)X(r) = min{X(s), X(r)}, where
s<t<r.
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If there exists tR such that X(t0) =1, then X is called normal.
A fuzzy real number X is said to be upper semi continuous if for each >0,
X-1([0, a+]), for all aJ is open in the usual topology of R.
The class of all upper semi continuous normal convex fuzzy real numbers is denoted
by R(J).
The absolute value of XR(J) is defined as
max { X (t ), X (t )}, for t  0;
|X| (t) = 
otherwise.
0,
Let D be the set of all closed and bounded intervals X = [XL, XR]. Then we write
XY if and only if XLYL and XRYR and
d (X,Y) = max {|XL-YL|, |XR-YR|}.
It is known that (D, d) is a complete metric space and “” is a partial order on D.
Let  : R(J) R(J)R be defined by
 (X, Y) = sup d ( X  , Y  ) , for all X, Y R(J).
0  1
Then (R(J),  ) is a metric space. We define XY if and only if X  Y , for J.
The additive identity and multiplicative identity in R(J) are denoted by 0 and 1
respectively.
Let X, Y R(J) and the -level sets be X = [ a1 , a 2 ], Y = [ b1 , b2 ], for [0,1].
The arithmetic operations on R(J) are defined as follows:
(XY)(t) = sup{X(s)Y(t-s)}, tR,
(XӨY)(t) = sup{X(s)Y(s-t)}, tR,

 t 
(XY)(t) = sup  X ( s )  Y   , tR,
 s 

(X/Y) (t) = sup{X (st)Y(s)}, tJ.
These operations can be defined in terms of -level sets as follows:
[XY]  = [ a1  b1 , a 2  b2 ] ,
[XӨY]  = [ a1  b1 , a 2  b2 ] ,
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[XY]  =  max ai b j , max ai b j  ,
i , j{1, 2}

i , j{1, 2}
-1 
 1
 1


[X ] = (a 2 ) , (a1 ) , a1 , a 2 >0.


Applying the notion of fuzzy real numbers, different fuzzy real-valued sequence
spaces were introduced and studied by Nanda [], Nuray and Savas [], Das and Choudhury
[], Tripathy and Dutta [] and many others.
A fuzzy real-valued sequences denoted by ( X n ), where X n R (J), for all nN.
A fuzzy real-valued sequence ( X n ) is said to convergent to the fuzzy real numbers X0,
if for every >0, there exists an integer n0 >0 such that
( X n , X0) <  , for all n  n0.
The works on double sequences of real and complex numbers is found in Bromwich
[]. The works on double sequences was further investigated by Basarir and Solankan [],
Tripathy [], Tripathy and Tripathy [] and many others.
The notion of I-convergence initially introduced by Kostyrko, Šalát and Wilczyñski [].
Later on it was further investigated from sequence space point of view and linked with
summability theory by Salat, Tripathy and Ziman [, ], Tripathy and Hazarika [, ] and
many other authors.
Let S be a non-empty set. Then a family of sets I 2S (the class of all subsets of S) is
called an ideal if and only if for each A, BI, we have ABI and for each AI and for
each B A, we have BI.
A non-empty family of sets  2S is called a filter on S if and only if , for each
A, B, we have AB and for each A and each B A, we have B.
An ideal I is called a non trivial ideal if I   and XI. Clearly I2S is a non trivial
ideal if and only if  = (I) = {X-A: AI} is a filter on S.
2. Definitions and Preliminaries
A fuzzy real-valued double sequence is a double infinite array of fuzzy real numbers.
We denote a fuzzy real-valued double sequence by ( X mn ), where X mn are fuzzy real
numbers, for each m, nN.
A subset E of NN (see Tripathy []) is said to have density (E) if
1
(E) = lim

  E (n, k ) exists.
r , s  r s
n r k  s
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n
Let tn =
1
 k , for all nN. Then a subset EN is said to have logarithmic density d(E) if
k 1
n
 E (k )
exists.
(2.1)
k
k 1
n
1
1
Since tn =  = log n +  + O   , where  is the Euler’s constant, so if (2.1) holds,
n
k 1 k
then it is equivalent to the following expression:
1 n  E (k )
d(E)= lim
.

n   log n
k
k 1
1
n t
n
d(E) = lim

The notion of logarithmic density for subsets of NN defined as follows:
A subset E NN is said to have logarithmic density *(E) if
1 r s  E ( n, k )
*(E) = lim
  nk exists.
r , s  t t
r s n 1 k 1
As the above expression if exists is equivalent to the following:
r
s
 E ( n, k )
1
*(E) = lim
exists.


r , s  log r log s
nk
n 1 k 1
Let A  N. For integers p0 and q1, let A (p+1,p+q) = card{nA: p+1 n  p+q}. Put
 q = lim inf A(p+1, p+q),  q = lim sup A(p+1`, p+q). It can be shown that u (A)=
p 
lim
q 
q
q
p 
q

exist. If u (A) = u (A) , then u (A) = u (A) =u(A) is called the
q  q
, u (A) = lim
uniform density of the set A (see Kostyrko, Šalát and Wilczyñski []).
The notion of uniform density for subsets of NN defined as:
Let p, q0 and s, t1 be integers. Let B  NN and B(p+1, q+t; q+1, q+s)
= card {(n, k)B: p+1  n  p+t and q+1 k  q+s}. Put  t , s = lim inf p, q  B (p+1, q+t;
 t ,s
, u  (B) =
t , s  t s
q+1, q+s) and  t , s = lim sup p,q  B(p+1, q+t ; q+1, q+s). u (B) = lim

 t ,s


exist. If u (B) = u  (B), then u (B) = u  (B)= u  (B) is called the uniform
t , s t s
density of the set B.
lim
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In order to distinguish between the ideals of 2N and 2N  N we shall denote the ideals of
2N by I and that of 2N  N by I2 respectively. In general there is no connection between I
and I2.
Definition 2.1. A fuzzy real-valued sequence space EF is said to be normal (or solid) if
(  mn X mn ) EF, whenever ( X mn ) EF and for all sequences (  mn ) of scalars with
|  mn | 1, for all m, nN.
The notion of step spaces for double sequences as follows:
Let K= {( ni , k i ): iN; n1<n2< - - - and k1<k2< - - -}NN and E be a double sequence
space. A K-Step space of E is a sequence space
 EK = {( x ni ki ) w2 : ( x nk ) E}.
A canonical preimages of a sequence ( x ni ki )E is a sequence ( y nk )w2 defined as
follows:
 x , if (n, k )  K ;
y nk   nk
otherwise.
0,
A fuzzy real-valued sequence space EF is said to be monotone if EF contains the
canonical preimages of all its step spaces.
Remark 2.1: From the above two definitions if follows that if a fuzzy real-valued
sequence space EF is solid then it is monotone.
Definition 2.2. A fuzzy real-valued sequence space EF is said to be symmetric if (X(m),
X(n))  EF, whenever ( X mn ) EF, when  is a permutation of N.
Definition 2.3. A fuzzy real-valued sequence space EF is said to be a sequence algebra if
( X mn  Ymn ) EF, whenever ( X mn ). ( Ymn )EF .
Definition 2.4. A fuzzy real-valued sequence space EF is said to be convergence free if
( Ymn )EF, whenever ( X mn )EF and Ymn = 0 implies X mn = 0 .
We introduce the following definitions:
Definition 2.5. Let I2 be an ideal of 2N  N. A fuzzy real-valued double sequence ( X mn ) is
said to be I-convergent in Pringsheim’s sense to a fuzzy real number X0, if for every  >0,
there exist two positive integers m0, n0 such that the set
{(m, n)NN: ( X mn , X0) , for all m  m0 and n n0}I2.
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Definition 2.6. Let I2 be an ideal of 2N  N. A fuzzy real-valued double sequence ( X mn ) is
said to be I-null in Pringsheim’s sense, if for every >0, there exist two positive integers
m0, n0 such that the set
{(m, n)NN: ( X mn , 0 ) , for all m  m0 and n  n0}I2.
Definition 2.7. Let I2 be an ideal of 2N  N. A fuzzy real-valued double sequence ( X mn ) is
said to be I-bounded if there exist M>0 such that the set
{(m, n)NN: ( X mn , 0) > M}I2,
Definition 2.8. Let I2 be an ideal of 2N  N and I be an ideal of 2N, then a fuzzy real-valued
double sequence ( X mn ) is said to be regularly I-convergent if it is I-convergent in
Pringsheim’s sense and for every >0, such that the following sets:
{mN: ( X mn , Ln ) , for some Ln R(J), for each nN}I2.
and
{nN: ( X mn , M m ) , for some M m R(J), for each mN}I2.
Definition 2.9. Let I2 be an ideal of 2N  N and I be an ideal of 2N, then a fuzzy real-valued
double sequence ( X mn ) is said to be regularly I-null if it is I-null in Pringsheim’s sense
and for every >0, such that the following sets:
{mN: ( X mn , 0 ) , for each nN}I2.
and
{nN: ( X mn , 0 ) , for each mN}I2.
Definition 2.10. A fuzzy real-valued double sequence ( X mn ) is said to be I-Cauchy if for
every >0, there exist m0 = m0(), n0 = n0()N such that the set
{(m, n)NN: ( X mn , X m0 n0 )  , for all m  m0 and n  n0}I2.
Definition. 2.11. Let ( X mn ) and ( Ymn ) be two fuzzy real-valued double sequences. Then
we say X mn = Ymn , for almost all m and n relatively to I2 (in short a. a. m & n. r. I2) if
{(m, n)NN: X mn  Ymn }I2.
Example 2.1. Let I2(P) be the class of all subsets of NN such that G I2(P) implies that
there exists n0, k0N such that G  NN – {(n, k) NN: nn0, k k0}. Then I2(P) is an
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7
ideal of 2NN. It corresponds to the usual Pringsheim’s sense convergence of double
sequences. With I2(P), if one considers the ideal I(f), the class of all finite subsets of N,
then one will get the usual regular convergence for double sequences.
Example 2.2. Let us consider I2()2NN i.e. the class of all subsets of NN of zero
natural density. Then I2() is an ideal of 2NN. On considering I() along with I2(), one
will get the different types of statistically convergent double sequences.
Example 2.3. Let us consider I2(*)2N  N i.e. the class of all subsets of NN of zero
logarithmic density. Then it can be easily verified that I2(*) is an ideal of 2N  N. With
this ideal if we consider I(d), then we will get the different definitions of logarithmic
convergence for double sequences.
Example 2.4. Let us consider I2(u*)2N  N i.e. the class of all subsets of NN of uniform
density zero. Then it can be easily verified that I2(u*) is an ideal of 2N  N. With this ideal
if we consider I(u), then we will get the different definitions of uniform convergence for
double sequences.
Throughout ( 2 w) F , ( 2   ) F , ( 2 c) F , ( 2 c 0 ) F , ( 2  I ) F , ( 2 c I ) F , ( 2 c 0I ) F , ( 2 c I ) BF ,
( 2 c 0I ) BF , ( 2 c I ) RF , ( 2 c 0I ) RF will denote the class of all, bounded, convergent in
Pringsheim’s sense, null in Pringshiem’s sense, I-bounded, I-convergent in Pringshiem’s
sense, I-null in Pringshiem’s sense, bounded and I- convergent, bounded and I-null,
regularly I-convergent and regularly I-null fuzzy real-valued double sequence spaces
respectively.
We define the following sequence spaces:
I P
I BP
( 2 c I ) BP
= ( 2 c 0I ) PF  ( 2   ) F
F = ( 2 c ) F  ( 2   ) F ; ( 2 c0 ) F
and
I R
= ( 2 c I ) RF  ( 2   ) F ; ( 2 c 0I ) BR
( 2 c I ) BR
F = ( 2 c0 ) F  ( 2   ) F .
F
3. Main Results
Theorem 3.1. Let I2 be the given ideal of 2N  N. Then the class of sequences ( 2 c I ) F ,
I BP
( 2 c 0I ) F , ( 2 c I ) RF , ( 2 c 0I ) RF , ( 2 c I ) BP
F and ( 2 c 0 ) F are linear spaces.
Proof. The proof of the result is easy, so omitted.
Theorem 3.2. A fuzzy double sequence ( X mn ) is I-convergent to X0 if and only if for
every  > 0, there exist m0 = m0 (), n0 = n0 () in N such that
(m, n)  N  N : ( X
mn
, X m0 n0 )    (I).

(3.1)
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Proof. Let I-lim X mn = X0, then for every  > 0, the set


A1 = (m, n)  N  N : ( X mn , X 0 )    (I).
2

Fix m0 , n0 . Then
 ( X mn , X m n )   ( X mn , X 0 ) +  ( X 0 , X m n )
0 0
0 0
< , for all (m, n)  A1 .
Hence the condition (3.1) holds.
Let the condition (3.1) holds for all  > 0. Let us fix  = 1. Then there exists
Theorem 3.2. The class of sequences ( 2  I ) F is a complete metric space.
Proof. Let ( X k ) be a Cauchy sequence in ( 2  I ) F . Define the metric  on ( 2  I ) F by
 (X, Y)= sup ( X mn , Ymn ).
m ,n
For a given >0, there exists an integer t0 such that
k
l
 ( X mn
, X mn
) < , for all k, l  t0.
k
l
 sup ( X mn
, X mn
) < , for all k, l  t0.
m ,n
k
l
 ( X mn
, X mn
) < , for all k, l  t0.
k
 ( X mn
) is a Cauchy sequence in R(J), for all m, nN.
k
Then ( X mn
) converges in R(J).
k
= X mn , for all m, nN.
Let lim X mn
k 
Hence for each >0, there exists t0 = t0(m ,n), such that
k
, X mn ) < , for all k  t0.
sup ( X mn
m ,n
Thus ( X mn ) ( 2  I ) F .
Then we have
k
k
)+ sup ( X mn
,0)
sup ( X mn , 0 )  sup ( X mn , X mn
m ,n
m ,n
m ,n
  +M < .
I
Therefore ( X mn )  ( 2   ) F .
Hence ( 2  I ) F is a complete metric space.
Theorem 3.2. The class of sequences ( 2 c I ) BF , ( 2 c 0I ) BF , ( 2 c I ) RF and ( 2 c 0I ) RF are complete
metric space.
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Proof. We prove the result only for the space ( 2 c 0I ) BF and rest of the cases can be prove
in similar way.
Let ( X k ) be a Cauchy sequence in ( 2 c 0I ) BF . Then ( X k ) is a Cauchy sequence in ( 2  I ) F .
Therefore ( X k ) converges in ( 2  I ) F . Let lim X k = X(say).
k 
We prove that X = ( X mn ) converges to 0 . Since lim X k = X then for given >0, we have
k 
s0N such that
k
) , for all k s0} I 2 .
{(m, n)NN: ( X mn , X mn
(3.2)
Again since ( X k )  ( 2 c 0I ) BF , so for >0, we have for each kN, k0 = k0(k), l0= l0(k) in N
such that
k
{(m, n)NN: ( X mn
, 0 ) , for all m k0 and n l0} I 2 .
(3.3)
For a fixed ks0, we have k0= k0(k) and l0= l0(k) in N and s0 such that for all m  k0 and
n  l0, we have
{(m, n)NN: ( X mn , 0 )}
k
k
{(m, n)NN: ( X mn , X mn
) }{(m, n)NN: ( X mn
, 0 ) } I 2 .
[from (3.2) and (3.3)]
Therefore for all m  k0 and n  l0, we have
{(m, n)NN: ( X mn , 0 ), } I 2 .
Thus ( X mn ) ( 2 c 0I ) BF .
Hence the space ( 2 c 0I ) BF is a complete metric space.
and ( 2 c 0I ) BP
are solid as well as
Theorem 3.3. The spaces ( 2 c 0I ) F , ( 2 c 0I ) RF , ( 2 c 0I ) BR
F
F
monotone, but the spaces ( 2 c I ) F , ( 2 c I ) RF , ( 2 c I ) BR
and ( 2 c I ) BP
F
F are neither solid nor
monotone in general.
Proof. Let ( X mn ) ( 2 c 0I ) F and let (  mn ) be a sequence of scalars such that |  mn |1, for
all m ,n N.
Let >0 be given. Then the proof of the results follows from the following inclusion
relation.
{(m, n) NN: ( X mn , 0 ) }{(m, n) NN: (  mn X mn , 0 ) }.
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The proof for the cases ( 2 c 0I ) RF , ( 2 c 0I ) BR
and ( 2 c 0I ) BP
can be established similar way.
F
F
If we considers I2 = I2(P) or I2() or I2(*) or I2(u*), then it is clear that the spaces
I BP
( 2 c I ) F , ( 2 c I ) RF , ( 2 c I ) BR
F and ( 2 c ) F are neither solid nor monotone.
are symmetric for I2 = I2(f) and
Theorem 3.4. (a) The spaces ( 2 c 0I ) RF and ( 2 c 0I ) BR
F
I2 = I2(P), but not in general.
I BP
I BP
are not
(b) The spaces ( 2 c I ) F , ( 2 c 0I ) F , ( 2 c I ) RF , ( 2 c I ) BR
F , ( 2 c ) F and ( 2 c 0 ) F
symmetric in general.
I R
R
Proof. (a) Let I2 =I2(P) then ( 2 c 0I ) BR
= ( 2 c 0 ) BR
F
F and ( 2 c 0 ) F = ( 2 c 0 ) F .
Let >0 be given. Then
{(m, n)NN: ( X mn , 0 )  }
is a finite subset of NN.
Hence the result follows.
The rest part of the proof of (a) and that of (b) follows from the following example.
Example 3.1. Let I2=I2(*) and consider the sequence ( X mn ) defined by
1  (1  t )m 2
1 

, for  1  2   t  0;

2
 m 
 1 m
2
1 
1  (1  t )m

, for 0  t  1  2 ;
Xm1(t) = 
2
 m 
 1 m
0,
otherwise.


for all mN
and
X mn = 0 , for n 1 and for mN.
I BR
I BP
Then ( X mn ) Z, for Z = ( 2 c I ) F , ( 2 c 0I ) F , ( 2 c 0I ) RF , ( 2 c I ) RF , ( 2 c I ) BR
F , ( 2 c0 ) F , ( 2 c ) F
and ( 2 c 0I ) BP
.
F
Consider the rearrangement ( Ymn ) be defined by
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1  (1  t )m 2
1 

, for  1  2   t  0;

2
 m 
 1 m
2
1 
1  (1  t )m

Ymm (t) = 
, for 0  t  1  2 ;
2
 m 
 1 m
0,
otherwise.


and Ymn = 0 , for m n.
I BR
I BP
and
Then ( Ymn ) Z, for Z = ( 2 c I ) F , ( 2 c 0I ) F , ( 2 c 0I ) RF , ( 2 c I ) RF , ( 2 c I ) BR
F , ( 2 c0 ) F , ( 2 c ) F
( 2 c 0I ) BP
F .
I BR
I BP
Hence the spaces ( 2 c I ) F , ( 2 c 0I ) F , ( 2 c 0I ) RF , ( 2 c I ) RF , ( 2 c I ) BR
F , ( 2 c 0 ) F , ( 2 c ) F and
( 2 c 0I ) BP
are not symmetric.
F
, ( 2 c 0I ) BR
, ( 2 c I ) BP
Theorem 3.5. The spaces ( 2 c I ) F , ( 2 c 0I ) F , ( 2 c 0I ) RF , ( 2 c I ) RF , ( 2 c I ) BR
F
F
F
and ( 2 c 0I ) BP
are sequence algebra.
F
Proof. We prove the result only for the space ( 2 c 0I ) F and rest of the results follows in
similar way.
Let ( X mn ) and ( Ymn )  ( 2 c 0I ) F . Let > 0 be given. Then there exist m1, n1, m2, n2N such
that
{(m ,n) NN: ( X mn , 0 ) , for all mm1 , nn1}I2.
and
{(m ,n) NN: ( Ymn , 0 ) , for all mm2 , nn2}I2.
Let A1 = {(m , n) NN: ( X mn , 0 ) , for all mm1 , nn1}I2.
and
A2 = {(m , n) NN: ( Ymn , 0 ) , for all mm2 , nn2}I2.
Let m0 = max{m1,m2} and n0 = max{n1,n2}. Then we have
A3 = A1A2 = {(m , n) NN: ( X mn  Ymn , 0 ) , for all m m0 , n n0}I2.
Thus ( X mn  Ymn ) ( 2 c 0I ) F .
Hence the space ( 2 c 0I ) F is sequence algebra.
Theorem 3.6. Let I2 be an ideal of 2N  N. Then the spaces ( 2 c I ) F , ( 2 c 0I ) F , ( 2 c 0I ) RF ,
I BR
I BP
I BP
are not convergence free.
( 2 c I ) RF , ( 2 c I ) BR
F , ( 2 c 0 ) F , ( 2 c ) F and ( 2 c 0 ) F
Proof. The proof of the result follows from the following example.
Example 3.2. Consider the sequence ( X mn ) be defined by
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12
1

1  (n  1)t , for  n  1  t  0;

1

X nn (t) = 1  (n  1)t , for 0  t 
;
n 1

otherwise.
0,


and X mn = 0 , for m n.
I BR
I BP
Then ( X mn )Z, for Z = ( 2 c I ) F , ( 2 c 0I ) F , ( 2 c 0I ) RF , ( 2 c I ) RF , ( 2 c I ) BR
F , ( 2 c 0 ) F , ( 2 c ) F and
( 2 c 0I ) BP
F .
Consider the sequence ( Ymn ) be defined by
Ynn
t

1  n  1 , for  (n  1)  t  0;

t

(t) = 1 
, for 0  t  (n  1);
 n 1
otherwise.
0,


and Ymn = 0 , for m n.
, ( 2 c 0I ) BR
, ( 2 c I ) BP
and
Thus ( Ymn )Z, for Z = ( 2 c I ) F , ( 2 c 0I ) F , ( 2 c 0I ) RF , ( 2 c I ) RF , ( 2 c I ) BR
F
F
F
( 2 c 0I ) BP
F .
I BR
I BP
Hence the spaces ( 2 c I ) F , ( 2 c 0I ) F , ( 2 c 0I ) RF , ( 2 c I ) RF , ( 2 c I ) BR
F , ( 2 c 0 ) F , ( 2 c ) F and
are not convergence free.
( 2 c 0I ) BP
F
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13
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