This document presents research on the η-dual of generalized difference sequence spaces. It defines the sequence spaces vsm(l∞), vsm(c), and vsm(c0) using a fixed sequence vk of non-zero complex numbers. It then proves the η-dual of these spaces are equal to the η-dual of other known sequence spaces. Specifically, it shows the η-dual of vsm(l∞), vsm(c), and vsm(c0) are equal to the η-dual of l∞, c, and c0 respectively. It also concludes these sequence spaces are not perfect spaces based on the definition of a perfect space.

1. International Journal of Research in Advent Technology, Vol.7, No.6, June 2019
E-ISSN: 2321-9637
Available online at www.ijrat.org
70
doi: 10.32622/ijrat.76201946

Abstract. The notion of   Köthe Toeplitz dual was
generalized by Tripathy and Chandra [11] to introduce
  dual. In this paper we give   dual of sequence spaces
, ( )m
v s l , , ( )m
v s c and , 0( )m
v s c .
Keywords: Difference sequence space, duals.
2000 MATHEMATICS SUBJECT CLASSIFICATION: 40A05,
46A45.
1. INTRODUCTION
Let l , c and 0c be the linear spaces of bounded convergent
and null sequences ( )kx x with complex term respectively
norm by
|| || sup | |k
k
x x 
where {1,2,3, }k N   the set of positive integer.
In 1981, Kizmaz [8] introduce the concept of difference
sequence and have defined  bounded,  convergent and
 null sequence spaces. Using the concept of difference
sequence Cólak [3] has defined m
 bounded
m
convergent and m
null sequence spaces. Further this
notion was generalized by Et. and Esi [5] and have defined
m
v  bounded, m
v  convergent and m
v  null sequence
spaces, where ( )kv v be any fixed sequence of non-zero
complex number’s. Later on Bektas and Cólak [2] have
defined m
r  bounded, m
r  convergent and m
r  null
sequence spaces.
Recently Ansari and Chaudhary [1] have defined the
following sequence spaces.
Let ( )kv v be any fixed sequence of non-zero complex
number, then
, ( ) { ( ):( ) }m s m
v s k v kl x x k x l   
Manuscript revised June 9, 2019 and published on July 10, 2019
A. A. Ansari, and, Department of Mathematics & Statistics, DDU
Gorakhpur University,Gorakhpur, India, aaansari@in.com.
S. K. Srivastava, Department of Mathematics & Statistics, DDU Gorakhpur
University,Gorakhpur, India, sudhirpr66@rediffmail.com.
N. K. Yadav, Department of Mathematics & Statistics, DDU Gorakhpur
University,Gorakhpur, India, nandu85gkp@gmail.com.
, ( ) { ( ):( ) }m s m
v s k v kc x x k x c  
, 0 0( ) { ( ):( ) }m s m
v s k v kc x x k x c  
where m N , s R ,
1 1
, 1( ) ( ) ( )m s m s m m
v s v k v k v kx k x k x x 
  
and
1
( 1) ( ) .
m
m j m
v k k j k j
j
x j v x 

 
These are Banach spaces with norm
,
1
|| || | | sup | |.
m
r m
v s i i v k
kc
x v x k x

 
It is trivial that 1
0 0( ) ( )m m
s sc c 
 , 1
( ) ( )m m
s sc c 
 ,
1
( ) ( )m m
s sl l 
  and 0( ) ( ) ( )m m m
s s sc c l  .
Lemma 1.1. [1] sup | |s m
v k
k
k x   iff
(i) 1 1
sup | |s m
v k
k
k x 
 
(ii) 1 1 1
1sup | ( 1) |s m m
v k v k
k
k x k k x  
   
Corollary 1.2. [1] , ( )m
v sx l implies sup | | .s m
k k
k
k v x
 
2. MAIN RESULTS
Definition 2.1. [11] Let E be a sequence space, then the
  dual of E is defined as
{ ( ) : | | , 1}.r
k k kE a a a x r
    
Definition 2.2. [11] Let E be a sequence space. Then E is
called a perfect space iff .E E

Lemma 2.3. [11]
(i) E
is a linear subspace of w for E w
η−dual of Generalized Difference Sequence Spaces
A. A. Ansari, S. K. Srivastava and N. K. Yadav

2. International Journal of Research in Advent Technology, Vol.7, No.6, June 2019
E-ISSN: 2321-9637
Available online at www.ijrat.org
71
doi: 10.32622/ijrat.76201946
(ii) E F implies E F 
 for every ,E F w
(iii) ( )E E E  
  for every E w
(iv) ( )j j
j j
E E 
 for every family { }jE with jE w for
all j N .
Theorem 2.4. Let m be a positive integer and s R , we put
1
1
( , ) { ( ) : ( ) | | }.m s r r
k k k
k
M v s a a k a v

 

   
Then,
, , , 0[ ( )] [ ( )] [ ( )] ( , ).m m m
v s v s v sl c c M v s  
    (2.1)
Proof. First we assume that ( , )a M v s . Then
1 1
| | | | ( ) | | | |r m s s m r m s r r s m r
k k k k k k k k k ka x k a v k x v k a v k x v     
 
or
1
1 1
| | ( ) | | | |r m s r r s m r
k k k k k k
k k
a x k a v k x v
 
  
 
    for each
, ( )m
v sx l , by corollary 1.2. Thus, we have to shown
,( , ) [ ( )] ,m
v sM v s l 
  (2.2)
conversely, let ( , )a M v s , then for some k , we have
1
1
( ) | | .m s r r
k k
k
k a v

 

 
So, there is a strictly increasing sequence ( )in of positive
integer in , such that
11
( ) | | .
1
r r
k k
ni m s rk a v i
k ni
  
 
We defined as a sequence ( )kx x by
1
1
0 (1 )
( 1 : 1,2, ).
k
m s
k k
i ir
k n
x v k
n k n i
i
 

 

 
    

Then, we see that
1
!
| | ( 1 , 1,2, ).s m
k k i ir
m
k v x n k n i
i
     
Hence,
, 0
1
( ) and | | 1 .m
v s k k
k
x c a x


    
Thus, ,[ ( )]m
v sa l 
 , and hence, we have shown
, 0[ ( )] ( , ).m
v s c M v s
 (2.3)
Since
, 0 , ,( ) ( ) ( )m m m
v s v s v sc c l 
implies
, , , 0[ ( )] [ ( )] [ ( )]m m m
v s v s v sl c c  
  
(2.1) follows from (2.2) and (2.3).
Theorem 2.5. Let m be a positive integer and s R , we put
{ ( ) :sup( ) | | }.s m r r
k k k
k
M a a k a v

    Then
, , , 0[ ( )] [ ( )] [ ( )] ( , ).m m m
v s v s v sl c c M v s  
    (2.4)
Proof. First we assume that ( , )a M v s . Then
1
1
| | | |
( ) | | ( ) | |
r s m m s r
k k k k k k
s m r r m s r r
k k k k
a x k a v k x v
k a v k x v
  
  


or,
1
1 1
| | sup( ) | | ( ) | |r s m r r m s r r
k k k k k k
kk k
a x k a v k x v
 
  
 
    for
each , 0[ ( )] ( , )m
v sx c M v s
  by using (2.1). Thus, we
have shown
, 0( , ) [ ( )] .m
v sM v s c 
  (2.5)
Conversely, let ( , )a M v s . Then, we have
sup( ) | | .s m r r
k k
k
k a v
 
Hence, there is strictly increasing sequence ( ( ))k i of positive
integer ( )k i such that
( ) ( ){[ ( )] } | | , 1.s m r r mr
k i k ik i a v i r
 
Then, we see that

3. International Journal of Research in Advent Technology, Vol.7, No.6, June 2019
E-ISSN: 2321-9637
Available online at www.ijrat.org
72
doi: 10.32622/ijrat.76201946
1
( ) ( )
1 1
1
( ) | | {[ ( )] } | |
.
m s r r m s r r
k k k i k i
k i
mr
i
k x v k i a v
i
 
  
 




  
 

Hence, ,[ ( )]m
v sx l 
 and
1 1
| | 1r
k k
k i
a x
 
 
    .
Thus , ,[ ( )]m m
v s v sa l 
 and hence we have to shown
,[ ( )] ( , ).m
v r l M v s
  (2.6)
Since,
, , , 0[ ( )] [ ( )] [ ( )]m m m
v s v s v sl c c  
  
implies
, 0 , ,[ ( )] [ ( )] [ ( )] ,m m m
v s v s v sc c l  
 
(2.4) follows from (2.5) and (2.6).
By definition 2.2., we also have
Corollary 2.6. The sequence spaces , ( )m
v s l , , ( )m
v s c and
, 0( )m
v s c are not perfect.
REFERENCES
[1] Ansari, A. A.; Chaudhary, V. K. (2012): On Köthe toeplitz duals of
some new and generalized difference sequence spaces, Italian J. Pure
and Appl. Maths, 29, pp. 135-148.
[2] Bektas, C. A.; Còlak, R. (2005): On some generalized difference
sequence spaces, Thai J. of Math., 3(1), pp. 83-89.
[3] Còlak, R. (1989): On some generalized sequence spaces, Commu. Fac.
Sci. Unir. Anic Series A1 38, pp. 35-46.
[4] Còlak, R.; Et., M. (1997): On some generalized difference sequence
spaces and related matrix transformation, Hokk. Math. J., 26, pp.
483-492.
[5] Et. Mikail; Esi, A. (2000): On Köthe-Toeplitz duals of generalized
difference sequence spaces, Bull. Malaysian Math. Sc. Soc., 23(2), pp.
25-32.
[6] Et. Mikail (1993): On some difference sequence spaces, Doğa- Tr. J. of
Math., 17, pp. 18-24.
[7] Et. Mikail; Còlak, R. (1995): On some difference sequence spaces,
Soochow J. of Math., 25(4), pp. 377-386.
[8] Kizmaz, H. (1981): On certain sequence spaces, Canadian Math. Bull.,
24(2), pp. 169-176.
[9] Kamthan, P. K.; Gupta, M. (1981): Sequence spaces and series, Marcel
Dekker Inc., New York.
[10] Sarigol, M. A. (1987): On difference sequence spaces, J. Kara deniz
Tec. Univ. Fac. Arts Sci. Ser. Math. Phys., 10, pp. 63-71.
[11] Tripathy, B. C.; Chandra, P. (2002): On Generalized Köthe-Toeplitz
duals of some sequence spaces, Ind. J. Pure and Appl. Math., 33(8), pp.
1301-1306.
[12] Wilansky, A. (1984): Summability through Functional Analysis, North
Holland.