Issuu on Google+

IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-3008, p-ISSN:2319-7676. Volume 10, Issue 1 Ver. III. (Feb. 2014), PP 15-21 www.iosrjournals.org

Paranormed Sequence spaces w(u, v; p) , w0 (u, v; p) and w (u, v; p) generated by weighted mean Shailendra Kumar Mishra, Vinod Parajuli, Geeta Bhakta Joshi Pulchowk Campus, Institute of Engineering, Tribhuvan University; Nepal

Abstract: The sequence spaces w( p), w0 ( p) and w ( p) were introduced and studied by I.J. Maddox

c0 (u, v, p), c(u, v, p), l (u, v, p) and l (u, v, p) and established some properties. In this paper we introduce the sequence spaces w(u, v; p) , w0 (u, v; p) and w (u, v; p) ; study some properties, find  - dual of w(u, v; p) . We also characterize the

[1,2,3,4]. In [8,9,10] , the authors have introduced sequence spaces

 

matrix classes w  u, v; p  , l , w  u, v; p  , c and w  u, v; p  , c0 . Key Words: Paranormed sequence spaces, AMS classification : 40

 - dual, matrix transformation, generalized weighted mean.

I. Introduction By  we mean the space of all real valued sequences. A vector subspace of  is called a sequence space. We shall write , with usual notation , l , c and c0 for the spaces of all bounded, convergent and null sequence respectively. A linear topological space X over the field ℝ is said to be a paramormed space if there is a sub-additive function g : X  such that g ( )  0 ,

g ( x)  g ( x) and scalar multiplication is continuous i.e.

 n    0 and g ( xn  x)  0

imply g ( n xn   x)  0 , where  is the zero vector in the linear space X .

If p   pk  be a bounded sequence of strictly positive real numbers, I.J. Maddox defined the sequence spaces w( p), w0 ( p) and w ( p) as:

1 n p   w( p)   x  ( xk )   :  xk  l k  0; for somel  , n    n k 1   n 1 p   w0 ( p)   x  ( xk )   :  xk k  0 , n    and n k 1   1 n p   w ( p)   x  ( xk )   :  xk k   , n    . n k 1   The spaces w( p) and w0 ( p) are paranormed spaces paranormed by

1 n g ( x)  sup   xk  n k 1

1

pk

M  or 

equivalently

r

g ( x)  sup r 2  xk r

pk

1 M

(1.1) where  is the sum over the range 2r  r  2r 1 and M  (1,sup pk ) . Further w ( p) is the r

paranorm space paranormed by (1.1) if and only if 0  inf pk  sup pk   [ 1]. Let X and Y be any two sequence spaces and A  (ank ); n, k 

be infinite matrix of complex

numbers ank . Then we say that A defines a matrix mapping X into Y

; and it is denoted by

writing A: X  Y if for every sequence x  ( xk ) X , the sequence  ( Ax) n  is in Y , where www.iosrjournals.org

15 | Page


Paranormed Sequence spaces w(u, v; p) , w0 (u, v; p) and w (u, v; p) generated by weighted mean 

( Ax)n   ank xk ; (n  k 1

)

(1.2) By ( X , Y ) we denote the class of all matrices A such that A: X  Y . Thus, A ( X , Y ) if and only if the series on right side of (1.2) converges for each n  and every x X ; and we write,

Ax   Ax n

n

Y for all x X .

We denote by U for the set of all sequences u  (un ) such that un  0

for all n 

. For u  U ,

  . Let us define the matrix G(u, v)  ( gnk ) as:  0k n u v ; g nk   n k k n  0;

let

1  1  u  un

(1.3) for all n, k  ,where un depends only on n and vk only on k . The matrix G(u, v)  ( gnk ) is called generalized weighted mean or factorable matrix. The main purpose of the present paper is to introduce the sequence spaces w(u, v; p), w0 (u, v; p) and

w (u, v; p) ; which are the set of all sequences whose G(u, v) - transforms are in the spaces w( p),w0 ( p) and w ( p) respectively, where G(u, v) denotes the matrix as defined in (1.3). We have discussed some topological properties of w(u, v; p), w0 (u, v; p) and w (u, v; p) ; investigated β –dual for the new space w(u, v; p) . Moreover we have characterized the matrix

 

classes w  u, v; p  , l , w  u, v; p  , c and w  u, v; p  , c0 . II. The paranormed sequence spaces w(u, v; p) , w0 (u, v; p) and w (u, v; p) . Before introducing these sequence spaces we would like to present some remarks. Malkowsky and Savas [10 ] have defined the sequence spaces Z(u,v,X) which consists of all sequences whose G(u,v)transforms are in X l , c , c0 , l ( p) where u, v U . Chaudhary B. and Mishra S.K. [6] have defined the sequence space l ( p) which consists of all sequences whose S- transforms are in

0k n k n Moreover I.J. Maddox [1] introduced the sequence space w( p), w0 ( p) and w ( p) which consists of 1; l ( p) ;where S  (snk ) is defined by snk   0;

all strongly summable , strongly summable to zero and bounded sequences respectively whose Cl ( p), c0 ( p) and l ( p) respectively transforms are in the spaces ; where

1 1 k  n  ; C  (cnk )   n 0; k n and C  (cnk ) is called the Ceasaro matrix of order 1 or the matrix of arithmetic mean. The matrix domain X A of an infinite matrix A in a sequence space X is defined by

X A  x  ( xk )   : Ax  X 

(2.1) , which is a sequence space. www.iosrjournals.org

16 | Page


Paranormed Sequence spaces w(u, v; p) , w0 (u, v; p) and w (u, v; p) generated by weighted mean With the notation as in (2.1) , we can have the following representations: X (u, v, p)   X Z , for X l , c , c0 , l ( p)

l ( p )  l ( p )  S .

Following the works of the authors [1,6,9,10] , for p   pk  is a bounded sequence of a strictly positive

real

numbers

,

we

now

 w( p), w0 ( p), w ( p) by

define

the

new

sequence

spaces

 (u, v; p) for

 

n

 (u, v; p)  x  ( xk )   :  unvk xk   k 1

(2.2) We may write, using (2.1),  (u, v; p)    G (u ,v ) ; for  w( p), w0 ( p), w ( p) If pk  1 for all k  , we write  (u, v) instead of  (u, v; p) We shall first establish following some simple properties. Proposition 2.1 The sequence spaces  (u, v; p) are complete paranorm space paramormed by

1

1

pk M 1 n  M p h( x)  sup   un vk xk  ; or equivalently g ( x)  sup r 2 r  un vk xk k r n  n k 1  where  is the sum over r in the range 2r  k  2r 1 . For the space w (u, v; p) , h( x) is a

r

paranorm if and only if 0  inf pk  sup pk   . Proof: The proof of this proposition follows from the similar arguments as in the theorems 5,6 in [4 ] and theorem 2.1 in [9]. If x n is a Cauchy sequence in  (u, v; p) ; then G(u, v) x n is a Cauchy

 

sequence in  . Now it is a routine work to show  (u, v; p) is complete paranorm space under the usual paranorm. Proposition 2.2 The sequence spaces  (u, v; p) are linearly isomorphic to  w( p), w0 ( p), w ( p) . Proof: We define the transformation T :  (u, v; p)   by, x y  T ( x) . Linearity of T is obvious. Further, if Tx  , then x   . Hence T is injective. Now, let y   and define the sequence x  ( xk ) by xk 

1 n Then, h( x)  sup   un vk xk n  n k 1

1 vk

 yk yk 1    ; k   uk uk 1 

.

1

pk

M  

1

p M 1 n = sup   yk k  n  n k 1  = g ( y) . Thus , we deduce that x   (u, v; p) and as a consequence we conclude that T is surjective and is a paranorm preserving. Hence T is a linear bijection and showing that the sequence spaces  (u, v; p) are linearly isomorphic to  .

www.iosrjournals.org

17 | Page


Paranormed Sequence spaces w(u, v; p) , w0 (u, v; p) and w (u, v; p) generated by weighted mean III.

Duals In this section we find  - dual of w(u.v; p) . If X be a sequence space , we define   dual of X as:

X   a  (ak ):  ak xk is convergent for each x  X . k 1

Theorem 3.1 Let 0  pk  1for every k 

     a  (ak ) :  ak r  

  1  vk  

. Then w (u.v; p)   where

  2r N 1   uk  

1 p k

1

2 N  1

r

p

k 1

uk 1

    convrges and lim 2r N 1 m    

1 p m

  am   O(1)  um vm  

. Proof : We first assume that the conditions hold. Let a  and x w(u, v; p) .Then for y  w( p) , there exists a positive integer N  1Such that

1 n  yk n k 1

pk



1  yk 2r r

or equivalently

yk  2r N 1

1 pk

. Now,

m m 1 1  ak xk   ak  k 1 k 1  vk m 1

ak k 1 v k

 

  , where sum over r runs from 2r  k  2r 1 . It follows that ,

pk

 yk yk 1   am ym      uk uk 1   um vm

 yk yk 1  am ym    u u u v k k  1 m m  

2

a  k r v k

r

N

1

1 p k

1

2 

uk

r

N

1

uk 1

p

k 1

am  um vm

2

r

N 1

1 p m

. 

It follows that  ak xk converges for each x w(u, v; p) . k 1

Hence,   w (u.v; p) . 

On the other hand , let a w (u, v; p) . Then ,  ak xk converges for each x w(u, v; p) . Since , k 1

    r 1  1  2 N x uk  vk     

   1 a  k  vk k 1  

  2r N 1   uk  

1 p k

1

2  1 p k

r

N 1

p

k 1

uk 1

     w(u , v; p )   

1

2 N  r

1

uk 1

p

k 1

;it

follows

that

    converges. We need to show that    www.iosrjournals.org

18 | Page


Paranormed Sequence spaces w(u, v; p) , w0 (u, v; p) and w (u, v; p) generated by weighted mean

lim 2r N 1

m 

1 p m

am  O(1) um vm

As a contrary let,

lim 2r N 1 n 

1 p m

 am  O(1) , which is immediately against the fact that  ak xk converges for k 1 um vm

   1 each x w(u, v; p) and  ak   vk k 1  

  2r N 1   uk  

Hence we must have, lim 2r N 1 m 

1 p m

1 p k

1

2 N  1

r

p

k 1

uk 1

    converges.   

am  O(1) um vm

So, we arrive at the result w (u.v; p)   ; thereby proving w (u.v; p)   . IV. Matrix Transformation this section we give characterization for (w(u, v; p), l ) , (w(u, v; p), c) and (w(u, v; p), c0 ) . Theorem 4.1 Let 0  pk  1for every k  . Then A  ( w(u, v; p), l ) if and only if In

the

matrix

classes

i) there exists an integer N  1 such that

    1 sup n  max r  ank   r vk       ii) lim  2r N 1 m  

1 p m

  2r N 1   uk  

1 p k

        and      

1

2 N  1

r

p

k 1

uk 1

 anm   O(1)  um vm  n

Proof: Let the conditions be satisfied. Since, m m 1 1  ank xk   ank  k 1 k 1  vk

 yk yk 1   ym anm      uk uk 1   um vm

m 1  1   ank  k 1   vk

  ank xk  k 1

 yk yk 1   am  ym     u u u v  k 1   m m  k

  1  max r ank  r  vk 

  2r N 1   uk  

1 p k

1

2 N   r

1

uk 1

p

k 1

     anm 2r N 1   um vm  

www.iosrjournals.org

1 p m

19 | Page


Paranormed Sequence spaces w(u, v; p) , w0 (u, v; p) and w (u, v; p) generated by weighted mean 1   p k   2r N 1 2r N 1 1   sup n  max r ank   r  v u uk 1 k k      , by using conditions (i) and (ii).

1 p

k 1

     2r N 1   

1 p m

anm um vm

It follows that An  and hence  ank xk  An ( x) converges for each x w(u, v; p) and n  k 1

.

Thus Ax  l . On the other hand , let A  ( w(u, v; p), l ) . Since ,

  1   vk 

  2 N   w(u, v; p) , the condition (i) holds. In order to see that condition   uk 1   (ii) is necessary, we assume that for N  1, 1   p m anm   lim  2r N 1  O(1) ,  m u v m m    n   2r N 1   uk  

1 p k

1

  that is,  2r N 1 

1 p m

r

1

p

k 1

 anm  l .  um vm  n

Now, therefore, there exists a sequence  N r    such that

    1 sup n  max r  ank    vk r    

  2r N 1 r   uk  

1 p k

1

2 N  1

r

p

k 1

r

uk 1

       o(1) and      

  p m anm   lim  2r N r 1  o(1) .  m u v m m    n o(w(u, v; p)) but xk l (w(u, v; p)) . So, we arrive at the contradiction to our Hence , xk assumption A  ( w(u, v; p), l ) . Thus , condition (ii) is necessary ; thereby completing the proof for 1

the theorem. By using the arguments as in theorem (4.1) it is straight forward matter to prove the following theorems: Theorem 4.2 Let 0  pk  1for every k  . Then A  (w(u, v; p), c) if and only if i) there exists an integer N  1 such that

    1 sup n  max r  ank    vk r    

  2r N 1   uk  

1 p k

1

2 N  r

1

uk

p

k 1

        and      

www.iosrjournals.org

20 | Page


Paranormed Sequence spaces w(u, v; p) , w0 (u, v; p) and w (u, v; p) generated by weighted mean

  p m anm   ii) lim  2r N 1  o(1)  m u v m m    n iii) lim ank   k exists for every fixed k. 1

n

Theorem 4.3 Let 0  pk  1for every k 

. Then A  ( w(u, v; p), c0 ) if and only if

i) there exists an integer N  1 such that

    1 sup n  max r  ank    vk r    

  2r N 1   uk  

1 p k

1

2 N  1

r

uk

p

k 1

             

  p m anm   ii) lim  2r N 1  o(1) and  m u v m m    n iii) lim ank   k with  k  0 for every fixed k. 1

n

References [1] [2] [3] [4]

I.J. Maddox, Paranormed sequence spaces generated by infinite matrices, Proc. Cambridge Philos. Soc.64(1968) 335-340 I.J. Maddox, Spaces of Strongly summable sequences, Quart. J. Math. Oxford 18(2) (1967) 345-355 C.G. Lascarides, I.J. Maddox, Matrix transformation between some classes of sequences, Proc. Cambrige Philos. Soc. (1970) 99104 I.J. Maddox, Some properties of paranormed sequence spaces, London J. Math. Soc.2, 1(1969) 316-322

[5]

S. Simons , The sequence spaces

[6]

B. Choudhary, S.K. Mishra, On Kothe-Toeplitz duals of certain sequence spaces and their matrix transformations, Indian J. Pure Appl. Math 24(5) (1993) 291-301 B. Altay, F. Basar, Some paramormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull.Math.30(2006) 591-608 B. Altay, F. Basar, Some paramormed sequence spaces of non-absolute type derived by weighted mean , J. Math.Anal. Appl. 319(2006) 454-508

[7] [8]

l ( p ) and m( p ) Proc. Lond. Math. Soc. 15 (3) (1965) 422-436

l ( p) derived by weighted mean, J.Math. Anal. 330(2007) 174-185

[9]

B. Altay, F. Basar, Generalization of the sequence space

[10]

E. Malkowsky, Es. Savas, Matrix transformation between sequence spaces of generalized weighted means, Appl. Math. Compt. 147(2004) 333-345

www.iosrjournals.org

21 | Page


B010131521