Available at
http://pvamu.edu/aam
Appl. Appl. Math.
ISSN: 1932-9466
Applications and Applied
Mathematics:
An International Journal
(AAM)
Vol. 8, Issue 2 (December 2013), pp. 756 – 766
Some Geometric Properties of a New Type Metric Space
Muhammed Çınar1, Murat Karakaş2 and Mikail Et3
1
Department of Mathematics
Muş Alparslan University
49100; Muş; TURKEY
muhammedcinar23@gmail.com
2
Department of Statistics
Bitlis Eren University
13000; Bitlis; TURKEY
m.karakas33@hotmail.com
3
Department of Mathematics
Firat University
23119; Elazig ; TURKEY
mikailet68@gmail.com
Received:August 8, 2013 ; Accepted: August 8, 2013
Abstract
In this paper, we define a metric on our new space and then show that this linear metric space is
k-nearly uniform convex and has property beta where p   p k  is a bounded sequence of
positive real numbers. Finally, we give a result about property (H ) by using k-nearly uniform
convexity.
Keywords: Cesàro difference sequence space, Luxemburg norm, extreme point, convex
modular, property (H)
AMS-MSC 2010 No: 40A05, 46A45, 46B20
1. Introduction
By a Lacunary sequence   k r  where
k 0  0, we'll mean an increasing sequence of
756
AAM: Intern. J., Vol. 8, Issue 2 (December 2013)
757
nonnegative integers with k r  k r 1   as r  . The intervals determined by  will be
kr
denoted by I r  k r 1 , k r . We write hr  k r  k r 1 . The ratio
will be denoted by q r .
k r 1
Freedman (1978) defined the space of lacunary strongly convergent sequences as follows:

1
N    x   x k  : lim
r  h

r
x

 l  0, for some l .

k
kI r
It is well known that there exists a very close connection between the space of lacunary strongly
convergent sequences and the space of strongly Cesaro summable sequences. This connection
can be found in Das and Patel (1989), Mursaleen and Chishti (1996). Because of this, a lot of
geometric properties of Cesaro sequence spaces can be generalized to the lacunary sequence
spaces.
Let  be the space of all real sequences. Let p   p r  be a bounded sequence of positive real
numbers. Karakaya (2007) defined the space  p,   as follows:

 p,     x   x k  :

1



r 1  hr


kI r

x k 

pr

  .

The paranorm on  p,   is given by
x
 p, 
  1
   
 r 1 h
  r

x k 


kI r
pr




1/ H
,
where H  sup p r . If p r  p for all r , we use the notation  p   in place of  p,  . The
norm on  p   is given by
x
 p  
  1
   
 r 1 h
  r

kI r

x k 

 
p




1/ p
.
 
If   2 r , then  p,    ces p . If   2 r
For x   p,  , let
1
  x    
r 1  hr


kI r

x k 

pr
and p r  p for all r , then  p,   ces p .
758
Muhammed Cinar et al.

x 
   1. The


inf
0
:



L
 

can be reduced to the usual norm on  p   , that is,
 p,  
and define the Luxemburg norm on
Luxemburg norm on
x
L
 x
 p  
 p  
by
x
.
Ahuja et al. (1977) introduced the notions of strict convexity and U.C.I (uniform convexity) in
linear metric spaces which are generalizations of the corresponding concepts in linear normed
spaces. Later, Sastry and Naidu (1979) introduced the notions of U.C.II and U.C.III in linear
metric spaces and showed that these three forms are not always equivalent. Further, Junde and
Chen (1994), Junde et al. (1995) showed that if a linear metric space is complete and U.C.I , then
it is reflexive.
Let X be a vector space over the scalar field of real numbers and d be an invariant metric on
X . We denote Bd  X  and S d  X  as follows:
Bd  X   x  X : d  x, 0  r and S d  X   x  X : d  x, 0  r.
Let  X , d  be a linear metric space and Bd  X ( resp., S d  X ) be the closed unit ball (resp.,
the unit sphere) of X . A linear metric space  X , d  has property   if and only if for each
r  0 and   0 there exists   0 such that for each element x  Bd 0, r  and each
sequence  x n  in Bd 0, r  with sepx n    , there is an index k for which
 x  xk 
, 0  1   ,
d
 2

where
sep x n   inf d  x n , x m  : n  m   , Sanhan and Mongkolkeha (2011). If for each
x  S d 0, r  and x n   S d 0, r ,
said to have property H  .
w
x n  x implies x n  x, a linear metric space
X , d 
is
Let k  2 be an integer. A linear metric space  X , d  is said to be k-nearly uniform convex
(k-NUC) if for every   0 and r  0 , there exists   0 such that for any sequence
x n   Bd 0, r  with sepx n    , there are s1 , s 2 ,..., s k such that
 x s  x s2  ...  x sk 
, 0   r   ,
d  1
k


Junde and Narang (2000).
AAM: Intern. J., Vol. 8, Issue 2 (December 2013)
759
2. Main Results
In this section, our goal is to define a metric on  p,   and show that  p,   has property
  and k  NUC under the metric. Let  be the space of all real sequences and p   p r 
be a bounded sequence of real numbers with p r  1 for all r  N. The mapping
pr

d x, y      h1r  xk   y k  

 r 1 kI r




1/ H
is a metric on the space  p,   , where H  sup p r , since the function t
p
is convex for
p  1. First, we will show that the space  p,   has property   under the above metric. To
do this, we need the following two lemmas.
Lemma 2.1.
Let y, z   p,  , d . If   0,1, then
d  y  z, 0
M
 d  y, 0   2  d  y, 0 
M
2M
M
M

M 1
d z, 0M .
Proof:
Let y, z   p,  , d  and 0    1. Then,
d  y  z, 0
M
1
  
r 1  hr


kI r

y k   z k  


1
   1   
hr
r 1 

1
 1     
r 1  hr

1
  
r 1  hr


kI r
1
2  
r 1  hr

M

1
y k   
hr


y k  

kI r
kI r

y k  


kI r
pr
pr
pr

kI r
1
   
r 1  hr

1
 2   
r 1  hr
z k  
 

M
pr
z k  
y k  
 

kI r

kI r
pr
z k  
y k  
 

y k  

pr
pr
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Muhammed Cinar et al.
1
  
r 1  hr


kI r
pr
 

1
y k    2 M   
r 1  hr

1
 M 1  

r 1  hr

2M

kI r
M
M
pr
pr

z k  

 d  y, 0   2  d  y, 0  
M

kI r

y k  

2M

d z, 0M .
M 1
Lemma 2.2.
Let y, z   p, , d . Then, for any   0 and L  0 there exists   0 such that
d  y  z, 0M  d  y, 0M
 ,
where
d  y, 0M
 L and d  z , 0   .
M
Proof:
Let   0 and L  0. For  

2 M 1 L   
, we take  
 M 1
2 M 1
. By using previous Lemma
2.1, we have
d  y  z, 0M  d  y, 0M  2 M  d  y, 0M 
 d  y, 0  2 M  L 
M
 d  y, 0   2 M

M
 d  y, 0 
M

2
 d  y, 0  
M
and

2 M 1

2
2M
 M 1
2M

M 1
d z, 0M

L
2 M  M 1
 M 1 M 1
L  
2
(2.1)
AAM: Intern. J., Vol. 8, Issue 2 (December 2013)
761
d  y, 0M  d  y  z, 0M  2 M  d  y  z, 0M 


2M

 d  y  z , 0  2 M  d  y, 0   
M
M
 d  y  z , 0  2 M  L    
M
 d  y  z , 0  2 M

M
 d  y  z , 0 
M

2
 d  y  z , 0  

2
M 1
L   
2M
M 1
d  z, 0M
2M
 M 1
 M 1

(2.2)
 M 1 2 M 1
L     
2

2
M
From 2.1 and 2.2  , we obtain that
d  y  z, 0M  d  y, 0M
 .
Theorem 2.3.
The space  p,  , d  has property  .
Proof:
Let   0 and x n   B p, , d  such that sep x n    and x  B p,  , d . We take
N


y N   0, 0,..., 0,  y k , y  N  1, y  N  2 ,... .
k 1


By using diagonal method, we can find a subsequence


x 
nj
of
that x n j k  converges for each k  N with 1  k  N , since
x n  for each N  N such
x n k k 1 is bounded for each
     .
k  N. Therefore, there is rN  N for each N  N such that sep x nN
sequence of positive integers
rN n1
1



r  N  hr

kI r

x rN 



with r1  r2  r3 ... such that d x rNN , 0 
N  N. Then, there exists   0 such that for all N  N ,

So, there is a
j  rN

2
for all
pr
 .
By Lemma 2.2, there exists  0 such that
(2.3)
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Muhammed Cinar et al.
d  y  z, 0M  d  y, 0M


,
2m
(2.4)
where
d  y, 0M
 r M and d  z , 0   0 .
M

There exists N 1  N such that d x N1 , 0

M
d x, 0M
  0 if x  B p,   and
  0 . Let
us take y  x rN1 and z  x . Hence, we have
N1
N1
1



r  N1  hr

From

kI r
xk   x rN1 k  


2

2.3, 2.4, 2.5
pr
1
 

r  N1  hr


kI r
x rN1 k  

2 
pr


2M
.
(2.5)
and by using convexity of the function f t   t
pr
, for all r  N, we
obtain that
  y  z 
, 0  
 d 

  2
M
1
 

r 1  hr

1
 

r 1  hr
N1 1
1
 

r 1  hr
N1 1
1 N1 1  1
  
2 r 1  hr
1
 M
2
pr

xk   x rN1 k  


2

pr

xk   xrN1 k  


2


xk   x rN1 k  


2



xk  

kI r
kI r
kI r
kI r
1



r  N1  h r


kI r
pr


r  N1
pr
pr


r kI r
1
 

r  N1  hr

1 N1 1  1
  
2 r 1  hr

x rN k  
1

1
  h 

2M

kI r
xk   xrN1 k  


2


kI r

x rN k  
1

x rN1 k  

2 
pr
pr


2M
AAM: Intern. J., Vol. 8, Issue 2 (December 2013)
1 N1 1  1
  
2 r 1  hr

kI r
763
pr

1  1
xk     
2 r 1  hr

2 M  2   1
 M 1  
2
r  N1  hr

kI r

xrN k  
1

pr
pr


xrN k    M
1
2


kI r
r M r M 2M  2



 M 1   M
2
2
2
2
 rM 

2
.

 

1/ M
Therefore, we have d y 2 z , 0  r M  2
the space  p,  , d  posses property  .


whenever   0, r  r M  2

1/ M
. Consequently,
Now, we'll examine k  NUC property of the space  p,  , d .
Theorem 2.4.
The space  p,   is k  NUC for any integer k  2.
Proof:
Let   0 and x n   Bd  p,  with sep x n    . For each m  N, let
x nm  0, 0,..., x n m , x n m  1,....
Since the sequence
find a subsequence
x n i i1
x 
nj
of
(2.6)
is bounded for each i  N, by using diagonal method, we can
xn 
x k  converges for each k  N. Therefore,
sep x     . Hence, there exists a sequence of


such that
there is an increasing sequence t m with
nj
m
nj
j t m


positive integers rm m1 with r1  r2  r3  ... such that d x rmm , 0 

there is   0 such that
1



r  m  hr


iI r

x rm 


2
, for all m  N. Then,
pr
.
(2.7)
764
Muhammed Cinar et al.
Let   0 such that 1    lim inf p r . Let  1 
r 
k  1  1 
for k  2. From Lemma 2.2,
k  1 k  2
there is a   0 such that
d  y  z, 0M  d  y, 0M
 1 ,
(2.8)
d  y, 0  r M and d z, 0   . Then, there exist positive integers
where
mi i  1, 2,.., k  1
with
m1  m2  ...  mk 1
such that
d ximi , 0   . Now, define
mk  mk 1  1. Then, we have d x rmmkk , 0   for all m  N. For 1  i  k  1, let t i  i and
M
M

t k  rmk . By using


2.6, 2.7 , 2.8

f i u   u
and the convexity of function
pi
i  N ,
we
obtain
  x s1  x s2  ...  x sk  
d
, 0  
 

k

 
M
pr

x s1 i   ...  x sk i  


k

pr

x s1 i   ...  x sk i  


k

  h 
x s1 i   ...  x sk i  


k

pr
1
 

r 1  hr

1
 

r 1  hr
m1

m1
1
r 1

iI r
iI r
r iI r
1
  

r  m1 1  hr

1 k  1
  
r 1 k j 1  hr
m1

1
  
r  m2 1  hr
1 k  1
  
r 1 k j 1  hr

iI r

iI r
1 k 1
   
r  m2 1 k j 3  hr
m3
1
 ...  
r  mk 1 1 k
mk
x s2 i   ...  x sk i  


k


x s j i  

iI r

m1

iI r

1
  

r  m1 1  hr
m2
iI r
pr
 1

iI r
x s2 i   x s3 i   ...  x sk i  


k

pr
pr
pr
1 k  1
  
r  m1 1 k j  2  hr
m2

x s j i  

1



j  k 1  hr
k
pr

x s1 i   ...  x sk i  


k

xs3 i   xs4 i   ...  xsk i  
  2 1

k


x s j i  

iI r
1
  

r  m1 1  hr


iI r

iI r

x s j i  

pr
pr

x s j i  

pr
1
  

r  mk 1  hr


iI r
x sk i  

k 
pr
 k  1 1
pr
AAM: Intern. J., Vol. 8, Issue 2 (December 2013)

 d x s ,
1




M
1 mk  1
  
k r 1  hr

 d x s2 , 
r
1
 
k

M

 ...  d x sk , 

iI r

x sk i  


iI r
pr
1
  

r  mk 1  hr


x sk i  


1
r M 1  M


r   
k
k
r  mk 1  hr

1



r  mk 1  hr

 
M


k
k  1 M 1 mk  1

r  
k
k r 1  hr
M
765

iI r
x sk i  

k 
pr
pr

iI r

iI r
1
 
k
x sk i  

k 
1



r  mk 1  hr

x sk i  


pr
pr
 k  1 1

iI r
x sk i  

k 
pr
 k  1 1




 k  1 1
 k  1  1 
 
 r M  k  1 1  

 k

r
M
 k  1  1    
k  1  1     k  1  1 
M
  .


r



 k  1 
   

k k  1  2   k  
 2 
 k
Thus, we have
 x s i   x s2 i   ...  x sk i  
d  1
, 0   r M 
k





1/ M
k  1 1 
2
k

 r 

for   0, r  r M 


1/ M
k  1 1 
2
k
.
Hence,  p,  , d  is k  NUC.
Since k  NUC implies NUC and NUC implies property H , by using previous theorem,
we give the following result:
Corollary 2.5.
The space  p,  , d  has property H .
766
Muhammed Cinar et al.
3. Conclusion
Many mathematicians are interested in  p  -type sequence spaces. Then, some geometric
properties on these spaces were considered equipped with the Luxemburg norm. In linear metric
spaces, the notion of uniform convexity,strict convexity or rotundity was introduced in 1977.
Later, the relation between these properties and property (H) was investigated. From this point of
view, we defined a metric on the space  p,   as an  p  -type sequence space. Then, we
studied the geometric structure of this space and showed that the linear metric space  p,  , d 
is k-NUC, has property (H) and possesses property  .
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