Gen. Math. Notes, Vol. 27, No. 2, April 2015, pp. 47-58
ISSN 2219-7184; Copyright © ICSRS Publication, 2015
www.i-csrs.org
Available free online at http://www.geman.in
Some New Sequence Spaces Defined by
Musielak-Orlicz Functions on a Real nNormed Space
Tanweer Jalal1 and Mobin Ahmad Sayed2
1
Department of Mathematics, National Institute of Technology
Hazratbal, Srinagar- 190006, India
E-mail: tjalal@rediffmail.com
2
Department of Mathematics, Faculty of Science
Jazan University, P.O. Box 2097, Jazan 45142, Saudi Arabia
E-mail: profmobin@yahoo.com
(Received: 23-8-14 / Accepted: 14-3-15)
Abstract
The purpose of this paper is to introduce the sequence
space E nq ( B , Μ , p , s , . , ... , . ) defined by using an infinite matrix and
Musielak-Orlicz function. We also study some topological properties and prove
some inclusion relations involving this space.
Keywords: Paranorm, Infinite matrix, n-norm, Musielak-Orlicz functions,
Euler transform.
1
Introduction and Preliminaries
The concept of 2-normed spaces was initially developed by Gähler [1] in the mid1960s, while one can see that of n -normed spaces in Misiak [2]. Since then, many
others have studied this concept and obtained various results; see Gunawan [3, 4]
48
Tanweer Jalal et al.
and Gunawan and Mashadi [5]. Let n be a non-negative integer and X be a
real vector space of dimension d, where d ≥ n ≥ 2 . A real-valued function
. , ..., .
on X
n
satisfying the following conditions:
(2)
(x1 , x 2 , ... , x n )
(x1 , x 2 , ... , x n )
(3)
α x 1 , x 2 , ... , x n
(4)
(x1
(1)
= 0 if and only if x1 , x 2 , ... , x n are linearly dependent.
is invariant under permutation,
(
= α
+ x , x 2 , ... , x n )
≤
(
x1 , x 2 , ... , x n ) , for any α ∈ R ,
x1 , x 2 , ... , x n ) +
is called an n − norm on X and the pair
space.
(X
(
, ., ..., .
x , x 2 , ..., x n
)
)
is called an n − normed
A trivial example of an n − normed space is X = R n , equipped with the
Euclidean n − norm
(x1 , x 2 , ... , x n ) E = volume of the n-dimensional
parallelepiped spanned by the vectors x1 , x 2 , ... , x n which may be given explicitly
by the formula
(x1 , x 2 , ... , x n )
= det ( x i j ) = a bs (det ( ⟨ xi , x j ⟩ ))
x i = ( x i 1 , x i 2 , ... , x i n ) ∈ R n
where
(X
E
, ., ..., .
)
be an
for
on X
i = 1 , 2 , 3 ..., n .
n − normed space of dimension
{ a1 , a 2 , ... , a n } be a linearly independent set in
n −1
each
(1)
Let
d ≥ n ≥ 2 and
X . Then the function
., ..., .
∞
is defined by
( x1 , x 2 , ... , x n )
∞
= max
1 ≤ i ≤n
{
x1 , x 2 ,... , x n −1 , a i
}
(2)
defines an (n − 1) -norm on X with respect to { a1 , a 2 , ... , a n } and this is known as
the derived (n − 1) -norm.
The standard n -norm on X a real inner product space of dimension d ≥ n is as
follows:
(x1 , x 2 , ... , x n )
[
]
s = det (⟨ x i , x j ⟩ ) 2 ,
1
where ⟨ , ⟩ denotes the inner product on X . If we take X = R n then this n norm is exactly the same as the Euclidean n -norm
(x1 , x 2 , ... , x n ) E
Some New Sequence Spaces Defined by…
mentioned
x1
=
earlier.
For
n = 1 this
49
n -norm
is
the
usual
norm
⟨ x1 , x1 ⟩ for further details (see Gunawan [4]).
We first introduce the following definitions:
(x )
A sequence
)
is said to be
= 0 , for every z1 , z 2 , ... , z n − 1 ∈ X .
(3)
in an n -normed space
convergent to some L ∈ X if
lim
k→ ∞
k
x k − L , z 1 , z 2 , ... , z n − 1
(X
, ., ..., .
A sequence (x k ) in an n -normed space ( X , ., ..., .
lim
k →∞
x k − x p , z 1 , z 2 , ... , z n − 1
) is said to be Cauchy if
= 0 , for every z1 , z 2 , ... , z n − 1 ∈ X .
(4)
p → ∞
If every Cauchy sequence space in X converges to some L ∈ X , then X is
said to be complete with respect to the n -norm. A complete n -normed space is
said to be a n - Banach space.
An Orlicz function is a function M : [ 0 , ∞ ) → [ 0 , ∞ ) , which is continuous,
non-decreasing and convex with
M ( 0 ) = 0 , M ( x ) > 0 as x > 0 a
M ( x ) → ∞ , as x → ∞ .
Lindenstrauss and Tzafriri [6] studied some Orlicz type sequence spaces defined
as follows:


∞
 xk 




(5)
ℓ M =  (x k ) ∈ w : ∑ M
< ∞ , for some ρ > 0  .


ρ
k =1





The space ℓ M with the norm


x = inf  ρ > 0 :

∞
∑
k =1
 xk
M 
 ρ


 ≤1






(6)
becomes a Banach space which is called an Orlicz sequence space. The space
ℓ M is closely related to the space ℓ p which is an Orlicz sequence space with
M (t ) = t
p
, for 1 ≤ p < ∞ . An
∆ 2 − condition for all values of
Orlicz function M
is said to satisfy
u , if there exists a constant K such that
M ( 2 u ) ≤ K M ( u ) , u ≥ 0 (see [7]).
50
Tanweer Jalal et al.
A sequence space Μ = (M k ) of Orlicz functions is called a Musielak-Orlicz
function see ([8], [9]). A sequence space Ν = (N k ) defined by
N k ( v ) = sup
{ v u − M ( u ) : u ≥ 0 },
k
k = 1 , 2 , ...
(7)
is called the complimentary function of a Musielak-Orlicz function Μ . For a
given Musielak-Orlicz function Μ , the Musielak-Orlicz sequence space function
t Μ and its subspace h Μ are defined as follows
tΜ =
{ x∈w:
IΜ ( c x
)< ∞
for some c > 0 },
hΜ =
{ x∈w:
IΜ ( c x
)< ∞
for all c > 0 },
(8)
where I Μ is a convex modular defined by
IΜ(x
)
∞
=
∑
k =1
M
k
(x ), x = (x ) ∈t
k
k
M
.
(9)
We consider t Μ equipped with the Luxemberg norm


 x
x = inf  k > 0 : I Μ   ≤ 1 
k


(10)
or equipped with the Orlicz norm
x
0
 1
= inf 
 k
( 1 + I Μ ( k x )) : k

> 0 .

(11)
Let X be a linear metric space. A function p : X → R is called a paranorm, if
(1) p ( x ) ≥ 0 , for all x ∈ X ;
(2) p ( − x ) = p ( x ), for all x ∈ X ;
(3) p ( x + y ) ≤ p ( x ) + p ( y ), for all x y ∈ X ;
(4) If (σ n ) is a sequence of scalars with σ n → σ as n → ∞ and (x n ) is a
sequence
of
vectors
with
p ( σ n x n − σ x ) → 0 as n → ∞ .
p ( x n − x )→ 0
as
n → ∞ , then
A paranorm p for which p ( x ) = 0 implies x = 0 is called total paranorm, and
the pair ( X , p ) is called a total paranormed space. It is well known that the
metric of any linear metric space is given by some total paranorm (see [10],
Some New Sequence Spaces Defined by…
51
Theorem 10. 4.2, P-183). For more details about sequence spaces see [11-24] and
the references therein.
∞
Let ( s k ) denotes the sequence of partial sums of the infinite series
be any positive real number. The Euler transform
s = ( s n ) is defined by
E nq ( s ) =
∞
The series
∑a
n=0
E nq ( s ) =
1
( 1+ q ) n
n
 n
v=o
 
∑  v  q
n−v
∑a
k =0
k
and q
( E , q ) of the sequence
sv .
(12)
is said to be summable ( E, q ) to the number s if
n
n
n
1
 q
( 1 + q ) ∑  v 
n
n−v
v=o
s v → s as n → ∞
and is said to be absolutely summable ( E, q ) or summable E , q
∑
E kq ( s ) − E kq − 1 ( s
)
(13)
if
<∞.
(14)
k
Let x = ( x k ) be a sequence of scalars we write N n ( x ) = E nq ( x ) − E nq− 1 ( x ) ,
where E nq ( x ) is defined by (12). After applications of Abel’s transformation,
we have
n−2
s n −1 A n −1
sn
1
q n −1
+
−
N n ( x )= −
x k +1 A k +
s ,
(15)
(1 + q ) n −1 (1 + q ) n (1+ q ) n 0
( 1+ q ) n −1 k∑
=o
where
Ak =
k
 q  n
 n − 1 n −i −1
 q
.
 i 
∑ 1+ q  i  − 
i =0

 
(16)
Note that for any sequences x = ( x n ) , y = ( y n ) and scalar λ , we have
N n( x + y ) = N n(x ) + N n ( y
Let
)
and
N n ( λ x ) = λ N n ( x ).
Μ = (M k ) be a sequence of Musielak-Orlicz functions, p = ( p k ) be a
bounded sequence of positive real numbers, " B = b n k " be an infinite matrix, and
(X
, ., ..., .
)
be an n − normed space, we define the sequence space:
52
Tanweer Jalal et al.
E nq ( B , Μ , p , s, . , ... , .

∞ b
x = (x k ): ∑ n k
s

k =1 k

 for some ρ > 0
)=

 Nk (x)
, z 1 , z 2 , ... , z n −1
 M k 
ρ


and for every z 1 , z 2 , ... z n −1 ∈ X





< ∞ , s ≥ 0 , 
.


pk
(17)
If we take p = p k = 1 for all k ∈ N , we have
E nq ( B , Μ , s, ., ... , .
)=
∞ b


 Nk (x)
nk
x = (x k ): ∑ s  M k 
, z 1 , z 2 , ... , z n −1

ρ
k =1 k




 for some ρ > 0 , and for every z 1 , z 2 , ... z n −1 ∈ X


 < ∞ , s ≥ 0 , 

.



(18)
If we take s = 0 , we have
E nq ( B , Μ , p, ., ... , .
)=

∞

 Nk ( x)
x = (x k ): ∑b n k  M k 
, z 1 , z 2 , ... , z n −1

ρ

k =1



 for some ρ > 0 , and for every z 1 , z 2 , ... z n −1 ∈ X




pk

< ∞ , 
.


(19)
The following well known inequality will be used throughout the article. Let
p = ( pk )
be
any
sequence
of
positive
real
numbers
with
{
}
0 ≤ p k ≤ sup p k = H , D = max 1, 2 H −1 then
k
ak + b k
pk
for all k ∈ N and a k , bk ∈ C .
≤ D  a k

Also
a
pk
pk
+ bk
pk
{


≤ max 1, a
(20)
H
}
for all a ∈ C
(see [25]).
The main object of the paper is to examine some topological properties and
inclusion relations between the above defined sequence spaces.
Some New Sequence Spaces Defined by…
2
53
Some Properties of the Sequence Space
E
q
n
( B, Μ,
p , s , . , ... , .
)
Theorem 2.1: Let Μ = (M k ) be a Musielak-Orlicz function and p = ( p k ) be a
bounded sequence of positive real numbers, then the space
E nq ( B , Μ , p , s , . , ... , . ) is linear over the real field.
Proof: Let x , y ∈ E nq ( B , Μ , p , s , . , ... , . ) and α , β ∈ ℜ (the set of real
numbers). Then there exists numbers ρ 1 and ρ 2 such that
∞
∑
k =1
b nk 
M
ks 

k
 N k (x )

, z 1, z 2 , ... , z n −1

ρ
1





pk
 N k (y)

, z 1, z 2 , ... , z n −1

ρ2




 
pk
<∞,
and
∞
∑
k =1
b nk 
M
k s 
k
< ∞.
(21)
Define ρ 3 = max (2 α ρ 1 , 2 β ρ 2 ).
Since Μ = ( M
have
∞
∑
k =1
≤
∑
k =1
) is non-decreasing, convex and so by using inequality (20), we
 N k (α x + β y )

, z 1, z 2 , ... , z n −1

ρ
3

b nk 
M k
k s 
∞
k
bnk 
M
ks 

k
 N k (α x )

, z 1, z 2 , ... , z n −1

ρ
3




 
pk
  N k (β y )
+ 
, z , z , ... , z n − 1
 
ρ
3
 




pk
(22)
bnk 
≤ D ∑ s M

k =1 k
∞
b nk 
D ∑ s M

k =1 k
∞
k
k
 N k (x )

, z 1, z 2 , ... , z n −1

ρ
3

 N k (y)

, z 1, z 2 , ... , z n −1

ρ
3









pk
+
pk
< ∞.
54
Tanweer Jalal et al.
Therefore, α x + β y ∈ E nq ( B , Μ , p , s , . , ... , . ).
Hence, E nq ( B , Μ , p , s , . , ... , .
)
is a linear space.
Theorem 2.2: Let Μ = (M k ) be a sequence of Musielak-Orlicz functions, p = ( p k )
be a bounded sequence of positive real numbers. Then the space
E nq ( B , Μ , p , s , . , ... , . ) is a paranormed space with the paranorm defined by


g (x ) = inf  ρ


pn
H
 ∞ b

nk
: ∑
s
k = 1 k


 Nk(x)
M k
, z 1 , z 2 , ... , z n − 1

ρ






pk
1

H


 ≤ 1, n = 1,2,3,... ,




(23)
where H = max  1 , sup p k  .
k


Proof: It is clear that g ( x ) = g ( − x ) and g ( x + y ) ≤ g ( x ) + g ( y ). Since
p H
= 0 for
x = 0 . Finally, we prove that
M k ( 0 ) = 0 , we get inf ρ
multiplication is continuous. Let λ ≠ 0 be any complex number, then by
definition, we have
{


g (λ x ) = inf ρ


pn
H
 ∞ b

nk
: ∑ s
k = 1 k

n
}

 λNk(x)
M k
, z 1 , z 2 , ..., z n −1

ρ






pk
1

H


 ≤ 1, n = 1,2,3,...  .




(24)
Thus, we have

pn

g (λ x) = inf ( λ s ) H


where s =
 ∞ b

nk
: ∑ s
k = 1 k

ρ
. Since λ
λ
pk


M k



(
Nk(x)
s
≤ max 1 , λ
H

, z 1 , z 2 , ..., z n − 1 


), we have
pk
1

H


≤
1
,
n
=
1
,
2
,
3
,...
,





(25)
Some New Sequence Spaces Defined by…
( (
g (λx) ≤ max1, λ
H
)

 p n
1
H
inf (s) H


 ∞ b

nk
: ∑ s
k = 1 k

55
  N k (x)
M k 
, z 1 , z 2 , ..., z n −1
  ρ




pk




≤
1
,
n
=
1
,
2
,
3
,...
,





(26)
1
H
and therefore, g (λ x ) converges to zero when g(x) converges to zero in
E nq ( B , Μ , p , s , . , ... , . ). Now, suppose that λ n → 0 as n → ∞ and x is in
E nq ( B , Μ , p , s , . , ... , . ). For arbitrary ε > 0 , let n 0 be a positive integer
such that
∞
∑
k =n 0 +1
b nk 
M k
k s 
 N k (x )

, z 1, z 2 , ... , z n −1

ρ




 
pk
<
ε
(27)
2
for some ρ > 0 . This implies that
 ∞ b
nk

 ∑ k s
k =n +1
 0

M

 N k (x )

, z 1, z 2 , ... , z n −1
k 
ρ




 




pk
1
H
≤
ε
2
(28)
.
Let 0 < λ < 1 , then using convexity of (M k ), we get
bnk 
M
∑
s 
k =n 0 +1 k

∞
 λN k ( x )

, z1 , z 2 , ... , z n −1
k 
ρ

b nk 
< λ ∑
M
s 
k =n 0 +1 k

∞
Since (M
k
h ( t )= ∑
k =1
pk
 N k (x )

, z1 , z 2 , ... , z n −1
k 
ρ

) is continuous everywhere on
n0



 



 
pk
ε 
<  .
2
H
(29)
[0, ∞ ) , then
 t N k (x )
bnk 
Mk
, z 1, z 2 , ... , z n −1
s 

ρ
k 




 
pk
(30)
is continuous at 0 . So there is 0 < δ < 1 such that h (t ) < ε 2 for 0 < t < δ . Let
K be such that λ n < δ for n > K , we have
 n0 b
nk

 ∑ k s
k =1


M

 λ n N k (x )

, z 1, z 2 , ... , z n −1
k 
ρ




 
pk




1
H
<
ε
2
.
(31)
56
Tanweer Jalal et al.
Thus,
 ∞ b 
 λ n N k (x )
nk


M
, z 1, z 2 , ... , z n −1

∑
k
s


ρ
k =1 k







 
pk




1
H
<ε,
for n > k .
Hence g (λ x ) → 0 as λ → 0 . This completes the proof of the theorem.
(32)
Theorem 2.3: If Μ ′ = (M ′k ) and Μ ′′ = (M k′′ ) are two sequences of MusielakOrlicz functions and s , s 1 , s 2 are nonnegative real numbers, then
(i) E nq ( B , Μ ′ , p , s , . , ... , . ) ∩ E nq ( B , Μ ′′ , p , s , . , ... , .
⊆E
q
n
( B , Μ ′ + M ′′ ,
p , s , . , ... , . ) .
(ii) If s 1 ≤ s 2 , then E nq ( B , Μ ′ , p , s 1 , . , ... , .
)⊆
)
E nq ( B , Μ ′ , p , s 2 , . , ... , . ).
Proof: It is obvious, so we omit the details.
Theorem 2.4: Suppose that 0 < r k ≤ p k < ∞ , for each k ∈ N . Then
E nq ( B , M , r , s , . , ... , .
)
⊆ E nq ( B , M , p , s , . , ... , . ) .
Proof: Let x ∈ E nq ( B , Μ , r , s , . , ... , . ). Then there exists some ρ > 0 such that
∞
∑
k =1
bnk 
M
ks 

k
 N k (x )

, z 1, z 2 , ... , z n −1

ρ
1

(
this implies that , M k N k ( x ) ρ , z1 , z 2 ,... , z n −1




∑
k≥ k 0
∞
∑
k≥ k 0
b nk 
M k
ks 

 N (x )
k

, z 1 , z 2 , ... , z n −1

ρ
1





b nk 
M k
ks 

 N (x )
k

, z 1 , z 2 , ... , z n −1

ρ
1





Hence x ∈ E nq ( B , Μ , p , s , . , ... , . ).
< ∞.
(33)
) ≤ 1 for sufficiently large value
of k , say k ≥ k 0 , for some fixed k 0 ∈ N . Since ( M k
get
∞
rk
)
is non decreasing, we
pk
≤
rk
< ∞.
(34)
Some New Sequence Spaces Defined by…
57
Theorem 2.5:
(i)
If 0 < p k ≤ 1 for each k , then
(ii)
If
E nq ( B , M , p , s , . , ... , .
p k ≥ 1 for
)
all
E nq ( B , M , p , s , . , ... , . ) .
⊆ E nq ( B , M , s , . , ... , . ) .
k,
then
E nq ( B , M , s , . , ... , . ) ⊆
Proof: It is easy to prove by using Theorem 2.4, so we omit the details.
Acknowledgements: The Authors expresses his heartfelt gratitude to the
anonymous reviewers for their valuable suggestions which have enormously
enhanced the quality and improved the presentation of the paper.
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