SOME NEW PARANORMED SEQUENCE SPACES DEFINED BY ORLICZ FUNCTION 20

International Journal of Science, Environment
and Technology, Vol. 1, No 2, 49-55, 2012

SOME NEW PARANORMED SEQUENCE SPACES DEFINED BY
ORLICZ FUNCTION
AYHAN ESI

AdiyamanUniversity,
University, Science
Faculty
Adiyaman
Scienceand
andArtArt
Faculty
Department of Mathematics, 02040, Adiyaman, Turkey
E-mail: aesi23@hotmail.com

Abstract. In this paper we present new classes of sequence spaces using the
concept of n-norm and to investigate these spaces for some linear topological
structures as well as examine these spaces with respect to derived (n-1) norms.
We use an Orlicz function, a bounded sequence of positive real numbers and

Λm operator to construct these spaces so that they became more generalized.
This investigations will enhance the acceptability of the notion of n-norm by
giving a way to contruct different sequence spaces with elements in n-normed
space.

Key words and phrases: n-norm, paranorm, completeness, Orlicz function.
AMS Subject Classification: 40A05, 46A45, 46B70

1. INTRODUCTION
Recall in [6] that an Orlicz function M is continuous, convex, nondecreasing
function define for x > 0 such that M (0) = 0 and M (x) > 0. If convexity of
Orlicz function is replaced by M (x + y) ≤ M (x) + M (y) then this function is
called the modulus function and characterized by Ruckle [7] .An Orlicz function M
is said to satisfy ∆2 − condition for all values u, if there exists K > 0 such that
M (2u) ≤ KM (u), u ≥ 0.
Lemma. Let M be an Orlicz function which satisfies ∆2 − condition and
let 0 < δ < 1. Then for each t ≥ δ, we have M (t) < Kδ −1 M (2) for some constant
K > 0.
A sequence space X is said to be solid or normal if (αk xk ) ∈ X, and for all
double sequences α = (αk ) of scalars with |αk | ≤ 1 for all k ∈ N.

The concept of 2-normed spaces was initially developed by Gahler [5] in the
mid of 1960’s, while that of n-normed spaces can be found in Misiak [4]. Since then,
many others have studied this concept and obtained various results, see for instance
Gunawan [2 − 3], Gunawan and Mashadi [1], Esi [9 − 10], Esi and Ozdemir [11],
Fistikci and et al.[12] and many others.
Let n ∈ N and X be a real vector space of dimension d, where n ≤ d. A
real-valued function k., ..., .k on X satisfying the following four condition:
(i) kx1 , x2 , ..., xn k = 0 if and only if x1 , x2 , ..., xn are linearly dependent,
(ii) kx1 , x2 , ..., xn k is invariant under permutation,
(iii) kαx1 , x2 , ..., xn k = |α| kx1 , x2 , ..., xn k , α ∈ R,
(iv) kx1 + xı1 , x2 , ..., xn k ≤ kx1 , x2 , ..., xn k+kxı1 , x2 , ..., xn k
called an n-norm on X, and the pair (X, k., ..., .k) is called an n-normed space [2] .
Let (X, k., ..., .k) be an n-normed space of dimension d ≥ n ≥ 2 and {a1 , a2 , ..., an }
be a linearly independent set in X. Then the following function k., ..., .k∞ on X n−1

AYHAN ESI

50

defined by

kx1 , x2 , ..., xn−1 k∞ = max {kx1 , x2 , ..., xn−1 , ai k : i = 1, 2, ..., n}

defines an (n-1)-norm on X with respect to {a1 , a2 , ..., an } .
Let n ∈ N and (X, h., .i) be a real inner product space of dimension d ≥ n.
Then the following function k., ..., .kS on X × X × ... × X (n factors) defined by
1

kx1 , x2 , ..., xn kS = [det (hxi , xj i)] 2

is an n-norm on X, which is known as standard n-norm on X. If we take X = Rn ,
then this n-norm is exactly the same as Euclidean n-norm such as


 x11 ...x1n 



...
kx1 , x2 , ..., xn kE = abs 


 xn1 ...xnn 

where xi = (xi1 , ..., xin ) ∈ Rn for each i=1,2,...,n.
We procure the following results those will help in establishing some results
of this article.
Lemma 1.[1] A standard n-normed space is complete if and only if it is
1
complete with respect to the usual norm k.k = h., .i 2 .
Lemma 2.[1] On a standard n-normed space X, the derived (n-1)-norms
k., ..., .k∞ , defined with respect to orthonormal set {e1 , e2 , ..., en }, is equivalent to
the standard (n-1)-norms k., ..., .kS . Precisely, we have for all x1 , x2 , ..., xn−1
√
kx1 , x2 , ..., xn−1 k∞ ≤ kx1 , x2 , ..., xn−1 kS ≤ n kx1 , x2 , ..., xn−1 k∞

where kx1 , x2 , ..., xn−1 k∞ = max {kx1 , x2 , ..., xn−1 , ei k : i = 1, 2, ..., n} .
In paper [8] , Mursaleen and Noman introduced the notion of λ−convergent
∞
and λ − bounded sequences as follows: Let λ = (λk )k=0 be a strictly increasing
sequence of positive real numbers tending to infinity, that is
0 < λo < λ1 < ... and λk → ∞ as k → ∞

and said that a sequence x = (xk ) ∈ w is λ − convergent to the number L, called
a the λ−limit of x, if Λm (x) → L as m → ∞, where
m
1 X
Λm (x) =
(λk − λk−1 ) xk .
λm
k=1

The sequence x = (xk ) ∈ w is λ − bounded if supm |Λm (x)| < ∞. It is well known
[8] that if limm xm = a in the ordinary sense of convergence, then
!
m
1 X
lim
(λk − λk−1 ) |xk − a| = 0.
m
λm
k=1

This implies that



m
 1 X



(λk − λk−1 ) (xk − a) = 0
lim |Λm (x) − a| = lim 
m
m  λm

k=1

which yields that limm Λm (x) = a and hence x = (xk ) ∈ w is λ − convergent to a.
2. MAIN RESULTS

SOME NEW PARANORMED SEQUENCE

.......

51

Let (X, k., ..., .k) be real n-normed space and w (n − X) denotes the space
of X-valued sequences. Let M be an Orlicz function and p = (pk ) be any bounded
sequence of strictly positive real numbers. Now, we define the following sequence
spaces:
[M, Λ, p, k., ..., .k]o =
ipm
)
(
h 


=0
x = (xk ) ∈ w (n − X) : limm M
Λmρ(x) , z1 , z2 , ..., zn−1

,
f or some ρ > 0 and for every z1 , z2 , ..., zn−1 ∈ X

[M, Λ, p, k., ..., .k] =
ipm
)
(
h 


=
0
,
z
,
z
,
...,
z
x = (xk ) ∈ w (n − X) : limm M

Λm (x)−L

1
2
n−1
ρ
,
f or some ρ > 0, L ∈ X and for every z1 , z2 , ..., zn−1 ∈ X

and

[M, Λ, p, k., ..., .k]∞
ipm
(
)
h 


x = (xk ) ∈ w (n − X) : supm M
Λmρ(x) , z1 , z2 , ..., zn−1

0 and for every z1 , z2 , ..., zn−1 ∈ X

The following well-known inequality will be used

in this study: If 0 ≤
inf k pk = Ho ≤ pk ≤ supk = H < ∞, D = max 1, 2H−1 , then
pk

≤ D {|xk |

pk

p

+ |yk | k }


p
H
for all k ∈ N and xk , yk ∈ C. Also |xk | k ≤ max 1, |xk |
for all xk ∈ C.

In this section we investigate some linear topological structures of the sequence spaces [M, Λ, p, k., ..., .k]o , [M, Λ, p, k., ..., .k] and [M, Λ, p, k., ..., .k]∞ .
It is clear from the definition that [M, Λ, p, k., ..., .k]o ⊂ [M, Λ, p, k., ..., .k].
Further [M, Λ, p, k., ..., .k] ⊂ [M, Λ, p, k., ..., .k]∞ follows from the following inequality
pm
 


Λm (x)

,
z
,
z
,
...,
z
M
1 2
n−1

2ρ

pm






Λm (x) − L
L
pm

1
1



≤
+ M
, z1 , z2 , ..., zn−1
.
M
, z1 , z2 , ..., zn−1

2
ρ
2
ρ
|xk + yk |

Similarly, we have [M, Λ, p, k., ..., .k]o ⊂ [M, Λ, p, k., ..., .k] ⊂ [M, Λ, p, k., ..., .k]∞ .
Theorem 2.1. If {Λm (x) , z1 , z2 , ..., zn−1 } is a linearly dependent set in
(X, k., ..., .k) for all but finite m, where x = (xk ) ∈ w (n − X) and inf m pm > 0,
then
ipm
h 


= 0, for every ρ > 0,
(i) limm M
Λmρ(x) , z1 , z2 , ..., zn−1

ipm
h 


(ii) supm M
Λmρ(x) , z1 , z2 , ..., zn−1
< ∞, for every ρ > 0.
Proof.(i). Suppose that {Λm (x) , z1 , z2 , ..., zn−1 } is linearly dependent set
in (X, k., ..., .k) for all but finite m. Then we have
kΛm (x) , z1 , z2 , ..., zn−1 k → 0 as m → ∞.

Since M is continuous and inf m pm > 0 for all m, we have
pm
 


Λm (x)

= 0, f or every ρ > 0.
,
z
,
z
,
...,
z
lim M
1 2
n−1

m
ρ
(ii) The proof is similar to part (i).

52

AYHAN ESI

Theorem 2.2. The classes of sequences [M, Λ, p, k., ..., .k]o , [M, Λ, p, k., ..., .k]
and [M, Λ, p, k., ..., .k]∞ are linear spaces.
Proof. We will prove only for [M, Λ, p, k., ..., .k]∞ and the others can be
proved similarly. Let x, y ∈ [M, Λ, p, k., ..., .k]∞ . Then there exist ρ1 > 0 and ρ2 > 0
such that
pm
 


Λm (x)

0 such that

 

Λm (x)


≤1
,
z
,
z
,
...,
z
sup M
1 2
n−1

ρ1
m
and


 

Λm (y)


sup M
≤ 1.
, z1 , z2 , ..., zn−1

ρ
m
2
Let ρ = ρ1 + ρ2 . Then by the convexity of M, we have

 

Λm (x + y)


sup M
,
z
,
z
,
...,
z
1 2
n−1

ρ
m

 


Λm (x)

ρ1

sup M
, z1 , z2 , ..., zn−1
≤

ρ1 + ρ2 m
ρ1

 



Λm (y)
ρ2
sup M
, z1 , z2 , ..., zn−1
≤ 1.
+


ρ1 + ρ2 m
ρ2
Hence we have


 

Λm (x + y)

pm

h (x + y) = inf ρ H : sup M
≤
1
, z1 , z2 , ..., zn−1

ρ
m

 

 p
Λm (x)

m
H


≤1
, z1 , z2 , ..., zn−1
≤ inf ρ1 : sup M
ρ1
m

SOME NEW PARANORMED SEQUENCE

....

53



 
 p

Λm (y)
m

≤
1
.
,
z
,
z
,
...,
z
+ inf ρ2H : sup M
1 2
n−1

ρ2
m
This implies h (x + y) ≤ h (x) + h (y) . The continuity of the scalar multiplication
follows from the following equality:


 


Λm (αx)
pm


H
≤1
, z1 , z2 , ..., zn−1
: sup M
h (αx) = inf ρ
ρ
m


 

Λm (x)

pm

= inf (t |α|) H : sup M
≤
1
,
,
z
,
z
,
...,
z
1 2
n−1

t
m


ρ
where t = |α|
. Now let xi be any Cauchy sequence in any one of the spaces


(i)
(i)
(i)
[M, Λ, p, k., ..., .k]o , [M, Λ, p, k., ..., .k] and [M, Λ, p, k., ..., .k]∞ , where xi = xo , x1 , x2 , ... .
Let xo > 0 be fixed and t > 0 be such that for a given ε (0 < ε < 1), xεo t > 0 and
xo t ≥ 1. Then there exists a positive integer no (ε) such that


ε
for all i, j ≥ no .
h xi − xj <
xo t
Using the definition of paranorm, we get

!#
"
(
)
Λ xi − xj


pm
m


inf ρ H : sup M
≤1
, z1 , z2 , ..., zn−1


ρ
m
<

Then we have
(
"
inf

ρ

pm
H

: sup M
m

Hence, we have
"

sup M
m

ε
for all i, j ≥ no .
xo t

!#

)

Λ xi − xj


m
, z1 , z2 , ..., zn−1
≤ 1 < ε for all i, j ≥ no .



ρ

!#


Λ xi − xj

m

≤ 1 for all i, j ≥ no .
,
z
,
z
,
...,
z

1 2
n−1


h (xi − xj )

It follows that

!
Λ xi − xj


m

M
, z1 , z2 , ..., zn−1
≤ 1 for each m ≥ 1 and for all i, j ≥ no .
i
j
h (x − x )



For t > 0 with M x2o t ≥ 1, we have
!




Λ xi − xj

xo t

m
, z1 , z2 , ..., zn−1
≤ M
.
M


h (xi − xj )
2

Then we have




Λm xi − xj , z1 , z2 , ..., zn−1
≤ xo t . ε = ε , for all i, j ≥ no ,
2 xo t
2


o
1
2
which leads to the fact that Λm x , Λm x , Λm x , ... is a Cauchy sequence

in X for
i
all m ∈ N. Since X is complete then it is convergent. Let limi Λm x = Λm (x) .
Now we have for all i, j ≥ no
!#

)
(
"

Λ xi − xj

pm

m
H
≤ 1 < ε.
, z1 , z2 , ..., zn−1
inf ρ
: sup M


ρ
m

54

AYHAN ESI

This implies that
(
lim inf
j

ρ

pm
H

"

: sup M
m


!#
)
Λ xi − xj


m

, z1 , z2 , ..., zn−1
≤ 1 < ε.



ρ

Since M and n-norms are continuous functions, we have
!#

)
(
"

Λ xi − x

pm
m


, z1 , z2 , ..., zn−1
≤ 1 < ε, for all i ≥ no .
inf ρ H : sup M


ρ
m


It follows that xi − x belongs to any one of the spaces [M, Λ, p, k., ..., .k]o , [M, Λ, p,

k., ..., .k]
i
i
and [M, Λ, p, k., ..., .k]∞ . Since these spaces are linear, so we have x = x − x − x
belongs to any one of the spaces. This completes the proof.
We state the following Theorem in view of Lemma 2.
Theorem 2.4. Let X be a standard n-norm space and {e1 , e2 , ..., en } be an
orthonormal set in X. Then
h
i
[M, Λ, p, k., ..., .k∞ ]o = M, Λ, p, k., ..., .k(n−1) ,
o

and

h
i
[M, Λ, p, k., ..., .k∞ ] = M, Λ, p, k., ..., .k(n−1)

h
i
[M, Λ, p, k., ..., .k∞ ]∞ = M, Λ, p, k., ..., .k(n−1)

∞

where k., ..., .k∞ is the derived (n-1)-norm defined with respect to {e1 , e2 , ..., en }
and k., ..., .k(n−1) is the standard (n-1)-norm on X.

References

[1] H.Gunawan and Mashadi M., On n-normed spaces, Int.J.Math.Math.Sci., 27(10)(2001), 631639.
[2] H.Gunawan, On n-inner product, n-norms and the Cauchy-Schwarz Inequality, Scientiae
Mathematicae Japonicae Online, 5(2001), 47-54
[3] H.Gunawan,
The space of p-summable sequences and its natural n-norm,
Bull.Aust.Math.Soc., 64(1)(2001), 137-147.
[4] A.Misiak, n-inner product spaces, Math.Nachr.,140(1989), 299-319.
[5] S.Gahler, Linear 2-normietre Rume, Math.Nachr., 28(1965), 1-43.
[6] M.A.Krasnoselski and Y.B.Rutickii, Convex function and Orlicz spaces, Groningen, Nederland, 1961.
[7] W.H.Ruckle, FK-spaces in which the sequence of coordinate vectors is bounded,
Canad.J.Math.,25(1973), 973-978.
[8] M.Mursaleen and A.K.Noman, On the spaces of λ − convergent and bounded sequences, Thai
J.Math.8(2)(2010), 311-329.
[9] A.Esi, Strongly almost summable sequence spaces in 2-normed spaces defined by ideal convergence and an Orlicz function, Stud.Univ.Babe¸s-Bolyai Math.27(1)(2012), 75-82.
[10] A.Esi, Strongly lacunary summable double sequence spaces in n-normed spaces defined
by ideal convergence and an Orlicz function, Advanced Modeling and Optimization,
14(1)(2012),79-86.
¨

SOME NEW PARANORMED SEQUENCE

....

55

[11] A.Esi and M.K.Ozdemir, Λ−Strongly summable sequence spaces in n-normed spaces defined
by ideal convergence and an Orlicz function, Mathematica Slovaca (2012) (to appear).
[12] N.Fistikci, M.Acikgoz and A.Esi, I-lacunary generalized difference convergent sequences in
n-normed spaces, Journal of Mathematical Analysis, 2(1)(2011), 18-24.

Received 11
Received
11April,
April,2012
2012