AYHAN ESI
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 Λmoperator 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.
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−1M(2) for some constant K >0.
A sequence space X is said to be solid or normal if (αkxk) ∈ 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, ..., xnk = 0 if and only if x1, x2, ..., xn are linearly dependent, (ii) kx1, x2, ..., xnk is invariant under permutation,
(iii) kαx1, x2, ..., xnk = |α| kx1, x2, ..., xnk , α ∈ R,
(iv) kx1+ xı1, x2, ..., xnk ≤ kx1, x2, ..., xnk+kxı1, x2, ..., xnk
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 Xn−1
Adiyaman University, Science and Art FacultyAdiyaman University, Science and Art Faculty Department of Mathematics, 02040, Adiyaman, Turkey
E-mail: [email protected]
AMS Subject Classification: 40A05, 46A45, 46B70
Key words and phrases: n-norm, paranorm, completeness, Orlicz function.
SOME NEW PARANORMED SEQUENCE SPACES DEFINED BY
ORLICZ FUNCTION
defined by
kx1, x2, ..., xn−1k∞= max {kx1, x2, ..., xn−1, aik : 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 kx1, x2, ..., xnkS = [det (hxi, xji)]12
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
kx1, x2, ..., xnkE= abs
x11...x1n
...
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 complete with respect to the usual norm k.k = h., .i12.
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−1k∞≤ kx1, x2, ..., xn−1kS≤√
nkx1, x2, ..., xn−1k∞
where kx1, x2, ..., xn−1k∞= max {kx1, x2, ..., xn−1, eik : 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
Λm(x) = 1 λm
m
X
k=1
(λk− λk−1) xk.
The sequence x = (xk) ∈ w is λ − bounded if supm|Λm(x)| < ∞. It is well known [8] that if limmxm= a in the ordinary sense of convergence, then
limm
1 λm
m
X
k=1
(λk− λk−1) |xk− a|
!
= 0.
This implies that
limm |Λm(x) − a| = lim
m
1 λm
m
X
k=1
(λk− λk−1) (xk− a)
= 0
which yields that limmΛm(x) = a and hence x = (xk) ∈ w is λ − convergent to a.
2. MAIN RESULTS
50 AYHAN ESI
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= (
x= (xk) ∈ w (n − X) : limm
h M
Λm(x)
ρ , z1, z2, ..., zn−1
ipm
= 0 f or some ρ > 0 and for every z1, z2, ..., zn−1∈ X
) ,
[M, Λ, p, k., ..., .k] = (
x= (xk) ∈ w (n − X) : limm
h M
Λm(x)−L
ρ , z1, z2, ..., zn−1
ipm
= 0 f orsome ρ > 0, L ∈ X and for every z1, z2, ..., zn−1∈ X
) , and
[M, Λ, p, k., ..., .k]∞
= (
x= (xk) ∈ w (n − X) : supm
h M
Λm(x)
ρ , z1, z2, ..., zn−1
ipm
<∞ f or some ρ > 0 and for every z1, z2, ..., zn−1∈ X
)
The following well-known inequality will be used in this study: If 0 ≤ infkpk= Ho≤ pk ≤ supk = H < ∞, D = max 1, 2H−1 , then
|xk+ yk|pk≤ D {|xk|pk+ |yk|pk} for all k ∈ N and xk, yk∈ C. Also |xk|pk ≤ max
1, |xk|H
for all xk∈ C.
In this section we investigate some linear topological structures of the se- quence 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 inequal- ity
M
Λm(x)
2ρ , z1, z2, ..., zn−1
pm
≤ 1 2M
Λm(x) − L
ρ , z1, z2, ..., zn−1
pm
+ 1 2M
L
ρ, z1, z2, ..., zn−1
pm
. 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 infmpm >0, then
(i) limm
h M
Λm(x)
ρ , z1, z2, ..., zn−1
ipm
= 0, for every ρ > 0, (ii) supmh
M
Λm(x)
ρ , z1, z2, ..., zn−1
ipm
<∞, 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−1k → 0 as m → ∞.
Since M is continuous and infmpm>0 for all m, we have limm
M
Λm(x)
ρ , z1, z2, ..., zn−1
pm
= 0, f or every ρ > 0.
(ii) The proof is similar to part (i).
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
M
Λm(x) ρ1
, z1, z2, ..., zn−1
pm
<∞
and
M
Λm(y) ρ2
, z1, z2, ..., zn−1
pm
<∞
for all m ≥ 1. Let α, β be any scalars and let ρ3= max (2 |α| ρ1,2 |β| ρ2) . Then we
have
M
Λm(αx + βy) ρ3
, z1, z2, ..., zn−1
pm
≤
M
Λm(αx) ρ3
, z1, z2, ..., zn−1
+ M
Λm(βy) ρ3
, z1, z2, ..., zn−1
pm
≤ D
M
Λm(x) ρ1
, z1, z2, ..., zn−1
pm
+
M
Λm(y) ρ2
, z1, z2, ..., zn−1
pm for all m ≥ 1. Hence [M, Λ, p, k., ..., .k]∞ is a linear space.
Theorem 2.3. The spaces [M, Λ, p, k., ..., .k]o,[M, Λ, p, k., ..., .k] and [M, Λ, p, k., ..., .k]∞
are complete paranormed spaces, paranormed by h defined by h(x) = inf
ρpmH : sup
m
M
Λm(x)
ρ , z1, z2, ..., zn−1
≤ 1
, where H = max (1, supmpm) .
Proof. Clearly h (x) = h (−x) and h (θ) = 0. Let x = (xk) and y = (yk) be any two sequences belong to any one of the spaces [M, Λ, p, k., ..., .k]o,[M, Λ, p, k., ..., .k]
and [M, Λ, p, k., ..., .k]∞. Then we have ρ1>0 and ρ2>0 such that sup
m
M
Λm(x) ρ1
, z1, z2, ..., zn−1
≤ 1 and
sup
m
M
Λm(y) ρ2
, z1, z2, ..., zn−1
≤ 1.
Let ρ = ρ1+ ρ2.Then by the convexity of M, we have sup
m
M
Λm(x + y)
ρ , z1, z2, ..., zn−1
≤
ρ1
ρ1+ ρ2
sup
m
M
Λm(x) ρ1
, z1, z2, ..., zn−1
+
ρ2
ρ1+ ρ2
sup
m
M
Λm(y) ρ2
, z1, z2, ..., zn−1
≤ 1.
Hence we have h(x + y) = inf
ρpmH : sup
m
M
Λm(x + y)
ρ , z1, z2, ..., zn−1
≤ 1
≤ inf
ρ1pmH : sup
m
M
Λm(x)
ρ1 , z1, z2, ..., zn−1
≤ 1
52 AYHAN ESI
+ inf
ρ2pmH : sup
m
M
Λm(y) ρ2
, z1, z2, ..., zn−1
≤ 1
.
This implies h (x + y) ≤ h (x) + h (y) . The continuity of the scalar multiplication follows from the following equality:
h(αx) = inf
ρpmH : sup
m
M
Λm(αx)
ρ , z1, z2, ..., zn−1
≤ 1
= inf
(t |α|)pmH : sup
m
M
Λm(x)
t , z1, z2, ..., zn−1
≤ 1
, where t = |α|ρ . Now let xi
be any Cauchy sequence in any one of the spaces [M, Λ, p, k., ..., .k]o,[M, Λ, p, k., ..., .k] and [M, Λ, p, k., ..., .k]∞, where xi=
x(i)o , x(i)1 , x(i)2 , ... . Let xo>0 be fixed and t > 0 be such that for a given ε (0 < ε < 1), xε
ot >0 and xot≥ 1. Then there exists a positive integer no(ε) such that
h xi− xj < ε
xot for all i, j ≥ no. Using the definition of paranorm, we get
inf (
ρpmH : sup
m
"
M
Λm xi− xj
ρ , z1, z2, ..., zn−1
!#
≤ 1 )
< ε
xot for all i, j ≥ no. Then we have
inf (
ρpmH : sup
m
"
M
Λm xi− xj
ρ , z1, z2, ..., zn−1
!#
≤ 1 )
< ε for all i, j ≥ no. Hence, we have
sup
m
"
M
Λm xi− xj
h(xi− xj) , z1, z2, ..., zn−1
!#
≤ 1 for all i, j ≥ no. It follows that
M
Λm xi− xj
h(xi− xj) , z1, z2, ..., zn−1
!
≤ 1 for each m ≥ 1 and for all i, j ≥ no. For t > 0 with M x2ot ≥ 1, we have
M
Λm xi− xj
h(xi− xj) , z1, z2, ..., zn−1
!
≤ M xot 2
. Then we have
Λm xi− xj , z1, z2, ..., zn−1
≤xot
2 . ε xot = ε
2, for all i, j ≥ no,
which leads to the fact that Λmxo,Λmx1,Λmx2, ... is a Cauchy sequence in X for all m ∈ N. Since X is complete then it is convergent. Let limiΛm xi = Λm(x) . Now we have for all i, j ≥ no
inf (
ρpmH : sup
m
"
M
Λm xi− xj
ρ , z1, z2, ..., zn−1
!#
≤ 1 )
< ε.
AYHAN ESI
This implies that limj inf
(
ρpmH : sup
m
"
M
Λm xi− xj
ρ , z1, z2, ..., zn−1
!#
≤ 1 )
< ε.
Since M and n-norms are continuous functions, we have inf
(
ρpmH : sup
m
"
M
Λm xi− x
ρ , z1, z2, ..., zn−1
!#
≤ 1 )
< ε, for all i ≥ no.
It follows that xi− x belongs to any one of the spaces [M, Λ, p, k., ..., .k]o,[M, Λ, p, k., ..., .k]
and [M, Λ, p, k., ..., .k]∞.Since these spaces are linear, so we have x = xi− xi− 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
[M, Λ, p, k., ..., .k∞]o=h
M,Λ, p, k., ..., .k(n−1)
i
o, [M, Λ, p, k., ..., .k∞] =h
M,Λ, p, k., ..., .k(n−1)
i and
[M, Λ, p, k., ..., .k∞]∞=h
M,Λ, p, k., ..., .k(n−1)
i
∞
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
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¨ 54
[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).
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Received 11 April, 2012Received 11 April, 2012
