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 functionM 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 functionM
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 eacht≥δ, we haveM(t)< Kδ−1M(2) for some constant
K >0.
A sequence spaceX is said to be solid or normal if (αkxk)∈X, and for all double sequencesα= (αk) of scalars with |αk| ≤1 for allk∈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 functionk., ..., .k onX satisfying the following four condition:
(i)kx1, x2, ..., xnk= 0 if and only ifx1, x2, ..., xn are linearly dependent,
(ii)kx1, x2, ..., xnkis 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 dimensiond≥n≥2 and{a1, a2, ..., an} be a linearly independent set in X. Then the following functionk., ..., .k∞ onXn−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.
defined by
kx1, x2, ..., xn−1k∞= max{kx1, x2, ..., xn−1, aik:i= 1,2, ..., n} defines an (n-1)-norm onX with respect to{a1, a2, ..., an}.
Letn∈Nand (X,h., .i) be a real inner product space of dimensiond≥n.
Then the following functionk., ..., .kS onX×X×...×X (n factors) defined by
kx1, x2, ..., xnkS = [det (hxi, xji)]
1 2
is an n-norm onX, which is known as standard n-norm onX. If we takeX =Rn, then this n-norm is exactly the same as Euclidean n-norm such as
kx1, x2, ..., xnkE=abs
x11...x1n
... xn1...xnn
wherexi= (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 normk.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)-normsk., ..., .kS.Precisely, we have for allx1, x2, ..., xn−1
kx1, x2, ..., xn−1k∞≤ kx1, x2, ..., xn−1kS≤√nkx1, x2, ..., xn−1k∞ wherekx1, x2, ..., xn−1k∞= max{kx1, x2, ..., xn−1, eik:i= 1,2, ..., n}.
In paper [8],Mursaleen and Noman introduced the notion ofλ−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< ...andλk→ ∞as k→ ∞
and said that a sequencex= (xk)∈wis λ−convergentto the numberL, called a theλ−limit ofx, if Λm(x)→Las m→ ∞,where
Λm(x) = 1
λm m X
k=1
(λk−λk−1)xk.
The sequencex= (xk)∈w isλ−boundedif supm|Λm(x)|<∞. It is well known [8] that if limmxm=ain the ordinary sense of convergence, then
lim m
1
λm m X
k=1
(λk−λk−1)|xk−a| !
= 0.
This implies that
lim
m |Λm(x)−a|= limm
1
λm m X
k=1
(λk−λk−1) (xk−a)
= 0
which yields that limmΛm(x) =aand hencex= (xk)∈wisλ−convergenttoa.
2. MAIN RESULTS
Let (X,k., ..., .k) be real n-normed space and w(n−X) denotes the space ofX-valued sequences. LetM be an Orlicz function andp= (pk) be any bounded sequence of strictly positive real numbers. Now, we define the following sequence spaces:
The following well-known inequality will be used in this study: If 0 ≤ infkpk=Ho≤pk ≤supk =H <∞, D= max 1,2H−1
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]∞.
Theorem 2.2. The classes of sequences [M,Λ, p,k., ..., .k]o,[M,Λ, p,k., ..., .k] are complete paranormed spaces, paranormed byhdefined by
h(x) = inf
Hence we have
h(x+y) = inf
+ inf follows from the following equality:
h(αx) = inf
Using the definition of paranorm, we get
inf
Then we have
inf
Hence, we have
sup
It follows that
M
Then we have
AYHAN ESI
This implies that
lim j 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 alli≥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 havex=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 inX. 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∞]∞=hM,Λ, p,k., ..., .k(n−1)i
∞
where k., ..., .k∞ is the derived (n-1)-norm defined with respect to {e1, e2, ..., en} andk., ..., .k(n−1) is the standard (n-1)-norm onX.
References
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¨
[11] A.Esi and M.K.Ozdemir, Λ−Strongly summable sequence spaces in n-normed spaces defined
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