An Orlicz extension of difference modular sequence

Bol. Soc. Paran. Mat. c

SPM –ISSN-2175-1188 on line SPM: www.spm.uem.br/bspm

(3s.) v. 33 2 (2015): 31–57. ISSN-00378712 in press doi:10.5269/bspm.v33i2.23272

An Orlicz extension of difference modular sequence spaces Seema Jamwal and Kuldip Raj

abstract:

In this paper we construct some new difference modular sequence spaces defined by a sequence of Orlicz functions over n-normed spaces. We also study several properties relevant to topological structures and interrelationship between these spaces.

Key Words: sequence space, difference sequence space, modular sequence space, paranormed space, Orlicz function, n-normed space, BK-space. Contents 1 Introduction and Preliminaries

31

m λ 2 Some topological properties of the spaces lλM ∆m 36 n , p and lN ∆n , p 3 Dual spaces of h(M), l(M, λ, p) and l(N, λ, p)

44

4 Some new sequence spaces over n-normed space

46

1. Introduction and Preliminaries Let X be a linear metric space. A function p : X → R is called 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 (λn ) is a sequence of scalars with λn → λ as n → ∞ and (xn ) is a sequence of vectors with p(xn − x) → 0 as n → ∞, then p(λn xn − λx) → 0 as n → ∞. 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 [32], Theorem 10.4.2, pp. 183). The notion of difference sequence spaces was introduced by Kızmaz [16], who studied the difference sequence spaces l∞ (∆), c(∆) and c0 (∆). The notion was further generalized by Et and Çolak [7] by introducing the spaces l∞ (∆n ), c(∆n ) and c0 (∆n ). Later the concept have been studied by Bektaş et al. [3] and Et et al. [8]. Another type of generalization of the difference sequence spaces is due to Tripathy and Esi [29] who studied the spaces l∞ (∆v ), c(∆v ) and c0 (∆v ). Recently, Esi et 2000 Mathematics Subject Classification: 40A05, 46A45

31

Typeset by BSP style. M c Soc. Paran. de Mat.

32

Seema Jamwal and Kuldip Raj

al. [9] and Tripathy et al. [30] have introduced a new type of generalized difference operators and unified those as follows. Let v, n be non-negative integers, then for Z a given sequence space, we have Z(∆nv ) = {x = (xk ) ∈ w : (∆nv xk ) ∈ Z} for Z = c, c0 and l∞ where ∆nv x = (∆nv xk ) = (∆vn−1 xk −∆vn−1 xk+v ) and ∆0v xk = xk for all k ∈ N, which is equivalent to the following binomial representation n X n n m xk+vm . ∆v xk = (−1) m m=0

Taking v = 1, we get the spaces l∞ (∆n ), c(∆n ) and c0 (∆n ) studied by Et and Çolak [7]. Taking v = n = 1, we get the spaces l∞ (∆), c(∆) and c0 (∆) introduced and studied by Kızmaz [16]. For more details about difference sequence spaces (see [1], [4], [5], [19], [20], [27]) and references therein. Let ω be the family of all real or complex sequences, which is a vector space with the usual pointwise addition and scalar multiplication. We write en (n ≥ 1) for the nth unit vector in ω, i.e en = {δnj }∞ j=1 where δ nj is the Kronecker delta, and ϕ for the subspace of ω generated by en ’s, n ≥ 1, i.e, ϕ = span{en : n ≥ 1}. A sequence space η is subspace of ω containing ϕ. The sequence space η is said to be solid if (αk xk ) ∈ η whenever (xk ) ∈ η for all sequences (αk ) of scalars such that |αk | ≤ 1 for all k ∈ N. A sequence space η is said to be monotone if η contains the canonical pre images of all its step spaces. A Banach sequence space (η, S) is called a BKspace if the topology S of η is finer than the co-ordinatewise convergence topology, or equivalently, the projection maps Pi : η → K, Pi (x) = xi , i ≥ 1 are continuous, where K is the scalar field R or C. For x = (x1 , ..., xn , ...) and n ∈ N, we write the nth section of x as x(n) = (x1 , ..., xn , 0, 0, ...). If x(n) → x in (η, S) for each x ∈ η, we say that (η, S) is an AK-space. The norm k.kη generating the topology S of η is said to be monotone if kxkη ≤ kykη for x = {xi }, y = {yi } ∈ η with |xi | ≤ |yi |, for all i ≥ 1 (see [14]). An Orlicz function M is a function, which is continuous, non-decreasing and convex with M (0) = 0, M (x) > 0 for x > 0 and M (x) −→ ∞ as x −→ ∞. Lindenstrauss and Tzafriri [17] used the idea of Orlicz function to define the following sequence space: ∞ o n |x | X k < ∞, for some ρ > 0 ℓM = x ∈ ω : M ρ k=1

which is called as an Orlicz sequence space. The space ℓM is a Banach space with the norm ∞ o n |x | X k ≤1 . ||x|| = inf ρ > 0 : M ρ k=1

It is shown in [15] that every Orlicz sequence space ℓM contains a subspace isomorphic to ℓp (p ≥ 1). In the later stage different Orlicz sequence spaces were

An Orlicz extension of difference

33

introduced and studied by Parashar and Choudhary [25], Esi and Et [6], Tripathy and Mahanta [31], Mursaleen [21] and many others. The ∆2 −condition is equivalent to M (Lx) ≤ kLM (x) for all values of x ≥ 0 and for L > 1. A sequence M = (Mk ) of Orlicz functions is called a Musielak-Orlicz function (see [22], [23]). A sequence N = (Nk ) defined by Nk (v) = sup{|v|u − Mk (u) : u ≥ 0}, k = 1, 2, · · · is called the complementary function of a Musielak-Orlicz function M. For a given Musielak-Orlicz function M, the Musielak-Orlicz sequence space tM and its subspace hM are defined as follows; n o tM = x ∈ ω : IM (cx) < ∞ for some c > 0 , n o hM = x ∈ ω : IM (cx) < ∞ for all c > 0 ,

where IM is a convex modular defined by IM (x) =

∞ X

Mk (xk ), x = (xk ) ∈ tM .

k=1

We consider tM equipped with the Luxemburg norm o n x ≤1 ||x|| = inf k > 0 : IM k

or equipped with the Orlicz norm o n1 1 + IM (kx) : k > 0 . ||x||0 = inf k

Any Orlicz function Mk can always be represented in the following integral form Z x Mk (x) = η k (t)dt, 0

where η k is known as the kernel of Mk , is a right differentiable for t ≥ 0, η k (0) = 0, η k (t) > 0, η k is non-decreasing and η k (t) → ∞ as t → ∞. Given an Orlicz function Mk with kernel η k (t), define ν k (s) = sup{t : η k (t) ≤ s, s ≥ 0}. Then ν k (s) possesses the same properties as η k (t) and the function Nk defined as Z x Nk (x) = ν k (s)ds 0

is an Orlicz function. The functions Mk and Nk are called mutually complementary Orlicz functions.

34


For a sequence M = (Mk ) of Orlicz functions, the modular sequence class ˜l(M) is defined by ∞ X ˜l(M) = {x = (xk ) ∈ ω : Mk (|xk |) < ∞}. k=1

Using the sequence N = (Nk ) of Orlicz functions, similarly we define ˜l(N). The class l(M) is defined by l(M) = {x = (xk ) ∈ ω :

∞ X

xk yk converges, for all y ∈ ˜l(N)}.

k=1

For a sequence M = (Mk ) of Orlicz functions, the modular sequence space l(M) is also defined as ∞ |x | X k l(M) = {x = (xk ) ∈ ω : Mk < ∞, for some ρ > 0}. ρ k=1

The space l(M) is a Banach space with respect to the norm kxkM defined as kxkM = inf{ρ > 0 :

∞ X

k=1

Mk

|x | k ≤ 1}. ρ

These spaces were introduced by Woo [33] around the year 1973 and generalizes the Orlicz sequence space lM and the modulared sequence spaces considered earlier by Nakano [24]. For more details about modular sequence spaces (see [15], [28]) and references therein. An important subspace of l(M), which is an AK-space, is the space h(M) defined as ∞ |x | X k h(M) = {x ∈ l(M) : Mk < ∞, for some ρ > 0}. ρ k=1

A sequence (Mk ) of Orlicz functions is said to satisfy uniform ∆2 − condition at 0 if there exist p > 0 and k0 ∈ N such that for all x ∈ (0, 1) and k > k0 , we have

′ ′

′

xMk (x) Mk (x) Mk (2x) Mk (x)

≤ p, or equivalently, there exists a constant K > 1 and k0 ∈ N such that

≤ K for all x ∈ (0, 12 ]. If the sequence (Mk ) satisfy uniform ∆2 − condition, then h(M) = l(M) and vice-versa (see [33]). Let Mk and Nk be mutually complementary Orlicz functions for each k and λ = (λk ) be a sequence of strictly positive real numbers. Bektaş and Atici [2] define the following sequence spaces: |∆m x | X k < ∞, for some ρ > 0 lλM (∆m ) = x = (xk ) : Mk λk ρ k≥1

and λ lN (∆m ) =

x = (xk ) :

X

k≥1

Nk

λ |∆m x | k k < ∞, for some ρ > 0 . ρ

35


Let M = (Mk ) and N = (Nk ) be two sequences of Orlicz functions, p = (pk ) be any bounded sequence of positive real numbers and λ = (λk ) be a sequence of strictly positive real numbers. In this paper we define the following sequence spaces: o X h |∆m xk | ipk n n lλM ∆m Mk < ∞, for some ρ > 0 n , p = x = (xk ) ∈ ω : λk ρ k≥1

and o X h λk |∆m xk | ipk m n n λ < ∞, for some ρ > 0 . lN ∆n , p = x = (xk ) ∈ ω : Nk ρ k≥1

If we take (pk ) = 1, for all k then o X h |∆m xk | i n n = x = (x ) ∈ ω : M lλM ∆m < ∞, for some ρ > 0 k k n λk ρ k≥1

and o X h λk |∆m xk | i m n n λ lN Nk ∆n = x = (xk ) ∈ ω : < ∞, for some ρ > 0 . ρ k≥1

If (λk ) = 1 for all k ∈ N, then o X h |∆m xk | ipk n n < ∞, for some ρ > 0 lM ∆m Mk n , p = x = (xk ) ∈ ω : ρ k≥1

and o X h |∆m xk | ipk n n lN ∆m Nk < ∞, for some ρ > 0 . n , p = x = (xk ) ∈ ω : ρ k≥1

If we take (pk ) = 1, for all k and n=1 we get the spaces defined by Bektaş and Atici [2]. The following inequality will be used throughout the paper. Let p = (pk ) be a sequence of positive real numbers with 0 < pk ≤ supk pk = H, and let D = max 1, 2H−1 . Then, for the factorable sequences (ak ) and (bk ) in the complex plane, we have |ak + bk |pk ≤ D(|ak |pk + |bk |pk ).

(1.1)

Throughout the paper we write Mk (1) = 1 and Nk (1) = 1 for all k ∈ N. The main purpose of this paper is to study some difference new modular sequence spaces defined by a sequence of Orlicz functions over n−normed spaces. mWe shall M study some topological, algebraic properties of the sequence spaces l λ ∆n , p and m λ lN ∆n , p in the second section of the paper. In the third section we shall determine the dual spaces of h(M), l(M, λ, p) and l(N, λ, p). Finally, we shall study some sequence spaces over n− normed spaces in the fourth section of the paper. We have also made an attempt to study some topological, m algebraic proper M ties and inclusion relations between the sequence spaces l λ ∆n , p, k·, · · · , ·k and λ lN ∆m n , p, k·, · · · , ·k .

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m λ 2. Some topological properties of the spaces lλM ∆m n , p and lN ∆n , p

The purpose of this section is to study the propertieslike linearity, paranorm, λ m solidity and relevant inclusion relations in the spaces lλM ∆m n , p and lN ∆n , p . Theorem 2.1. Let M = (Mk ) and N = (Nk ) be two sequences of Orlicz functions, p = (pk ) be a bounded sequence of positive real numbers and λ = (λk ) be a M m sequence m of strictly positive real numbers. Then the sequence spaces lλ ∆n , p and λ lN ∆n , p are linear spaces over the complex field C.

Proof: Let x = (xk ) and y = (yk ) ∈ lλM ∆m n , p and α, β ∈ C. Then there exist positive real numbers ρ1 and ρ2 such that X h |∆m xk | ipk n Mk 0 and ρ2 > 0 be such that ! H1

≤1

! H1

≤ 1.

X h |∆m xk | ipk n Mk λk ρ1 k≥1

and X h |∆m xk | ipk n Mk λ k ρ2

k≥1

Let ρ = ρ1 + ρ2 . Then by Minkowski’s inequality, we have

38


X h |∆m xk | ipk n Mk λk ρ

k≥1

! H1

Xh Mk

≤

k≥1

X

≤

k≥1

"

ipk |∆m n xk | λk (ρ1 + ρ2 )

! H1

|∆m x | ρ1 n k Mk ρ1 + ρ2 λk ρ1

# !1 |∆m x | pk H ρ2 n k + Mk ρ1 + ρ2 λ k ρ2 ! # !1 " |∆m x | pk H X ρ1 n k Mk ≤ ρ1 + ρ2 λk ρ1 k≥1

ρ1 + ρ1 + ρ2

!

≤ 1. Since ρ’s are non-negative, so we have ( g(x + y) =

≤

+

inf

inf

inf

(ρ1 + ρ2 )

(

(ρ1 )

pk H

pk H

:

X

k≥1

k≥1

k≥1

(

(ρ2 )

pk H

X h |∆m yk | ipk n Mk λk ρ2

:

|∆m x | n k Mk λk ρ2

X h |∆m (xk + yk )| ipk n Mk λk (ρ1 + ρ2 )

X h |∆m xk | ipk n Mk λ k ρ1

:

"

k≥1

! H1

! H1

≤1

! H1

#pk ! H1

≤1

)

)

)

≤1 .

Therefore, g(x + y) ≤ g(x) + g(y). Finally, we prove that the scalar multiplication is continuous. Let µ be any complex number, therefore, by definition !1 ) ( X h |∆m µxk | ipk H pk n ≤1 and g(µx) = inf (ρ) H : Mk λk ρ k≥1

thus, g(µx) = inf where t =

ρ |µ| .

(

(|µ|t)

pk H

:

X h |∆m xk | ipk n Mk λk t

k≥1

! H1

≤1

)

Since |µ|pk ≤ max(1, |µ| sup pk ). Hence,

g(µx) = max(1, |µ| sup pk ) inf

(

(t)

pk H

:

Xh

k≥1

|∆m x | ipk n k Mk λk t

! H1

)

≤1 .


39

So, the fact that scalar multiplication is continuous follows from the above inequality. This completes the proof. ✷ Theorem 2.3. Suppose M = (Mk ) be a sequence of Orlicz functions, p = (pk ) be a bounded sequence of positive real numbers and λ = (λk ) be a sequenceof strictly M m positive real numbers. If 0 < p ≤ q < ∞, for each k ∈ N, then l ∆ , p ⊆ k k n λ m M lλ ∆n , q . Proof: Suppose that x = (xk ) ∈ lλM ∆m n , p . This implies that X h |∆m xk | ipk n ≤1 Mk λk ρ

k≥1

for sufficiently large value of k say k ≥ k0 , for some fixed k0 ∈ N. Since M = (Mk ) is non decreasing, we have ∞ h |∆m x | iqk X n k Mk λk ρ

≤

k=k0

M

Hence, x = (xk ) ∈ lλ

∞ h |∆m x | ipk X n k Mk λk ρ

k=k0

0 Mk λ k ρ1

k≥1

and

X h ′ |∆m xk | ipk n Mk < ∞, for some ρ2 > 0. λk ρ2 k≥1

Let ρ = max(ρ1 , ρ2 ). The result follows from the inequality |∆m x | ipk Xh ′ n k (Mk + Mk ) λk ρ1

=

k≥1

≤

|∆m x | ipk |∆m x | ′ n k n k + Mk λk ρ1 λk ρ1 k≥1 X h |∆m xk | ipk n D Mk λ k ρ1 k≥1 X h ′ |∆m xk | ipk n +D . Mk λk ρ1 Xh

Mk

k≥1

This completes the proof. The proof of the following theorems are easy so omitted.

✷

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Theorem 2.8. The sequence space lλM ∆m n , p is a normed space with norm kxkM λ =

m X

n o X h |∆m xk | ipk n |xi | + inf ρ > 0 : Mk ≤1 . λk ρ i=1 k≥1

m λ Theorem 2.9. The sequence space lN ∆n , p is a normed space with norm kxkλN =

m X

n o X h λk |∆m xk | ipk n |xi | + inf ρ > 0 : Nk ≤1 . ρ i=1 k≥1

M λ λ Theorem 2.10. The spaces lλM ∆m and lN ∆m are Ban , p , k.kλ n , p , k.kN nach spaces. M Theorem 2.11. The space lλM ∆m n , p equipped with the norm k.kλ and the space λ m λ lN ∆n , p equipped with the norm k.kN are BK-spaces. M Proof: The space lλM ∆m , p , k.k is a Banach space by the Theorem 2.10. n λ Now let kxl − xkM λ → 0 as l → ∞. Then

|xlk − xk | → 0 as l → ∞, for each k ≤ m and inf

(

X h |∆m xl − ∆m xk | ipk n k n ≤1 ρ>0: Mk λk ρ k≥1

)

→0

as l →∞ for all k ∈N. pk l m |∆m |∆m xl −∆m x | n xk −∆n xk | If Mk ≤ 1 then nλ kkxkMn k ≤ 1 for all k. Therefore, we also M λk kxkλ k λ obtain l m l M |∆m n xk − ∆n xk | ≤ λk kx − xkλ . m l m Since kxl − xkM λ → 0, then |∆n xk − ∆n xk | → 0 and m X m v (xlk+nv − xk+nv ) → 0 (−1) v v=0

as l → ∞ for all k ∈ N. On the other hand, since we may write m X m m l v (xk+nv − xk+nv ) + − xk+nv | ≤ (−1) (xlk − xk ) v 0 v=0 m +....... + (xlk+n(m+1) − xk+n(m+1) ) . m−1 M Then |xlk − xk | → 0 as l → ∞ for each k ∈ N. Hence, lλM ∆m n , p , k.kλ λ λ is a BK-space. This is a BK-space. Similarly we can prove lN ∆m , p , k.k n N completes the proof. ✷ |xlk+nv

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Theorem 2.12. If Z is a normal sequence space containing λ, then lλM ∆m n ,p is a proper subspace of Z. In addition, ifZ is equipped with the monotone norm m (quasi-norm) k.kZ . The inclusion R : lλM ∆m , p → Z ∆ , p is continuous with n n kRk ≤ k{λk }kZ . Proof: Let x ∈ lλM ∆m n , p , then ∞ h |∆m x | ipk X n k < ∞, for some ρ > 0. Mk λk ρ k=1

So there exists a constant K > 0 such that |∆m n xk | ≤ K for all k ∈ N. λk ρ m pk ∈ Z and so Since Z is a normal sequence space containing λ, we have ∆n xk m m m M that x ∈ Z ∆n , p . Hence, lλ ∆n , p ⊆ Z ∆n , p . Further, since Mk (1) = 1 for all k ∈ N then X h |∆m xk | ipk n ≤1 Mk M λ kxk k λ k≥1 M m so that |∆m n xk | ≤ λk kxkλ . As k.kZ is monotone, kRxkZ = k∆n xk kZ ≤ M k{λk }kZ kxkλ and hence, kRk ≤ k{λk }kZ . This completes the proof.

✷

Theorem 2.13. If Y is a normal sequence space containing λ−1 ≡ { λ1k }, then m λ lN ∆n , p is a proper subspace of Y. In addition, if Y is equipped mwith the monotone λ norm (quasi-norm) k.kY . The inclusion S : lN ∆m , p → Y ∆n , p is continuous n −1 with kSk ≤ k{λk }kY . Proof: The proof of the theorem is similar to that of Theorem 2.12 and so is omitted. ✷ Theorem 2.14. If λ = (λk ) is a bounded sequence m such that minf λk > 0 (i.e both λ M λ and λ−1 are in l∞ ). Then lM ∆m , p = l ∆ , p = l M ∆n , p . n n λ Proof: Let x = (xk ) ∈ lM ∆m n , p , then

X h |∆m xk | ipk n < ∞, for some ρ > 0. Mk ρ

k≥1

Since λ = (λk ) is bounded, we can write a ≤ λk ≤ b for some b > a ≥ 0. Define ρ1 = ρb. Also since Mk′ s are increasing, it follows that X h λk |∆m xk | ipk n Mk ρ1 k≥1

≤

X h |∆m xk | ipk n Mk ρ

k≥1

< ∞.


43

m m m λ λ Hence, lM ∆m n , p ⊆ lM ∆n , p . The other inclusion lM ∆n , p ⊆ lM ∆n , p follows from the inequality X h |∆m xk | ipk n Mk ρ

Xh

≤

a

k≥1

Mk

k≥1

∞. m λ m m M Therefore, m lM ∆n , p = lM ∆n , p . Similarly one can prove that lλ ∆n , p = lM ∆n , p . This completes the proof. ✷

0 and ρ′ = ρc2 , we have X h λk |∆m xk | ipk n Mk ρ′

≤

X h |∆m xk | ipk n Mk λk ρ

k≥1

k≥1

< ∞. λ , p ⊂ lM for x = {xk }. Hence, lλM ∆m ∆m , p . We now show that the containment n n m −1 λ lλM ∆m n , p ⊂ lM ∆n , p is proper. From the unboundedness of the sequence {λk }, −1 choose a sequence {kl } of positive integers such that λkl ≥ l. Now define 1 l , k = kl , l = 1, 2, ...; ∆m x = n k 0, otherwise. λ Then x ∈ lM ∆m / lλM ∆m n , p , but x ∈ n , p . To prove the continuity of the inclusion map U, let us first consider the case obtained for c = 1. For x ∈ lλM ∆m n , p , we write ) ( X h |∆m xk | ipk m n M ≤1 Mk Aλ ∆n , p = ρ > 0 : λk ρ k≥1

and

(

) X h λk |∆m xk | ipk n = ρ>0: Mk ≤1 . ρ k≥1 m m λ Since Mk′ s are increasing and c = 1, we get AM λ ∆n , p ⊂ BM ∆n , p . Hence, m M λ (2.1) ∆n , p ≤ inf AλM ∆m kxkλM = inf BM n , p = kxkλ λ BM

∆m n ,p

2 i.e kU (x)kλM ≤ kxkM λ . Thus, U is continuous with kU k ≤ 1 = c . If c 6= 1, define λk δ k = c , k ∈ N. Then δ k ≤ 1 and from (2.1), it follows that (2.2) kxkδM ≤ kxkM for x ∈ lλM ∆m n ,p . δ

44


Hence, from (2.2) kU (x)kλM = kxkλM ≤ c2 kxkM λ i.e. U is continuous with kU k ≤ c2 . This completes the proof.

✷

Theorem 2.16. If {λk } is unbounded with sup λk−1 = d ≥ 1, λk > 0 for all k≥1 m m λ M k, then lM ∆ , p is properly contained in l n λ ∆n , p and the inclusion map V : λ M m 2 lM ∆m n , p → lλ ∆n , p is continuous with kV k ≤ d .

Proof: The proof of the theorem is similar to that of Theorem 2.15 and so is omitted. ✷ 3. Dual spaces of h(M), l(M, λ, p) and l(N, λ, p)

Let η be a sequence space and defined ∞ X η α = a = (ak ) : |ak xk | < ∞, for all x ∈ η , k=1

∞ X η β = a = (ak ) : ak xk k=1

converges for all x ∈ η ,

∞ X ak xk < ∞, for all x ∈ η (see [13]). η γ = a = (ak ) : sup k=1

α

β

γ

Then η , η , η are called α−, β−, γ− dual spaces of η respectively. It is easy to show that φ ⊂ η α ⊂ η β ⊂ η γ . If η ⊂ ν, then ν σ ⊂ η σ for σ = α, β, γ. We shall write η αα = (η α )α . Let η be a sequence space. Then η is called perfect if η = η αα (see [15]). For m = n = 0 we write l(M, λ, p) and l(N, λ, p) instead of lλM ∆m n , p and λ lN ∆m n , p repectively which we define as: n o X h |xk | ipk l(M, λ, p) = x = (xk ) ∈ ω : Mk < ∞, for some ρ > 0 λk ρ k≥1

n o X h λk |xk | ipk l(N, λ, p) = x = (xk ) ∈ ω : < ∞, for some ρ > 0 . Nk ρ k≥1

In this section we shall obtain α−, β− and γ− duals of the sequence space h(M) and α− duals of l(M, λ, p) and l(N, λ, p). Proposition 3.1. η is perfect ⇒ η is normal ⇒ η is monotone (see [15]). Proposition 3.2. Let η be a sequence space. If η is monotone, then η α = η β and if η is normal, then η α = η γ .


45

Proposition 3.3. The sequence space h(M) is normal for any sequence (Mk ) of Orlicz functions. Proof: Let x ∈ h(M) and |yk | ≤ |xk |, for each k ∈ N. Since Mk′ s are non-decreasing we have ∞ ∞ |y | X |x | X k k Mk ≤ Mk < ∞. ρ ρ k=1

k=1

Hence, y ∈ h(M). Thus, h(M) is normal.

✷

Theorem 3.1. Let (Mk ) and (Nk ) for each k be mutually complementary Orlicz functions. Then [h(M)]β = [h(M)]α = [h(M)]γ = l(N). The proof is seen from Proposition 3.1, Proposition 3.2 and Proposition 3.3. Theorem 3.2. If the sequence (Mk ) satisfies uniform ∆2 − condition, then [l(M, λ, p)]α = l(N, λ, p) Proof: Let the sequence (Mk ) satisfies uniform ∆2 − condition, Then for any x ∈ l(M, λ, p) and a ∈ l(N, λ, p), we have ∞ X

k=1

|ak xk | ≤

∞ h ∞ h |x | ipk X λ |a | ipk X k k k Mk + Nk 0. Thus, a ∈ [l(M, λ, p)]α . Hence, l(N, λ, p) ⊂ [l(M, λ, p)]α . To prove the [l(M, λ, p)]α ⊂ l(N, λ, p), let a ∈ [l(M, λ, p)]α . Then for all inclusion {xk } with λxkk ∈ l(M) we have ∞ X

|ak xk | < ∞.

(3.1)

k=1

Since the sequence satisfies uniform ∆2 −condition, then l(M) = h(M) and so for ∞ X (yk ) ∈ h(M) we have |λk yk ak | < ∞ by (3.1). Thus, (λk ak ) ∈ [h(M)]α = l(N) k=1

and hence, (ak ) ∈ l(N, λ, p). Therefore, [l(M, λ, p)]α = l(N, λ, p).

✷

Theorem 3.3. If the sequence (Mk ) satisfies uniform ∆2 − condition, then [l(N, λ, p)]α = l(M, λ, p). Proof: Immediate from Theorem 3.5.

✷

46


4. Some new sequence spaces over n-normed space The concept of 2-normed spaces was initially developed by Gähler [10] in the mid of 1960’s, while that of n-normed spaces one can see in Misiak [18]. Since then, many others have studied this concept and obtained various results, see Gunawan ([11], [12]) and Gunawan and Mashadi [13]. Let n ∈ N and X be a linear space over the field K, where K is field of real or complex numbers of dimension d, where d ≥ n ≥ 2. A real valued function ||·, · · · , ·|| on X n satisfying the following four conditions: 1. ||x1 , x2 , · · · , xn || = 0 if and only if x1 , x2 , · · · , xn are linearly dependent in X, 2. ||x1 , x2 , · · · , xn || is invariant under permutation, 3. ||αx1 , x2 , · · · , xn || = |α| ||x1 , x2 , · · · , xn || for any α ∈ K, and 4. ||x + x′ , x2 , · · · , xn || ≤ ||x, x2 , · · · , xn || + ||x′ , x2 , · · · , xn || is called an n-norm on X, and the pair (X, ||·, · · · , ·||) is called a n-normed space over the field K. For example, we may take X = Rn being equipped with the n-norm ||x1 , x2 , · · · , xn ||E = the volume of the n-dimensional parallelopiped spanned by the vectors x1 , x2 , · · · , xn which may be given explicitly by the formula ||x1 , x2 , · · · , xn ||E = | det(xij )|, where xi = (xi1 , xi2 , · · · , xin ) ∈ Rn for each i = 1, 2, · · · , n. Let (X, ||·, · · · , ·||) be an n-normed space of dimension d ≥ n ≥ 2 and {a1 , a2 , · · · , an } be linearly independent set in X. Then the following function ||·, · · · , ·||∞ on X n−1 defined by ||x1 , x2 , · · · , xn−1 ||∞ = max{||x1 , x2 , · · · , xn−1 , ai || : i = 1, 2, · · · , n} defines an (n − 1)-norm on X with respect to {a1 , a2 , · · · , an }. A sequence (xk ) in a n-normed space (X, ||·, · · · , ·||) is said to converge to some L ∈ X if lim ||xk − L, z1 , · · · , zn−1 || = 0 for every z1 , · · · , zn−1 ∈ X.

k→∞

A sequence (xk ) in a n-normed space (X, ||·, · · · , ·||) is said to be Cauchy if lim ||xk − xp , z1 , · · · , zn−1 || = 0 for every z1 , · · · , zn−1 ∈ X.

k,p→∞

If every Cauchy sequence in X converges to some L ∈ X, then X is said to be complete with respect to the n-norm. Any complete n-normed space is said to be n-Banach space. For more details about n− normed space (see [26]) and references therein.


47

Let (X, ||·, · · · , ·||) be a n-normed space and W (n − X) denotes the space of Xvalued sequences. Let p = (pk ) be a bounded sequence of positive real numbers, λ = (λk ) be a sequence of strictly positive real numbers. Let M = (Mk ) be a sequence of Orlicz functions and N = (Nk ) is a complementary function of Orlicz function M = (Mk ). In this section of the paper we define the following sequences: n = x = (xk ) ∈ W (n − X) : lλM ∆m n , p, k·, · · · , ·k

ipk o X h

∆m xk

Mk n , z1 , · · · , zn−1 < ∞, for some ρ > 0 λk ρ k≥1

and

n m λ lN ∆n , p, k·, · · · , ·k = x = (xk ) ∈ W (n − X) :

ipk o X h

λk ∆m n xk < ∞, for some ρ > 0 . , z1 , · · · , zn−1 Nk ρ k≥1

If we take (pk ) = 1 for all k then

n = x = (xk ) ∈ W (n − X) : lλM ∆m n , k·, · · · , ·k

i o X h

∆m xk Mk n , z1 , · · · , zn−1 < ∞, for some ρ > 0 λk ρ k≥1

and

n m λ ∆n , k·, · · · , ·k = x = (xk ) ∈ W (n − X) : lN

i o X h

λk ∆m

n xk Nk , z1 , · · · , zn−1 < ∞, for some ρ > 0 . ρ k≥1

If (λk ) = 1 for all k ∈ N, then

n = x = (xk ) ∈ W (n − X) : lM ∆m n , p, k·, · · · , ·k

ipk o X h

∆m xk < ∞, for some ρ > 0 Mk n , z1 , · · · , zn−1 ρ k≥1

and

n lN ∆m , p, k·, · · · , ·k = x = (xk ) ∈ W (n − X) : n

ipk o ∆m x h X

k < ∞, for some ρ > 0 Nk n , z1 , · · · , zn−1 ρ k≥1

48


respectively. The main aim of this section is to study some topological properties and inclu m λ sion relations between the spaces lλM ∆m , p, k·, · · · , ·k and l ∆ , p, k·, · · · , ·k . n n N

Theorem 4.1. Let M = (Mk ) and N = (Nk ) be two sequences of Orlicz functions, p = (pk ) be a bounded sequence of positive real numbers and λ = (λ ) be a sequence km M of strictly positive real numbers. Then the sequence spaces l ∆ n , p, k·, · · · , ·k λ m λ and lN ∆n , p, k·, · · · , ·k are linear spaces over the field C of complex numbers. Proof: Let x = (xk ) and y = (yk ) ∈ lλM ∆m n , p, k·, · · · , ·k and α, β ∈ C. Then there exist positive real numbers ρ1 and ρ2 such that

ipk X h

∆m xk 0 and ρ2 > 0 be such that ! H1

≤1

! H1

≤ 1.

ipk X h

∆m xk Mk n , z1 , · · · , zn−1 λk ρ1 k≥1

and

ipk X h

∆m xk Mk n , z1 , · · · , zn−1 λ k ρ2

k≥1

50


Let ρ = ρ1 + ρ2 . Then by Minkowski’s inequality, we have !1

ipk H X h

∆m

n xk , z1 , · · · , zn−1 Mk λk ρ k≥1

X h

Mk

≤

k≥1

X

≤

k≥1

≤

"

ipk ∆m

n xk , z1 , · · · , zn−1 λk (ρ1 + ρ2 )

! H1

∆m x ρ1

k Mk n , z1 , · · · , zn−1 ρ1 + ρ2 λ k ρ1

# !1

pk H ∆m x ρ2

k + Mk n , z1 , · · · , zn−1 ρ1 + ρ2 λk ρ2 ! # !1 "

pk H ∆m x X ρ1

n k

, z1 , · · · , zn−1 Mk ρ1 + ρ2 λk ρ1 k≥1

ρ1 ρ1 + ρ2

+ ≤

!

X

k≥1

1.

# !1

pk H ∆m x

n k

, z1 , · · · , zn−1 Mk λk ρ2

"

Since ρ’s are non-negative, so we have

g(x + y) = inf

(

pk

(ρ1 + ρ2 ) H : Xh

k≥1

≤ inf

+ inf

( (

(ρ1 )

pk H

ipk ∆m (x + y )

k k , z1 , · · · , zn−1 Mk n λk (ρ1 + ρ2 ) :

! H1

ipk X h

∆m xk Mk n , z1 , · · · , zn−1 λk ρ1 k≥1

(ρ2 )

pk H

:

ipk X h

∆m yk

Mk n , z1 , · · · , zn−1 λ k ρ2 k≥1

≤1

! H1 ! H1

)

≤1

)

)

≤1 .

Therefore, g(x + y) ≤ g(x) + g(y). Finally, we prove that the scalar multiplication is continuous. Let µ be any complex number, therefore, by definition

g(µx) = inf

(

(ρ)

pk H

:

ipk X h

∆m µxk , z1 , · · · , zn−1 Mk n λk ρ k≥1

! H1

≤1

)

and

51


thus,

g(µx) = inf where t =

ρ |µ| .

g(µx)

(

(|µ|t)

pk H

:

ipk X h

∆m xk

Mk n , z1 , · · · , zn−1 λk t

k≥1

! H1

≤1

)

Since |µ|pk ≤ max(1, |µ| sup pk ). Hence,

=

max(1, |µ| sup pk ) inf

(

pk

(t) H :

ipk X h

∆m xk Mk n , z1 , · · · , zn−1 λk t

k≥1

! H1

)

≤1 .

So, the fact that scalar multiplication is continuous follows from the above inequality. This completes the proof. ✷ Theorem 4.3. Suppose M = (Mk ) be a sequence of Orlicz functions, p = (pk ) be a bounded sequence of positive real numbers and λ = (λk ) be a sequence of strictly positive real numbers. m If 0 < pk ≤ qk < ∞, for each k ∈ N, then M lλM ∆m , p, k·, · · · , ·k ⊆ l n λ ∆n , q, k·, · · · , ·k . Proof: Suppose that x = (xk ) ∈ lλM ∆m n , p, k·, · · · , ·k , this implies that

ipk X h

∆m xk ≤1 Mk n , z1 , · · · , zn−1 λk ρ

k≥1

for sufficiently large value of k say k ≥ k0 , for some fixed k0 ∈ N. Since M = (Mk ) is non decreasing, we have ∞ h ∞ h

iqk

ipk ∆m x ∆m x X X

n k

k Mk , z1 , · · · , zn−1 Mk n , z1 , · · · , zn−1 ≤ λk ρ λk ρ

k=k0

k=k0

0 λ k ρ1 k≥1

and

ipk X h ′

∆m xk

Mk n , z1 , · · · , zn−1 < ∞, for some ρ2 > 0. λk ρ2 k≥1

Let ρ = max(ρ1 , ρ2 ). The result follows from the inequality

ipk ∆m x Xh ′

k (Mk + Mk ) n , z1 , · · · , zn−1 λk ρ1

k≥1

ipk ∆m x X h ′

∆m xk

k Mk n , z1 , · · · , zn−1 + Mk n , z1 , · · · , zn−1 λk ρ1 λ k ρ1 k≥1

ipk X h

∆m xk ≤ D Mk n , z1 , · · · , zn−1 λk ρ1 k≥1

ipk X h ′

∆m xk . +D Mk n , z1 , · · · , zn−1 λ k ρ1 =

k≥1

This completes the proof.

✷

Theorem 4.8. If Z is a normal sequence space containing λ, then lλM ∆m n , p, k·, · · · , ·k is a proper subspace of Z. In addition, if Z is equipped with the monotone m norm (quasi-norm) k.kZ . The inclusion R : lλM ∆m n , p, k·, · · · , ·k → Z ∆n , p, k·, · · · , ·k is continuous with kRk ≤ k{λk }kZ . Proof: Let x ∈ lλM ∆m n , p, k·, · · · , ·k , then

∞ h

ipk ∆m x X

k < ∞, for some ρ > 0. Mk n , z1 , · · · , zn−1 λk ρ k=1

So there exists a constant K > 0 such that

∆m x

n k , z1 , · · · , zn−1 ≤ K for all k ∈ N.

λk ρ Since Z is a normal sequence space containing λ, we have ∆m x ,z ,··· ,

pk m n mk 1 M

zn−1 ∈ Z and so that x ∈ Z ∆ , p, k·, · · · , ·k . Hence, l ∆ n n , p, k·, · · · , ·k ⊆ λ m Z ∆n , p, k·, · · · , ·k . Further, since Mk (1) = 1 for all k ∈ N then

ipk X h

∆m xk

Mk n M , z1 , · · · , zn−1 ≤1 λ kxk k λ k≥1

54


M m so that k∆m n xk , z1 , · · · , zn−1 k ≤ λk kxkλ . As k.kZ is monotone, kRxkZ = k∆n xk , M z1 , · · · , zn−1 kZ ≤ k{λk }kZ kxkλ and hence, kRk ≤ k{λk }kZ . This completes the proof. ✷

Theorem 4.9. If Y is a normal sequence space containing λ−1 ≡ { λ1k }, then m λ lN ∆n , p, k·, · · · , ·k is a proper subspace of Y. In addition, if Y is equipped with λ the monotone norm (quasi-norm) k.kY . The inclusion S : lN ∆m n , p, k·, · · · , ·k → m Y ∆n , p, k·, · · · , ·k is continuous with kSk ≤ k{λ−1 k }kY .

Proof: The proof of the theorem is similar to that of Theorem 4.8 and so is omitted. ✷ Theorem 4.10. If λ = (λk ) is a bounded sequence such that m m inf λk > 0 (i.e −1 λ M both λ and λ are in l ). Then l ∆ , p, k·, · · · , ·k = l ∞ n λ ∆n , p, k·, · · · , ·k = M lM ∆m , p, k·, · · · , ·k . n Proof: Let x = (xk ) ∈ lM ∆m n , p, k·, · · · , ·k . Then

ipk X h

∆m xk < ∞, for some ρ > 0. Mk n , z1 , · · · , zn−1 ρ

k≥1

Since λ = (λk ) is bounded, we can write a ≤ λk ≤ b for some b > a ≥ 0. Define ρ1 = ρb. Also since Mk′ s are increasing, it follows that

ipk X h ∆m x

ipk X h

λk ∆m k n xk , z1 , · · · , zn−1 ≤ Mk n , z1 , · · · , zn−1 Mk ρ1 ρ

k≥1

k≥1

< ∞. m m λ λ · · · , ·k ⊆ lM Hence, lM ∆n , p, ∆m n , p, k·, · · · , ·k . The other inclusion lM ∆n , p, k·, k·, · · · , ·k ⊆ lM ∆m n , p, k·, · · · , ·k follows from the inequality

ipk X h λ ∆m x

ipk X h

∆m xk

k n k

, z1 , · · · , zn−1 ≤ Mk nρ , z1 , · · · , zn−1 Mk ρ a

k≥1

k≥1

< ∞. m λ Therefore, lM ∆n , p, k·, · · · , ·k = lM ∆m n , p, k·, · · · , ·k . Similarly one can prove that m lλM ∆m ✷ n , p, k·, · · · , ·k = lM ∆n , p, k·, · · · , ·k . This completes the proof. Theorem 4.11. If {λk } ∈ l∞ with c = sup λk ≥ 1 and {λ−1 k } is unbounded, k≥1 λ then lλM ∆m in lM ∆m n , p, k·, · · ·, ·k is properly contained n , p, k·, · · · , ·k and the λ m inclusion map U : lλM ∆m n , p, k·, · · · , ·k → lM ∆n , p, k·, · · · , ·k is continuous with 2 kU k ≤ c .

55


Proof: For any ρ > 0 and ρ′ = ρc2 , we have

ipk X h ∆m x

ipk X h

λk ∆m

n k n xk Mk ≤ M , z , · · · , z , z , · · · , z

k 1 n−1 1 n−1 ρ′ λk ρ

k≥1

k≥1

0 : , z1 , · · · , zn−1 ≤1 λk ρ

∆m n xk

k≥1

and

λ BM ∆m n , p, k·, · · · , ·k =

′ M ks

(

)

ipk X h x

λk ∆m k n ≤1 . , z1 , · · · , zn−1 ρ>0: Mk ρ k≥1

m m λ Since are increasing and c = 1, we get AM λ ∆n , p, k·, · · · , ·k ⊂ BM ∆n , p, k·, · · · , ·k . Hence, m M λ (4.1) ∆n , p, k·, · · · , ·k ≤ inf AλM ∆m kxkλM = inf BM n , p, k·, · · · , ·k = kxkλ 2 i.e kU (x)kλM ≤ kxkM λ . Thus, U is continuous with kU k ≤ 1 = c . If c = 1, define λk δ k = c , k ∈ N. Then δ k ≤ 1 and from (4.1), it follows that (4.2) kxkδM ≤ kxkM for x ∈ lλM ∆m n , p, k·, · · · , ·k . δ

Hence, from (4.2)

kU (x)kλM = kxkλM ≤ e2 kxkM λ ,

thus, U is continuous with kU k ≤ c2 . This completes the proof.

✷

Theorem 4.12. If {λk } is unbounded with sup λk−1 = d ≥ 1, λk > 0 for all k≥1 m λ k, then lM ∆n , p, k·, ·· · , ·k is properly contained in lλM ∆m k·, · · · , ·k and the n , p, λ M m inclusion map V : lM ∆m n , p, k·, · · · , ·k → lλ ∆n , p, k·, · · · , ·k is continuous with 2 kV k ≤ d . Proof: The proof of the theorem is similar to that of Theorem 4.11 and so is omitted. ✷

56


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Seema Jamwal and Kuldip Raj School of Mathematics Shri Mata Vaishno Devi University KATRA-182320, J&K, INDIA. E-mail address: [email protected] E-mail address: [email protected]