1Introductionandpreliminaries KuldipRaj SunilK.Sharma Somesequencespacesin2-normedspacesdefinedbyMusielak-Orliczfunction

Some sequence spaces in 2-normed spaces defined by Musielak-Orlicz function

Kuldip Raj

School of Mathematics Shri Mata Vaishno Devi University

Katra-182320, J&K, India email:[email protected]

Sunil K. Sharma

School of Mathematics Shri Mata Vaishno Devi University

Katra-182320, J&K, India email:[email protected]

Abstract. In the present paper we introduce some sequence spaces com- bining lacunary sequence, invariant means in 2-normed spaces defined by Musielak-Orlicz functionM = (Mk). We study some topological prop- erties and also prove some inclusion results between these spaces.

1 Introduction and preliminaries

The concept of 2-normed space was initially introduced by Gahler [2] as an interesting linear generalization of a normed linear space which was subse- quently studied by many others see ([3], [9]). Recently a lot of activities have started to study sumability, sequence spaces and related topics in these linear spaces see ([4], [10]).

Let X be a real vector space of dimension d, where 2 ≤d < ∞. A 2-norm on Xis a function||., .||:X×X→Rwhich satisfies

(i) kx, yk=0 if and only if xand yare linearly dependent (ii) kx, yk=ky, xk

(iii) kαx, yk=|α|kx, yk, α∈R

(iv) kx, y+zk ≤ kx, yk+kx, zkfor all x, y, z∈X.

2010 Mathematics Subject Classification:40A05, 40A30

Key words and phrases: Paranorm space, Difference sequence space, Orlicz function, Musielak-Orlicz function, Lacunary sequence, Invariant mean

97

The pair (X,k., .k) is then called a 2-normed space see [3]. For example, we may takeX=R²equipped with the 2-norm defined askx, yk= the area of the parallelogram spanned by the vectors xand y which may be given explicitly by the formula

||x₁, x₂||_E=abs

x₁₁ x₁₂ x₂₁ x₂₂

.

Then, clearly(X,k., .k)is a 2-normed space. Recall that(X,k., .k)is a 2-Banach space if every cauchy sequence inXis convergent to some xinX.

Letσbe the mapping of the set of positive integers into itself. A continuous linear functional ϕ on l_∞,is said to be an invariant mean or σ-mean if and only if

(i) ϕ(x)≥0when the sequence x= (x_k) hasx_k≥0 for all k, (ii) ϕ(e) =1, wheree= (1, 1, 1, . . .) and

(iii) ϕ(x_σ(k)) =ϕ(x) for all x∈l_∞.

If x= (x_n), write Tx=Txn= (x_σ(n)). It can be shown in [11] that V_σ=

x∈l_∞ |lim

k t_kn(x) =l, uniformly in n, l=σ−limx , where

t_kn(x) = x_n+x_σ1n+...+x_σkn

k+1 .

In the case σis the translation mapping n→n+1,σ-mean is often called a Banach limit and Vσ, the set of bounded sequences all of whose invariant means are equal, is the set of almost convergent sequences see [6].

By a lacunary sequence θ= (k_r)wherek₀=0, we shall mean an increasing sequence of non-negative integers withkr−kr−1→ ∞asr→ ∞. The intervals determined by θ will be denoted by I_r = (kr−1, k_r]. We write h_r = k_r− kr−1. The ratio k_r

k_r−1 will be denoted by q_r. The space of lacunary strongly convergent sequence was defined in [1].

LetX be a linear metric space. A functionp:X→ Ris called paranorm, if (i) p(x)≥0, for all x∈X

(ii) p(−x) =p(x), for all x∈X

(iii) p(x+y)≤p(x) +p(y), for all x, y∈X

(iv) if (σ_n) is a sequence of scalars with σ_n→ σ as n → ∞ and (x_n) is a sequence of vectors withp(x_n−x)→0 as n→ ∞, thenp(σ_nxn−σx)→ 0 as n→ ∞.

A paranormpfor whichp(x) =0impliesx=0is 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 [12], Theorem 10.4.2, P-183).

An orlicz function M: [0,∞)→[0,∞) is a continuous, non-decreasing and convex function such thatM(0) =0,M(x)> 0 forx > 0 andM(x)−→ ∞ as x−→ ∞.

Lindenstrauss and Tzafriri [5] used the idea of Orlicz function to define the following sequence space. Letw be the space of all real or complex sequences x= (x_k), then

lM=

x∈w| X∞

k=1

M |x_k|

ρ

<∞

which is called a Orlicz sequence space. Also lM is a Banach space with the norm

kxk=inf

ρ > 0| X∞

k=1

M |x_k|

ρ

≤1

.

Also, it was shown in [5] that every Orlicz sequence space l_M contains a subspace isomorphic to lp(p ≥ 1). The ∆₂− condition is equivalent to M(Lx) ≤ LM(x), for all L with 0 < L < 1. An Orlicz function M can al- ways be represented in the following integral form

M(x) = Zx

0

η(t)dt

where η is known as the kernel of M, is right differentiable for t≥0, η(0) = 0, η(t)> 0,ηis non-decreasing and η(t)→ ∞ ast→ ∞.

A sequenceM= (M_k)of Orlicz function is called a Musielak-Orlicz function see ([7], [8]). A sequence N = (N_k) is called a complementary function of a Musielak-Orlicz function M

N_k(v) =sup

|v|u−M_k|u≥0 , k=1, 2, . . .

For a given Musielak-Orlicz function M, the Musielak-Orlicz sequence space tM and its subspacehM are defined as follows

tM =

x∈w|I_M(cx)<∞, for some c > 0 ,

hM =

x∈w|IM(cx)<∞, for all c > 0

,

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

X∞

k=1

M_k(x_k), x= (x_k)∈tM. We considertM equipped with the Luxemburg norm

kx||=inf

k > 0|IM

x k

≤1

or equipped with the Orlicz norm

||x||⁰=inf 1

k(1+IM(kx))|k > 0

.

Let M= (M_k) be a Musielak-Orlicz function, (X,||., .||) be a 2-normed space and p= (P_k) be any sequence of strictly positive real numbers. ByS(2−X) we denote the space of all sequences defined over(X,||., .||). We now define the following sequence spaces:

w^o_σ[M, p,k., .k]_θ=

x∈S(2−X)| lim

r→∞

1 h_r

X

k∈Ir

M_k

t_kn(x) ρ , z

pk

=0, ρ > 0, uniformly in n

, w_σ[M, p,||., .||]_θ=

x∈S(2−X)| lim

r→∞

1 h_r

X

k∈Ir

M_k

t_kn(x−l) ρ , z

pk

=0, ρ > 0, uniformly in n

, and w^∞_σ [M, p,||., .||]_θ=

x∈S(2−X)|sup

r,n

1 hr

X

k∈Ir

M_k

t_kn(x) ρ , z

p_k

<∞, for some ρ > 0

.

When M(x) = x for all k, the spaces w^o_σ

M, p,||., .||

θ, wσ

M, p,||., .||

θ

and w^∞_σ

Mk, p,||., .||

θreduces to the spaces w^o_σ

p,||., .||

θ,wσ

, p,||., .||

θand w^∞_σ

p,||., .||

θrespectively.

If p_k = 1 for all k, the spaces w^o_σ

M, p,||., .||

θ, w_σ

M, p,||., .||

θ and w^∞_σ

M, p,||., .||

θ reduces to w^o_σ

M,||., .||

θ, w_σ

M,||., .||

θ and w^∞_σ

M,||., .||

θrespectively.

The following inequality will be used throughout the paper. If 0 ≤ p_k ≤ supp_k=H,K=max(1, 2^H−1) then

|a_k+bk|^pk ≤K

|a_k|^pk +|b_k|^pk (1)

for all kand a_k, b_k∈C. Also |a|^pk ≤max(1,|a|^H) for all a∈C.

In the present paper we study some topological properties of the above sequence spaces.

2 Main results

Theorem 1 LetM= (M_k)be Musielak-Orlicz function,p= (p_k) be a boun- ded sequence of positive real numbers, then the classes of sequences w^o_σ

M, p,||., .||

θ,w_σ

M, p,||., .||

θandw^∞_σ

M, p,||., .||

θare linear spaces over the field of complex numbers.

Proof.Let x, y∈w^o_σ

M, p,||., .||

θand α, β∈C. In order to prove the result we need to find someρ₃ such that

rlim→∞

1 h_r

X

k∈Ir

M_k

tkn(αx+βy) ρ₃ , z

pk

=0, uniformly in n.

Since x, y∈w^o_σ

M, p,||., .||

θ, there exist positiveρ₁, ρ₂such that

rlim→∞

1 hr

X

k∈Ir

Mk

tkn(x) ρ₁ , z

pk

=0, uniformly in n and

rlim→∞

1 hr

X

k∈Ir

M_k

t_kn(y) ρ2

, z

pk

=0, uniformly in n.

Define ρ₃=max(2|α|ρ1, 2|β|ρ₂). Since (M_k) is non-decreasing and convex

1 hr

X

k∈Ir

Mk

tkn(αx+βy) ρ3

, z

p_k

≤ 1 hr

X

k∈Ir

Mk

tkn(αx) ρ3

, z

+

tkn(βy) ρ3

, z

≤ 1 hr

X

k∈Ir

Mk

tkn(x) ρ1

, z

+

tkn(y) ρ2

, z

≤K 1 hr

X

k∈Ir

Mk

tkn(x) ρ1

, z

+

+K 1 hr

X

k∈Ir

Mk

tkn(y) ρ2

, z

→0 as r→ ∞, uniformly in n.

So that αx+βy∈ w^o_σ

M, p,||., .||

θ. This completes the proof. Similarly, we can prove that w_σ

M, p,||., .||

θ andw^∞_σ

M, p,||., .||

θare linear spaces.

Theorem 2 Let M = (M_k) be Musielak-Orlicz function, p = (p_k) be a bounded sequence of positive real numbers. Then w^o_σ

M, p,||., .||

θ is a topo- logical linear spaces paranormed by

g(x) =inf



ρ^prH : 1 hr

X

k∈Ir

Mk

tkn(x) ρ , z

p_k!_H¹

≤1, r=1, 2, . . . , n=1, 2, . . .



,

where H=max(1,sup_kp_k<∞).

Proof.Clearlyg(x)≥0forx= (x_k)∈w^o_σ

M, p,||., .||

θ. SinceMk(0) =0, we getg(0) =0.

Conversely, suppose thatg(x) =0, then

inf



ρ^prH : 1 h_r

X

k∈Ir

M_k

t_kn(x) ρ , z

pk!_H¹

≤1, r≥1, n≥1



=0.

This implies that for a given ǫ > 0, there exists some ρǫ(0 < ρ_ǫ < ǫ) such that

1 h_r

X

k∈Ir

M_k

t_kn(x) ρ_ǫ , z

pk!_H¹

≤1.

Thus 1 h_r

X

k∈Ir

M_k

t_kn(x) ǫ , z

pk!_H¹

≤ 1 h_r

X

k∈Ir

M_k

t_kn(x) ρ_ǫ , z

pk!_H¹

≤1, for each r and n. Suppose that xk6=0 for each k∈N. This implies that t_kn(x)6=0,for eachk, n∈N.Letǫ→0,then

tkn(x) ǫ , z

→ ∞.It follows that 1

h_r X

k∈Ir

"

M_k

t_kn(x) ǫ , z

#p_k!_H¹

→ ∞

which is a contradiction.

Therefore,t_kn(x) =0for each k and thusx_k=0 for eachk∈N. Letρ₁> 0 and ρ₂> 0be such that

1 h_r

X

k∈Ir

M_k

t_kn(x) ρ₁ , z

pk!_H¹

≤1

and

1 h_r

X

k∈Ir

M_k

t_kn(y) ρ₂ , z

pk!_H¹

≤1

for each r . Letρ=ρ₁+ρ₂.Then, we have

1 hr

X

k∈Ir

M_k

t_kn(x+y)

ρ , z

p_k!_H¹

≤ 1 hr

X

k∈Ir

M_k

t_kn(x) +t_kn(y) ρ1+ρ2

, z

pk!_H¹

≤ 1 hr

X

k∈Ir

ρ1

ρ₁+ρ₂Mk

t_kn(x) ρ₁ , z

+ ρ₂ ρ₁+ρ₂Mk

t_kn(y) ρ₂ , z

p^k!_H¹

≤

ρ1

ρ₁+ρ₂ 1

hr

X

k∈Ir

Mk

tkn(x) ρ₁ , z

pk!_H¹

+

ρ2

ρ₁+ρ₂ 1

h_r X

k∈Ir

M_k

tkn(y) ρ₂ , z

pk!_H¹

≤1.

(by Minkowski’s inequality)

Since ρ^′s are non-negative, so we have g(x+y) =

=inf



ρ^prH | 1 h_r

X

k∈Ir

M_k

t_kn(x) +t_kn(y)

ρ , z

pk!_H¹

≤1, r≥1, n≥1





≤inf



ρ₁^prH | 1 h_r

X

k∈Ir

M_k

t_kn(x) ρ₁ , z

pk!_H¹

≤1, r≥1, n≥1



+ +inf



ρ

pr H

2 | 1 hr

X

k∈Ir

Mk

t_kn(x) ρ₂ , z

pk!¹

H

≤1, r≥1, n≥1



. Therefore,

g(x+y)≤g(x) +g(y).

Finally, we prove that the scalar multiplication is continuous. Let λ be any complex number. By definition,

g(λx)=inf



ρ^prH | 1 hr

X

k∈Ir

Mk

tkn(λx) ρ , z

pk!_H¹

≤1, r≥1, n≥1



. Then

g(λx)=inf



(|λ|t)^prH | 1 h_r

X

k∈Ir

M_k

t_kn(x) t , z

pk!_H¹

≤1, r≥1, n≥1



. wheret= ρ

|λ|.Since |λ|^pr ≤max(1,|λ|^suppr),we have g(λx) ≤ max(1,|λ|^suppr)

inf



t^prH | 1 h_r

X

k∈Ir

M_k

t_kn(x) t , z

p_k!_H¹

≤1, r≥1, n≥1



. So, the fact that scalar multiplication is continuous follows from the above inequality.

This completes the proof of the theorem.

Theorem 3 Let M= (M_k) be Musielak-Orlicz function. If sup

k

M_k(t)

p_k<∞ for allt > 0, then

w_σ[M, p,||., .||]_θ⊂w^∞_σ [M, p,||., .||]_θ. Proof.Let x∈wσ

M, p,||., .||

θ. By using inequality (1), we have 1

h_r X

k∈Ir

M_k

t_kn(x) ρ , z

pk

≤ K h_r

X

k∈Ir

M_k

t_kn(x−l) ρ , z

pk

+ K h_r

X

k∈Ir

M_k

l ρ, z

pk

. Since sup_k

M_k(t)pk

< ∞, we can take that sup_k

M_k(t)pk = T. Hence we getx∈w^∞_σ

M, p,||., .||

θ.

Theorem 4 Let M= (M_k) be Musielak-Orlicz function which satisfies ∆2- condition for all k, then

w_σ[p,||., .||]_θ⊂w_σ[M, p,||., .||]_θ. Proof.Let x∈wσ[p,||., .||]_θ. Then we have

Tr= 1 hr

X

k∈Ir

||t_kn(x−l), z||^pk → ∞ as r→ ∞ uniformly in n, for some l.

Letǫ > 0 and choose δwith 0 < δ < 1such that M_k(t)< ǫfor0≤t≤δfor all k. So that

1 h_r

X

k∈Ir

M_k

t_kn(x−l) ρ , z

pk

= 1 h_r

X1

k∈I_r ktkn(x−l)zk ≤δ

M_k

t_kn(x−l) ρ , z

pk

+ 1 h_r

X2

k∈I_r kt_kn(x−l)zk ≤δ

M_k

tkn(x−l)

ρ , z

pk

.

For the first summation in the right hand side of the above equation, we have X1

≤ǫ^Hby using continuity of M_k for allk. For the second summation, we write

||t_kn(x−l), z||≤1+||t_kn(x−l) δ , z||.

Since M_kis non-decreasing and convex for all k, it follows that M_k(||t_kn(x−l), z||)< M_k

1+

t_kn(x−l) δ , z

≤ 1

2M_k(2) + 1 2M_k

(2)

t_kn(x−l) δ , z

. Since Mksatisfies ∆2-condition for allk, we can write

Mk

||t_kn(x−l), z||

≤ 1 2L

tkn(x−l) δ , z

Mk(2) + 1 2L

tkn(x−l) δ , z

Mk(2)

=L

t_kn(x−l) δ , z

M_k(2).

So we write 1 hr

X

k∈Ir

M_k

t_kn(x−l)

ρ , z

pk

≤ǫ^H+ [max(1, LMk(2))δ]^HTr. Letting r→ ∞, it follows that x∈wσ

M, p,||., .||

θ.

This completes the proof.

Theorem 5 Let M= (M_k) be Musielak-Orlicz function. Then the following statements are equivalent:

(i) w^∞_σ

p,||., .||

θ⊂w^o_σ

M, p,||., .||

θ, (ii) w^o_σ

p,||., .||

θ⊂w^o_σ

M, p,||., .||

θ, (iii) sup

r

1 hr

X

k∈Ir

Mk(t)pk

<∞ for all t > 0.

Proof. (i) =⇒ (ii) We have only to show thatw^o_σ

p,||., .||

θ⊂w^∞_σ

p,||., .||

θ. Letx∈w^o_σ

p,||., .||

θ. Then there existsr≥r_o, forǫ > 0, such that 1

hr

X

k∈Ir

t_kn(x) ρ , z

pk

< ǫ.

Hence there exists H > 0 such that sup

r,n

1 h_r

X

k∈Ir

t_kn(x) ρ , z

pk

< H

for all nand r. So we getx∈w^∞_σ

p,||., .||

θ.

(ii) =⇒(iii) Suppose that (iii) does not hold. Then for somet > 0 sup

r

1 h_r

X

k∈Ir

[M_k(t)]^pk =∞

and therefore we can find a subintervalI_r(m)of the set of intervalIrsuch that 1

h_r(m) X

k∈I_r(m)

M_k(1

m) pk

> m, m=1, 2, (2)

Let us define x = (x_k) as follows, x_k = _m¹ if k ∈ I_r(m) and x_k = 0 if k 6∈

I_r(m). Then x ∈ w^o_σ

p,||., .||

θ but by eqn. (2), x 6∈ w^∞_σ

M, p,||., .||

θ. which contradicts (ii). Hence (iii) must hold. (iii) =⇒ (i) Suppose (i) not holds, then forx∈w^∞_σ

p,||., .||

θ, we have sup

r,n

1 hr

X

k∈Ir

Mk

tkn(x) ρ , z

pk

= (3)

Lett=

tkn(x) ρ , z

for each kand fixed n, so that eqn. (3) becomes sup

r

1 h_r

X

k∈Ir

[M_k(t)]^pk =∞

which contradicts (iii). Hence (i) must hold.

Theorem 6 Let M= (M_k) be Musielak-Orlicz function. Then the following statements are equivalent:

(i) w^o_σ

M, p,||., .||

θ⊂w^o_σ

p,||., .||

θ, (ii) w^o_σ

M, p,||., .||

θ⊂w^∞_σ

p,||., .||

θ, (iii) inf

r

X

k∈Ir

Mk(t)pk

> 0 for allt > 0.

Proof.(i) =⇒ (ii) : It is easy to prove.

(ii) =⇒ (iii) Suppose that (iii) does not hold. Then infr

1 h_r

X

k∈Ir

[M_k(t)]^pk =0 for some t > 0,

and we can find a subintervalI_r(m)of the set of intervalI_rsuch that 1

h_r X

k∈I_r(m)

[M_k(m)]^pk < 1

m, m=1, 2, . . . (4) Let us define x_k = m if k ∈ I_r(m) and x_k = 0 if k 6∈ I_r(m). Thus by eqn.(4), x∈w^o_σ

M, p,||., .||

θ butx6∈w^∞_σ

p,||., .||

θ which contradicts (ii). Hence (iii) must hold.

(iii) =⇒ (i) It is obvious.

Theorem 7 Let M = (M_k) be Musielak-Orlicz function. Then w^∞_σ M, p,

||., .||

θ ⊂w^o_σ

p,||., .||

θ if and only if

rlim→∞

1 h_r

X

k∈Ir

M_k(t)pk =∞ (5) Proof. Letw^∞_σ

M, p,||., .||

θ⊂w^o_σ

p,||., .||

θ. Suppose that eqn. (5) does not hold. Therefore there is a subinterval I_r(m) of the set of interval I_r and a number to> 0, whereto=

tkn(x) ρ , z

for allk and n, such that 1

h_r(m) X

k∈I_r(m)

[M_k(t_o)]^pk ≤M <∞, m=1, 2, . . . (6) Let us define x_k=t_o ifk ∈I_r(m)and x_k= 0 ifk6∈I_r(m). Then, by eqn. (6),

x∈w^∞_σ [M_k, p,||., .||]_θ. Butx6∈w^o_σ[p,||., .||]_θ. Hence eqn. (5) must hold.

Conversely, suppose that eqn. (5) hold and thatx∈w^∞_σ [M_k, p,||., .||]_θ. Then for each rand n

1 h_r

X

k∈Ir

M_k

t_kn(x) ρ , z

pk

≤M <∞. (7)

Now suppose that x6∈ w^o_σ[p,||., .||]_θ. Then for some number ǫ > 0 and for a subinterval I_ri of the set of intervalI_r, there isk_o such that||t_kn(x), z||^pk > ǫ fork≥ko. From the properties of sequence of Orlicz functions, we obtain

Mk

ǫ ρ

pk

≤

Mk

tkn(x) ρ , z

pk

which contradicts eqn.(6), by using eqn. (7). This completes the proof.

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Received: October 20, 2010