Vector Valued Multiple Sequence Spaces Defined by

Bol. Soc. Paran. Mat. c

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(3s.) v. 33 1 (2015): 67–79. ISSN-00378712 in press doi:10.5269/bspm.v33i1.21602

Vector Valued Multiple Sequence Spaces Defined by Orlicz Function Binod Chandra Tripathy and Rupanjali Goswami

abstract: In this article we define some vector valued multiple sequence spaces defined by Orlicz function. We study some of their properties like solidness, symmetry, completeness etc and some inclusion results.

Key Words: Orlicz function; completeness; semi-norm; regular convergence; solid space. Contents 1 Introduction

67

2 Definition and Preliminaries.

68

3 Main Results

70

4 Particular cases

78 1. Introduction

Throughout this article the space of all, bounded, convergent in Pringsheim’s sense, null in Pringsheim’s sense, regularly convergent and regularly null multiple sequences defined over a semi-normed space (X, q), semi-normed by q will be denoted by k ω (q), k ℓ∞ (q), k c (q), k c0 (q), k cR (q), k cR 0 (q). For X = C, the field of complex numbers, these spaces represent the corresponding scalar sequence spaces. Throughout this article θ represents the zero element of X. The zero element of a single sequence space is denoted by θ = (θ, θ, ....). The zero element of a multiple sequence is denoted by k θ, a multiple infinite array of θ’s. An Orlicz function M is a mapping M : [0, ∞) → [0, ∞) such that it is continuous, non-decreasing and convex with M (0) = 0, M (x) > 0 for x > 0 and M (x) → ∞ as x → ∞. The idea of Orlicz function was used by Lindenstrauss and Tzafriri [6] to construct the sequence space.     ∞ P ℓM = (xk ) : M |xρk | , for some ρ > 0 , k=1

which is a Banach space normed by

2000 Mathematics Subject Classification: 40A05, 40B05, 46E30

67

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

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Binod Chandra Tripathy and Rupanjali Goswami

    ∞ P M xρk ≤ 1 ||xk || = inf ρ > 0 : k=1

The space ℓM is an Orlicz sequence space with M (x) = |x|p for 1 ≤ p < ∞, which is closely related to the sequence space ℓp . An Orlicz function M is said to satisfy the ∆2 -condition for all values of u, if there exists a constant K > 0, such that M (2u) ≤ K (M u), where u ≥ 0. Recently Tripathy [9], Tripathy and Mahanta [11], Altin et. al. [1] and many others investigated the Orlicz sequence spaces from sequence point of view and related with the summability theory. Remark 1.1. Let 0 < λ < 1, then M (λx) ≤ λM (x), for all x ≥ 0. 2. Definition and Preliminaries. Throughout this article a multiple sequence is denoted by A =< an1 n2 ...nk >, a multiple infinite array of elements an1 n2 ...nk ∈ X for all n1 , n2 , .., nk ∈ N . Initial works on double sequences is found in Bromwich [3]. Hardy [5] introduced the notion of regular convergence for double sequences. Moricz [7] studied some properties of double sequences of real and complex numbers. Recently different types of double sequence have been introduced and investigated from different aspects by Basarir and Sonalcan [2], Colak and Turkmenoglu [4], Turkmenoglu [14], Moricz and Rhodes [8], Tripathy [10], Tripathy and Sarma ( [12], [13])and many others. Definition 2.1. A multiple sequence space E is said to be solid if < αn1 n2 ...nk an1 n2 ...nk >∈ E, whenever < an1 n2 ...nk >∈ E for all multiple sequences < αn1 n2 ...nk > of scalars with |αn1 n2 ...nk | ≤ 1, for all n1 , n2 , ..., nk ∈ N . Definition 2.2. A multiple sequence space E is said to be symmetric if < an1 n2 ...nk >∈ E, implies < aπ(n1 ,n2 ,...,nk ) >∈ E, where π(n1 , n2 , ..., nk ) are permutations of N × N... × N . Definition 2.3. A multiple sequence space E is said to be monotone if it contains the canonical pre-images of all its step spaces. Definition 2.4. A multiple sequence space E is said to be convergence free if < bn1 n2 ...nk >∈ E, whenever < an1 n2 ...nk >∈ E and bn1 n2 ...nk = θ whenever an1 n2 ...nk = θ. Remark 2.5. A sequence space E is solid implies E is monotone.

Vector Valued Multiple Sequence Spaces Defined by Orlicz Function 69

Let M be an Orlicz function. Now we introduce the following multiple sequence spaces:

k ℓ∞ (M, q)

=



< an1 n2 ...nk

sup >∈ k ω(q) : M n1 n2 ...nk

   an1 n2 ...nk q 0

 k c(M, q) = < an1 n2 ...nk >∈

}

   an1 n2 ...nk − L →0 k ω(q) : M q ρ as n1 , n2 , .., nk → ∞ for some ρ > 0

}

R

A =< an1 n3 ...nk >∈ k c (M, q), i.e., regularly convergent if < an1 n2 ...nk >∈ and the following limits hold:

k c(M, q)

There exists Ln2 n3 ...nk , Ln1 n3 ...nk , Ln1 n2 n4 ...nk , ....., Ln1 n2 ...ni−1 ni+1 ...nk , ....., Ln1 n2 .....nk−1 ∈ X such that    −L a → 0 as n1 → ∞ for some ρ > 0 and n2 , n3 , ..., nk ∈ M q n1 n2 ...nk ρ n2 n3 ...nk N.    −L a M q n1 n2 ...nk ρ n1 n3 ...nk → 0 as n2 → ∞ for some ρ > 0 and n1 , n3 , ..., nk ∈ N.    −L a → 0 as n3 → ∞ for some ρ > 0 and n1 , n2 , n4 , M q n1 n2 ...nk ρ n1 n2 n4 ....nk ..., nk ∈ N . ........................................................ ........................................................  a  n1 n2 ...nk −Ln1 n2 ...ni−1 ni+1 ...nk M q → 0 as ni → ∞ for some ρ > 0 and ρ n1 , n2 , ..., ni−1 , ni+1 , .., nk ∈ N . ........................................................ ........................................................   a n n ...n −Ln n ...n M q( 1 2 k ρ 1 2 k−1 ) → 0 as nk → ∞ for some ρ > 0 and n1 , n2 , ..., nk−1 ∈ N .

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Binod Chandra Tripathy and Rupanjali Goswami

Without loss of generality, ρ can be chosen to be same for all the above cases. The definition of k c0 (M, q) and k cR 0 (M, q) follows from the above definition on taking L = Ln2 n3 ...nk = Ln1 n3 n4 ...nk = ... = Ln1 n2 ...ni−1 ni+1 ...nk = ... = Ln1 n2 ...nk−1 = θ for all n1 , n2 , ..., nk ∈ N Remark 2.6. The space k cR 0 (M, q) has the following definition too. n    an1 n2 ...nk R → 0 as max{n1 , n2 , k c0 (M, q) = < an1 n2 ...nk >∈ k ω(q) : M q ρ ..., nk } → ∞ for some ρ > 0

}

.

We also define kc

B

(M, q) =

k c (M, q) ∩ k ℓ∞ (M, q)

and k cB 0 (M, q) =

k c0

(M, q) ∩ k ℓ∞ (M, q).

3. Main Results Theorem 3.1. The classes Z (M, q) for Z = multiple sequences are linear spaces.

k ℓ∞ , k c, k c0 , k c

B

, k cR , and k cR 0 of

Proof: We prove it for the case k ℓ∞ (M, q) and other cases can also be proved similarly. Let < an1 n2 ...nk >, < bn1 n2 ...nk >∈

k ℓ∞

(M, q).

Then we have    an1 n2 ...nk q < ∞, for some ρ1 > 0 ρ1    sup an1 n2 ...nk < ∞ for some ρ2 > 0. M q n1 , n2 , ..., nk ρ2

sup M n1 , n2 , ..., nk

Let α, β be scalars and ρ = max {2|α|ρ1 , 2|β|ρ2 }. Then    αan1 n2 ...nk +βbn1 n2 ...nk q ρ       an1 n2 ...nk bn1 n2 ...nk sup 1 ≤ 21 n1 ,nsup M q M q + < ∞. ,...,n ρ 2 n ,n ,...,n ρ 2 1 2 k k sup n1 ,n2 ,...,nk M

1

2

(3.1) (3.2)

Vector Valued Multiple Sequence Spaces Defined by Orlicz Function 71

Hence < αan1 n2 ...nk + βbn1 n2 ...nk >∈

k ℓ∞

(M, q).

Thus k ℓ∞ (M, q) is a linear space.

✷

Theorem 3.2. The spaces Z (M, q) for Z = normed spaces, semi-normed by

f (< an1 n2 ...nk >) = inf



ρ>0:

k ℓ∞ , k c

sup M n1 , n2 , ..., nk

B

R R , k cB 0 , k c , k c0 are semi-

    an1 n2 ...nk q ≤ 1 . (3.3) ρ

Proof: Clearly f (k θ) = 0 and f (− < an1 n2 ...nk >) = f (< an1 n2 ...nk >) for all < an1 n2 ...nk >∈ k ℓ∞ (M, q) . Let λ ∈ C, then we have n    o an1 n2 ...nk f (< an1 n2 ...nk >) = inf ρ > 0 : n1 ,nsup M q ≤ 1 ρ 2 ,...,nk n    o λan1 n2 ...nk f (λ < an1 n2 ...nk >) = inf ρ > 0 : n1 ,nsup M q ≤ 1 ,...,n ρ 2 k o n an1 n2 ...nk

ρ ≤ 1 where r = |λ| = |λ| inf r > 0 : n1 ,nsup M q r ,...,n 2 k = |λ|f (< an1 n2 ...nk >). Next let < an1 n2 ...nk >, < bn1 n2 ...nk >∈ k ℓ∞ (M, q) . Then we have  o n   an1 n2 ...nk ≤ 1 , f (< an1 n2 ...nk >) = inf ρ1 > 0 : n1 ,nsup M q ρ1 2 ,...,nk n    o bn1 n2 ...nk f (< bn1 n2 ...nk >) = inf ρ2 > 0 : n1 ,nsup M q ≤ 1 . ,...,n ρ 2 k 2

Let ρ1 > 0 and ρ2 > 0 be such that    an1 n2 ...nk sup ≤1 M q ρ n1 ,n2 ,...,nk 1

and sup n1 ,n2 ,...,nk M

   b q n1 nρ2 ...nk ≤ 1. 2

Let ρ = ρ1 + ρ2 , then we have    an1 n2 ...nk +bn1 n2 ...nk sup M q n1 n2 ...nk ρ    an1 n2 ...nk sup 1 + M q ≤ ρ ρ+ρ n1 ,n2 ,...,nk ρ 1

2

1

ρ2 sup ρ1 +ρ2 n1 ,n2 ,...,nk M

Since ρ1 and ρ2 are non-negative, so we have

   b q n1 nρ2 ...nk . 2

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Binod Chandra Tripathy and Rupanjali Goswami

f (< an1 n2 ...nk > + < bn1 n2 ...nk >)      sup an1 n2 ...nk + bn1 n2 ...nk = inf ρ = ρ1 + ρ2 > 0 : ≤1 M q n1 , n2 , ..., nk ρ      sup an1 n2 ...nk ≤ inf ρ1 > 0 : M q ≤1 n1 , n2 , ..., nk ρ1      sup bn1 n2 ...nk ≤1 + inf ρ2 > 0 : M q ρ2 n1 , n2 , ..., nk = f (< an1 n2 ...nk > + < bn1 n2 ...nk >) Hence f is a semi-norm on Z(M, q) for Z =

k ℓ∞ , k c

B

R R , k cB 0 , k c , k c0 .

✷ Theorem 3.3. The spaces k ℓ∞ (M, q) and k cR 0 (M, q) are symmetric where as the R spaces Z(M, q) for Z = k c, k c0 , k cB , k cB , c , k cR 0 are not symmetric. 0 k Proof: The space k ℓ∞ (M, q) is symmetric is obvious. We prove it for k cR 0 (M, q). Let < an1 n2 ...nk >∈ k cR 0 (M, q). Then for a given ε > 0 there exists positive integers k1 , k2 , k3 , ..., kk , kk+1 such that    a M q n1 nρ2 ...nk < ε for all n1 > k1 for all n2 , n3 , ..., nk ∈ N    a < ε for all n2 > k2 for all n1 , n3 , ..., nk ∈ N M q n1 nρ2 ...nk .............................................. ..............................................    a M q n1 nρ2 ...nk < ε for all nk > kk for all n1 , n2 , n3 , ..., nk−1 ∈ N    a < ε for all n1 > kk+1 , n2 > kk+1 , ...., nk > kk+1 M q n1 nρ2 ...nk and without loss of generality ρ can be chosen to be same for the (k + 1) cases. Let k0 = max {k1 , k2 , k3 , ..., kk , kk+1 }. Let < bn1 n2 ...nk > be a rearrangement of < an1 n2 ...nk >. Then we have ai1 i2 ...ik = bni1 ni2 ni3 ...nik for all i1 , i2 , ..., ik ∈ N . Let

Vector Valued Multiple Sequence Spaces Defined by Orlicz Function 73

 kk+2 = max n11 , n21 , ..., nk1 , n(k+1)1 , n(k0 )1 , n12 , ...., nk2 , n(k+1)2 , n(k0 )2 , ..., n1k , n2k , ..., nnk , n(k+1)k , n(k0 )k . Then we have    b M q n1 nρ2 ...nk < ε for all n1 > kk+2 , n2 > kk+2 , ...., nk > kk+2 . Thus < bn1 n2 ...nk >∈

R k c0 (M, q).

Hence k cR 0 (M, q) is a symmetric space.

✷

To show that k cR (M, q) is not symmetric, we consider the following example. Example 3.4. Let X = C and define < an1 n2 ...nk > by an1 n2 ...nk = 1, for all n1 = 1 and for all n2 , n3 , ..., nk ∈ N . = 0, otherwise. Then < an1 n2 ...nk >∈

kc

R

(M, q)

Now consider the rearranged sequence < bn1 n2 ...nk > of < an1 n2 ...nk > defined by bn1 n2 ...nk = 1 for all n1 = n2 = ... = nk = 0, otherwise. Then < bn1 n2 ...nk >∈ /

kc

R

(M, q). Hence k cR (M, q) is not symmetric.

Examples similar to the above can be constructed to establish that the other spaces are also not symmetric. B Theorem 3.5. The spaces k cR 0 (M, q),k c0 (M, q),k c0 (M, q) and k ℓ∞ (M, q) are solid, B but the spaces k c(M, q), k c (M, q) and k cR (M, q) are not solid.

R Proof: The spaces Z(M, q) for Z = k ℓ∞ , k c0 , k cB 0 and k c0 , are solid follows from the following inequality.       a α a M q n1 n2 ...nkρ n1 n2 ...nk ≤ M q n1 nρ2 ...nk for all n1 , n2 , ..., nk ∈ N and scalars < αn1 n2 ...nk > with |αn1 n2 ...nk | ≤ 1, for all n1 , n2 , ..., nk ∈ N . ✷

To show that k c(M, q), k cB (M, q) and k cR (M, q) are not solid we consider the following example. Example 3.6. Let X = C, M (x) = x, q(x) = |x|. Define the sequence < an1 n2 ...nk > by an1 n2 ...nk = 1 for all n1 , n2 , ..., nk ∈ N . Consider the sequence

74

Binod Chandra Tripathy and Rupanjali Goswami

n1 +n2 +...+nk

< αn1 n2 ...nk > of scalars defined by αn1 n2 ...nk = (−1) ..., nk ∈ N .

for all n1 , n2 ,

Then < an1 n2 ...nk >∈ Z(M, q), but < αn1 n2 ...nk an1 n2 ...nk >∈ / Z(M, q) for Z = k c, k cB and k cR . Hence Z(M, q) is not solid for Z = k c, k cB , k cR . Theorem 3.7. The spaces Z(M, q) are monotone for Z = but are not monotone for Z = k c, k cB , k cR .

R B k c0 , k ℓ ∞ , k c0 , k c0 ,

Proof: The first part follows from the Remark 2.5 and Theorem 3.5. For the second part, we consider the following example. ✷ Example 3.8. Let X = C, M (x) = x, q(x) = |x|. Consider the sequence < an1 n2 ...nk > defined by an1 n2 ...nk = 1 for all n1 , n2 , ..., nk ∈ N . We consider its pre-images on the step space E defined by < bn1 n2 ...nk >∈ E, implies bn1 n2 ...nk = 0, for n2 , n3 , , ..., nk even and for all n1 ∈ N . Then the pre-image of < an1 n2 ...nk >∈ / Z(M, q), for Z =

k c, k c

B

, k cR .

Theorem 3.9. Let X be a complex semi-normed space, then the spaces Z(M, q) for R B B Z = k cR are complete semi-normed spaces semi-normed 0 , k ℓ∞ , k c , k c0 and k c f defined by (3.3).

Proof: We prove it for the space k ℓ∞ (M, q) and other cases can be established following similar technique. Let Ai =< ain1 n2 ...nk > be a Cauchy sequence in k ℓ∞ (M, q). Let ε > 0 be fixed
rx0 and r > 0, we choose x0 such that M 2 ≥ 1 and rx0 ≥ 1. Then there exists a positive integer m0 such that
f ain1 n2 ...nk − ajn2 ...nk
0 :

sup M n1 , n2 , ..., nk

    i an1 n2 ...nk − ajn1 n2 ...nk ε , ≤1 < q r rx0

(3.4)

for all i, j ≥ m0 and sup n1 n2 ...nk M

  i  a −aj q f ani1 n2 ...nk −anj1 n2 ...nk ≤ 1, for all i, j ≥ m0 . ( n1 n2 ...nk n1 n2 ...nk )

Vector Valued Multiple Sequence Spaces Defined by Orlicz Function 75

It follows that    i a −aj ≤ 1, for all i, j ≥ m0 . M q f ani1 n2 ...nk −anj1 n2 ...nk ( n1 n2 ...nk n1 n2 ...nk )
For r > 0 with M rx2 0 ≥ 1 we have    i
an1 n2 ...n −ajn1 n2 ...n k k ≤ M rx2 0 , for all i, j ≥ m0 . M q f ai ( n1 n2 ...nk −ajn1 n2 ...nk ) Since M is continuous, so we have   i
an1 n2 ...n −ajn1 n2 ...n k k ≤ rx2 0 , , for all i, j ≥ m0 . q f ai ( n1 n2 ...nk −ajn1 n2 ...nk )
 ε  ε
⇒ q ain1 n2 ...nk − ajn2 ,...,nk ≤ rx2 0 rx0 = 2 , for all i, j ≥ m0 . ⇒ < ain1 n2 ...nk > is a Cauchy sequence in X. Since X is complete, there exists an1 n2 ...nk ∈ X such that lim i i→∞ an1 n2 ...nk

= an1 n2 ...nk for all n1 , n2 , ...., nk ∈ N .

So we have from (3.4) for all i, j ≥ m0 ,      i an1 n2 ...n −ajn1 n2 ...n k k ≤ 1 < ε, inf r > 0 : n1 ,nsup M q r 2 ,..,nk      i an1 n2 ...n −ajn1 n2 ...n lim sup k k ≤ 1 < ε, for all i ≥ inf r > 0 : M q r j→∞ n1 ,n2 ,..,nk m0 . On taking the infimum of such r’s we have    i an1 n2 ...n −an1 n2 ...nk sup k ≤ 1} < ε, for all i ≥ m0 . inf{r > 0 : n1 ,n2 ,...,nk M q r ⇒< ain1 n2 ...nk − an1 n2 ...nk >∈

k ℓ∞ (M, q).

Since k ℓ∞ (M, q) is a linear space, we have for all i ≥ m0 < an1 n2 ...nk >=< ain1 n2 ...nk > − < ain1 n2 ...nk − an1 n2 ...nk >∈k ℓ∞ (M, q). Thus k ℓ∞ (M, q) is a complete semi-normed space.

✷

We state the following result without proof. Theorem 3.10. The spaces Z(M, q) for Z = K-spaces.

k ℓ∞ , k c

B

R , k cB and k cR 0 , kc 0 are

76

Binod Chandra Tripathy and Rupanjali Goswami

Since Z(M, q) ⊂ k ℓ∞ (M, q) for Z = result is a consequence of Theorem 3.9.

kc

B

Theorem 3.11. The spaces Z(M, q) for Z = dense subsets of k ℓ∞ (M, q).

R R , k cB 0 , k c and k c0 , so the following

kc

B

R R , k cB 0 , k c and k c0 are nowhere

Theorem 3.12. Let M1 and M2 be Orlicz functions. Then we have (i) Z(M1 , q) ⊆ Z(M2 ◦ M1 , q), for Z =

k ℓ∞ , k c, k c0 , k c

B

R R , k cB 0 , k c and k c0 . B

, k cB 0 ,

B

, k cB 0 ,

(ii) Z(M1 , q) ∩ Z(M2 , q) ⊆ Z(M1 + M2 , q), for Z = R R k c and k c0 .

k ℓ∞ , k c, k c0 , k c

(iii) Z(M1 , q1 ) ∩ Z(M1 , q) ⊆ Z(M1 , q1 + q2 ), for Z = R c and k cR k 0 , q1 , q2 are two semi-norms on X.

k ℓ∞ , k c, k c0 , k c

(iv) If q1 is stronger than q2 , then Z(M1 , q1 ) ⊆ Z(M1 , q2 ), for Z = B B R R c , k 0 k c , k c0 , k c and k c0 .

k ℓ∞ , k c,

Proof: (i) We prove this for the space k cR (M1 , q) and the other cases can be established in a similar way. Let < an1 n2 ...nk >∈ k cR (M1 , q). Then there exists ρ > 0 such that for a given ε > 0 with 0 < M2ε(1) < 1. We have n10 , n20 , n30 , ..., nk0 ∈ N such that

   an1 n2 ...nk − L ε < M1 q for all n1 > n10 , n2 > n20 , ...., nk > nk0 . ρ M2 (1) (3.5)    ε an1 n2 ...nk − Ln1 < for all n1 > n10 and for all n2 , n3 , ..., nk ∈ N. M1 q ρ M2 (1) (3.6)    an1 n2 ...nk − Ln2 ε M1 q < for all n2 > n20 and for all n1 , n3 , ..., nk ∈ N. ρ M2 (1) (3.7) ................................... ...................................

   an1 n2 ...nk − Lnk ε < , for all nk > nk0 and for all n1 , n2 , ..., nk−1 ∈ N, M1 q ρ M2 (1) (3.8)

Vector Valued Multiple Sequence Spaces Defined by Orlicz Function 77

Thus from Remark 1.1 and from (3.5),(3.6), (3.7) and (3.8) we have    −L a k (M2 ◦ M1 ) q n1 n2 ...n < ε for all n1 > n10 , n2 > n20 , ...., nk > nk0 . ρ    −Ln1 a k < ε for all n1 > n10 and for all n2 , n3 , ..., nk ∈ N (M2 ◦M1 ) q n1 n2 ...n ρ    −Ln2 a k (M2 ◦ M1 ) q n1 n2 ...n < ε for all n2 > n20 and for all n1 , n3 , ..., nk ∈ ρ N.

N.

...........................    −Lnk a k < ε for all nk > nk0 and for all n1 , n2 , ..., nk−1 ∈ (M2 ◦M1 ) q n1 n2 ...n ρ Hence < an1 n2 ...nk >∈ Thus k cR (M1 , q) ⊆

kc

kc

R

R

(M2 ◦ M1 , q).

(M2 ◦ M1 , q).

(ii) We prove the result for the case k ℓ∞ . Other cases will follow similarly. Let < an1 n2 ...nk >∈ k ℓ∞ (M1 , q) ∩ k ℓ∞ (M2 , q). Then there exists ρ1 > 0 and ρ2 > 0, such that    an1 n2 ...nk sup ∈

k ℓ∞ (M1

2

+ M2 , q).

(iii) Let < an1 n2 ...nk >∈ k ℓ∞ (M1 , q1 ) ∩ k ℓ∞ (M1 , q2 ). Then there exists ρ1 > 0 and ρ2 > 0 such that    an1 n2 ...nk sup ∈

2

k ℓ∞ (M1 , q1

+ q2 ).

✷

The following result is a consequence of Theorem 3.12(i). Proposition 3.13. Let M be an Orlicz function, then Z(q) ⊂ Z(M, q), for Z = k ℓ∞ , B B R R k c, k c0 , k c , k c0 , k c , k c0 . 4. Particular cases If we take X to be normed linear space, instead of a semi-normed space, then all the results of Section 3 will follow immediately. In that case spaces Z(M, ||.||),where R R Z = k ℓ ∞ , k cB , k cB 0 , k c , k c0 will be normed linear spaces, normed by n   o an1 n2 .....nk f (< an1 n2 .....nk >) = inf ρ > 0 : n1 ,nsup M k k ≤ 1 . ,...,n ρ 2 k These spaces would be Banach spaces under f when X is a Banach space. References 1. Altin, Y.; Et, M. and Tripathy, B.C. : The sequence space|Np | (M, r, q, s) on seminormed spaces; Appl. Math. Comput.; 154(2004), 423-430. 2. Basarir, M. and Sonalcan, O.: On some double sequence spaces: J. Ind. Acad. Math.; 21(2) (1999), 193-200. 3. Bromwich, T. J. IA : An introduction to the theory of Infinite Series; MacMillan & Co. Ltd. Newyork (1965). 4. Colak, R. and Turkmenoglu, A. : The double sequence spaces ℓ2∞ (p), c20 (p)and c2 (p), (to appear). 5. Hardy, G.H.: On the convergence of certain multiple series; Proc. Camb.Phil. Soc.; 19 (1917) 86-95. 6. Lindenstrauss, J. and Tzafriri, L.: On Orlicz sequence spaces: Israel J. Math.; 10(1971), 379-390. 7. Moricz, F.: Extension of the spaces c and c0 from single to double sequences: Acta. Math. Hung.; 57 (1-2), (1991), 129-136.

Vector Valued Multiple Sequence Spaces Defined by Orlicz Function 79

8. Moricz, F and Rhoades, B.E.: Almost convergence of double sequences and strong regularity of summability matrics; Math. Proc. Camb. Phil. Soc.; 104 (1988), 283-294. 9. Tripathy B.C.: Statistically convergent double sequence spaces; Tamkang J. Math.; 34(3) (2003), 231-237. 10. Tripathy, B. C.: Generalized difference paranormed statistically convergent sequences defined by Orlicz functions in a locally convex space; Soochow J. Math.; 30(4)(2004), 431-446. 11. Tripathy, B.C. and Mahanta, S. : On a class of generalized lacunary difference sequence spaces defined by Orlicz functions; Acta Math. Appl. Sinica (Eng. Ser.); 20(2)(2004),231-238. 12. Tripathy, B. C. and Sarma,B. : Vector valued double sequence spaces defined by Orlicz function; Math. Slovaca; 59(6)(2009), 767-776. 13. Tripathy, B. C. and Sarma, B. : Double sequence spaces of fuzzy numbers defined by Orlicz function; Acta Math. Sci., 31B(1)(2011), 134-140. 14. Turkmenoglu, A. : Matrix transformation between some classes of double sequences; Jour. Inst. Math. & Comp. Sci. (Math. Ser.) ; 12(1) (1999), 23-31.

Binod Chandra Tripathy Mathematical Science Division, Institute of Advanced Study in Science and Technology, Pachim Boragaon; Gorchuk. Guwahati-781035; India. E-mail address: [email protected], [email protected] and Rupanjali Goswami Department of Mathematics, Raha Higher Secondary School, Raha, Nagaon-782103, ASSAM. E-mail address: [email protected]