Some inclusion relations of vector valued sequence spaces defined by

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Dec 24, 2018 - Some inclusion relations of vector valued sequence spaces defined by musielak-φ-function. To cite this article: D A Harahap and E Herawati ...
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Some inclusion relations of vector valued sequence spaces defined by musielak-φ-function To cite this article: D A Harahap and E Herawati 2018 J. Phys.: Conf. Ser. 1116 022013

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SEMIRATA- International Conference on Science and Technology 2018 IOP Conf. Series: Journal of Physics: Conf. Series 1116 (2018) 022013

IOP Publishing doi:10.1088/1742-6596/1116/2/022013

Some inclusion relations of vector valued sequence spaces defined by musielak--function D A Harahap and E Herawati* Department. of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Sumatera Utara, Medan, Indonesia *

E-mail: [email protected]

Abstract. Let ℳ be a sequence of Musielak--function and  be a vector space. For  is the complementary ofℳ, in this paper, we constuct the new class ℎ∀ () andℓ∃ℳ (  ) and investigate the inclusion relations of its Köthe-Töeplitz dual.

1. Introduction and Preliminaries Let  be a vector space. The idea of  - valued sequence space was introduced by Pietcsh who investigated about nuclear spaces and summing operators [1]. Some authors have been studied the various type of - valued sequence spaces such as, Walsh [2], studied K-absolutely summing operators of a particular type of - valued sequence spaces. Another paper of him told about Banach lattices in [3]. Furthermore, Srivastava and Ghosh [4] studied about the properties of topology and inclusion relations of ℎ ( ), ℓ ( ), and ℓ  ′ spaces by using Örlicz function. Kolk [5] introduced -valued sequence spaces defined by - function that the generalization of Örlicz function and investigated some topologies of spaces equipped with seminorm. Using - function, Gultom and Herawati [6] studied some topological properties and inclusion relations of four -valued sequence spaces ℓ (),  (, ), (, ) and ∞ (, ) that the generalization of Parashar and Choudhary’s spaces [7]. The approach constructing the new sequence spaces defined by special function such that Musielak-Örlicz function and order -function have been employed by several authors, e.g., Mursaleen et al [8] and Herawati et al [9] respectively. They studied some topological properties , its Köthe-Töeplitz dual spaces and some basic geometries. A function : ℕ  ℝ → [0, ∞) defined  () for every  ∈ ℕ and  ∈ ℝ is called a Musielak-function, if for every ,  (∙) is even, vanishing at zero, continuous and nondecreasing on [0,∞). A sequence  = ( ) is the complementary of Musielak  − function ℳ = ( ) defined by  () = sup{|| −  ():  ≥ 0}

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1

SEMIRATA- International Conference on Science and Technology 2018 IOP Conf. Series: Journal of Physics: Conf. Series 1116 (2018) 022013

IOP Publishing doi:10.1088/1742-6596/1116/2/022013

For any normed linear space , we denote () as a collection of all linear operators from  to ℝ and subspace of () consisting of all order bounded operators by  ≡  () . Since ℝ is a complete we have  with the operator norm defined by ‖#‖ =sup{|#$ |: $ ∈ %()} where # ∈  and %() is a closed unit ball of . 2. The space &∀ (', *) and +∃(-, *. ) Let  be any normed linear space and  be any convex Musielak- function for every . The space of all  − valued sequences, denoted by Ω(), we define on Ω() the following convex modular 2 ∶ Ω() → [0,∞),

2 () = ∑6

78  (‖ ‖5 ).

In this section, we introduce the new type of  − valued sequence spaces are given as follows ? λ

ℎ∀ (, ) = 9 = ( ) ∈ Ω(), (∀; > 0) 2 < @ < ∞B,

(1)

and E η

ℓ∃ ℳ,  = 9# = (# ) ∈ Ω C , (∃D > 0) 2ℳ < @ < ∞B .

(2)

The space ℎ∀ (, ) become an AK- vector BK space under the luxemberg norm ? H

‖ ‖ = inf 9λ > 0 ∶ 2 < @ ≤ 1B.

3. Köthe-Töeplitz duals K and L of &∀ (', *) and +∃ (-, *. ) The Köthe-Töeplitz duals M and N of an - valued sequence space () denoted as [()]O and [()]P respectively, are given byas : 6

[()]O

= Q = ( ) ∶  ∈ , ∀ ≥ 1 and R ‖ S ‖ < ∞ , ∀ S = (ST ) ∈ ()U,

(3)

78 6

[()]P = Q = ( ) ∶  ∈ , ∀ ≥ 1 and WR  S W < ∞, ∀ S = (ST ) ∈ ()U.

(4)

78

The Köthe-Töeplitz (M-dual) and its generalized (N-dual) of the space ℎ∀ (, ) are denoted O

P

by Yℎ∀ (, )Z and Yℎ∀ (, )Z respectively, are given by as: ∞ ∀

O

Yℎ (, )Z = Q# = (# ): # ∈  ∶ R |# ( )| < ∞ , ∀ = ( ) ∈ ℎ∀ (, ) U ; 

(5)

78 ∞ P

Yℎ∀ (, )Z = Q# = (# ): # ∈  ∶ _R # ( )_ < ∞, ∀ = ( ) ∈ ℎ∀ (, ) U.

78

2

(6)

SEMIRATA- International Conference on Science and Technology 2018 IOP Conf. Series: Journal of Physics: Conf. Series 1116 (2018) 022013

by

IOP Publishing doi:10.1088/1742-6596/1116/2/022013

The Köthe-Töeplitz (M-dual) and its generalized (N-dual) of the space ℓ∃(ℳ,   ) are denoted and [ℓ∃(ℳ,  )]P respectively, are given by as:

[ℓ∃(ℳ,  )]O



[ℓ∃(ℳ,  )]O

= Q = ( ):  ∈  ∶ R |# ( )| < ∞ , ∀# = (# ) ∈ ℓ∃ ℳ,  U ;

(7)

78 ∞

[ℓ∃ (ℳ,  )]P = Q = ( ):  ∈  ∶ _R # ( )_ < ∞, ∀# = # ) ∈ ℓ∃ ℳ,  U.

(8)

78

Corollary 1. P

O ∗ = ℎ = ℎ = ℓ ℎ

4. Main Results This results containts 2 subsection. The first section introduce the new type of -valued sequence spaces and the second section we study their inclusion relations of its Köthe-Töeplitz duals. Theorem 2.1. Let the sequence ℳ and  be complementary Musielak--function , then O

ℎ∀ (, ) ⊂ Yℓ∃ ℳ,  Z . Proof. Suppose  = ( ) ∈ (ℎ∀ (, ). So, for every k > 0 such that (1)  q 0. Let us now consider ∞



R |# ( )| ≤ R v

78

78

‖# ‖5 w ‖ ‖5   xv x ≤ 2ℳ l q + 2 l q 0 such that # 2ℳ l q < ∞  If we take  = ( ) ∈ ℎ∀ (, ), we get 2
0.

Therefore

3

SEMIRATA- International Conference on Science and Technology 2018 IOP Conf. Series: Journal of Physics: Conf. Series 1116 (2018) 022013





R |# ( )| ≤ R v

78

78

IOP Publishing doi:10.1088/1742-6596/1116/2/022013

‖# ‖5 w ‖ ‖5 #  xv x ≤ 2ℳ l q + 2 l q