New Difference Sequence Spaces Defined by Musielak-Orlicz Function

Abstract

We introduce new sequence spaces by using Musielak-Orlicz function and a generalized 𝐵^𝜇_∧-difference operator on 𝑛-normed space. Some topological properties and inclusion relations are also examined.

1. Introduction and Preliminaries

The notion of the difference sequence space was introduced by Kızmaz [1]. It was further generalized by Et and Çolak [2] as follows: 𝑍(Δ^𝜇)={𝑥=(𝑥_𝑘)∈𝜔∶(Δ^𝜇𝑥_𝑘)∈𝑧} for 𝑧=_∞,𝑐, and 𝑐₀, where 𝜇 is a nonnegative integer and Δ^𝜇𝑥_𝑘=Δ^𝜇−1𝑥_𝑘−Δ^𝜇−1𝑥_𝑘+1,Δ⁰𝑥_𝑘=𝑥_𝑘∀𝑘∈ℕ (1) or equivalent to the following binomial representation: Δ^𝜇𝑥_𝑘=^𝜇_𝑣=0(−1)^𝑣𝜇𝑣𝑥_𝑘+𝑣. (2) These sequence spaces were generalized by Et and Basarir [3] taking 𝑧=_∞(𝑝), 𝑐(𝑝), and 𝑐₀(𝑝).

Dutta [4] introduced the following difference sequence spaces using a new difference operator: 𝑍Δ_(𝜂)=𝑥=𝑥_𝑘∈𝜔∶Δ_(𝜂)𝑥∈𝑧for𝑧=_∞,𝑐,𝑐₀, (3) where Δ_(𝜂)𝑥=(Δ_(𝜂)𝑥_𝑘)=(𝑥_𝑘−𝑥_𝑘−𝜂) for all 𝑘,𝜂∈ℕ.

In [5], Dutta introduced the sequence spaces 𝑐(‖⋅,⋅‖,Δ^𝜇_(𝜂),𝑝), 𝑐₀(‖⋅,⋅‖,Δ^𝜇_(𝜂),𝑝), _∞(‖⋅,⋅‖,Δ^𝜇_(𝜂),𝑝), 𝑚(‖⋅,⋅‖,Δ^𝜇_(𝜂),𝑝), and 𝑚₀(‖⋅,⋅‖,Δ^𝜇_(𝜂),𝑝), where 𝜂,𝜇∈ℕ and Δ^𝜇_(𝜂)𝑥_𝑘=(Δ^𝜇_(𝜂)𝑥_𝑘)=(Δ^𝜇−1_(𝜂)𝑥_𝑘−Δ^𝜇−1_(𝜂)𝑥_𝑘−𝜂) and Δ⁰_(𝜂)𝑥_𝑘=𝑥_𝑘 for all 𝑘,𝜂∈ℕ, which is equivalent to the following binomial representation: Δ^𝜇_(𝜂)𝑥_𝑘=^𝜇_𝑣=0(−1)^𝑣𝜇𝑣𝑥_{𝑘−𝜂𝑣}. (4) The difference sequence spaces have been studied by authors [6–14] and references therein. Başar and Altay [15] introduced the generalized difference matrix 𝐵=(𝑏_𝑚𝑘) for all 𝑘,𝑚∈ℕ, which is a generalization of Δ₍₁₎-difference operator by 𝑏_𝑚𝑘=()().𝑟,𝑘=𝑚𝑠,𝑘=𝑚−10,𝑘>𝑚or0≤𝑘<𝑚−1 (5)

Başarir and Kayikçi [16] defined the matrix 𝐵^𝜇(𝑏^𝜇_𝑚𝑘) which reduced the difference matrix Δ^𝜇₍₁₎ in case 𝑟=1, 𝑠=−1. The generalized 𝐵^𝜇-difference operator is equivalent to the following binomial representation: 𝐵^𝜇𝑥=𝐵^𝜇𝑥_𝑘=^𝜇_𝑣=0𝜇𝑣𝑟^𝜇−𝑣𝑠^𝑣𝑥_𝑘−𝑣. (6)

Let ∧=(∧_𝑘) be a sequence of nonzero scalars. Then, for a sequence space 𝐸, the multiplier sequence space 𝐸_∧, associated with the multiplier sequence ∧, is defined as 𝐸_∧=𝑥=𝑥_𝑘∈𝜔∶∧_𝑘𝑥_𝑘∈𝐸. (7) An Orlicz function 𝑀 is a function, 𝑀∶[0,∞)→[0,∞), which is continuous, nondecreasing, and convex with 𝑀(0)=0, 𝑀(𝑥)>0 for 𝑥>0, and 𝑀(𝑥)→∞ as 𝑥→∞.

We say that an Orlicz function 𝑀 satsfies the Δ₂-condition if there exists 𝐾>2 and 𝑥₀≥0 such that 𝑀(2𝑥)≤𝐾𝑀(𝑥) for all 𝑥≥𝑥₀. The Δ₂-condition is equivalent to 𝑀(𝐿𝑥)≤𝐾𝐿𝑀(𝑥) for all 𝑥>𝑥₀>0 and for 𝐿,𝐾>1.

Lindenstrauss and Tzafriri [17] used the idea of Orlicz function to define the following sequence space: _𝑀=𝑥∈𝜔∶^∞_𝑘=1𝑀𝑥_𝑘𝜌<∞ (8) which is called an Orlicz sequence space. The space 𝑀 is a Banach space with the norm ‖𝑥‖=inf𝜌>0∶^∞_𝑘=1𝑀𝑥_𝑘𝜌≤1. (9) It is shown in [17] that every Orlicz sequence space 𝑀 contains a subspace isomorphic to _𝑝(𝑝≥1).

A sequence ℳ=(𝑀_𝑘) of Orlicz function is called a Musielak-Orlicz function; see [18, 19]. A sequence 𝒩=(𝑁_𝑘) defined by 𝑁_𝑘(𝑣)=sup|𝑣|𝑢−𝑀_𝑘(𝑢)∶𝑢≥0,𝑘=1,2,…, (10) is called the complimentary function of a Musielak-Orlicz function ℳ. For a given Musielak-Orlicz function ℳ, the Musielak-Orlicz sequence space 𝑡_ℳ and its subspace ℎ_ℳ are defined as follows: 𝑡_ℳ=𝑥∈𝜔∶𝐼_𝑀(),ℎ𝑐𝑥<∞forsome𝑐>0_ℳ=𝑥∈𝜔∶𝐼_𝑀(𝑐𝑥)<∞∀𝑐>0, (11) where 𝐼_ℳ is a convex modular defined by 𝐼_ℳ(𝑥)=^∞_𝑘=1𝑀_𝑘𝑥_𝑘,𝑥=𝑥_𝑘∈𝑡_𝑀. (12) We consider 𝑡_ℳ equipped with the Luxemburg norm ‖𝑥‖=inf𝑘>0∶𝐼_𝑀𝑥𝑘≤1 (13) or equipped with the Orlicz norm ‖𝑥‖⁰1=inf𝑘1+𝐼_𝑀(𝑘𝑥)∶𝑘>0. (14) By a lacunary sequence 𝜃=(𝑖_𝑟), 𝑟=0,1,2,…, where 𝑖₀=0, we will mean an increasing sequence of nonnegative integers ℎ_𝑟=(𝑖_𝑟−𝑟_𝑟−1)→∞(𝑟→∞). The intervals determined by 𝜃 are denoted by 𝐼_𝑟=(𝑖_𝑟−1,𝑖_𝑟] and the ratio 𝑖_𝑟/𝑖_𝑟−1 will be denoted by 𝑞_𝑟. The space of lacunary strongly convergent sequences 𝑁_𝜃 was defined by Freedman et al. [20] as follows: 𝑁_𝜃=𝑥=𝑥_𝑘∶lim_𝑟→∞1ℎ_{𝑟𝑘∈𝐼𝑟}𝑥_𝑘−𝐿=0,forsome𝐿. (15) The concept of 2-normed spaces was initially developed by Gähler [21] in the mid of 1960’s, while that of 𝑛-normed spaces one can see in Misiak [22]. Since then, many others have studied this concept and obtained various results; see Gunawan [23, 24] and Gunawan and Mashadi [25]. For more details about sequence spaces see [26–33] and references therein. Let 𝑛∈ℕ and 𝑋 be linear space over the field 𝕂, where 𝕂 is the 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,…,𝑥_𝑛)‖=0 if and only if 𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑛 are linearly dependent in 𝑋;(2)‖(𝑥₁,𝑥₂,…,𝑥_𝑛)‖ is invariant under permutation;(3)‖(𝛼𝑥₁,𝑥₂,…,𝑥_𝑛)‖=|𝛼|‖(𝑥₁,𝑥₂,…,𝑥_𝑛)‖ for any 𝛼∈𝕂;(4)‖(𝑥+𝑥^′,𝑥₂,…,𝑥_𝑛)‖≤‖(𝑥,𝑥₂,…,𝑥_𝑛)‖+‖(𝑥^′,𝑥₂,…,𝑥_𝑛)‖is called an 𝑛-norm on 𝑋 and the pair (𝑋,‖⋅,…,⋅‖) is called an 𝑛-normed space over the field 𝕂. For example, we may take 𝑋=ℝ^𝑛 being equipped with the Euclidean 𝑛-norm ‖(𝑥₁,𝑥₂,…,𝑥_𝑛)‖_𝐸= the volume of the 𝑛-dimensional parallelepiped spanned by the vectors 𝑥₁,𝑥₂,…,𝑥_𝑛 which may be given explicitly by the formula 𝑥₁,𝑥₂,…,𝑥_𝑛𝐸=det𝑥_𝑖𝑗, (16) where 𝑥_𝑖=(𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑛)∈ℝ^𝑛 for each 𝑖=1,2,3,…,𝑛 and ‖⋅‖_𝐸 denotes the Euclidean norm. Let (𝑋,‖⋅,…,⋅‖) be an 𝑛-normed space of dimension 𝑑≥𝑛≥2 and {𝑎₁,𝑎₂,…,𝑎_𝑛} linearly independent set in 𝑋. Then the following function ‖(⋅,…,⋅)‖_∞ on 𝑋^𝑛−1 defined by 𝑥₁,𝑥₂,…,𝑥_𝑛∞=max𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑎_𝑖∶𝑖=1,2,…,𝑛 (17) defines an (𝑛−1) norm on 𝑋 with respect to {(𝑎₁,𝑎₂,…,𝑎_𝑛)}.

A sequence (𝑥_𝑘) in an 𝑛-normed space (𝑋,‖⋅,…,⋅‖) is said to converge to some 𝐿∈𝑋 if lim_𝑘→∞𝑥_𝑘−𝐿,𝑧₁,…,𝑧_𝑛−1=0,forevery𝑧₁,…,𝑧_𝑛−1∈𝑋. (18) A sequence (𝑥_𝑘) in a normed space (𝑋,‖⋅,…,⋅‖) is said to be Cauchy if lim_{𝑘→∞𝑝→∞}𝑥_𝑘−𝑥_𝑝,𝑧₁,…,𝑧_𝑛−1=0,forevery𝑧₁,…,𝑧_𝑛−1∈𝑋. (19) 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.

Let (𝑋,‖⋅,…,⋅‖) be an 𝑛-normed space and let 𝑠(𝜔−𝑥) denote the space of 𝑋-valued sequences. Let 𝑝=(𝑝_𝑘) be any bounded sequence of positive real numbers and ℳ=(𝑀_𝑘) a Musielak-Orlicz function. We define the following sequence spaces in this paper: 𝑤^𝜃₀ℳ,𝐵^𝜇_∧‖‖=𝑥,𝑝,⋅,…,⋅𝑥=_𝑘∈𝑠(𝑤−𝑥)∶lim_𝑟→∞1ℎ_𝑟×_{𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘,𝑤=0,𝜌>0_𝜃ℳ,𝐵^𝜇_∧‖‖=𝑥,𝑝,⋅,…,⋅𝑥=_𝑘∈𝑠(𝑤−𝑥)∶lim_𝑟→∞1ℎ_𝑟×_{𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘−𝐿𝜌,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘,𝑤=0,forsome𝐿,𝜌>0_𝜃∞ℳ,𝐵^𝜇_∧‖‖=𝑥,𝑝,⋅,…,⋅𝑥=_𝑘∈𝑠(𝑤−𝑥)∶lim_𝑟→∞1ℎ_𝑟×_{𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘<∞,𝜌>0; (20) when ℳ(𝑥)=𝑥, we get 𝑤^𝜃₀𝐵^𝜇_∧‖‖=𝑥,𝑝,⋅,…,⋅𝑥=_𝑘∈𝑠(𝑤−𝑥)∶lim_𝑟→∞1ℎ_𝑟×_{𝑘∈𝐼𝑟}𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘,𝑤=0,𝜌>0_𝜃𝐵^𝜇_∧‖‖=𝑥,𝑝,⋅,…,⋅𝑥=_𝑘∈𝑠(𝑤−𝑥)∶lim_𝑟→∞1ℎ_𝑟×_{𝑘∈𝐼𝑟}𝐵^𝜇_∧𝑥_𝑘−𝐿𝜌,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘,𝑤=0,forsome𝐿,𝜌>0_𝜃∞𝐵^𝜇_∧‖‖=𝑥,𝑝,⋅,…,⋅𝑥=_𝑘∈𝑠(𝑤−𝑥)∶lim_𝑟→∞1ℎ_𝑟×_{𝑘∈𝐼𝑟}𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘<∞,𝜌>0; (21) when 𝑝_𝑘=1, for all 𝑘, we get 𝑤^𝜃₀ℳ,𝐵^𝜇_∧,‖‖=𝑥⋅,…,⋅𝑥=_𝑘∈𝑤(𝑠−𝑥)∶lim_𝑟→∞1ℎ_𝑟×_{𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1,𝑤=0,𝜌>0_𝜃ℳ,𝐵^𝜇_∧,‖‖=𝑥⋅,…,⋅𝑥=_𝑘∈𝑤(𝑠−𝑥)∶lim_𝑟→∞1ℎ_𝑟×_{𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘−𝐿𝜌,𝑧₁,…,𝑧_𝑛−1,𝑤=0,forsome𝐿,𝜌>0_𝜃∞ℳ,𝐵^𝜇_∧,‖‖=𝑥⋅,…,⋅𝑥=_𝑘∈𝑤(𝑠−𝑥)∶lim_𝑟→∞1ℎ_𝑟×_{𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1<∞,𝜌>0. (22) 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+𝑏_𝑘^𝑝𝑘≤𝐾𝑎_𝑘|^𝑝𝑘+𝑏_𝑘|^𝑝𝑘 (23) for all 𝑘 and 𝑎_𝑘,𝑏_𝑘∈ℂ. Also |𝑎|^𝑝𝑘≤max(1,|𝑎|^𝐻) for all 𝑎∈ℂ.

2. Main Results

Theorem 1. Let ℳ=(𝑀_𝑘) be a Musielak-Orlicz function and 𝑝=(𝑝_𝑘) a bounded sequence of positive real numbers; the spaces 𝑤^𝜃₀(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖), 𝑤^𝜃(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖), and 𝑤^𝜃_∞(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖) are linear over the field of complex numbers ℂ.

Proof. Let 𝑥=(𝑥_𝑘), 𝑦=(𝑦_𝑘)∈𝑤^𝜃₀(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖), and 𝛼,𝛽∈ℂ. Then there exist positive real numbers 𝜌₁ and 𝜌₂ such that lim_𝑟→∞1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘𝜌₁,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘=0,lim_𝑟→∞1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑦_𝑘𝜌₂,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘=0. (24) Define 𝜌₃=max(2|𝛼|𝜌₁,2|𝛽|𝜌₂). Since ‖⋅,…,⋅‖ is an 𝑛-norm on 𝑋 and 𝑀^′_𝑘𝑠 are nondecreasing and convex functions so by using inequality (23) we have lim_𝑟→∞1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝛼𝑥_𝑘+𝛽𝑦_𝑘𝜌₃,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘≤lim_𝑟→∞1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝛼𝑥_𝑘𝜌₃,𝑧₁,…,𝑧_𝑛−1+𝐵^𝜇_∧𝛽𝑦_𝑘𝜌₃,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘≤𝐾lim_𝑟→∞1ℎ_{𝑟𝑘∈𝐼𝑟}12^𝑝𝑘𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘𝜌₁,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘+𝐾lim_𝑟→∞1ℎ_{𝑟𝑘∈𝐼𝑟}12^𝑝𝑘𝑀_𝑘𝐵^𝜇_∧𝑦_𝑘𝜌₂,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘=0. (25) Thus, we have 𝛼𝑥+𝛽𝑦∈𝑤^𝜃₀(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖). Hence 𝑤^𝜃₀(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖) is a linear space. Similarly, we can prove that 𝑤^𝜃(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖) and 𝑤^𝜃_∞(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖) are linear spaces. This completes the proof of the theorem.

Theorem 2. Let ℳ=(𝑀_𝑘) be a Musielak-Orlicz function and 𝑝=(𝑝_𝑘) a bounded sequence of positive real numbers; the space 𝑤^𝜃₀(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖) is a topological linear space paranormed by 𝑔(𝑥)=inf𝜌^{𝑝𝑟/𝑀}∶1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1^{𝑝𝑘1/𝑀}≤1, (26) where 𝑀=max(1,sup_𝑘𝑝_𝑘<∞).

Proof. Clearly 𝑔(𝑥)≥0 for 𝑥=(𝑥_𝑘)∈𝑤^𝜃₀(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖). Since 𝑀_𝑘(0)=0, we get 𝑔(0)=0. Again, if 𝑔(𝑥)=0, then 𝑔(𝑥)=inf𝜌^{𝑝𝑟/𝑀}∶1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1^{𝑝𝑘1/𝑀}≤1=0. (27) This implies that, for a given 𝜀>0, there exist some 𝜌_𝜀(0<𝜌_𝜀<𝜀) such that 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘𝜌_𝜀,𝑧₁,…,𝑧_𝑛−1^{𝑝𝑘1/𝑀}≤1. (28)
Thus 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘𝜀,𝑧₁,…,𝑧_𝑛−1^{𝑝𝑘1/𝑀}≤1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘𝜌_𝜀,𝑧₁,…,𝑧_𝑛−1^{𝑝𝑘1/𝑀}≤1 (29) for each 𝑟, and suppose that 𝑥_𝑘≠0 for each 𝑘∈ℕ. This implies that 𝐵^𝜇_∧𝑥_𝑘≠0 for each 𝑘∈ℕ. Let 𝜀→0, then 𝐵^𝜇_∧𝑥_𝑘𝜌_𝜀,𝑧₁,…,𝑧_𝑛−1^{𝑝𝑘1/𝑀}⟶∞; (30) which is a contradiction. Therefore, 𝐵^𝜇_∧𝑥_𝑘=0 for each 𝑘 and thus 𝑥_𝑘=0 for each 𝑘∈ℕ. Let 𝜌₁>0 and 𝜌₂>0 be such that 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘𝜌₁,𝑧₁,…,𝑧_𝑛−1^{𝑝𝑘1/𝑀}1≤1,ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑦_𝑘𝜌₂,𝑧₁,…,𝑧_𝑛−1^{𝑝𝑘1/𝑀}≤1foreach𝑟. (31) Let 𝜌=𝜌₁+𝜌₂; then by using Minkowski’s inequality, we have 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘+𝑦_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1^{𝑝𝑘1/𝑀}≤1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘+𝐵^𝜇_∧𝑦_𝑘𝜌₁+𝜌₂,𝑧₁,…,𝑧_𝑛−1^{𝑝𝑘1/𝑀}≤1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘×𝜌₁𝜌₁+𝜌₂𝐵^𝜇_∧𝑥_𝑘𝜌₁,𝑧₁,…,𝑧_𝑛−1+𝜌₂𝜌₁+𝜌₂𝐵^𝜇_∧𝑦_𝑘𝜌₂,𝑧₁,…,𝑧_𝑛−1^{𝑝𝑘1/𝑀}≤𝜌₁𝜌₁+𝜌₂×1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘𝜌₁,𝑧₁,…,𝑧_𝑛−1^{𝑝𝑘1/𝑀}+𝜌₂𝜌₁+𝜌₂×1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑦_𝑘𝜌₂,𝑧₁,…,𝑧_𝑛−1^{𝑝𝑘1/𝑀}≤1. (32)
Since 𝜌^′𝑠 are nonnegative, we have 𝑔𝜌𝑥+𝑦=inf^{𝑝𝑟/𝑀}∶1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘×𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1^{𝑝𝑘1/𝑀}𝜌≤1≤inf₁^{𝑝𝑟/𝑀}∶1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘×𝐵^𝜇_∧𝑥_𝑘𝜌₁,𝑧₁,…,𝑧_𝑛−1^{𝑝𝑘1/𝑀}𝜌≤1+inf₂^{𝑝𝑟/𝑀}∶1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘×𝐵^𝜇_∧𝑦_𝑘𝜌₂,𝑧₁,…,𝑧_𝑛−1^{𝑝𝑘1/𝑀}≤1. (33) Therefore, 𝑔(𝑥+𝑦)≤𝑔(𝑥)+𝑔(𝑦).
Finally, we prove that the scalar multiplication is continuous. Let 𝜈 be any complex number. By definition, 𝑔()𝜌𝜈𝑥=inf^{𝑝𝑟/𝑀}∶1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘×𝜈𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1^{𝑝𝑘1/𝑀}≤1. (34) Then 𝑔()𝜈|𝑡)𝜈𝑥=inf(|^{𝑝𝑟/𝑀}∶1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘×𝐵^𝜇_∧𝑥_𝑘𝑡,𝑧₁,…,𝑧_𝑛−1^{𝑝𝑘1/𝑀}≤1, (35) where 𝑡=𝜌/|𝜈|. Since |𝜈|^𝑝𝑟≤max(1,|𝜈|sup𝑝_𝑘), we have 𝑔(𝜈𝑥)=max1,|𝜈|sup𝑝_𝑘×(𝑡)inf_{𝑝𝑟/𝑀}∶1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘×𝐵^𝜇_∧𝑥_𝑘𝑡,𝑧₁,…,𝑧_𝑛−1^{𝑝𝑘1/𝑀}≤1. (36) So, the fact that scalar multiplication is continuous follows from the above inequality. This completes the proof of theorem.

Theorem 3. Let ℳ=(𝑀_𝑘) be a Musielak-Orlicz function. If sup_𝑘(𝑀_𝑘(𝑥))^𝑝𝑘<∞ for all fixed 𝑥>0, then 𝑤^𝜃(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖)⊂𝑤^𝜃_∞(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖).

Proof. Let 𝑥=(𝑥_𝑘)∈𝑤^𝜃(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖). Then there exists some positive number 𝜌₁ such that lim_𝑟→∞1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘−𝐿𝜌₁,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘=0. (37) Define 𝜌=2𝜌₁. Since 𝑀_𝑘 is nondecreasing and convex and by using inequality (23), we have lim_𝑟→∞1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘=lim_𝑟→∞1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘−𝐿+𝐿𝜌,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘≤𝐾lim_𝑟→∞1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘12^𝑝𝑘𝐵^𝜇_∧𝑥_𝑘−𝐿𝜌₁,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘+lim_𝑟→∞1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘12^𝑝𝑘𝐿𝜌₁,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘<𝐾lim_𝑟→∞1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘−𝐿𝜌₁,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘+lim_𝑟→∞1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐿𝜌₁,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘. (38) Hence 𝑥=(𝑥_𝑘)∈𝑤^𝜃_∞(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖). This completes the proof of the theorem.

Theorem 4. Let ℳ=(𝑀_𝑘) be a Musielak-Orlicz function and 0<ℎ=inf𝑝_𝑘. Then 𝑤^𝜃_∞ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖⊂𝑤^𝜃₀ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖ (39) if and only if lim_𝑟→∞1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘(𝑡)^𝑝𝑘=∞,𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝑡>0. (40)

Proof. Let 𝑤^𝜃_∞(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖)⊂𝑤^𝜃₀(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖). Suppose (40) does not hold. Therefore there are a subinterval 𝐼_𝑟(𝑚) of the set of intervals 𝐼_𝑟 and a number 𝑛₀, where 𝑛₀=‖(𝐵^𝜇_∧𝑥_𝑘/𝜌,𝑧₁,…,𝑧_𝑛−1)‖ for all 𝑘, such that 1ℎ_𝑟(𝑚)_{𝑘∈𝐼𝑟(𝑚)}𝑀𝑛₀^𝑝𝑘≤𝑁<∞,𝑚=1,2,3,…. (41) Let us define 𝑥=(𝑥_𝑘) as follows: 𝐵^𝜇_∧𝑥_𝑘=𝜌𝑛₀,𝑘∈𝐼_𝑟(𝑚)0,𝑘∉𝐼_𝑟(𝑚). (42) Thus by (41), 𝑥=(𝑥_𝑘)∈𝑤^𝜃_∞(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖). But 𝑥=(𝑥_𝑘)∉𝑤^𝜃₀(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖). Hence (40) must hold.
Conversely, suppose that (40) holds and let 𝑥=(𝑥_𝑘)∈𝑤^𝜃_∞(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖). Then, 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘≤𝑁<∞. (43) Suppose that 𝑥=(𝑥_𝑘)∉𝑤^𝜃₀(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖). Then for some number 𝜀, 1>𝜀>0, there is a number 𝑁₀ such that, for a subinterval 𝐼_𝑟(𝑚) of the set of intervals 𝐼_𝑟, 𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1>𝜀for𝑁≥𝑁₀. (44) We have 𝑀_𝑘(‖(𝐵^𝜇_∧𝑥_𝑘/𝜌,𝑧₁,…,𝑧_𝑛−1)‖)≥𝑀(𝜀)^𝑝𝑘, which contradicts (40) by using (43). Hence we get 𝑤^𝜃_∞ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖⊂𝑤^𝜃₀ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖. (45) This completes the proof.

Theorem 5. Let 0<ℎ=inf𝑝_𝑘≤sup𝑝_𝑘=𝐻<∞. For any Musielak-Orlicz function ℳ=(𝑀_𝑘) which satisfies Δ₂-condition, one has(i)𝑤^𝜃₀(𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖)⊂𝑤^𝜃₀(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖)(ii)𝑤^𝜃(𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖)⊂𝑤^𝜃(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖)(iii)𝑤^𝜃_∞(𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖)⊂𝑤^𝜃_∞(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖).

Proof. (i) Let 𝑥=(𝑥_𝑘)∈𝑤^𝜃₀(𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖). Then, we have 1ℎ_{𝑟𝑘∈𝐼𝑟}𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘⟶0as𝑟⟶∞. (46) Let 𝜀>0, and choose 𝛿 with 0<𝛿<1 such that 𝑀_𝑘<𝜀 for 0≤𝑡≤𝛿. We can write 1ℎ_{𝑟𝑘∈𝐼𝑟}𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘=1ℎ_{𝑟𝑘∈𝐼𝑟‖(𝐵𝜇∧𝑥𝑘/𝜌,𝑧1,…,𝑧𝑛−1)‖≤𝛿}𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘+1ℎ_{𝑟𝑘∈𝐼𝑟‖(𝐵𝜇∧𝑥𝑘/𝜌,𝑧1,…,𝑧𝑛−1)‖>𝛿}𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘. (47) For the first summation above, we can write 1ℎ_{𝑟𝑘∈𝐼𝑟‖𝐵𝜇∧𝑥𝑘/𝜌,𝑧1,…,𝑧𝑛−1‖≤𝛿}𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘<max𝜀,𝜀^ℎ. (48) By using continuity of 𝑀_𝑘, for the second summation we can write 𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1𝐵<1+^𝜇∧𝑥_𝑘/𝜌,𝑧₁,…,𝑧_𝑛−1𝛿. (49) Since each 𝑀_𝑘 is nondecreasing and convex and satisfies Δ₂-condition, it follows that 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘≤max𝜀,𝜀^ℎ+max1,2𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘/𝜌,𝑧₁,…,𝑧_𝑛−1𝛿^ℎ×1ℎ_{𝑟𝑘∈𝐼𝑟}𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘. (50) Taking limit as 𝜀→0 and 𝑟→∞, it follows that 𝑥=(𝑥_𝑘)∈𝑤^𝜃₀(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖). Hence 𝑤^𝜃₀(𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖)⊂𝑤^𝜃₀(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖). Similarly, we can prove (ii) and (iii). This completes the proof of the theorem.

Theorem 6. Let ℳ=(𝑀_𝑘) be a Musielak-Orlicz function. Then the following statements are equivalent: (i)𝑤^𝜃_∞(𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖)⊂𝑤^𝜃_∞(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖); (ii)𝑤^𝜃₀(𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖)⊂𝑤^𝜃_∞(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖); (iii)sup_𝑟(1/ℎ_𝑟)∑_{𝑘∈𝐼𝑟}𝑀_𝑘(𝑡)^𝑝𝑘<∞ for all 𝑡>0, where 𝑡=‖𝐵^𝜇_∧𝑥_𝑘/𝜌,𝑧₁,…,𝑧_𝑛−1‖.

Proof. (i) ⇒ (ii) Suppose (i) holds. In order to prove (ii) we have to show that 𝑤^𝜃𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖⊂𝑤^𝜃_∞ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖. (51) Let 𝑥=(𝑥_𝑘)∈𝑤^𝜃₀(𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖). Then for a given 𝜀>0 there exists 𝑠>𝑠_𝜀 such that 1ℎ_{𝑟𝑘∈𝐼𝑟}𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘<𝜀. (52) Hence there exists 𝐾>0 such that sup_𝑟1ℎ_{𝑟𝑘∈𝐼𝑟}𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘<𝐾. (53) This shows that 𝑥=(𝑥_𝑘)∈𝑤^𝜃_∞(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖).
(ii) ⇒ (iii) Suppose (ii) holds and (iii) fails to hold. Then for some 𝑡>0, sup_𝑟1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘(𝜀)^𝑝𝑘=∞, (54) and, therefore, we can find a subinterval 𝐼_𝑟(𝑚) of the set of intervals 𝐼_𝑟 such that 1ℎ_𝑟(𝑚)_{𝑘∈𝐼𝑟(𝑚)}𝑀_𝑘1𝑚^𝑝𝑘≥𝑚,𝑚=1,2,3,…. (55) Let us define 𝑥=(𝑥_𝑘) as follows: 𝐵^𝜇_∧𝑥_𝑘=𝜌𝑚,𝑘∈𝐼_𝑟(𝑚)0,𝑘∉𝐼_𝑟(𝑚). (56) Thus 𝑥=(𝑥_𝑘)∈𝑤^𝜃₀(𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖). But by (55), 𝑥=(𝑥_𝑘)∉𝑤^𝜃_∞(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖) which contradicts (ii). Hence (iii) must hold.
(iii) ⇒ (i) Let (iii) hold. Suppose that 𝑥=(𝑥_𝑘)∉𝑤^𝜃_∞(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖). Then for 𝑥=(𝑥_𝑘)∈𝑤^𝜃_∞(𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖). sup_𝑟1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘=∞. (57) Let 𝑡=‖𝐵^𝜇_∧𝑥_𝑘/𝜌,𝑧₁,…,𝑧_𝑛−1‖ for each 𝑘, and then by (57) sup_𝑟(1/ℎ_𝑟)∑_{𝑘∈𝐼𝑟}𝑀_𝑘(𝑡)^𝑝𝑘=∞, which contradicts (iii). Hence (i) must hold. This completes the proof of the theorem.

Theorem 7. Let ℳ=(𝑀_𝑘) be a Musielak-Orlicz function. Then the following statements are equivalent: (i)𝑤^𝜃₀(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖)⊂𝑤^𝜃₀(𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖); (ii)𝑤^𝜃₀(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖)⊂𝑤^𝜃_∞(𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖); (iii)inf_𝑟(1/ℎ_𝑟)∑_{𝑘∈𝐼𝑟}𝑀_𝑘(𝑡)^𝑝𝑘>0 for all 𝑡>0.

Proof. (i) ⇒ (ii) is obvious
(ii) ⇒ (iii) Let (ii) hold and let (iii) fail to hold. Then inf_𝑟1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘(𝑡)^𝑝𝑘=0forsome𝑡>0, (58) and we can find a subinterval 𝐼_𝑟(𝑚) of the set of intervals 𝐼_𝑟 such that 1ℎ_𝑟(𝑚)_{𝑘∈𝐼𝑟(𝑚)}𝑀_𝑘(𝑚)^𝑝𝑘<1𝑚,𝑚=1,2,3,…. (59) Let us define 𝑥=(𝑥_𝑘) as follows: 𝐵^𝜇_∧𝑥_𝑘=𝜌𝑚,𝑘∈𝐼_𝑟(𝑚)0,𝑘∉𝐼_𝑟(𝑚). (60) Thus by (iii), 𝑥=(𝑥_𝑘)∈𝑤^𝜃₀(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖). But 𝑥=(𝑥_𝑘)∉𝑤^𝜃_∞(𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖) which contradict (ii). Hence (iii) must hold.
(iii) ⇒ (i) Let (iii) hold. Suppose that 𝑥=(𝑥_𝑘)∈𝑤^𝜃₀(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖). Therefore, 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘⟶0as𝑟⟶∞. (61) Again suppose 𝑥=(𝑥_𝑘)∉𝑤^𝜃₀(𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖) for some number 𝜀>0 and a subinterval 𝐼_𝑟(𝑚) of the set of intervals 𝐼_𝑟, we have 𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1≥𝜀∀𝑘. (62) Then, from properties of the Orlicz function, we can write 𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘𝜌,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘≥𝑀_𝑘(𝜀)^𝑝𝑘. (63) Consequently, by (61), we have lim_𝑟→∞(1/ℎ_𝑟)∑_{𝑘∈𝐼𝑟}𝑀_𝑘(𝜀)^𝑝𝑘=0, which contradicts (iii). Hence (i) must hold. This completes the proof of the theorem.

Theorem 8. (i) If 0<inf𝑝_𝑘≤𝑝_𝑘≤1 for all 𝑘, then 𝑤^𝜃(ℳ,𝐵^𝜇_∧,‖⋅,…,⋅‖)⊆𝑤^𝜃(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖).
(ii) If 1≤𝑝_𝑘≤sup𝑝_𝑘=𝐻<∞, then 𝑤^𝜃(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖)⊆𝑤^𝜃(ℳ,𝐵^𝜇_∧,‖⋅,…,⋅‖).

Proof. (i) Let 𝑥∈𝑤^𝜃(ℳ,𝐵^𝜇_∧,‖⋅,…,⋅‖). Since 0<inf𝑝_𝑘≤1, we get 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘−𝐿𝜌,𝑧₁,…,𝑧_𝑛−1≤1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘−𝐿𝜌,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘, (64) and hence 𝑥∈𝑤^𝜃(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖).
(ii) 1≤𝑝_𝑘≤sup𝑝_𝑘=𝐻<∞ and 𝑥=(𝑥_𝑘)∈𝑤^𝜃(ℳ,𝐵^𝜇_∧,𝑝,‖⋅,…,⋅‖). Then for each 0<𝜀<1 there exists a positive integer 𝑠₀ such that 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘−𝐿𝜌,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘≤𝜀<1∀𝑟>𝑠₀. (65)
This implies that 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘−𝐿𝜌,𝑧₁,…,𝑧_𝑛−1^𝑝𝑘≤1ℎ_{𝑟𝑘∈𝐼𝑟}𝑀_𝑘𝐵^𝜇_∧𝑥_𝑘−𝐿𝜌,𝑧₁,…,𝑧_𝑛−1. (66) Therefore 𝑥=(𝑥_𝑘)∈𝑤^𝜃(ℳ,𝐵^𝜇_∧,‖⋅,…,⋅‖). This completes the proof of the theorem.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors gratefully acknowledge that this research was partially supported by the University Putra Malaysia under the ERGS Grant Scheme having Project no. ERGS 1-2013/5527179. The authors are grateful also to the anonymous referees for a careful checking of the details and for helpful comments that improved the paper.