An Extension of Modular Sequence Spaces

Abstract

The main aim of this paper is to present an extension of the modular sequence spaces by means of Cesàro mean of order one, to investigate several relevant algebraic and topological properties, and derive some other spaces in the sequel.

1. Introduction

Throughout the paper 𝑤(𝑋) will represent the spaces of all 𝑋 valued sequences spaces, where (𝑋,𝑞) is a seminormed space, seminormed by 𝑞. For 𝑋=𝐶, the space of complex numbers, these represent the corresponding scalar valued sequence spaces. The zero sequence is denoted by 𝜃=(𝜃,𝜃,𝜃,…), where 𝜃 is the zero element of 𝑋.

An Orlicz function 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→∞.

If convexity of Orlicz function 𝑀 is replaced by 𝑀(𝑥+𝑦)≤𝑀(𝑥)+𝑀(𝑦), then this function is called a modulus function introduced by Nakano [1].

Lindenstrauss and Tzafriri [2] used the idea of Orlicz function to construct sequence space ℓ_𝑀=𝑥∈𝑤∶^∞_𝑘=1𝑀𝑥_𝑘𝜌<∞,forsome𝜌>0. (1)

The space ℓ_𝑀 becomes a Banach space, with the norm ‖𝑥‖=inf𝜌>0∶^∞_𝑘=1𝑀𝑥_𝑘𝜌≤1 (2) which is called an Orlicz space. The space ℓ_𝑀 is closely related to the space ℓ_𝑝 which is an Orlicz sequence space with 𝑀(𝑥)=𝑥^𝑝 for 1≤𝑝<∞.

Another generalization of Orlicz sequence spaces is due to Woo [3]. Let {𝑀_𝑘} be a sequence of Orlicz functions. Define the vector space ℓ{𝑀_𝑘} by ℓ𝑀_𝑘=𝑥∈𝑤∶^∞_𝑘=1𝑀_𝑘𝑥_𝑘𝜌<∞,forsome𝜌>0 (3) and equip this space with the norm ‖𝑥‖=inf𝜌>0∶^∞_𝑘=1𝑀_𝑘𝑥_𝑘𝜌≤1. (4)

Then ℓ{𝑀_𝑘} becomes a Banach space and is called a modular sequence space. The space ℓ{𝑀_𝑘} also generalizes the concept of modular sequence space introduced earlier by Nakano [4], who considered the space ℓ{𝑀_𝑘} when 𝑀_𝑘(𝑥)=𝑥^𝛼𝑘, where 1≤𝛼_𝑘<∞ for 𝑘≥1.

An Orlicz function 𝑀 is said to satisfy the Δ₂-condition for all values of 𝑢, if there exists a constant 𝐾>0, such that 𝑀(2𝑢)≤𝑘𝑀(𝑢)(𝑢≥0). The Δ₂-condition is equivalent to the satisfaction of inequality 𝑀(𝑙𝑢)≤𝑘𝑙𝑢𝑀(𝑢) for all values of 𝑢 and for 𝑙>1 (see [5]).

The previous Δ₂-condition implies 𝑀(𝑙𝑢)≤𝐾𝑙^log2𝐾𝑀(𝑢), for all 𝑢>0,𝑙>1.

Bektaş and Altin [6], Parasar and Choudhary [7], Mursaleen et al. [8], Dutta and Başar [9], Dutta and Bilgin [10], Karakaya and Dutta [11], Tripathy and Dutta [12], Jebril [13], and many others have studied different summable spaces and other sequence spaces using Orlicz functions.

A 𝐵𝐾-space (introduced by Zeller [14]) (𝑋,‖⋅‖) is a Banach space of complex sequences 𝑥=(𝑥_𝑘) in which the coordinate maps are continuous; that is, |𝑥^𝑛_𝑘−𝑥_𝑘|→0, whenever ‖𝑥^𝑛−𝑥‖→0 as 𝑛→∞, where 𝑥^𝑛=(𝑥^𝑛_𝑘), for all 𝑛∈𝑁 and 𝑥=(𝑥_𝑘).

Let 𝐴 denote the set of all complex sequences which have only a finite number of nonzero coordinates, and let 𝜆 denote a 𝐵𝐾-space of sequences 𝑥=(𝑥_𝑘) which contains 𝐴. An element 𝑥=(𝑥_𝑘) of 𝜆 will be called sectionally convergent if 𝑥^(𝑛)=^𝑛_𝑘=1𝑥_𝑘𝑒_𝑘⟶𝑥,as𝑛⟶∞, (5) where 𝑒_𝑘=(𝛿_𝑘𝑖), where 𝛿_𝑘𝑘=1, 𝛿_𝑘𝑖=0 for 𝑘≠𝑖.

𝜆 will be called 𝐴𝐾-space if and only if each of its elements is sectionally convergent.

Let 𝐌=(𝑀_𝑘) be a sequence of Orlicz functions, let 𝑋 be a seminormed space with seminorm 𝑞, let 𝑝=(𝑝_𝑘) be sequence of positive real numbers, and let 𝐶=(𝑐_𝑛𝑘)^∞_𝑛,𝑘=0 be the Cesàro matrix of order one with 𝑐_𝑛𝑘=1/(𝑛+1) if 0≤𝑘≤𝑛 and 𝑐_𝑛𝑘=0, otherwise. Then for nonnegative real numbers 𝑠 we define ℓ𝑀_𝑘=(𝑋)∶,𝑝,𝑞,𝑠,𝐶𝑥∈𝑤^∞_𝑘=0𝑘^−𝑠𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜌(𝑘+1)^𝑝𝑘<∞,forsome𝜌>0. (6)

The following inequality will be used throughout the paper. Let 𝑝=(𝑝_𝑘) be a positive sequence of real numbers with 0<𝑝_𝑘≤sup𝑝_𝑘=𝐺, 𝐷=max{1,2^𝐺−1}. Then for all 𝑎_𝑘,𝑏_𝑘∈𝐶 for all 𝑘∈𝑁, we have 𝑎_𝑘+𝑏_𝑘^𝑝𝑘≤𝐷𝑎_𝑘^𝑝𝑘+𝑏_𝑘^𝑝𝑘. (7)

2. Main Results

In this section we give the theorems that characterize the structure of the class of sequences ℓ{𝑀_𝑘,𝑝,𝑞,𝑠,𝐶} and some other spaces which can be derived from this space.

Theorem 1. Let 𝑝=(𝑝_𝑘) be bounded sequence of positive reals; then ℓ{𝑀_𝑘,𝑝,𝑞,𝑠,𝐶} is a linear space over the field of complex numbers.

Proof. Let 𝑥,𝑦∈ℓ{𝑀_𝑘,𝑝,𝑞,𝑠,𝐶} and 𝛼,𝛽∈𝐶. Then there exist some 𝜌₁>0 and 𝜌₂>0 such that ^∞_𝑘=0𝑘^−𝑠𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜌₁(𝑘+1)^𝑝𝑘<∞,_∞𝑘=0𝑘^−𝑠𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑦_𝑗𝜌₂(𝑘+1)^𝑝𝑘<∞. (8) We consider 𝜌₃=max(2|𝛼|𝜌₁,2|𝛽|𝜌₂). Since each 𝑀_𝑘 is non-decreasing and convex, and since 𝑞 is a seminorm, ^∞_𝑘=0𝑘^−𝑠𝑀_𝑘𝑞∑^𝑘_𝑗=0𝛼𝑥_𝑘+𝛽𝑦_𝑘𝜌₃(𝑘+1)^𝑝𝑘≤^∞_𝑘=0𝑘^−𝑠𝑀_𝑘𝑞∑^𝑘_𝑗=0𝛼𝑥_𝑗𝜌₃()∑𝑘+1+𝑞^𝑘𝑗=0𝛽𝑦_𝑗𝜌₃(𝑘+1)^𝑝𝑘≤^∞_𝑘=0𝑘^−𝑠𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗2𝜌₁()∑𝑘+1+𝑞^𝑘𝑗=0𝑦_𝑗2𝜌₂(𝑘+1)^𝑝𝑘≤𝐷^∞_𝑘=0𝑘^−𝑠𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜌₁(𝑘+1)^𝑝𝑘+^∞_𝑘=0𝑘^−𝑠𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑦_𝑗𝜌₂(𝑘+1)^𝑝𝑘<∞. (9) This completes the proof.

Theorem 2. ℓ{𝑀_𝑘,𝑝,𝑞,𝑠,𝐶} is a paranormed space (need not total paranorm) space with paranorm 𝑔, defined as follows 𝑔(𝑥)=inf𝜌^{𝑝𝑛/𝐻}∶^∞_𝑘=0𝑘^−𝑠𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜌(),𝑘+1≤1,𝑛=1,2,… (10) where 𝐻=max{1,sup_𝑘𝑝_𝑘}.

Proof. Clearly 𝑔(𝑥)=𝑔(−𝑥). Since 𝑀_𝑘(0)=0, for all 𝑘∈𝑁 we get inf{𝜌^{𝑝𝑛/𝐻}}=0 for 𝑥=𝜃.
Now let 𝑥,𝑦∈ℓ{𝑀_𝑘,𝑝,𝑞,𝑠,𝐶}, and let us choose 𝜌₁>0 and 𝜌₂>0 such that ^∞_𝑘=0𝑘^−𝑠𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜌₁(𝑘+1)≤1,_∞𝑘=0𝑘^−𝑠𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑦_𝑗𝜌₂(𝑘+1)≤1. (11) Let 𝜌=𝜌₁+𝜌₂. Then we have ^∞_𝑘=0𝑘^−𝑠𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗+𝑦_𝑗𝜌()≤𝜌𝑘+1₁𝜌₁+𝜌₂^∞_𝑘=0𝑘^−𝑠𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜌₁()+𝜌𝑘+1₂𝜌₁+𝜌₂^∞_𝑘=0𝑘^−𝑠𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑦_𝑗𝜌₂(𝑘+1)≤1. (12) Hence 𝑔(𝑥+𝑦)≤𝑔(𝑥)+𝑔(𝑦).
Finally let 𝜆 be a given non-zero scalar; then the continuity of the scalar multiplication follows from the following equality 𝑔()𝜌𝜆𝑥=inf^{𝑝𝑛/𝐻}∶^∞_𝑘=0𝑘^−𝑠𝑀_𝑘𝑞∑^𝑘_𝑗=0𝜆𝑥_𝑗𝜌()𝜆|𝑠)𝑘+1≤1,𝑛=1,2,…=inf(|^{𝑝𝑛/𝐻}∶^∞_𝑘=0𝑘^−𝑠𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝑠()𝜌𝑘+1≤1,𝑛=1,2,…,where𝑠=|𝑠|. (13) This completes the proof.

The proof of the following theorem is easy, so omitted.

Theorem 3. Let 𝐌=(𝑀_𝑘) and 𝐓=(𝑇_𝑘) be sequences of Orlicz functions. For any two sequences 𝑝=(𝑝_𝑘) and 𝑡=(𝑡_𝑘) of bounded positive real numbers and for any two seminorms 𝑞₁ and 𝑞₂ one has(i)if 𝑞₁ is stronger than 𝑞₂, then ℓ{𝑀_𝑘,𝑝,𝑞₁,𝑠,𝐶}⊂ℓ{𝑀_𝑘,𝑝,𝑞₂,𝑠,𝐶},(ii)ℓ{𝑀_𝑘,𝑝,𝑞₁,𝑠,𝐶}∩ℓ{𝑀_𝑘,𝑝,𝑞₂,𝑠,𝐶}⊂ℓ{𝑀_𝑘,𝑝,𝑞₁+𝑞₂,𝑠,𝐶},(iii)ℓ{𝑀_𝑘,𝑝,𝑞,𝑠,𝐶}∩ℓ{𝑇_𝑘,𝑝,𝑞,𝑠,𝐶}⊂ℓ{𝑀_𝑘+𝑇_𝑘,𝑝,𝑞,𝑠,𝐶}, (iv)ℓ{𝑀_𝑘,𝑝,𝑞₁,𝑠,𝐶}∩ℓ{𝑀_𝑘,𝑡,𝑞₂,𝑠,𝐶}≠𝜑, (v)If 𝑠₁≤𝑠₂, then ℓ{𝑀_𝑘,𝑝,𝑞,𝑠₁,𝐶}⊂ℓ{𝑀_𝑘,𝑝,𝑞,𝑠₂,𝐶}.

Theorem 4. Let 𝐌=(𝑀_𝑘) and 𝐓=(𝑇_𝑘) be sequences of Orlicz functions which satisfy Δ₂-condition and 𝑠>1, then ℓ𝑀_𝑘,𝑝,𝑞,𝑠,𝐶⊆ℓ𝑇_𝑘∘𝑀_𝑘,𝑝,𝑞,𝑠,𝐶. (14)

Proof. Let 𝑥∈ℓ{𝑀_𝑘,𝑝,𝑞,𝑠,𝐶} and 𝜀>0. We choose 0<𝛿<1 such that each 𝑀_𝑘(𝑢)<𝜀 for 0≤𝑢≤𝛿. We write 𝑦_𝑘=𝑀_𝑘(𝑞((∑^𝑘_𝑗=0𝑥_𝑗)/𝜌(𝑘+1))) and consider ^∞_𝑘=0𝑘^−𝑠𝑇_𝑘𝑦_𝑘^𝑝𝑘=₁𝑘^−𝑠𝑇_𝑘𝑦_𝑘^𝑝𝑘+₂𝑘^−𝑠𝑇_𝑘𝑦_𝑘^𝑝𝑘, (15) where the first summation is over 𝑦_𝑘≤𝛿 and the second is over 𝑦_𝑘>𝛿. Now we have ₁𝑘^−𝑠𝑇_𝑘𝑦_𝑘^𝑝𝑘<max1,𝜀^𝐻∞_𝑘=1𝑘^−𝑠<∞,since𝑠>1. (16) For 𝑦_𝑘>𝛿, we use the fact that 𝑦_𝑘<𝑦_𝑘𝛿𝑦≤1+^𝑘𝛿. (17) Since each 𝑇_𝑘 is non-decreasing and convex, it follows that 𝑇_𝑘𝑦_𝑘<𝑇_𝑘𝑦1+^𝑘𝛿<12𝑇_𝑘(2)+12𝑇_𝑘2𝑦_𝑘𝛿,foreach𝑘in𝑁. (18) Since each 𝑇_𝑘 satisfies Δ₂-condition, we have 𝑇_𝑘𝑦_𝑘<12𝐾𝑦_𝑘𝛿𝑇_𝑘(2)+12𝐾𝑦_𝑘𝛿𝑇_𝑘(2)=𝐾𝑦_𝑘𝛿⁻¹𝑇_𝑘(2). (19) Hence ₂𝑘^−𝑠𝑇_𝑘𝑦_𝑘^𝑝𝑘≤max1,𝐾𝛿⁻¹𝑀(2)^𝐻×^∞_𝑘=1𝑘^−𝑠𝑦_𝑘^𝑝𝑘<∞. (20)
Thus ^∞_𝑘=0𝑘^−𝑠𝑇_𝑘𝑦_𝑘^𝑝𝑘=₁𝑘^−𝑠𝑇_𝑘𝑦_𝑘^𝑝𝑘+₂𝑘^−𝑠𝑇_𝑘𝑦_𝑘^𝑝𝑘≤max1,𝜀^𝐻×^∞_𝑘=1𝑘^−𝑠+max1,𝐾𝛿⁻¹𝑀(2)^𝐻×^∞_𝑘=1𝑘^−𝑠𝑦_𝑘^𝑝𝑘<∞. (21) Hence 𝑥∈ℓ{𝑇_𝑘∘𝑀_𝑘,𝑝,𝑞,𝑠,𝐶}.
This completes the proof.

Taking 𝑀_𝑘(𝑥)=𝑥,forall  𝑥∈[0,∞) and 𝑘 in 𝑁, in Theorem 4, we get the next corollary.

Corollary 5. Let 𝐌=(𝑀_𝑘) be any sequence of Orlicz functions which satisfy Δ₂-condition and 𝑠>1, then ℓ𝑝,𝑞,𝑠,𝐶⊆ℓ𝑀_𝑘,𝑝,𝑞,𝑠,𝐶. (22)

We will write 𝑓≈𝑔 for non-negative functions 𝑓and 𝑔 whenever 𝐶₁𝑓≤𝑔≤𝐶₂𝑓 for some 𝐶_𝑗>0,𝑗=1,2.

Theorem 6. Let 𝐌=(𝑀_𝑘) and 𝐓=(𝑇_𝑘) be sequences of Orlicz functions. If 𝑀_𝑘≈𝑇_𝑘 for each 𝑘∈𝑁, then ℓ{𝑀_𝑘,𝑝,𝑞,𝑠,𝐶}=ℓ{𝑇_𝑘,𝑝,𝑞,𝑠,𝐶}.

Proof. The proof is obvious.

Theorem 7. Let 𝐌=(𝑀_𝑘) be a sequence of Orlicz functions. If lim_𝑡→0(𝑀_𝑘(𝑡)/𝑡)>0 and lim_𝑡→0(𝑀_𝑘(𝑡)/𝑡)<∞, for each 𝑘∈𝑁, then ℓ{𝑀_𝑘,𝑝,𝑞,𝑠,𝐶}=ℓ{𝑝,𝑞,𝑠,𝐶}.

Proof. If the given conditions are satisfied, we have 𝑀_𝑘(𝑡)≈𝑡 for each 𝑘, and the proof follows from Theorem 6.

If we take 𝑠=0, the sequence space ℓ{𝑀_𝑘,𝑝,𝑞,𝑠,𝐶} reduces to the following sequence space: ℓ𝑀_𝑘=(𝑋)∶,𝑝,𝑞,𝐶𝑥∈𝑤^∞_𝑘=0𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜌(𝑘+1)^𝑝𝑘<∞,forsome𝜌>0. (23)

Theorem 8. Let 𝑝=(𝑝_𝑘) be bounded sequence of positive reals, and let (𝑋,𝑞) be a complete seminormed space, then ℓ{𝑀_𝑘,𝑝,𝑞,𝐶} is a complete paranormed space paranormed by ℎ, defined by ℎ(𝑥)=inf𝜌^{𝑝𝑛/𝐻}∶^∞_𝑘=0𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜌(),𝑘+1≤1,𝑛=1,2,… (24) where 𝐻=max{1,sup_𝑘𝑝_𝑘}.

Proof. Let (𝑥^𝑖) be a Cauchy sequence in ℓ{𝑀_𝑘,𝑝,𝑞,𝐶}. Let 𝛿>0 be fixed, and let 𝑟>0 be such that for a given 0<𝜀<1,𝜀/𝑟𝛿>0, and 𝑟𝛿≥1. Then there exists a positive integer 𝑛₀ such that ℎ𝑥^𝑠−𝑥^𝑡<𝜀𝑟𝛿,∀𝑠,𝑡≥𝑛₀.⟹inf𝜌^{𝑝𝑛/𝐻}∶^∞_𝑘=0𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥^𝑠_𝑗−𝑥^𝑡_𝑗𝜌()<𝜀𝑘+1≤1,𝑛=1,2,…𝑟𝛿,∀𝑠,𝑡≥𝑛₀. (25) Hence we have ^∞_𝑘=0𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥^𝑠_𝑗−𝑥^𝑡_𝑗ℎ(𝑥^𝑠−𝑥^𝑡)(𝑘+1)≤1,∀𝑠,𝑡≥𝑛₀. (26) It follows that 𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥^𝑠_𝑗−𝑥^𝑡_𝑗ℎ(𝑥^𝑠−𝑥^𝑡)(𝑘+1)≤1,∀𝑠,𝑡≥𝑛₀,𝑘∈𝑁. (27) For 𝑟>0 with 𝑀_𝑘(𝑟𝛿/2)≥1, we have 𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥^𝑠_𝑗−𝑥^𝑡_𝑗ℎ(𝑥^𝑠−𝑥^𝑡)(𝑘+1)≤𝑀_𝑘𝑟𝛿2,∀𝑠,𝑡≥𝑛₀,𝑘∈𝑁. (28)
Since 𝑀_𝑘 is non-decreasing for each 𝑘∈𝑁, we have 𝑞∑^𝑘_𝑗=0𝑥^𝑠_𝑗−𝑥^𝑡_𝑗≤𝑘+1𝑟𝛿2⋅𝜀=𝜀𝑟𝛿2. (29) Hence it follows that (𝑥^𝑠_𝑘) is a Cauchy sequence in (𝑋,𝑞) for each 𝑘∈𝑁. But (𝑋,𝑞) is complete, and so (𝑥^𝑠_𝑘) is convergent in (𝑋,𝑞) for each 𝑘∈𝑁.
Let lim_𝑠→∞𝑥^𝑠_𝑘=𝑥_𝑘 exists for each 𝑘∈𝑁.
Now we have for all 𝑠,𝑡≥𝑛₀, inf𝜌^{𝑝𝑛/𝐻}∶^∞_𝑘=0𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥^𝑠_𝑗−𝑥^𝑡_𝑗𝜌(𝑘+1)≤1,𝑛=1,2,…<𝜀. (30) Then we have lim_𝑡→∞inf𝜌^{𝑝𝑛/𝐻}∶^∞_𝑘=0𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥^𝑠_𝑗−𝑥^𝑡_𝑗𝜌(𝑘+1)≤1,𝑛=1,2,…<𝜀,∀𝑠≥𝑛₀. (31) Using the continuity of Orlicz functions, we have inf𝜌^{𝑝𝑛/𝐻}∶^∞_𝑘=0𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥^𝑠_𝑗−lim_𝑡→∞𝑥^𝑡_𝑗𝜌(𝑘+1)≤1,𝑛=1,2,…<𝜀,∀𝑠≥𝑛₀. (32) This implies inf𝜌^{𝑝𝑛/𝐻}∶^∞_𝑘=0𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥^𝑠_𝑗−𝑥_𝑗𝜌≤1,𝑛=1,2,…<𝜀,∀𝑠≥𝑛₀. (33) It follows that (𝑥^𝑠−𝑥)∈ℓ{𝑀_𝑘,𝑝,𝑞,𝐶}.
Since (𝑥^𝑠)∈ℓ{𝑀_𝑘,𝑝,𝑞,𝐶} and ℓ{𝑀_𝑘,𝑝,𝑞,𝐶} is a linear space, so we have 𝑥=𝑥^𝑠−(𝑥^𝑠−𝑥)∈ℓ{𝑀_𝑘,𝑝,𝑞,𝐶}.
This completes the proof.

If we take 𝑠=0 and 𝑝_𝑘=𝑙, 𝑎 constant the sequence space ℓ{𝑀_𝑘,𝑝,𝑞,𝑠,𝐶} reduces to the following sequence space: ℓ𝑀_𝑘=(𝑋)∶,𝑞,𝐶𝑥∈𝑤^∞_𝑘=0𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜌().𝑘+1<∞,forsome𝜌>0 (34)

Theorem 9. Let (𝑋,𝑞) be a complete normed space; then ℓ{𝑀_𝑘,𝑞,𝐶} is a Banach space normed by ‖⋅‖, defined by ‖𝑥‖=inf𝜌∶^∞_𝑘=0𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜌(𝑘+1)≤1. (35)

Proof. We prove that ‖⋅‖ is a norm on ℓ{𝑀_𝑘,𝑞,𝐶}. The completeness part can be proved using similar arguments as applied to prove the previous theorem.
If 𝑥=𝜃, then it is obvious that ‖𝑥‖=0. Conversely assume ‖𝑥‖=0. Then using the definition of norm, we have inf𝜌∶^∞_𝑘=0𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜌(𝑘+1)≤1=0. (36) This implies that for a given 𝜀>0, there exists some 𝜌_𝜀(0<𝜌_𝜀<𝜀) such that ^∞_𝑘=0𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜌_𝜀(𝑘+1)≤1. (37) It follows that 𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜌_𝜀(𝑘+1)≤1,∀𝑘∈𝑁. (38) Thus 𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜀(𝑘+1)≤𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜌_𝜀(𝑘+1)≤1,∀𝑘∈𝑁. (39) Suppose (∑^𝑛𝑖_𝑗=0𝑥_𝑗)/(𝑛_𝑖+1)≠0, for some 𝑖. Let 𝜀→0 then (∑^𝑛𝑖_𝑗=0𝑥_𝑗)/(𝜀(𝑛_𝑖+1))→∞.
It follows that 𝑀_𝑘(𝑞((∑^𝑘_𝑗=0𝑥_𝑗)/(𝜀(𝑘+1))))→∞ as 𝜀→0 for some 𝑛_𝑖∈𝑁. This is a contradiction.
Therefore (∑^𝑘_𝑗=0𝑥_𝑗)/(𝑘+1)=0,forall𝑘∈𝑁.
It follows that 𝑥_𝑘=0 for all 𝑘≥1. Hence 𝑥=𝜃.
Again proof of the properties ‖𝑥+𝑦‖≤‖𝑥‖+‖𝑦‖ and for any scalar 𝛼,‖𝛼𝑥‖=|𝛼|‖𝑥‖ are similar to that of Theorem 2.

It is easy to see that ‖𝑥^𝑖‖→0 implies that 𝑥^𝑖_𝑘→0 for each 𝑖≥1. Hence we have the following proposition.

Proposition 10. The space ℓ{𝑀_𝑘,𝑞,𝐶} is a BK-space.

Now we study the AK-characteristic of the space ℓ{𝑀_𝑘,𝑞,𝑠,𝐶}. Before that we give a new definition and prove some results which will be required.

Definition 11. For any sequence of Orlicz functions 𝐌=(𝑀_𝑘), we define ℎ𝑀_𝑘,𝑞,𝐶=𝑥∈𝑤(𝑋)∶^∞_𝑘=0𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜌().𝑘+1<∞,forevery𝜌>0 (40) Clearly ℎ{𝑀_𝑘,𝑞,𝐶} is a subspace of ℓ{𝑀_𝑘,𝑞,𝐶}. The topology of ℎ{𝑀_𝑘,𝑞,𝐶} is the one it inherits from ‖⋅‖.

Proposition 12. Let 𝐌=(𝑀_𝑘) be a sequence of Orlicz functions which satisfy Δ₂-condition. Then ℓ𝑀_𝑘,𝑞,𝐶=ℎ𝑀_𝑘,𝑞,𝐶. (41)

Proof. It is enough to prove that ℓ{𝑀_𝑘,𝑞,𝐶}⊆ℎ{𝑀_𝑘,𝑞,𝐶}.
Let 𝑥∈ℓ{𝑀_𝑘,𝑞,𝐶}, then for some 𝜌>0, ^∞_𝑘=0𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜌(𝑘+1)<∞. (42) Choose an arbitrary 𝜂>0. If 𝜌≤𝜂 then ^∞_𝑘=0𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜂()<𝑘+1_∞𝑘=0𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜌(𝑘+1)<∞. (43) Let now 𝜂<𝜌 and put 𝑙=𝜌/𝜂>1.
Since each 𝑀_𝑘 satisfies the Δ₂-condition, there exist constants 𝐾_𝑘 such that ^∞_𝑘=0𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜂()≤𝑘+1_∞𝑘=0𝐾_𝑘𝜌𝜂^log2𝐾𝑘𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜌().𝑘+1 (44) Let 𝑆=sup_𝑘𝐾_𝑘(𝜌/𝜂)^log2𝐾𝑘. Then for every 𝜂>0^∞_𝑘=0𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜂(𝑘+1)≤𝑆^∞_𝑘=0𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜌(𝑘+1)<∞. (45) This completes the proof.

Proposition 13. Let (𝑋,𝑞) be a complete normed space, then ℎ{𝑀_𝑘,𝑞,𝐶} is an AK-space.

Proof. Let 𝑥∈ℎ{𝑀_𝑘,𝑞,𝐶}. Then for each 𝜀,0<𝜀<1, we can find an 𝑠₀ such that _𝑘≥𝑠0𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜀(𝑘+1)≤1. (46) Hence for 𝑠≥𝑠₀, 𝑥−𝑥^(𝑠)=inf𝜌>0∶_{𝑘≥𝑠+1}𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜌(𝑘+1)≤1≤inf𝜌>0∶_𝑘≥𝑠𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜌(𝑘+1)≤1<𝜀. (47) Thus we can conclude that ℎ{𝑀_𝑘,𝑞,𝐶} is an AK space.

Combining Proposition 10 and Proposition 12, we have the following theorem.

Theorem 14. Let 𝐌=(𝑀_𝑘) be a sequence of Orlicz functions which satisfy Δ₂-condition, then ℓ{𝑀_𝑘,𝑞,𝐶} is an AK-space.

Proposition 15. The space ℎ{𝑀_𝑘,𝑞,𝐶} is a closed subspace of ℓ{𝑀_𝑘,𝑞,𝐶}.

Proof. Let {𝑥^𝑠} be a sequence in ℎ{𝑀_𝑘,𝑞,𝐶} such that ‖𝑥^𝑠−𝑥‖→0, where 𝑥∈ℓ{𝑀_𝑘,𝑞,𝐶}.
To complete the proof we need to show that 𝑥∈ℎ{𝑀_𝑘,𝑞,𝐶}; that is, _𝑘≥0𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜌(𝑘+1)<∞forevery𝜌>0. (48) To 𝜌>0 there corresponds an 𝑙 such that ‖𝑥^𝑙−𝑥‖≤𝜌/2. Then using convexity of each 𝑀_𝑘, _𝑘≥0𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜌(𝑘+1)=_𝑘≥0𝑀_𝑘𝑞2∑^𝑘_𝑗=0𝑥^𝑙_𝑗−2∑^𝑘_𝑗=0𝑥^𝑙_𝑗−∑^𝑘_𝑗=0𝑥_𝑗()≤12𝜌𝑘+12_𝑘≥0𝑀_𝑘𝑞2∑^𝑘_𝑗=0𝑥^𝑙_𝑗𝜌()+1𝑘+12_𝑘≥0𝑀_𝑘𝑞2∑^𝑘_𝑗=0𝑥^𝑙_𝑗−𝑥_𝑗𝜌()≤1𝑘+12_𝑘≥0𝑀_𝑘𝑞2∑^𝑘_𝑗=0𝑥^𝑙_𝑗𝜌()+1𝑘+12_𝑘≥0𝑀_𝑘𝑞2∑^𝑘_𝑗=0𝑥^𝑙_𝑗−𝑥_𝑗𝑥^𝑙).−𝑥(𝑘+1 (49) Now from Theorem 9, using the definition of norm ‖⋅‖, we have _𝑘≥0𝑀_𝑘𝑞2∑^𝑘_𝑗=0𝑥^𝑙_𝑗−𝑥_𝑗𝑥^𝑙−𝑥(𝑘+1)≤1. (50) It follows that _𝑘≥0𝑀_𝑘𝑞∑^𝑘_𝑗=0𝑥_𝑗𝜌(𝑘+1)<∞forevery𝜌>0. (51) Thus 𝑥∈ℎ{𝑀_𝑘,𝑞,𝐶}.

Hence we have the following corollary.

Corollary 16. The space ℎ{𝑀_𝑘,𝑞,𝐶} is a BK-space.

Acknowledgment

This research is supported by Deanship of Scientific Research, Taibah University, Almadinah Almunawwarah, Saudi Arabia.