On the Paranormed Nörlund Sequence Space of Nonabsolute Type

Abstract

Maddox defined the space ℓ(𝑝) of the sequences 𝑥=(𝑥_𝑘) such that ∑^∞_𝑘=0|𝑥_𝑘|^𝑝𝑘<∞, in Maddox, 1967. In the present paper, the Nörlund sequence space 𝑁^𝑡(𝑝) of nonabsolute type is introduced and proved that the spaces 𝑁^𝑡(𝑝) and ℓ(𝑝) are linearly isomorphic. Besides this, the alpha-, beta-, and gamma-duals of the space 𝑁^𝑡(𝑝) are computed and the basis of the space 𝑁^𝑡(𝑝) is constructed. The classes (𝑁^𝑡(𝑝)∶𝜇) and (𝜇∶𝑁^𝑡(𝑝)) of infinite matrices are characterized. Finally, some geometric properties of the space 𝑁^𝑡(𝑝) are investigated.

1. Introduction

We denote the space of all sequences of complex entries by 𝜔. Any vector subspace of 𝜔 is called a sequence space. We write ℓ_∞, 𝑐, and 𝑐₀ for the spaces of all bounded, convergent, and null sequences, respectively. Also by 𝑏𝑠, 𝑐𝑠, ℓ₁, and ℓ_𝑝, we denote the spaces of all bounded, convergent, absolutely and 𝑝-absolutely convergent series, respectively.

A linear topological space 𝑋 over the real field ℝ is said to be a paranormed space if there is a subadditive function 𝑔∶𝑋→ℝ such that 𝑔(𝜃)=0, 𝑔(𝑥)=𝑔(−𝑥) and scalar multiplication is continuous; that is, |𝛼_𝑛−𝛼|→0 and 𝑔(𝑥_𝑛−𝑥)→0 imply 𝑔(𝛼_𝑛𝑥_𝑛−𝛼𝑥)→0 for all 𝛼's in ℝ and all 𝑥's in 𝑋, where 𝜃 is the zero vector in the linear space 𝑋. Assume here and after that (𝑝_𝑘) is a bounded sequence of strictly positive real numbers with sup𝑝_𝑘=𝐻 and 𝑀=max{1,𝐻}. Then, the linear spaces ℓ(𝑝) and ℓ_∞(𝑝) were defined by Maddox in [1] (see also [2, 3]) as follows: ℓ𝑝=𝑥=𝑥_𝑘∈𝜔∶_𝑘𝑥_𝑘^𝑝𝑘<∞with0<𝑝_𝑘ℓ≤𝐻<∞,_∞𝑝=𝑥=𝑥_𝑘∈𝜔∶sup_𝑘∈ℕ𝑥_𝑘^𝑝𝑘<∞ (1) which are the complete spaces paranormed by 𝑔₁(𝑥)=_𝑘𝑥_𝑘^{𝑝𝑘1/𝑀},𝑔₂(𝑥)=sup_𝑘∈ℕ𝑥_𝑘^{𝑝𝑘/𝑀}iffinf_𝑘∈ℕ𝑝_𝑘>0, (2) respectively, where ℕ={0,1,2,…}. For simplicity in notation, here and in what follows, the summation without limits runs from 0 to ∞. We assume throughout that 𝑝_𝑘⁻¹+(𝑝^′_𝑘)⁻¹=1, provided 1<inf𝑝_𝑘≤𝐻<∞, and denote the collection of all finite subsets of ℕ by ℱ.

For the sequence spaces 𝜆 and 𝜇, define the set 𝑆(𝜆,𝜇) by 𝑆𝜆,𝜇=𝑧=𝑧_𝑘∈𝜔∶𝑥𝑧=𝑥_𝑘𝑧_𝑘∈𝜇∀𝑥∈𝜆. (3) With the notation of (3), the alpha-, beta-, and gamma-duals of a sequence space 𝜆, which are, respectively, denoted by 𝜆^𝛼, 𝜆^𝛽, and 𝜆^𝛾, are defined by 𝜆^𝛼=𝑆𝜆,ℓ₁,𝜆^𝛽=𝑆(𝜆,𝑐𝑠),𝜆^𝛾=𝑆(𝜆,𝑏𝑠). (4)

If a sequence space 𝜆 paranormed by 𝑔 contains a sequence (𝑏_𝑛) with the property that, for every 𝑥∈𝜆, there is a unique sequence of scalars (𝛼_𝑛) such that lim_𝑛→∞𝑔𝑥−^𝑛_𝑘=0𝛼_𝑘𝑏_𝑘=0, (5) then (𝑏_𝑛) is called a Schauder basis (or briefly basis) for 𝜆. The series ∑_𝑘𝛼_𝑘𝑏_𝑘 which has the sum 𝑥 is then called the expansion of 𝑥 with respect to (𝑏_𝑛) and written as 𝑥=∑_𝑘𝛼_𝑘𝑏_𝑘.

Let 𝜆, 𝜇 be any two sequence spaces, and let 𝐴=(𝑎_𝑛𝑘) be an infinite matrix of complex numbers 𝑎_𝑛𝑘, where 𝑘,𝑛∈ℕ. Then, we say that 𝐴 defines a matrix transformation from 𝜆 into 𝜇 and we denote it by writing 𝐴∶𝜆→𝜇, if for every sequence 𝑥=(𝑥_𝑘)∈𝜆, the sequence 𝐴𝑥={(𝐴𝑥)_𝑛}, the 𝐴-transform of 𝑥, is in 𝜇, where (𝐴𝑥)_𝑛=_𝑘𝑎_𝑛𝑘𝑥_𝑘foreach𝑛∈ℕ. (6) By (𝜆∶𝜇), we denote the class of all matrices 𝐴 such that 𝐴∶𝜆→𝜇. Thus, 𝐴∈(𝜆∶𝜇) if and only if the series on the right side of (6) converges for each 𝑛∈ℕ and every 𝑥∈𝜆, and we have 𝐴𝑥∈𝜇 for all 𝑥∈𝜆. Also, we write 𝐴_𝑛=(𝑎_𝑛𝑘)_𝑘∈ℕ for the sequence in the 𝑛th row of 𝐴.

Now, following Peyerimhoff [4, pp. 17–19] and Mears [5], we give short knowledge on the Nörlund means. Let (𝑡_𝑘) be a sequence of nonnegative real numbers with 𝑡₀>0 and write 𝑇_𝑛=∑^𝑛_𝑘=0𝑡_𝑘 for all 𝑛∈ℕ. Then, the Nörlund means with respect to the sequence 𝑡=(𝑡_𝑘) is defined by the matrix 𝑁^𝑡=(𝑎^𝑡_𝑛𝑘) which is given by 𝑎^𝑡_𝑛𝑘=𝑡_𝑛−𝑘𝑇_𝑛,0≤𝑘≤𝑛,0,𝑘>𝑛 (7) for all 𝑘,𝑛∈ℕ. It is known that the Nörlund matrix 𝑁^𝑡 is a Toeplitz matrix if and only if 𝑡_𝑛/𝑇_𝑛→0, as 𝑛→∞, and is reduced in the case 𝑡=𝑒=(1,1,1,…) to the matrix 𝐶₁ of arithmetic means. Additionally, for 𝑡_𝑛=𝐴_𝑛^𝑟−1 for all 𝑛∈ℕ, the method 𝑁^𝑡 is reduced to the Cesàro method 𝐶_𝑟 of order 𝑟>−1, where 𝐴^𝑟_𝑛=(𝑟+1)(𝑟+2)⋯(𝑟+𝑛)𝑛!,𝑛=1,2,3,…,1,𝑛=0. (8) Let 𝑡₀=𝐷₀=1 and define 𝐷_𝑛 for 𝑛∈{1,2,3,…} by 𝐷_𝑛=𝑡₁𝑡100⋯0₂𝑡₁𝑡10⋯0₃𝑡₂𝑡₁𝑡1⋯0⋮⋮⋮⋮⋱⋮_𝑛−1𝑡_𝑛−2𝑡_𝑛−3𝑡_𝑛−4𝑡⋯1_𝑛𝑡_𝑛−1𝑡_𝑛−2𝑡_𝑛−3⋯𝑡₁. (9) The inverse matrix 𝑈^𝑡=(𝑢^𝑡_𝑛𝑘) of the matrix 𝑁^𝑡=(𝑎^𝑡_𝑛𝑘) is given by Mears in [5] as follows: 𝑢^𝑡_𝑛𝑘=(−1)^𝑛−𝑘𝐷_𝑛−𝑘𝑇_𝑘,0≤𝑘≤𝑛,0,𝑘>𝑛 (10) for all 𝑘,𝑛∈ℕ. Also, one can derive by straightforward calculation for all 𝑘∈{1,2,3,…} that 𝐷_𝑘=^𝑘−1_𝑗=1(−1)^𝑗−1𝑡_𝑗𝐷_𝑘−𝑗+(−1)^𝑘−1𝑡_𝑘. (11)

The rest of this paper is organized as follows.

In Section 2, the complete paranormed Nörlund sequence space 𝑁^𝑡(𝑝) is introduced and proved that 𝑁^𝑡(𝑝) is linearly isomorphic to the space ℓ(𝑝) and the basis for the space 𝑁^𝑡(𝑝) is determined. Section 3 is devoted to the alpha-, beta-, and gamma-duals of the space 𝑁^𝑡(𝑝). In Section 4, the classes (𝑁^𝑡(𝑝)∶𝜇) and (𝜇∶𝑁^𝑡(𝑝)) of infinite matrices are characterized, where 𝜇 denotes any given sequence space. In Section 5, the rotundity of the space 𝑁^𝑡(𝑝) is characterized and some results related to this concept are given. In the final section of the paper, the significance of the space is mentioned and further suggestions are recorded.

2. The Nörlund Sequence Space 𝑁^𝑡(𝑝) of Nonabsolute Type

In this section, we define the Nörlund sequence space 𝑁^𝑡(𝑝) and prove that 𝑁^𝑡(𝑝) is linearly isomorphic to the space ℓ(𝑝), where 0<𝑝_𝑘≤𝐻<∞ for all 𝑘∈ℕ. Finally, we give the basis for the space 𝑁^𝑡(𝑝).

Let 𝜆 be any sequence space. Then, the matrix domain 𝜆_𝐴 of an infinite matrix 𝐴 in 𝜆 is defined by 𝜆_𝐴=𝑥=𝑥_𝑘∈𝜔∶𝐴𝑥∈𝜆. (12) In [6], Choudhary and Mishra defined the sequence space ℓ(𝑝) which consists of all sequences such that 𝐵-transforms of them are in ℓ(𝑝), where 𝐵=(𝑏_𝑛𝑘) is defined by 𝑏_𝑛𝑘=1,0≤𝑘≤𝑛0,𝑘>𝑛. (13) Başar and Altay [7] examined the space 𝑏𝑠(𝑝) which was formerly defined by Başar [8] as the set of all series whose sequences of partial sums are in the space ℓ_∞(𝑝). With the notation of (12), the spaces ℓ(𝑝) and 𝑏𝑠(𝑝) can be redefined by ℓ𝑝=ℓ𝑝_𝐵,𝑏𝑠𝑝=ℓ_∞𝑝_𝐵. (14)

In [9], Başar and Altay defined the sequence space 𝑟^𝑞(𝑝) which consists of all sequences such that 𝑅^𝑞-transforms of them are in ℓ(𝑝), where 𝑅^𝑞=(𝑟^𝑞_𝑛𝑘) is the matrix of Riesz mean; that is, 𝑟^𝑞𝑝=ℓ𝑝_𝑅𝑞,𝑟^𝑞_𝑝=ℓ_𝑝𝑅𝑞. (15)

In [10], Wang defined the sequence space 𝑋_𝑎(𝑝) consisting of all sequences whose 𝑁^𝑡-transforms are in ℓ_𝑝 which is a Banach space with the norm ‖𝑥‖_𝑝=_𝑘1𝑇_𝑘^𝑘_𝑗=0𝑡_𝑘−𝑗𝑥_𝑗^𝑝1/𝑝with1≤𝑝<∞. (16)

Now, we introduce the Nörlund sequence space 𝑁^𝑡(𝑝) defined by 𝑁^𝑡𝑝∶=𝑥=𝑥_𝑘∈𝜔∶_𝑘1𝑇_𝑘^𝑘_𝑗=0𝑡_𝑘−𝑗𝑥_𝑗^𝑝𝑘<∞with0<𝑝_𝑘≤𝐻<∞. (17) It is natural that the space 𝑁^𝑡(𝑝) can also be defined with the notation of (12) that 𝑁^𝑡(𝑝)={ℓ(𝑝)}_𝑁𝑡.

Define the sequence 𝑦=(𝑦_𝑘) by the 𝑁^𝑡-transform of a sequence 𝑥=(𝑥_𝑘); that is, 𝑦_𝑘=1𝑇_𝑘^𝑘_𝑗=0𝑡_𝑘−𝑗𝑥_𝑗∀𝑘∈ℕ. (18)

Theorem 1. 𝑁^𝑡(𝑝) is a complete linear metric space paranormed by 𝑔 defined by 𝑔(𝑥)=_𝑘1𝑇_𝑘^𝑘_𝑗=0𝑡_𝑘−𝑗𝑥_𝑗^{𝑝𝑘1/𝑀}𝑤𝑖𝑡ℎ0<𝑝_𝑘≤𝐻<∞. (19)

Proof. Since this can be shown by a routine verification, we omit the detail.

Remark 2. One can easily see that the absolute property does not hold on the space 𝑁^𝑡(𝑝); that is, 𝑔(𝑥)≠𝑔(|𝑥|) for at least one sequence in the space 𝑁^𝑡(𝑝), and this says that 𝑁^𝑡(𝑝) is a sequence space of nonabsolute type, where |𝑥|=(|𝑥_𝑘|).

Theorem 3. The Nörlund sequence space 𝑁^𝑡(𝑝) of nonabsolute type is linearly isomorphic to the space ℓ(𝑝), where 0<𝑝_𝑘≤𝐻<∞ for all 𝑘∈ℕ.

Proof. To prove the theorem, we should show the existence of a linear bijection between the spaces 𝑁^𝑡(𝑝) and ℓ(𝑝) for 0<𝑝_𝑘≤𝐻<∞. Consider the transformation 𝑇 defined, with the notation of (18), from 𝑁^𝑡(𝑝) to ℓ(𝑝) by 𝑥↦𝑦=𝑇𝑥=𝑁^𝑡𝑥. The linearity of 𝑇 is clear. Further, it is trivial that 𝑥=𝜃 whenever 𝑇𝑥=𝜃 and hence 𝑇 is injective.
Let us take any 𝑦∈ℓ(𝑝) and define the sequence 𝑥=(𝑥_𝑘) by 𝑥_𝑘=^𝑘_𝑖=0(−1)^𝑘−𝑖𝐷_𝑘−𝑖𝑇_𝑖𝑦_𝑖∀𝑘∈ℕ. (20) Therefore, we see from (19) that 𝑔(𝑥)=_𝑘1𝑇_𝑘^𝑘_𝑗=0𝑡_𝑘−𝑗𝑥_𝑗^{𝑝𝑘1/𝑀}=_𝑘1𝑇_𝑘^𝑘_𝑗=0𝑡^𝑗_{𝑘−𝑗𝑖=0}(−1)^𝑗−𝑖𝐷_𝑗−𝑖𝑇_𝑖𝑦_𝑖^{𝑝𝑘1/𝑀}=_𝑘𝑦_𝑘^{𝑝𝑘1/𝑀}=𝑔₁𝑦<∞. (21) This means that 𝑥∈𝑁^𝑡(𝑝). Consequently, 𝑇 is surjective and is paranorm preserving. Hence, 𝑇 is linear bijection and this says us that the spaces 𝑁^𝑡(𝑝) and ℓ(𝑝) are linearly isomorphic. Therefore, the proof is completed.

We determine the basis for the paranormed space 𝑁^𝑡(𝑝).

Theorem 4. Define the sequence 𝑏^(𝑘)(𝑡)={𝑏_𝑛^(𝑘)(𝑡)}_𝑛∈ℕ of the elements of the space 𝑁^𝑡(𝑝) for every fixed 𝑘∈ℕ by 𝑏_𝑛^(𝑘)(𝑡)=(−1)^𝑛−𝑘𝐷_𝑛−𝑘𝑇_𝑘,0≤𝑘≤𝑛,0,𝑘>𝑛. (22) Then, the sequence {𝑏^(𝑘)(𝑡)}_𝑘∈ℕ is a basis for the space 𝑁^𝑡(𝑝) and any 𝑥∈𝑁^𝑡(𝑝) has a unique representation of the form 𝑥=_𝑘𝜆_𝑘(𝑡)𝑏^(𝑘)(𝑡), (23) where 𝜆_𝑘(𝑡)=(𝑁^𝑡𝑥)_𝑘 for all  𝑘∈ℕ and 0<𝑝_𝑘≤𝐻<∞.

Proof. It is clear that {𝑏^(𝑘)(𝑡)}⊂𝑁^𝑡(𝑝), since 𝑁^𝑡𝑏^(𝑘)(𝑡)=𝑒^(𝑘)∈ℓ𝑝∀𝑘∈ℕ, (24) where 𝑒^(𝑘) is the sequence whose only nonzero term is a 1 in the 𝑘th place for each 𝑘∈ℕ and 0<𝑝_𝑘≤𝐻<∞.
Let 𝑥∈𝑁^𝑡(𝑝) be given. For every nonnegative integer 𝑚, we put 𝑥^[𝑚]=^𝑚_𝑘=0𝜆_𝑘(𝑡)𝑏^(𝑘)(𝑡). (25) Then, we obtain by applying 𝑁^𝑡 to (25) with (24) that 𝑁^𝑡𝑥^[𝑚]=^𝑚_𝑘=0𝜆_𝑘(𝑡)𝑁^𝑡𝑏^(𝑘)(𝑡)=^𝑚_𝑘=0𝑁^𝑡𝑥_𝑘𝑒^(𝑘),𝑁^𝑡𝑥−𝑥^[𝑚]_𝑖=𝑁0,0≤𝑖≤𝑚,_𝑡𝑥_𝑖,𝑖>𝑚, (26) where 𝑖,𝑚∈ℕ. Given 𝜖>0, then there is an integer 𝑚₀ such that ^∞_𝑖=𝑚+1𝑁^𝑡𝑥_𝑖^{𝑝𝑘1/𝑀}<𝜖 (27) for all (𝑚+1)≥𝑚₀. Hence, 𝑔𝑁^𝑡𝑥−𝑥^[𝑚]=^∞_𝑖=𝑚+1𝑁^𝑡𝑥_𝑖^{𝑝𝑘1/𝑀}≤^∞_𝑖=𝑚0𝑁^𝑡𝑥_𝑖^{𝑝𝑘1/𝑀}<𝜖 (28) for all (𝑚+1)≥𝑚₀ which proves that 𝑥∈𝑁^𝑡(𝑝) is represented as in (23).
Let us show the uniqueness of the representation for 𝑥∈𝑁^𝑡(𝑝) given by (23). Suppose, on the contrary, that there exists a representation 𝑥=∑_𝑘𝜇_𝑘(𝑡)𝑏^(𝑘)(𝑡). Since the linear transformation 𝑇, from 𝑁^𝑡(𝑝) to ℓ(𝑝), used in the proof of Theorem 3 is continuous, we have at this stage that 𝑁^𝑡𝑥_𝑛=_𝑘𝜇_𝑘(𝑡)𝑁^𝑡𝑏^(𝑘)(𝑡)_𝑛=_𝑘𝜇_𝑘(𝑡)𝑒_𝑛^(𝑘)=𝜇_𝑛(𝑡) (29) for all 𝑛∈ℕ which contradicts the fact that (𝑁^𝑡𝑥)_𝑛=𝜆_𝑛(𝑡) for all 𝑛∈ℕ. Hence, the representation (23) of 𝑥∈𝑁^𝑡(𝑝) is unique. This completes the proof.

3. The Alpha-, Beta-, and Gamma-Duals of the Space 𝑁^𝑡(𝑝)

In this section, we determine the alpha-, beta-, and gamma-duals of the space 𝑁^𝑡(𝑝). We will quote some lemmas which are needed in proving our theorems.

Lemma 5 (see [11], Theorem  5.1.0). The following statements hold.(i)Let 1<𝑝_𝑘≤𝐻<∞ for every 𝑘∈ℕ. Then, 𝐴=(𝑎_𝑛𝑘)∈(ℓ(𝑝)∶ℓ₁) if and only if there exists an integer 𝐵>1 such that sup_{𝑁∈ℱ𝑘𝑛∈𝑁}𝑎_𝑛𝑘𝐵^{−1𝑝′𝑘}<∞. (30)(ii)Let 0<𝑝_𝑘≤1 for every 𝑘∈ℕ. Then, 𝐴=(𝑎_𝑛𝑘)∈(ℓ(𝑝)∶ℓ₁) if and only if sup_𝑁∈ℱsup_{𝑘∈ℕ𝑛∈𝑁}𝑎_𝑛𝑘^𝑝𝑘<∞. (31)

Lemma 6 (see [12], Theorem  1). The following statements hold.(i)Let 1<𝑝_𝑘≤𝐻<∞ for every 𝑘∈ℕ. Then, 𝐴=(𝑎_𝑛𝑘)∈(ℓ(𝑝)∶ℓ_∞) if and only if there exists an integer 𝐵>1 such that sup_{𝑛∈ℕ𝑘}𝑎_𝑛𝑘𝐵^{−1𝑝′𝑘}<∞. (32)(ii)Let 0<𝑝_𝑘≤1 for every 𝑘∈ℕ. Then, 𝐴=(𝑎_𝑛𝑘)∈(ℓ(𝑝)∶ℓ_∞) if and only if sup_{𝑛,𝑘∈ℕ}𝑎_𝑛𝑘^𝑝𝑘<∞. (33)

Lemma 7 (see [12], Theorem  1). Let 0<𝑝_𝑘≤𝐻<∞ for every 𝑘∈ℕ. Then, 𝐴=(𝑎_𝑛𝑘)∈(ℓ(𝑝)∶𝑐) if and only if (32), (33) hold and there is 𝛽_𝑘∈ℂ such that 𝑎_𝑛𝑘→𝛽_𝑘 for each 𝑘∈ℕ.

Theorem 8. Let 1<𝑝_𝑘≤𝐻<∞ for every 𝑘∈ℕ. Define the sets 𝐷₁(𝑝), 𝐷₂(𝑝), and 𝐷₃(𝑝) as follows: 𝐷₁𝑝∶=𝑎=𝑎_𝑘∈𝜔∶sup_{𝑁∈ℱ𝑘𝑛∈𝑁}(−1)^𝑛−𝑘𝑎_𝑛𝐷_𝑛−𝑘𝑇_𝑘𝐵^{−1𝑝′𝑘},𝐷<∞₂𝑝∶=𝑎=𝑎_𝑘∈𝜔∶sup_{𝑛∈ℕ𝑘}^𝑛_𝑖=𝑘(−1)^𝑖−𝑘𝑎_𝑖𝐷_𝑖−𝑘𝑇_𝑘𝐵^{−1𝑝′𝑘}<∞,𝑎_𝑛𝑇_𝑛𝐵^{−1𝑝′𝑘}∈ℓ_∞,𝐷₃𝑝=𝑐𝑠. (34) Then, the following statements hold:(i){𝑁^𝑡(𝑝)}^𝛼=𝐷₁(𝑝);(ii){𝑁^𝑡(𝑝)}^𝛾=𝐷₂(𝑝);(iii){𝑁^𝑡(𝑝)}^𝛽=𝐷₂(𝑝)∩𝐷₃(𝑝).

Proof. (i) Let us take 𝑎=(𝑎_𝑘)∈𝜔. We easily derive with (20) that 𝑎_𝑛𝑥_𝑛=^𝑛_𝑘=0(−1)^𝑛−𝑘𝑎_𝑛𝐷_𝑛−𝑘𝑇_𝑘𝑦_𝑘=𝐶𝑦_𝑛∀𝑛∈ℕ, (35) where 𝐶=(𝑐_𝑛𝑘) is defined by 𝑐_𝑛𝑘=(−1)^𝑛−𝑘𝑎_𝑛𝐷_𝑛−𝑘𝑇_𝑘,0≤𝑘≤𝑛,0,𝑘>𝑛 (36) for all 𝑘,𝑛∈ℕ. Thus, we observe by combining (35) with Part (i) of Lemma 5 that 𝑎𝑥=(𝑎_𝑛𝑥_𝑛)∈ℓ₁ whenever 𝑥=(𝑥_𝑘)∈𝑁^𝑡(𝑝) if and only if 𝐶𝑦∈ℓ₁ whenever 𝑦=(𝑦_𝑘)∈ℓ(𝑝). This gives the result that {𝑁^𝑡(𝑝)}^𝛼=𝐷₁(𝑝).
(ii) Consider the equality ^𝑛_𝑘=0𝑎_𝑘𝑥_𝑘=^𝑛−1𝑛_{𝑘=0𝑖=𝑘}(−1)^𝑖−𝑘𝑎_𝑖𝐷_𝑖−𝑘𝑇_𝑘𝑦_𝑘+𝑎_𝑛𝑇_𝑛𝑦_𝑛=𝐸𝑦_𝑛∀𝑛∈ℕ, (37) where 𝐸=(𝑒_𝑛𝑘) is defined by 𝑒_𝑛𝑘=^𝑛_𝑖∑_=𝑘(−1)^𝑖−𝑘𝑎_𝑖𝐷_𝑖−𝑘𝑇_𝑘𝑎,0≤𝑘≤𝑛−1,_𝑛𝑇_𝑛,𝑘=𝑛,0,𝑘>𝑛 (38) for all 𝑘,𝑛∈ℕ. Thus, we deduce from Part (i) of Lemma 6 with (37) that 𝑎𝑥=(𝑎_𝑛𝑥_𝑛)∈𝑏𝑠 whenever 𝑥=(𝑥_𝑘)∈𝑁^𝑡(𝑝) if and only if 𝐸𝑦∈ℓ_∞ whenever 𝑦=(𝑦_𝑘)∈ℓ(𝑝). Therefore, we obtain from Part (i) of Lemma 6 that {𝑁^𝑡(𝑝)}^𝛾=𝐷₂(𝑝).
(iii) We see from Lemma 7 that 𝑎𝑥=(𝑎_𝑛𝑥_𝑛)∈𝑐𝑠 whenever 𝑥=(𝑥_𝑘)∈𝑁^𝑡(𝑝) if and only if 𝐸𝑦∈𝑐 whenever 𝑦=(𝑦_𝑘)∈ℓ(𝑝). Therefore, we derive from Lemma 7 that {𝑁^𝑡(𝑝)}^𝛽=𝐷₂(𝑝)∩𝐷₃(𝑝).
Therefore, the proof is completed.

Theorem 9. Let 0<𝑝_𝑘≤1 for every 𝑘∈ℕ. Define the sets 𝐷₄(𝑝) and 𝐷₅(𝑝) by 𝐷₄𝑝∶=𝑎=𝑎_𝑘∈𝜔∶sup_𝑁∈ℱsup_{𝑘∈ℕ𝑛∈𝑁}(−1)^𝑛−𝑘𝑎_𝑛𝐷_𝑛−𝑘𝑇_𝑘^𝑝𝑘,𝐷<∞₅𝑝∶=𝑎=𝑎_𝑘∈𝜔∶sup_{𝑛,𝑘∈ℕ}^𝑛_𝑖=𝑘(−1)^𝑖−𝑘𝑎_𝑖𝐷_𝑖−𝑘𝑇_𝑘^𝑝𝑘<∞,𝑎_𝑛𝑇_𝑛^𝑝𝑘∈ℓ_∞. (39) Then, the following statements hold:(i){𝑁^𝑡(𝑝)}^𝛼=𝐷₄(𝑝);(ii){𝑁^𝑡(𝑝)}^𝛾=𝐷₅(𝑝);(iii){𝑁^𝑡(𝑝)}^𝛽=𝐷₃(𝑝)∩𝐷₅(𝑝).

Proof. This is easily obtained by proceeding as in the proof of Theorem 8 by using Lemma 7 and the second parts of Lemmas 5 and 6 instead of the first parts. So, we omit the detail.

4. Some Matrix Transformations Related to the Sequence Space 𝑁^𝑡(𝑝)

In the present section, we characterize the matrix transformations from the space 𝑁^𝑡(𝑝) into any given sequence space 𝜇 and from a given sequence space 𝜇 into the space 𝑁^𝑡(𝑝). Since 𝜇_𝐴≅𝜇 for any triangle 𝐴 and any sequence space 𝜇, it is trivial that the equivalence “𝑥∈𝜇_𝐴 if and only if 𝑦=𝐴𝑥∈𝜇” holds.

Now, we can give the following theorem.

Theorem 10. Suppose that the elements of the infinite matrices 𝐴=(𝑎_𝑛𝑘) and 𝐹=(𝑓_𝑛𝑘) are connected with the relation 𝑓_𝑛𝑘∶=^∞_𝑗=𝑘(−1)^𝑗−𝑘𝐷_𝑗−𝑘𝑇_𝑘𝑎_𝑛𝑗 (40) for all 𝑘,𝑛∈ℕ and 𝜇 is any given sequence space. Then, 𝐴∈(𝑁^𝑡(𝑝)∶𝜇) if and only if 𝐴_𝑛∈{𝑁^𝑡(𝑝)}^𝛽 for all 𝑛∈ℕ and 𝐹∈(ℓ(𝑝)∶𝜇).

Proof. Let 𝜇 be any given sequence space. Suppose that (40) holds between the elements of the matrices 𝐴=(𝑎_𝑛𝑘) and 𝐹=(𝑓_𝑛𝑘), and take into account that the spaces 𝑁^𝑡(𝑝) and ℓ(𝑝) are linearly isomorphic.
Let 𝐴∈(𝑁^𝑡(𝑝)∶𝜇) and take any 𝑦∈ℓ(𝑝). Then 𝐹𝑁^𝑡_𝑛𝑘=^∞_𝑗=𝑘𝑓_𝑛𝑗𝑎^𝑡_𝑗𝑘=^∞∞_{𝑗=𝑘𝑖=𝑗}(−1)^𝑖−𝑗𝐷_𝑖−𝑗𝑎_𝑛𝑖𝑇_𝑗𝑡_𝑗−𝑘𝑇_𝑗=𝑎_𝑛𝑘. (41) That is, 𝐹𝑁^𝑡 exists and 𝐴_𝑛∈{𝑁^𝑡(𝑝)}^𝛽 which yields that 𝐹_𝑛∈ℓ₁ for each 𝑛∈ℕ. Hence, 𝐹𝑦 exists and thus _𝑘𝑓_𝑛𝑘𝑦_𝑘=_𝑘^∞_𝑖=𝑘(−1)^𝑖−𝑘𝐷_𝑖−𝑘𝑎_𝑛𝑖𝑇_𝑘×1𝑇_𝑘^𝑘_𝑗=0𝑡_𝑘−𝑗𝑥_𝑗=_𝑘𝑎_𝑛𝑘𝑥_𝑘 (42) for all 𝑛∈ℕ. So, we have that 𝐹𝑦=𝐴𝑥, which leads us to the consequence 𝐹∈(ℓ(𝑝)∶𝜇).
Conversely, let 𝐴_𝑛∈{𝑁^𝑡(𝑝)}^𝛽 for each 𝑛∈ℕ and 𝐹∈(ℓ(𝑝)∶𝜇), and take 𝑥=(𝑥_𝑘)∈𝑁^𝑡(𝑝). Then, 𝐴𝑥 exists. Therefore, we obtain from the equality _𝑘𝑎_𝑛𝑘𝑥_𝑘=_𝑘𝑎_𝑛𝑘^𝑘_𝑖=0(−1)^𝑘−𝑖𝐷_𝑘−𝑖𝑇_𝑖𝑦_𝑖=_𝑘𝑓_𝑛𝑘𝑦_𝑘∀𝑛∈ℕ (43) that 𝐴𝑥=𝐹𝑦 and this shows that 𝐴∈(𝑁^𝑡(𝑝)∶𝜇). This completes the proof.

By changing the roles of the spaces 𝑁^𝑡(𝑝) with 𝜇 in Theorem 10, we have the following.

Theorem 11. Suppose that 𝜇 is any given sequence space and the elements of the infinite matrices 𝐴=(𝑎_𝑛𝑘) and 𝐺=(𝑔_𝑛𝑘) are connected with the relation 𝑔_𝑛𝑘=∑^𝑛_𝑗=0(𝑡_𝑛−𝑗/𝑇_𝑛)𝑎_𝑗𝑘 for all 𝑘,𝑛∈ℕ. Then, 𝐴∈(𝜇∶𝑁^𝑡(𝑝)) if and only if 𝐺∈(𝜇∶ℓ(𝑝)).

Proof. Let 𝑥=(𝑥_𝑘)∈𝜇 and consider the following equality: ^𝑛_𝑗=0𝑡_𝑛−𝑗𝑇_𝑛^𝑚_𝑘=0𝑎_𝑗𝑘𝑥_𝑘=^𝑚_𝑘=0𝑔_𝑛𝑘𝑥_𝑘∀𝑛∈ℕ. (44) Then, by letting 𝑚→∞ in (44), we have {𝑁^𝑡(𝐴𝑥)}_𝑛=(𝐺𝑥)_𝑛 for all 𝑛∈ℕ. Since 𝐴𝑥∈𝑁^𝑡(𝑝), 𝑁^𝑡(𝐴𝑥)=𝐺𝑥∈ℓ(𝑝). This completes the proof.

5. The Rotundity of the Space 𝑁^𝑡(𝑝)

In functional analysis, the rotundity of Banach spaces is one of the most important geometric properties. For details, the reader may refer to [13–15]. In this section, we give the necessary and sufficient condition in order to the space 𝑁^𝑡(𝑝) be rotund and present some results related to this concept.

Definition 12. Let 𝑆(𝑋) be the unit sphere of a Banach space 𝑋. Then, a point 𝑥∈𝑆(𝑋) is called an extreme point if 2𝑥=𝑦+𝑧 implies 𝑦=𝑧 for every 𝑦,𝑧∈𝑆(𝑋). A Banach space 𝑋 is said to be rotund (strictly convex) if every point of 𝑆(𝑋) is an extreme point.

Definition 13. A Banach space 𝑋 is said to have Kadec-Klee property (or property (𝐻)) if every weakly convergent sequence on the unit sphere is convergent in norm.

Definition 14. A Banach space 𝑋 is said to have(i)the Opial property if every sequence (𝑥_𝑛) weakly convergent to 𝑥₀∈𝑋 satisfies liminf_𝑛→∞𝑥_𝑛−𝑥₀<liminf_𝑛→∞𝑥_𝑛+𝑥 (45) for every 𝑥∈𝑋 with 𝑥≠𝑥₀;(ii)the uniform Opial property if for each 𝜖>0, there exists an 𝑟>0 such that 1+𝑟≤liminf_𝑛→∞𝑥_𝑛+𝑥 (46) for each 𝑥∈𝑋 with ‖𝑥‖≥𝜖 and each sequence (𝑥_𝑛) in 𝑋 such that 𝑥−→_𝑛^𝑤0 and liminf_𝑛→∞‖𝑥_𝑛‖≥1.

Definition 15. Let 𝑋 be a real vector space. A functional 𝜎∶𝑋→[0,∞) is called a modular if(i)𝜎(𝑥)=0 if and only if 𝑥=𝜃;(ii)𝜎(𝛼𝑥)=𝜎(𝑥) for all scalars 𝛼 with |𝛼|=1;(iii)𝜎(𝛼𝑥+𝛽𝑦)≤𝜎(𝑥)+𝜎(𝑦) for all 𝑥,𝑦∈𝑋 and 𝛼,𝛽≥0 with 𝛼+𝛽=1;(iv)the modular 𝜎 is called convex if 𝜎(𝛼𝑥+𝛽𝑦)≤𝛼𝜎(𝑥)+𝛽𝜎(𝑦) for all 𝑥,𝑦∈𝑋 and 𝛼,𝛽>0 with 𝛼+𝛽=1. A modular 𝜎 on 𝑋 is called(a)right continuous if lim_𝛼→1⁺𝜎(𝛼𝑥)=𝜎(𝑥) for all 𝑥∈𝑋_𝜎;(b)left continuous if lim_𝛼→1⁻𝜎(𝛼𝑥)=𝜎(𝑥) for all 𝑥∈𝑋_𝜎;(c)continuous if it is both right and left continuous, where 𝑋_𝜎=𝑥∈𝑋∶lim_𝛼→0+𝜎(𝛼𝑥)=0. (47)

We define 𝜎_𝑝 on 𝑁^𝑡(𝑝) by 𝜎_𝑝(𝑥)=∑_𝑘|(1/𝑇_𝑘)∑^𝑘_𝑗=0𝑡_𝑘−𝑗𝑥_𝑗|^𝑝𝑘. If 𝑝_𝑘≥1 for all 𝑘∈ℕ₁={1,2,…}, by the convexity of the function 𝑡↦|𝑡|^𝑝𝑘 for each 𝑘∈ℕ, 𝜎_𝑝 is a convex modular on 𝑁^𝑡(𝑝). We consider 𝑁^𝑡(𝑝) equipped with Luxemburg norm given by ‖𝑥‖=inf𝛼>0∶𝜎_𝑝𝑥𝛼≤1. (48)𝑁^𝑡(𝑝) is a Banach space with this norm. This can be shown by the similar way used in the proof of Theorem 7 in [16].

We establish some basic properties for the modular 𝜎_𝑝.

Proposition 16. The modular 𝜎_𝑝 on 𝑁^𝑡(𝑝) satisfies the following properties with 𝑝_𝑘≥1 for all 𝑘∈ℕ.(i)If 0<𝛼≤1, then 𝛼^𝑀𝜎_𝑝(𝑥/𝛼)≤𝜎_𝑝(𝑥) and 𝜎_𝑝(𝛼𝑥)≤𝛼𝜎_𝑝(𝑥).(ii)If 𝛼≥1, then 𝜎_𝑝(𝑥)≤𝛼^𝑀𝜎_𝑝(𝑥/𝛼).(iii)If 𝛼≥1, then 𝛼𝜎_𝑝(𝑥/𝛼)≤𝜎_𝑝(𝑥).(iv)The modular 𝜎_𝑝 is continuous.

Proof. (i) Let 0<𝛼≤1. Then 𝛼^𝑀/𝛼^𝑝𝑘≤1 for all 𝑝_𝑘≥1. So, we have 𝛼^𝑀𝜎_𝑝𝑥𝛼=_𝑘𝛼^𝑀𝛼^𝑝𝑘1𝑇_𝑘^𝑘_𝑗=0𝑡_𝑘−𝑗𝑥_𝑗^𝑝𝑘≤_𝑘1𝑇_𝑘^𝑘_𝑗=0𝑡_𝑘−𝑗𝑥_𝑗^𝑝𝑘=𝜎_𝑝(𝑥),𝜎_𝑝(𝛼𝑥)=_𝑘𝛼^𝑝𝑘1𝑇_𝑘^𝑘_𝑗=0𝑡_𝑘−𝑗𝑥_𝑗^𝑝𝑘≤𝛼_𝑘1𝑇_𝑘^𝑘_𝑗=0𝑡_𝑘−𝑗𝑥_𝑗^𝑝𝑘=𝛼𝜎_𝑝(𝑥). (49)
(ii) Let 𝛼≥1. Then 1≤𝛼^𝑀/𝛼^𝑝𝑘 for all 𝑝_𝑘≥1. So, we have 𝜎_𝑝(𝑥)≤𝛼^𝑀𝛼^𝑝𝑘𝜎_𝑝(𝑥)=𝛼^𝑀𝜎_𝑝𝑥𝛼. (50)
(iii) Let 𝛼≥1. Then 𝛼/𝛼^𝑝𝑘≤1 for all 𝑝_𝑘≥1. Therefore, one can easily see that 𝛼𝜎_𝑝𝑥𝛼=_𝑘𝛼𝛼^𝑝𝑘1𝑇_𝑘^𝑘_𝑗=0𝑡_𝑘−𝑗𝑥_𝑗^𝑝𝑘≤_𝑘1𝑇_𝑘^𝑘_𝑗=0𝑡_𝑘−𝑗𝑥_𝑗^𝑝𝑘=𝜎_𝑝(𝑥). (51)
(iv) If 𝛼>1, then we have _𝑘𝛼1𝑇_𝑘^𝑘_𝑗=0𝑡_𝑘−𝑗𝑥_𝑗^𝑝𝑘≤_𝑘𝛼^𝑝𝑘1𝑇_𝑘^𝑘_𝑗=0𝑡_𝑘−𝑗𝑥_𝑗^𝑝𝑘≤_𝑘𝛼^𝑀1𝑇_𝑘^𝑘_𝑗=0𝑡_𝑘−𝑗𝑥_𝑗^𝑝𝑘; (52) that is, 𝛼𝜎_𝑝(𝑥)≤𝜎_𝑝(𝛼𝑥)≤𝛼^𝑀𝜎_𝑝(𝑥). (53) By passing to limit as 𝛼→1⁺ in (53), we have 𝜎_𝑝(𝛼𝑥)→𝜎_𝑝(𝑥). Hence, 𝜎_𝑝 is right continuous.
If 0<𝛼<1, we have _𝑘𝛼^𝑀1𝑇_𝑘^𝑘_𝑗=0𝑡_𝑘−𝑗𝑥_𝑗^𝑝𝑘≤_𝑘𝛼^𝑝𝑘1𝑇_𝑘^𝑘_𝑗=0𝑡_𝑘−𝑗𝑥_𝑗^𝑝𝑘≤_𝑘𝛼1𝑇_𝑘^𝑘_𝑗=0𝑡_𝑘−𝑗𝑥_𝑗^𝑝𝑘; (54) that is, 𝛼^𝑀𝜎_𝑝(𝑥)≤𝜎_𝑝(𝛼𝑥)≤𝛼𝜎_𝑝(𝑥). (55) By letting 𝛼→1⁻ in (55), we have 𝜎_𝑝(𝛼𝑥)→𝜎_𝑝(𝑥). Hence, 𝜎_𝑝 is left continuous. Since 𝜎_𝑝 is both right and left continuous, it is continuous.

Now, we give some relationships between the modular 𝜎_𝑝 and the Luxemburg norm on 𝑁^𝑡(𝑝).

Proposition 17. For any 𝑥∈𝑁^𝑡(𝑝), the following statements hold.(i)If ‖𝑥‖<1, then 𝜎_𝑝(𝑥)≤‖𝑥‖.(ii)If ‖𝑥‖>1, then 𝜎_𝑝(𝑥)≥‖𝑥‖.(iii)‖𝑥‖=1 if and only if 𝜎_𝑝(𝑥)=1.(iv)‖𝑥‖<1 if and only if 𝜎_𝑝(𝑥)<1.(v)‖𝑥‖>1 if and only if 𝜎_𝑝(𝑥)>1.(vi)If 0<𝛼<1 and ‖𝑥‖>𝛼, then 𝜎_𝑝(𝑥)>𝛼^𝑀.(vii)If 𝛼≥1 and ‖𝑥‖<𝛼, then 𝜎_𝑝(𝑥)<𝛼^𝑀.

Proof. Let 𝑥∈𝑁^𝑡(𝑝).(i)Let 𝜖>0 such that 0<𝜖<1−‖𝑥‖. By the definition of ‖⋅‖ in (48), there exists an 𝛼>0 such that ‖𝑥‖+𝜖>𝛼 and 𝜎_𝑝(𝑥/𝛼)≤1. So, we have 𝜎_𝑝(𝑥)≤_𝑘‖𝑥‖+𝜖𝛼^𝑝𝑘1𝑇_𝑘^𝑘_𝑗=0𝑡_𝑘−𝑗𝑥_𝑗^𝑝𝑘≤(‖𝑥‖+𝜖)𝜎_𝑝𝑥𝛼≤‖𝑥‖+𝜖. (56) Since 𝜖 is arbitrary, we have 𝜎_𝑝(𝑥)≤‖𝑥‖ from (56).(ii) If we choose 𝜖>0 such that 0<𝜖<1−1/‖𝑥‖, then 1<(1−𝜖)‖𝑥‖<‖𝑥‖. By the definition of ‖⋅‖ in (48) and Part (iii) of Proposition 16, we have 1<𝜎_𝑝𝑥(𝑥‖≤11−𝜖)‖(𝑥‖𝜎1−𝜖)‖^𝑝(𝑥). (57) So, (1−𝜖)‖𝑥‖<‖𝑥‖ for all 𝜖∈(0,1−(1/‖𝑥‖)). This implies that ‖𝑥‖<𝜎_𝑝(𝑥).(iii)Since 𝜎_𝑝 is continuous, by Theorem 1.4 of [15] we directly have (iii).(iv)This follows from Parts (i) and (iii).(v)This follows from Parts (ii) and (iii).(vi)This follows from Part (ii) and Part (i) of Proposition 16.(vii)This follows from Part (i) and Part (ii) of Proposition 16.

Theorem 18. The space 𝑁^𝑡(𝑝) is rotund if and only if 𝑝_𝑘>1 for all 𝑘∈ℕ.

Proof. Let 𝑁^𝑡(𝑝) be rotund and choose 𝑘∈ℕ such that 𝑝_𝑘=1 for all 𝑘<3. Consider the following sequences given by 𝑥=1,−𝐷₁,𝐷₂,−𝐷₃,𝐷₄,…,𝑦=0,𝑇₁,−𝑇₁𝐷₁,𝑇₁𝐷₂,−𝑇₁𝐷₃,…. (58) Then, obviously 𝑥≠𝑦 and 𝜎_𝑝(𝑥)=𝜎_𝑝𝑦=𝜎_𝑝𝑥+𝑦2=1. (59) By Part (iii) of Proposition 17, 𝑥,𝑦,(𝑥+𝑦)/2∈𝑆[𝑁^𝑡(𝑝)] which leads us to the contradiction that the sequence space 𝑁^𝑡(𝑝) is not rotund. Hence, 𝑝_𝑘>1 for all 𝑘∈ℕ.
Conversely, let 𝑥∈𝑆[𝑁^𝑡(𝑝)] and 𝑣,𝑧∈𝑆[𝑁^𝑡(𝑝)] with 𝑥=(𝑣+𝑧)/2. By convexity of 𝜎_𝑝 and Part (iii) of Proposition 17, we have 1=𝜎_𝑝(𝑥)≤𝜎_𝑝(𝑣)+𝜎_𝑝(𝑧)2=1, (60) which gives that 𝜎_𝑝(𝑥)=𝜎_𝑝(𝑣)+𝜎_𝑝(𝑧)2. (61) Also, since 𝑥=(𝑣+𝑧)/2 and from (61), we obtain that _𝑘1𝑇_𝑘^𝑘_𝑗=0𝑡_𝑘−𝑗(𝑣_𝑗+𝑧_𝑗)2^𝑝𝑘=12_𝑘1𝑇_𝑘^𝑘_𝑗=0𝑡_𝑘−𝑗𝑣_𝑗^𝑝𝑘+_𝑘1𝑇_𝑘^𝑘_𝑗=0𝑡_𝑘−𝑗𝑧_𝑗^𝑝𝑘. (62) This implies that 𝑣_𝑗+𝑧_𝑗2^𝑝𝑘=𝑣_𝑗^𝑝𝑘+𝑧_𝑗^𝑝𝑘2 (63) for all 𝑘∈ℕ. Since the function 𝑡→|𝑡|^𝑝𝑘 is strictly convex for all 𝑘∈ℕ, it follows by (63) that 𝑣_𝑘=𝑧_𝑘 for all 𝑘∈ℕ. Hence, 𝑣=𝑧. That is, 𝑁^𝑡(𝑝) is rotund.

Theorem 19. Let (𝑥_𝑛) be a sequence in 𝑁^𝑡(𝑝). Then, the following statements hold:(i)lim_𝑛→∞‖𝑥_𝑛‖=1 implies lim_𝑛→∞𝜎_𝑝(𝑥_𝑛)=1;(ii)lim_𝑛→∞𝜎_𝑝(𝑥_𝑛)=0 implies lim_𝑛→∞‖𝑥_𝑛‖=0.

Proof. The proof is similar to that of Theorem 10 in [16].

Theorem 20. Let 𝑥∈𝑁^𝑡(𝑝) and (𝑥^(𝑛))⊂𝑁^𝑡(𝑝). If 𝜎_𝑝(𝑥^(𝑛))→𝜎_𝑝(𝑥) as 𝑛→∞ and 𝑥_𝑘^(𝑛)→𝑥_𝑘 as 𝑛→∞ for all 𝑘∈ℕ, then 𝑥^(𝑛)→𝑥 as 𝑛→∞.

Proof. Let 𝜖>0 be given. Since 𝑥∈𝑁^𝑡(𝑝) and (𝑥^(𝑛))⊂𝑁^𝑡(𝑝), 𝜎_𝑝(𝑥^(𝑛)−𝑥)=∑_𝑘|{𝑁^𝑡(𝑥^(𝑛)−𝑥)}_𝑘|^𝑝𝑘<∞. So, there exists an 𝑘₀∈ℕ such that ^∞_𝑘=𝑘0+1𝑁^𝑡𝑥^(𝑛)−𝑥_𝑘^𝑝𝑘<𝜖2. (64) Also, since 𝑥_𝑘^(𝑛)→𝑥_𝑘 as 𝑛→∞, we have ^𝑘0_𝑘=1𝑁^𝑡𝑥^(𝑛)−𝑥_𝑘^𝑝𝑘<𝜖2. (65) Therefore, we obtain from (64) and (65) that 𝜎_𝑝(𝑥^(𝑛)−𝑥)<𝜖. This means that 𝜎_𝑝(𝑥^(𝑛)−𝑥)→0 as 𝑛→∞. This result implies ‖𝑥^(𝑛)−𝑥‖→0 as 𝑛→∞ from Part (ii) of Theorem 19. Hence, 𝑥_𝑛→𝑥 as 𝑛→∞.

Theorem 21. The sequence space 𝑁^𝑡(𝑝) has the Kadec-Klee property.

Proof. Let 𝑥∈𝑆[𝑁^𝑡(𝑝)] and (𝑥^(𝑛))⊂𝑁^𝑡(𝑝) such that ‖𝑥^(𝑛)‖→1 and 𝑥^𝑤(𝑛)−→𝑥 are given. By Part (i) of Theorem 19, we have 𝜎_𝑝(𝑥^(𝑛))→1 as 𝑛→∞. Also, 𝑥∈𝑆[𝑁^𝑡(𝑝)] implies ‖𝑥‖=1. By Part (iii) of Proposition 17, we obtain 𝜎_𝑝(𝑥)=1. Therefore, we have 𝜎_𝑝(𝑥^(𝑛))→𝜎_𝑝(𝑥) as 𝑛→∞.
Since 𝑥^𝑤(𝑛)−→𝑥 and 𝑞_𝑘∶𝑁^𝑡(𝑝)→ℝ (or ℂ) defined by 𝑞_𝑘(𝑥)=𝑥_𝑘 is continuous, 𝑥_𝑘^(𝑛)→𝑥_𝑘 as 𝑛→∞. Therefore, 𝑥^(𝑛)→𝑥 as 𝑛→∞. This completes the proof.

Theorem 22. For any 1<𝑝<∞, the space 𝑋_𝑎(𝑝) has the uniform Opial property.

Proof. Since the proof can be given by the similar way used in proving Theorem 13 of Nergiz and Başar [16], we omit the detail.

6. Conclusion

Wang introduced the sequence space 𝑋_𝑎(𝑝), in [10]. Although the domain of several triangle matrices in the classical sequence spaces ℓ_𝑝, 𝑐₀, 𝑐, and ℓ_∞ and in the Maddox spaces ℓ(𝑝), 𝑐₀(𝑝), 𝑐(𝑝), and ℓ_∞(𝑝) was investigated by researchers, the domain of Nörlund mean neither in a normed sequence space nor in a paranormed sequence space was not studied and is still as an open problem. So, we have worked on the domain of Nörlund mean in the Maddox space ℓ(𝑝). Additionally, we emphasize on some geometric properties of the new space 𝑁^𝑡(𝑝). It is obvious that the matrix 𝑁^𝑡 is not comparable with the matrices 𝐸^𝑟, 𝐴^𝑟, or 𝐵(𝑟,𝑠). So, the present results are new.

It is clear that by depending on the choice of the sequence space 𝜇, the characterization of several classes of matrix transformations from the space 𝑁^𝑡(𝑝) and into the space 𝑁^𝑡(𝑝) can be obtained from Theorems 10 and 11, respectively. As a natural continuation of this paper, we will study the domain of the Nörlund mean in Maddox's spaces ℓ_∞(𝑝), 𝑐(𝑝), and 𝑐₀(𝑝).

Conflict of Interests

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