Some ℓ(𝑝)-Type New Sequence Spaces and Their Geometric Properties

Abstract

We introduce an ℓ(𝑝)-type new sequence space and investigate its some topological properties including 𝐴𝐾 and 𝐴𝐷 properties. Besides, we examine some geometric properties of this space concerning Banach-Saks type 𝑝 and Gurarii's modulus of convexity.

1. Introduction

In general, the ℓ(𝑝)-type spaces have many useful applications because of the properties of the spaces ℓ(𝑝) and ℓ_𝑝. In [1], it was shown that the subspaces of Orlicz spaces, which have rich geometric properties, are isomorphic to the space ℓ_𝑝. Also since the space ℓ_𝑝 is reflexive and convex, it is natural to consider the geometric structure of these spaces.

Recently there has been a lot of interest in investigating geometric properties of sequence spaces besides topological and some other usual properties. In literature, there are many papers concerning the geometric properties of different sequence spaces. For example; in [2], Mursaleen et al. studied some geometric properties of normed Euler sequence space. Sanhan and Suantai [3] investigated the geometric properties of Cesáro sequence space ces(p) equipped with Luxemburg norm. Further information on geometric properties of sequence spaces can be found in [4–7].

The main purpose of our work is to introduce an ℓ_𝑝-type new sequence space together with matrix domain and its summability methods. Also we investigate some topological properties of this new space as the paranorm, 𝐴𝐾 and 𝐴𝐷 properties, and furthermore characterize geometric properties concerning Banach-Saks type 𝑝 and Gurarii's modulus of convexity.

2. Preliminaries and Notations

Let 𝑤 be the space of all real-valued sequences. Each linear subspace of 𝑤 is called a sequence space denoted by 𝜆. We denote by ℓ₁ and ℓ_𝑝 absolutely and 𝑝-absolutely convergent series, respectively.

A sequence space 𝜆 with a linear topology is called a 𝐾-space provided that each of the maps 𝑝_𝑖∶𝜆→ℂ defined by 𝑝_𝑖(𝑥)=𝑥_𝑖 is continuous for all 𝑖∈ℕ, where ℂ denotes the complex field, and ℕ={0,1,2,3,…}. A 𝐾-space 𝜆 is called 𝐹𝐾-space provided that 𝜆 is a complete linear metric space. An 𝐹𝐾-space whom topology is normable is called 𝐵𝐾-space. An 𝐹𝐾-space 𝜆 is said to have 𝐴𝐾 property, if 𝜙⊂𝜆 and {𝑒^(𝑘)} is a basis for 𝜆, where 𝑒^(𝑘) is a sequence whose only nonzero term is 1, 𝑘th place for each 𝑘∈ℕ, and 𝜙=span{𝑒^(𝑘)}, the set of all 𝐴𝐷-space, thus 𝐴𝐾 implies 𝐴𝐷.

A linear topological space 𝑋 over the real field ℝ is said to be a paranormed space if there is a subadditivity function 𝑔∶𝑋→ℝ such that 𝑔(𝜃)=0, 𝑔(−𝑥)=𝑔(𝑥) and scalar multiplication is continuous. It is well known that the space ℓ_𝑝 is 𝐴𝐾-space where 1≤𝑝<∞.

Throughout this work, we suppose that (𝑝_𝑘) is a bounded sequence of strictly positive real numbers with sup𝑝_𝑘=𝐻 and 𝑀=max{1,𝐻}. Also the summation without limits runs from 0 to ∞. In [8], the linear space ℓ(𝑝) was defined by Maddox (see also Simons [9] and Nakano [10]) as follows: ℓ(𝑝)={𝑥=(𝑥_𝑘)∈𝑤∶_𝑛|𝑥_𝑛|^𝑝𝑛<∞}(2.1) which is a complete space paranormed by 𝑔(𝑥)=(_𝑛|𝑥_𝑛|^𝑝𝑛)^1/𝑀.(2.2)

Let 𝜆, 𝜇 be any two sequence spaces, and let 𝐴=(𝑎_𝑛𝑘) be an infinite matrix of real numbers 𝑎_𝑛𝑘, where 𝑛,𝑘∈ℕ. Then we write 𝐴𝑥=((𝐴𝑥)_𝑛), the 𝐴-transform of 𝑥, if (𝐴𝑥)_𝑛=∑_𝑘𝑎_𝑛𝑘𝑥_𝑘 converges for each 𝑛∈ℕ.

A matrix 𝐴=(𝑎_𝑛𝑘) is called a triangle if 𝑎_𝑛𝑘=0 for 𝑘>𝑛 and 𝑎_𝑛𝑛≠0 for all 𝑛∈ℕ. It is trivial that 𝐴(𝐵𝑥)=(𝐴𝐵)𝑥 holds for the triangle matrices 𝐴, 𝐵 and a sequence 𝑥. Further, a triangle matrix 𝑃 uniquely has an invert 𝑃⁻¹=𝑄 which is also a triangle matrix. Then if 𝑃𝑥=𝑦,𝑥=𝑃(𝑄𝑥)=𝑄(𝑃𝑥),𝑥=𝑄𝑦(2.3) hold for all 𝑥∈𝑤.

By (𝜆,𝜇), we denote the class of all infinite matrices 𝐴 such that 𝐴∶𝜆→𝜇. The matrix domain 𝜆_𝐴 of an infinite matrix 𝐴 in a sequence space 𝜆 is defined by 𝜆_𝐴={𝑥=(𝑥_𝑘)∈𝑤∶𝐴𝑥∈𝜆} which is a sequence space. It is well known that the new sequence space 𝜆_𝐴 generated by the limitation matrix 𝐴 from a sequence space 𝜆 is the expansion or the contraction of original space 𝜆.

If 𝐴 is triangle, then one can easily observe that the sequence spaces 𝜆_𝐴 and 𝜆 are linearly isomorphic, that is, 𝜆_𝐴=𝜆. Let 𝜆 be a sequence space. Then the continuous dual 𝜆^′_𝐴 of the space 𝜆_𝐴 is defined by 𝜆^′_𝐴={𝑓∶𝑓=𝑔∘𝐴;𝑔∈𝜆^′}. Let 𝑋 be a seminormed space. A set 𝑌⊂𝑋 is called fundamental set if the span of 𝑌 is dense in 𝑋. An application of Hahn-Banach theorem on fundamental set is as follows: if 𝑌 is the subset of a seminormed space 𝑋 and 𝑓(𝑌)=0 implies 𝑓=0 for 𝑓∈𝑋^′, then 𝑌 is a fundamental set (see [11]).

By the idea mentioned above, let us give the definitions of some matrices to construct a new sequence space in sequel to this work. We denote Δ=(𝛿_𝑛𝑘) and 𝑆=(𝑠_𝑛𝑘) by 𝛿_𝑛𝑘=(−1)^𝑛−𝑘,if𝑛−1≤𝑘≤𝑛,0,otherwise,𝑠_𝑛𝑘=1,if0≤𝑘≤𝑛,0,otherwise.(2.4) Malkowsky and Savas [12], Choudhary and Mishra [13], and Altay and Basar [14] have defined the sequence spaces 𝑍(𝑢,𝑣;𝑋), ℓ(𝑝), and ℓ(𝑢,𝑣;𝑝), respectively. By using the matrix domain, the spaces 𝑍(𝑢,𝑣;𝑋),ℓ(𝑝), and ℓ(𝑢,𝑣;𝑝) may be redefined by 𝑍(𝑢,𝑣;𝑋)=𝑋_{𝐺(𝑢,𝑣)},ℓ(𝑝)=(ℓ_𝑝)_𝑆, and ℓ(𝑢,𝑣;𝑝)=(ℓ(𝑝))_{𝐺(𝑢,𝑣)}, respectively.

If 𝜆⊂𝑤 is a sequence space and 𝑥=(𝑥_𝑘)∈𝜆, (𝑆𝑥)-transform with (2.4) corresponds to 𝑛th partial sum of the series ∑_𝑛𝑥_𝑛 and it is denoted by 𝑠=(𝑠_𝑛).

By using (2.4) and any infinite lower triangular matrix 𝐴, we can define two infinite lower triangular matrices 𝐴 and 𝐴 as follows: 𝐴=𝐴𝑆 and 𝐴 =Δ𝐴. Let 𝑥=(𝑥_𝑘) be a sequence in 𝜆. By considering the multiplication of infinite lower triangular matrices, we have 𝐴(𝑆𝑥)=𝐴𝑥, that is, 𝑡_𝑛=^𝑛_𝑣=0𝑎_𝑛𝑣𝑠_𝑣=^𝑛_𝑣=0𝑎_𝑛𝑣𝑥_𝑣.(2.5) Also since 𝐴=Δ𝐴, we have 𝐴𝑥=(Δ𝐴)𝑥, that is, 𝑡_𝑛−𝑡_𝑛−1=^𝑛_𝑣=0𝑎_𝑛𝑣𝑥_𝑣.(2.6) Now let us write the following equality: 𝑧_𝑛=(𝐴𝑥)_𝑛=^𝑛_𝑣=0𝑎_𝑛𝑣𝑥_𝑣.(2.7) It can be seen that for any sequences 𝑥, 𝑦 and scalar 𝛼∈ℝ, (𝐴(𝑥+𝑦))_𝑛=(𝐴𝑥)_𝑛+(𝐴𝑦)_𝑛 and (𝐴(𝛼𝑥))_𝑛=𝛼(𝐴𝑥)_𝑛. We now define new sequence space as follows: ℓ(𝐴;𝑝)={𝑥=(𝑥_𝑘)∈𝑤∶_𝑛|(𝐴𝑥)_𝑛|^𝑝𝑛<∞}.(2.8) For some special cases of the infinite lower triangular matrix 𝐴 and the sequence (𝑝_𝑘), we obtain the following spaces.

(i) If 𝑝_𝑘=𝑝 for all 𝑘∈ℕ, the space ℓ(𝐴;𝑝) reduces to the normed space ℓ_𝑝(𝐴) denoted by

ℓ_𝑝(𝐴)={𝑥=(𝑥_𝑘)∈𝑤∶_𝑛|^𝑛_𝑣=0𝑎_𝑛𝑣𝑥_𝑣|^𝑝<∞}.(2.9)

(ii) If 𝐴=(𝐶,1), which is Cesáro matrix order 1, then the space ℓ(𝐴;𝑝) corresponds to the space ℓ(𝐶;𝑝) denoted by ℓ(𝐶;𝑝)={𝑥=(𝑥_𝑘)∈𝑤∶_𝑛|(𝐶𝑥)_𝑛|^𝑝𝑛<∞},(2.10) where (𝐶𝑥)_𝑛=(1/𝑛(𝑛+1))∑^𝑛_𝑘=1𝑘𝑥_𝑘 for 𝑛≥1 and (𝐶𝑥)₀=𝑥₀.

(iii) If 𝐴=(𝑁,𝑝_𝑛), which is Nörlund type matrix, then the space ℓ(𝐴;𝑝) reduces to the space ℓ(𝑁;𝑝)=|𝑁,𝑝_𝑛|(𝑟) (see [15, 16]) denoted by ℓ(𝑁;𝑝)={𝑥=(𝑥_𝑘)∈𝑤∶_𝑛|(𝑁𝑥)_𝑛|^𝑝𝑛<∞},(2.11) where (𝑁𝑥)_𝑛=(𝑝_𝑛/𝑃_𝑛𝑃_𝑛−1)∑^𝑛_𝑘=1𝑃_𝑘−1𝑥_𝑘 for 𝑛≥1 and (𝑁𝑥)₀=𝑥₀.

Also if 𝑝_𝑘=𝑝 for all 𝑘∈ℕ, then the spaces ℓ(𝐶;𝑝) and ℓ(𝑁;𝑝)=|𝑁,𝑝_𝑛|(𝑟) reduce to the spaces ℓ_𝑝(𝐶) and ℓ_𝑝(𝑁)=|𝑁_𝑝| (see [17]), respectively.

Now let us introduce some definitions of geometric properties of sequence spaces.

Let (𝑋,‖⋅‖) be a normed linear space, and let 𝑆(𝑋) and 𝐵(𝑋) be the unit sphere and unit ball of 𝑋 (for the brevity 𝑋=(𝑋,‖⋅‖)), respectively. Consider Clarkson's modulus of convexity (Clarkson [18] and Day [19]) defined by 𝛿_𝑋(𝜀)=inf{1−‖𝑥+𝑦‖2;𝑥,𝑦∈𝑆(𝑋),‖𝑥−𝑦‖=𝜀},(2.12) where 0≤𝜀≤2. The inequality 𝛿_𝑋(𝜀)>0 for all 𝜀∈(0,2] characterizes the uniformly convex spaces.

In [20], Gurariĭ's modulus of convexity is defined by 𝛽_𝑋(𝜀)=inf{1−inf_𝛼∈[0,1]‖𝛼𝑥+(1−𝛼)𝑦‖;𝑥,𝑦∈𝑆(𝑋),‖𝑥−𝑦‖=𝜀},(2.13) where 0≤𝜀≤2. It is easily shown that 𝛿_𝑋(𝜀)≤𝛽_𝑋(𝜀)≤2𝛿_𝑋(𝜀) for any 0≤𝜀≤2. Also if 0<𝛽_𝑋(𝜀)<1, then 𝑋 is uniformly convex, and if 𝛽_𝑋(𝜀)<1, then 𝑋 is strictly convex.

A Banach space 𝑋 is said to have the Banach-Saks property if every bounded sequence (𝑥_𝑛) in 𝑋 admits a subsequence (𝑧_𝑛) such that the sequence {𝑡_𝑘(𝑧)} is convergent in the norm in 𝑋 (see [21]), where 𝑡_𝑘1(𝑧)=𝑘+1(𝑧₀+𝑧₁+𝑧₂+⋯+𝑧_𝑘)(𝑘∈ℕ).(2.14)

Let 1<𝑝<∞. A Banach space is said to have the Banach-Saks type   𝑝 or property (𝐵𝑆_𝑝), if every weakly null sequence (𝑥_𝑘) has a subsequence (𝑥_𝑘𝑙) such that for some 𝐶>0‖^𝑛_𝑙=0𝑥_𝑘𝑙‖<𝐶(𝑛+1)^1/𝑝(2.15) for all 𝑛∈ℕ (see [22]).

3. Some Topological Properties of the Space ℓ(𝐴;𝑝)

In this section, we investigate some topological properties of the sequence space ℓ(𝐴;𝑝) as the paranorm 𝐴𝐾 property and 𝐴𝐷 property. Let us begin the following theorem.

Theorem 3.1. (i) The space ℓ(𝐴;𝑝) is complete linear metric space with respect to the paranorm defined by ℎ(𝑥)=(_𝑛|(𝐴𝑥)_𝑛|^𝑝𝑛)^1/𝑀.(3.1)
(ii) If the sequence (𝑝_𝑛) is constant sequence and 𝑝≥1, then ℓ_𝑝(𝐴) is a Banach space normed by ‖𝑧‖_ℓ𝑝=‖𝑥‖_ℓ𝑝(𝐴₎=(_𝑛|(𝐴𝑥)_𝑛|^𝑝)^1/𝑝.(3.2)

Proof. The proof of (ii) is routine verification by using standard techniques and hence it is omitted.
The proof of (i) is that the linearity of ℓ(𝐴;𝑝) with respect to the coordinatewise addition and scalar multiplication follows from the following inequalities which are satisfied for 𝑥,𝑦∈ℓ(𝐴;𝑝): (_𝑛|(𝐴(𝑥+𝑦))_𝑛|^𝑝𝑛)^1/𝑀≤(_𝑛|(𝐴𝑥)_𝑛|^𝑝𝑛)^1/𝑀+(_𝑛|(𝐴𝑦)_𝑛|^𝑝𝑛)^1/𝑀(3.3) and |𝛼|^𝑝𝑛≤max{1,|𝛼|^𝑀} for any 𝛼∈ℝ (see [23]). After this step, we must show that the space ℓ(𝐴;𝑝) holds the paranorm property and the completeness with respect to given paranorm. It is easy to show that ℎ(𝜃)=0, and ℎ(𝑥)=ℎ(−𝑥) for all 𝑥∈ℓ(𝐴;𝑝). Besides, from (3.3) we obtain ℎ(𝑥+𝑦)≤ℎ(𝑥)+ℎ(𝑦) for all 𝑥,𝑦∈ℓ(𝐴;𝑝). To complete the paranorm conditions for the space ℓ(𝐴;𝑝), it remains to show the continuity of the scalar multiplication. Let (𝑥^𝑚) be any sequence in ℓ(𝐴;𝑝) such that ℎ(𝑥^𝑚−𝑥)→0, and let (𝛼_𝑚) be also any sequence of scalars such that |𝛼_𝑚−𝛼|→0(𝑚→∞). From subadditivity of ℎ, we give the inequality ℎ(𝑥^𝑚)≤ℎ(𝑥)+ℎ(𝑥^𝑚−𝑥). Hence {ℎ(𝑥^𝑚)} is bounded and we have ℎ(𝛼_𝑚𝑥^𝑚−𝛼𝑥)=(_𝑛|(𝛼_𝑚−𝛼)^𝑛_𝑣=0𝑎_𝑛𝑣𝑥^𝑚_𝑣+𝛼^𝑛_𝑣=0𝑎_𝑛𝑣(𝑥^𝑚_𝑣−𝑥_𝑣)|^𝑝𝑛)^1/𝑀(3.4) which tends to zero as 𝑚→∞. Consequently we obtain that ℎ is a paranorm over the space ℓ(𝐴;𝑝). To prove the completeness of the space ℓ(𝐴;𝑝), let us take any Cauchy sequence (𝑥^𝑖) in the space ℓ(𝐴;𝑝). Then for a given 𝜀>0, there exists a positive integer 𝑛₀(𝜀) such that ℎ(𝑥^𝑖−𝑥^𝑗)<𝜀 for all 𝑖,𝑗≥𝑛₀(𝜀). By using the definitions of the Cauchy sequence and the paranorm, we have, for each fixed 𝑛, |(𝐴𝑥^𝑖)_𝑛−(𝐴𝑥^𝑗)_𝑛|≤(_𝑛|(𝐴(𝑥^𝑖)_𝑛−(𝐴(𝑥^𝑗)_𝑛|^𝑝𝑛)^1/𝑀<𝜀(3.5) for every 𝑖,𝑗≥𝑛₀(𝜀). Hence we obtain that the sequence {𝐴(𝑥⁰)_𝑛,𝐴(𝑥¹)_𝑛,𝐴(𝑥²)_𝑛,…} is a Cauchy sequence of real numbers for every fixed 𝑛∈ℕ. Since ℝ is complete, it converges, that is, (𝐴(𝑥^𝑗)_𝑛→(𝐴𝑥)_𝑛 as 𝑗→∞, where {(𝐴𝑥)_𝑛}={(𝐴𝑥)₀,(𝐴𝑥)₁,(𝐴𝑥)₂,…}. Now let us choose 𝑚∈ℕ such that ∑^𝑚_𝑛=0|(𝐴𝑥^𝑖)_𝑛−(𝐴𝑥^𝑗)_𝑛|^𝑝𝑛<𝜀^𝑀 for each 𝑚∈ℕ and 𝑖,𝑗≥𝑛₀(𝜀). By taking 𝑗→∞ and for every 𝑖≥𝑛₀(𝜀), we get ^𝑚_𝑛=0|(𝐴𝑥^𝑖)_𝑛−(𝐴𝑥)_𝑛|^𝑝𝑛<𝜀^𝑀.(3.6) Again taking 𝑚→∞ and for every 𝑖≥𝑛₀(𝜀), it is obtained that ℎ(𝑥^𝑖−𝑥)<𝜀. We write the following equality: |(𝐴𝑥)_𝑛|=|(𝐴𝑥)_𝑛+(𝐴𝑥^𝑖)_𝑛−(𝐴𝑥^𝑖)_𝑛|.(3.7) By using (3.7) and Minkowski's inequality, we get (_𝑛|(𝐴𝑥)_𝑛|^𝑝𝑛)^1/𝑀≤ℎ(𝑥^𝑖)+ℎ(𝑥^𝑖−𝑥)(3.8) which implies 𝑥∈ℓ(𝐴;𝑝). It follows 𝑥^𝑖→𝑥 as 𝑖→∞. Consequently, since (𝑥^𝑖) is any Cauchy sequence, we obtain that the space ℓ(𝐴;𝑝) is complete. This completes the proof.

Theorem 3.2. The space ℓ(𝐴;𝑝) is linearly isomorphic to the space ℓ(𝑝).

Proof. Let us define 𝐴-transform between the spaces ℓ(𝐴;𝑝) and ℓ(𝑝) such that 𝑥→𝑧=𝐴𝑥. We have to show that the transformation 𝐴 is linear, injective and surjective. The linearity of 𝐴 is obvious. Moreover it is injective because of 𝑥=𝜃 whenever 𝐴𝑥=𝜃. For the surjective property, let 𝑦∈ℓ(𝑝). From (2.3) and (2.7), there exists a matrix 𝐵 such that 𝑥_𝑛=(𝐵𝑦)_𝑛. We have ℎ(𝑥)=(_𝑛|(𝐴(𝐵𝑦))_𝑛|^𝑝𝑛)^1/𝑀=(_𝑛|𝑦_𝑛|^𝑝𝑛)^1/𝑀=𝑔(𝑦)<∞.(3.9) Hence we obtain that the transformation 𝐴 is surjective. Consequently, the spaces ℓ(𝐴;𝑝) and ℓ(𝑝) are linearly isomorphic spaces.

Theorem 3.3. The space ℓ_𝑝(𝐴) has 𝐴𝐷 property.

Proof. Let 𝑓∈(ℓ_𝑝(𝐴))^′. Then 𝑓(𝑥)=𝑔(𝐴𝑥) for some 𝑔∈ℓ^′_𝑝. Since ℓ_𝑝 has AK property and ℓ^′_𝑝≅ℓ_𝑞 where 1/𝑝+1/𝑞=1, 𝑓(𝑥)=_𝑛𝑎_𝑛(𝐴𝑥)_𝑛(3.10) for some 𝑎=(𝑎_𝑛)∈ℓ_𝑞. Also since ℓ_𝑝(𝐴)≅ℓ_𝑝 and the inclusion 𝜙⊂ℓ_𝑝 holds, we have 𝜙⊂ℓ_𝑝(𝐴). For any 𝑓∈(ℓ_𝑝(𝐴))^′ and 𝑒^(𝑘)∈𝜙, we have 𝑓(𝑒^(𝑘))=_𝑛𝑎_𝑛(𝐴𝑒^(𝑘))_𝑛=(𝐻𝑎)_𝑘,(3.11) where 𝐻 is transpose of the matrix 𝐴. Hence from Hahn-Banach theorem, 𝜙⊂ℓ_𝑝(𝐴) is dense in ℓ_𝑝(𝐴) if and only if 𝐻𝑎=𝜃 for 𝑎∈ℓ_𝑞 implies 𝑎=𝜃. Besides, since the null space of the operator 𝐻 on 𝑤 is {𝜃}, ℓ_𝑝(𝐴) has 𝐴𝐷 property. Hence the proof is completed.

4. Some Geometric Properties of the space ℓ_𝑝(𝐴)

In this section, we give some geometric properties for the space ℓ_𝑝(𝐴).

Theorem 4.1. The space ℓ_𝑝(𝐴) has the Banach-Saks of type 𝑝.

Proof. Let (𝜀_𝑛) be a sequence of positive numbers for which ∑^∞_𝑛=1𝜀_𝑛≤1/2. Let (𝑥_𝑛) be a weakly null sequence in 𝐵(ℓ_𝑝(𝐴)). Set 𝑧₀=𝑥₀=0 and 𝑧₁=𝑥_𝑛1=𝑥₁. Then there exists 𝑠₁∈ℕ such that ‖^∞_𝑖=𝑠1+1𝑧₁(𝑖)𝑒^(𝑖)‖_ℓ𝑝(𝐴⁾<𝜀_1.(4.1) Since (𝑥_𝑛) is a weakly null sequence implies that 𝑥_𝑛→0 with respect to the coordinatwise, there is an 𝑛₂∈ℕ such that ‖^𝑠1_𝑖=0𝑥_𝑛(𝑖)𝑒^(𝑖)‖_ℓ𝑝(𝐴₎<𝜀₁,(4.2) where 𝑛≥𝑛₂. Set 𝑧₂=𝑥_𝑛2. Then there exists an 𝑠₂>𝑠₁ such that ‖^∞_𝑖=𝑠2+1𝑧₂(𝑖)𝑒^(𝑖)‖_ℓ𝑝(𝐴⁾<𝜀₂.(4.3) By using the fact that 𝑥_𝑛→0 (coordinatwise), there exists an 𝑛₃>𝑛₂ such that ‖^𝑠2_𝑖=0𝑥_𝑛(𝑖)𝑒^(𝑖)‖_ℓ𝑝(𝐴₎<𝜀₂,(4.4) where 𝑛≥𝑛₃.
If we continue this process, we can find two increasing subsequences (𝑠_𝑖) and (𝑛_𝑖) such that ‖^𝑠𝑗_𝑖=0𝑥_𝑛(𝑖)𝑒^(𝑖)‖_ℓ𝑝(𝐴₎<𝜀_𝑗(4.5) for each 𝑛≥𝑛_𝑗+1 and ‖^∞_{𝑖=𝑠𝑗+1}𝑧_𝑗(𝑖)𝑒^(𝑖)‖_ℓ𝑝(𝐴₎<𝜀_𝑗,(4.6) where 𝑧_𝑗=𝑥_𝑛𝑗. Hence, ‖^𝑛_𝑗=0𝑧_𝑗‖_ℓ𝑝(𝐴₎=‖^𝑛_𝑗=0(^𝑠𝑗−1_𝑖=0𝑧_𝑗(𝑖)𝑒^(𝑖)+^𝑠𝑗_{𝑖=𝑠𝑗−1+1}𝑧_𝑗(𝑖)𝑒^(𝑖)+^∞_{𝑖=𝑠𝑗+1}𝑧_𝑗(𝑖)𝑒^(𝑖))‖_ℓ𝑝(𝐴₎≤‖^𝑛_𝑗=0(^𝑠𝑗_{𝑖=𝑠𝑗−1+1}𝑧_𝑗(𝑖)𝑒^(𝑖))‖_ℓ𝑝(𝐴₎+‖^𝑛_𝑗=0(^𝑠𝑗−1_𝑖=0𝑧_𝑗(𝑖)𝑒^(𝑖))‖_ℓ𝑝(𝐴⁾+‖^𝑛_𝑗=0(^∞_{𝑖=𝑠𝑗+1}𝑧_𝑗(𝑖)𝑒^(𝑖))‖_ℓ𝑝(𝐴₎≤‖^𝑛_𝑗=0(^𝑠𝑗_{𝑖=𝑠𝑗−1+1}𝑧_𝑗(𝑖)𝑒^(𝑖))‖_ℓ𝑝(𝐴₎+2^𝑛_𝑗=0𝜀_𝑗.(4.7) On the other hand, since 𝑥_𝑛∈𝐵(ℓ_𝑝(𝐴)) and ‖𝑥‖_ℓ𝑝(𝐴₎=(∑^∞_𝑖=0|∑^𝑖_𝑣=0𝑎_𝑖𝑣𝑥_𝑣|^𝑝)^1/𝑝, it can be seen that ‖𝑥‖_ℓ𝑝(𝐴⁾<1. Therefore ‖𝑥‖^𝑝_ℓ𝑝(𝐴₎<1. We have ‖^𝑛_𝑗=0(^𝑠𝑗_{𝑖=𝑠𝑗−1+1}𝑧_𝑗(𝑖)𝑒^(𝑖))‖^𝑝_ℓ𝑝(𝐴₎=^𝑛𝑠_𝑗=0^𝑗_{𝑖=𝑠𝑗−1+1}|^𝑖_𝑣=0𝑎_𝑖𝑣𝑧_𝑗(𝑣)|^𝑝≤^𝑛∞_𝑗=0𝑖=0|^𝑖_𝑣=0𝑎_𝑖𝑣𝑧_𝑗(𝑣)|^𝑝≤(𝑛+1).(4.8) Hence we obtain ‖^𝑛_𝑗=0(^𝑠𝑗_{𝑖=𝑠𝑗−1+1}𝑧_𝑗(𝑖)𝑒^(𝑖))‖_ℓ𝑝(𝐴₎≤(𝑛+1)^1/𝑝.(4.9) By using the fact 1≤(𝑛+1)^1/𝑝 for all 𝑛∈ℕ and 1≤𝑝<∞, we have ‖^𝑛_𝑗=0𝑧_𝑗‖_ℓ𝑝(𝐴⁾≤(𝑛+1)^1/𝑝+1≤2(𝑛+1)^1/𝑝.(4.10) Hence ℓ_𝑝(𝐴) has the Banach-Saks type 𝑝. This completes the proof.

Theorem 4.2. Gurarii's modulus of convexity for the normed space ℓ_𝑝(𝐴) is 𝛽_ℓ𝑝(𝐴₎𝜀(𝜀)≤1−(1−(2)^𝑝)^1/𝑝,(4.11) where 0≤𝜀≤2.

Proof. We have 𝑥∈ℓ_𝑝(𝐴). Then we have ‖𝑥‖_ℓ𝑝(𝐴₎=‖𝐴𝑥‖_𝑙𝑝=(_𝑛|(𝐴𝑥)_𝑛|^𝑝)^1/𝑝.(4.12) Let 0≤𝜀≤2 and consider the following sequences: 𝑥=(𝑥_𝑛𝜀)=(𝐵((1−(2)^𝑝)^1/𝑝𝜀),𝐵(2),0,0,…),𝑦=(𝑦_𝑛𝜀)=(𝐵((1−(2)^𝑝)^1/𝑝𝜀),𝐵(−2),0,0,…),(4.13) where 𝐵 is the inverse of the matrix 𝐴. Since 𝑧_𝑛=(𝐴𝑥)_𝑛 and 𝑡_𝑛=(𝐴𝑦)_𝑛, we have 𝑧=(𝑧_𝑛𝜀)=((1−(2)^𝑝)^1/𝑝𝜀,(2),0,0,…),𝑡=(𝑡_𝑛𝜀)=((1−(2)^𝑝)^1/𝑝𝜀,(−2),0,0,…).(4.14) By using sequences given above, we obtain the following equalities: ‖𝑥‖^𝑝_ℓ𝑝(𝐴₎=‖𝐴𝑥‖^𝑝_𝑙𝑝𝜀=|(1−(2)^𝑝)^1/𝑝|^𝑝𝜀+|2|^𝑝𝜀=1−(2)^𝑝𝜀+(2)^𝑝=1,‖𝑦‖^𝑝_ℓ𝑝(𝐴⁾=‖𝐴𝑦‖^𝑝_𝑙𝑝𝜀=|(1−(2)^𝑝)^1/𝑝|^𝑝𝜀+|−2|^𝑝𝜀=1−(2)^𝑝𝜀+(2)^𝑝=1,‖𝑥−𝑦‖_ℓ𝑝(𝐴₎=‖𝐴𝑥−𝐴𝑦‖_𝑙𝑝𝜀=(|(1−(2)^𝑝)^1/𝑝𝜀−(1−(2)^𝑝)^1/𝑝|^𝑝𝜀+|2𝜀−(−2)|^𝑝)^1/𝑝=𝜀.(4.15) To complete the conditions of 𝛽_ℓ𝑝(𝐴⁾(𝜀) for Gurarii's modulus of convexity, it remains to show the infimum of ‖𝛼𝑥+(1−𝛼)𝑡‖_ℓ𝑝(𝐴₎ for 0≤𝛼≤1. We have inf_0≤𝛼≤1‖𝛼𝑥+(1−𝛼)𝑦‖_ℓ𝑝(𝐴⁾=inf_0≤𝛼≤1‖𝛼𝐴𝑥+(1−𝛼)𝐴𝑦‖_𝑙𝑝=inf_0≤𝛼≤1𝜀[|𝛼(1−(2)^𝑝)^1/𝑝𝜀+(1−𝛼)(1−(2)^𝑝)^1/𝑝|^𝑝𝜀+|𝛼(2𝜀)+(1−𝛼)(−2)|^𝑝]^1/𝑝=inf_0≤𝛼≤1𝜀[1−(2)^𝑝+|2𝛼−1|^𝑝(𝜀2)^𝑝]^1/𝑝𝜀=(1−(2)^𝑝)^1/𝑝.(4.16) Consequently we get for 𝑝≥1𝛽_ℓ𝑝(𝐴₎𝜀(𝜀)≤1−(1−(2)^𝑝)^1/𝑝.(4.17) This is the desired result. Hence the proof is completed.

Corollary 4.3. (i) If 𝜀=2, then 𝛽_ℓ𝑝(𝐴⁾(𝜀)≤1 and hence ℓ_𝑝(𝐴) is strictly convex.
(ii) If 0<𝜀<2, then 0<𝛽_ℓ𝑝(𝐴⁾(𝜀)<1 and hence ℓ_𝑝(𝐴) is uniformly convex.

Corollary 4.4. If 𝛼=1/2, then 𝛿_ℓ𝑝(𝐴⁾(𝜀)=𝛽_ℓ𝑝(𝐴⁾(𝜀).