Journal of Inequalities and Applications
Volume 2009 (2009), Article ID 932734, 13 pages
doi:10.1155/2009/932734
Research Article

On Some Geometrical Properties of Generalized Modular Spaces of Cesáro Type Defined by Weighted Means

Necip Şimşek1 and Vatan Karakaya2

1Department of Mathematics, Faculty of Arts and Sciences, Adıyaman University, 02040 Adıyaman, Turkey
2Department of Mathematical Engineering, Yıldız Technical University, Davutpasa Campus, Esenler, 80750 İstanbul, Turkey

Received 4 April 2009; Accepted 19 June 2009

Academic Editor: A. Laforgia

Copyright © 2009 Necip Şimşek and Vatan Karakaya. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The main purpose of this paper is to introduce modular structure of the sequence space defined by Altay and Başar (2007), and to study Kadec-Klee ( 𝐻 ) and uniform Opial properties of this sequence space on Köthe sequence spaces.

1. Introduction

In [1], Malkowsky and Savaş defined a new sequence space by using generalized weighted means and they studied 𝛽 -dual and matrix transformations of this space.

Recently, Altay and Başar [2] constructed a new paranormed sequence space inspired by the sequence space defined in [1].

On the other hand, Shue [3] first defined the Cesáro sequence spaces with a norm. Many authors studied the Cesáro sequence spaces with several properties. In [4], it is shown that the Cesáro sequence spaces c e s 𝑝 ( 1 ≤ 𝑝 < ∞ ) have Kadec-Klee and local uniform rotundity properties.Cui et al. [5] showed that Banach-Saks of type 𝑝 property holds in these spaces.

Quite recently, Sanhan and Suantai [6] generalized normed Cesáro sequence spaces to paranormed sequence spaces by making use of Köthe sequence spaces. They also defined and investigated modular structure and some geometrical properties of these generalized sequence spaces. Besides, Petrot and Suantai [7] studied the uniform Opial property of these spaces.

Our goal is first to introduce modular sequence space 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) obtained from paranormed ones by generalized weighted means on Köthe sequence spaces.

In special cases, the sequence space 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) includes the well-known Cesàro and Nörlund sequence spaces that are normed and paranormed spaces having modular structure (see [8]). We also show that the modular space 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) is a Banach space when it is equipped with Luxemburg norm.

The main purpose of this study is to show that the Kadec-Klee and Opial properties hold in the 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) space.

The organization of this paper is as follows. In the first section, we introduce some definitions and the concepts that are used throughout the paper. In the second section, we construct the modular space 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) which was obtained by paranormed space 𝑙 ( 𝑢 , 𝑣 ; 𝑝 ) and we investigate the Kadec-Klee property of this space. We also show that the modular space 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) is a Banach space under the Luxemburg norm. Finally, in the third section, uniform Opial property of the space 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) is investigated by using some topological structures.

We denote by ℕ , ℝ , and 𝔽 the set of natural numbers, the set of real numbers and the scalar field, respectively. Let ( 𝑋 , ‖ ⋅ ‖ ) be a real Banach space and let 𝐵 ( 𝑋 ) ( 𝑆 ( 𝑋 ) ) be the closed unit ball (the unit sphere) of 𝑋 . The space of all real sequences 𝑥 = ( 𝑥 ( 𝑖 ) ) ∞ 𝑖 = 1 is denoted by ℓ 0 .

A Banach space 𝑋 = ( 𝑋 , ‖ ⋅ ‖ ) is said to be a Köthe sequence space if 𝑋 is a subspace of ℓ 0 such that (see [9]): (i) If 𝑥 ∈ ℓ 0 , 𝑦 ∈ 𝑋 , and | 𝑥 ( 𝑖 ) | ≤ | 𝑦 ( 𝑖 ) | for all 𝑖 ∈ ℕ , then 𝑥 ∈ 𝑋 and ‖ 𝑥 ‖ ≤ ‖ 𝑦 ‖ . (ii) There is an element 𝑥 ∈ 𝑋 such that 𝑥 ( 𝑖 ) > 0 for all 𝑖 ∈ ℕ .

We say that 𝑥 ∈ 𝑋 is order continuous if for any sequence ( 𝑥 𝑛 ) in 𝑋 such that 𝑥 𝑛 ( 𝑖 ) ≤ | 𝑥 ( 𝑖 ) | for each 𝑖 ∈ ℕ and 𝑥 𝑛 ( 𝑖 ) → 0 ( 𝑛 → ∞ ) , we have ‖ 𝑥 𝑛 ‖ → 0 holds.

A Köthe sequence space 𝑋 is said to be order continuous if all sequences in 𝑋 are order continuous. It is easy to see that 𝑥 ∈ 𝑋 is order continuous if and only if ‖ ( 0 , 0 , … , 0 , 𝑥 ( 𝑛 + 1 ) , 𝑥 ( 𝑛 + 2 ) , … ) ‖ → 0 as 𝑛 → ∞ .

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

Let 𝑋 be a real Banach space. We say that 𝑋 has the Opial property if for any weakly null sequence { 𝑥 𝑛 } in 𝑋 and any 𝑥 in 𝑋 ⧵ { 0 } , the inequality l i m i n f 𝑛 → ∞ ‖ ‖ 𝑥 𝑛 ‖ ‖ < l i m i n f 𝑛 → ∞ ‖ ‖ 𝑥 𝑛 ‖ ‖ + 𝑥 ( 1 . 1 ) holds (see [10]). Opial [10] has proved that ℓ 𝑝 space ( 1 < 𝑝 < ∞ ) has this property.

Franchetti [11] has shown that any infinite dimensional Banach space has an equivalent norm that satisfies the Opial property.

We say that 𝑋 has the uniform Opial property (see [10]) if for any 𝜀 > 0 there exists 𝑟 > 0 such that for any 𝑥 from 𝑋 with ‖ 𝑥 ‖ ≥ 𝜀 and any weakly null sequence { 𝑥 𝑛 } in the unit sphere 𝑆 ( 𝑋 ) of 𝑋 , the following inequality 1 + 𝑟 ≤ l i m i n f 𝑛 → ∞ ‖ ‖ 𝑥 𝑛 ‖ ‖ + 𝑥 ( 1 . 2 ) holds. It is well known that the Opial property of a Banach space 𝑋 plays an important role in the fixed point theory and in the theory of differential and integral equations (see e.g [12–15]). Also the geometrical properties of some modular sequence spaces have been studied in ([16, 17]).

For a real vector space 𝑋 , a function 𝜌 ∶ 𝑋 → [ 0 , ∞ ] is called a modular if it satisfies the following conditions:

(i) 𝜌 ( 𝑥 ) = 0 ⇔ 𝑥 = 0 , (ii) 𝜌 ( 𝛼 𝑥 ) = 𝜌 ( 𝑥 ) for all 𝛼 ∈ 𝔽 with | 𝛼 | = 1 , (iii) 𝜌 ( 𝛼 𝑥 + 𝛽 𝑦 ) ≤ 𝜌 ( 𝑥 ) + 𝜌 ( 𝑦 ) for all 𝑥 , 𝑦 ∈ 𝑋 and all 𝛼 , 𝛽 ≥ 0 with 𝛼 + 𝛽 = 1 .

Further, the modular 𝜌 is called  convex if

(iv) 𝜌 ( 𝛼 𝑥 + 𝛽 𝑦 ) ≤ 𝛼 𝜌 ( 𝑥 ) + 𝛽 𝜌 ( 𝑦 ) holds for all 𝑥 , 𝑦 ∈ 𝑋 and all 𝛼 , 𝛽 ≥ 0 with 𝛼 + 𝛽 = 1 .

For any modular 𝜌 on 𝑋 , the space 𝑋 𝜌 = 𝑥 ∈ 𝑋 ∶ 𝜌 ( 𝜆 𝑥 ) ⟶ 0 a s 𝜆 ⟶ 0 + ( 1 . 3 ) is called the  modular space.

A sequence ( 𝑥 𝑛 ) of elements of 𝑋 𝜌 is called  modular convergent to 𝑥 ∈ 𝑋 𝜌 if there exists a 𝜆 > 0 such that 𝜌 ( 𝜆 ( 𝑥 𝑛 − 𝑥 ) ) → 0 as 𝑛 → ∞ . If 𝜌 is a convex modular, then the following formulas: ‖ 𝑥 ‖ 𝐿 𝑥 = i n f 𝜆 > 0 ∶ 𝜌 𝜆 , ≤ 1 ‖ 𝑥 ‖ 𝐴 = i n f 𝜆 > 0 1 𝜆 ( 1 + 𝜌 ( 𝜆 𝑥 ) ) ( 1 . 4 ) define two norms on 𝑋 𝜌 which are called the  Luxemburg norm and the  Amemiya norm, respectively. In addition ‖ 𝑥 ‖ 𝐿 ≤ ‖ 𝑥 ‖ 𝐴 ≤ 2 ‖ 𝑥 ‖ 𝐿 ( 1 . 5 ) for all 𝑥 ∈ 𝑋 𝜌 holds (see [18]).

Proposition 1.1. Let ( 𝑥 𝑛 ) ⊂ 𝑋 𝜌 . Then | | 𝑥 𝑛 | | 𝐿 → 0 (or equivalently | | 𝑥 | | 𝐴 → 0 ) if and only if 𝜌 ( 𝜆 ( 𝑥 𝑛 ) ) → 0 as 𝑛 → ∞ , for every 𝜆 > 0 .

Proof. See [19, page 15, Theorem].

Throughout this paper, the sequence 𝑝 = ( 𝑝 𝑟 ) is a bounded sequence of positive real numbers with 𝑝 𝑟 > 1 , also 𝐻 = s u p 𝑝 𝑟 and 𝑀 = m a x ( 1 , 𝐻 ) .

For 𝑥 ∈ ℓ 0 , 𝑖 ∈ ℕ , we denote 𝑒 𝑖 = ( 0 , 0 , … 0 , 1 , 0 , 0 , … ) 1 s e a t s - i n 𝑖 t h p l a c e o f 𝑒 𝑖 , 𝑥 | 𝑖 𝑥 = ( 𝑥 ( 1 ) , 𝑥 ( 2 ) , … , 𝑥 ( 𝑖 ) , 0 , 0 , … ) , | ℕ − 𝑖 = ( 0 , 0 , … , 0 , 0 , 𝑥 ( 𝑖 + 1 ) , 𝑥 ( 𝑖 + 2 ) , … ) , ( 1 . 6 ) and s u p p ( 𝑥 ) = { 𝑖 ∈ ℕ ∶ 𝑥 ( 𝑖 ) ≠ 0 } . Also let 𝐸 be the set of all sequences with finite number of coordinates different from zero. Besides we will need the following inequality in the sequel: | | 𝑎 𝑟 + 𝑏 𝑟 | | 𝑝 𝑟 | | 𝑎 ≤ 𝐾 𝑟 | | 𝑝 𝑟 + | | 𝑏 𝑟 | | 𝑝 𝑟 , ( 1 . 7 ) where 𝐾 = m a x { 1 , 2 𝐻 − 1 } , with 𝐻 = s u p 𝑟 𝑝 𝑟 .

By using the sequence space defined in [1], Altay and Başar [2] defined the sequence space 𝑙 ( 𝑢 , 𝑣 ; 𝑝 ) as 𝑙 ⎧ ⎪ ⎨ ⎪ ⎩ 𝑥 ( 𝑢 , 𝑣 ; 𝑝 ) = 𝑥 = 𝑘 ∈ ℓ 0 ∶ ∞ 𝑘 = 0 | | | | | 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 𝑥 𝑗 | | | | | 𝑝 𝑘 ⎫ ⎪ ⎬ ⎪ ⎭ < ∞ . ( 1 . 8 )

We write 𝑈 for the set of all sequences 𝑢 such that 𝑢 𝑘 ≠ 0 for all 𝑘 ∈ ℕ . Let 𝑢 , 𝑣 ∈ 𝑈 and define the matrix 𝐺 ( 𝑢 , 𝑣 ) = ( 𝑔 𝑛 𝑘 ) by 𝑔 𝑛 𝑘 = 𝑢 𝑛 𝑣 𝑘 , 0 , 0 ≤ 𝑘 ≤ 𝑛 , 𝑘 > 𝑛 ( 1 . 9 ) for all 𝑘 , 𝑛 ∈ ℕ where 𝑢 𝑛 depends only on 𝑛 and 𝑣 𝑘 depends only on 𝑘 .

They also showed that the space 𝑙 ( 𝑢 , 𝑣 ; 𝑝 ) is a complete linear metric space paranormed by ⎛ ⎜ ⎜ ⎝ ℎ ( 𝑥 ) = ∞ 𝑘 = 0 | | | | | 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 𝑥 𝑗 | | | | | 𝑝 𝑘 ⎞ ⎟ ⎟ ⎠ 1 / 𝑀 . ( 1 . 1 0 ) We now introduce a generalized modular sequence space 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) by 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) = 𝑥 ∈ ℓ 0 ∶ 𝜌 ( 𝜆 𝑥 ) < ∞ , f o r s o m e 𝜆 > 0 , ( 1 . 1 1 ) where 𝜌 ( 𝑥 ) = ∞ 𝑘 = 0 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | 𝑥 ( 𝑗 ) | 𝑝 𝑘 . ( 1 . 1 2 ) It can be seen that 𝜌 ∶ 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) → [ 0 , ∞ ] is a modular on 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) .

Note that the Luxemburg norm on the sequence space 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) is defined as follows: ‖ 𝑥 ‖ 𝐿 𝑥 = i n f 𝜆 > 0 ∶ 𝜌 𝜆 ≤ 1 , ∀ 𝑥 ∈ 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) ( 1 . 1 3 ) or equally ‖ 𝑥 ‖ 𝐿 ⎧ ⎪ ⎨ ⎪ ⎩ 𝑥 = i n f 𝜆 > 0 ∶ 𝜌 𝜆 = ∞ 𝑘 = 0 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | | 𝑥 ( 𝑗 ) 𝜆 | | | 𝑝 𝑘 ⎫ ⎪ ⎬ ⎪ ⎭ ≤ 1 . ( 1 . 1 4 ) In the same way we can introduce the Amemiya norm on the sequence space 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) as follows: ‖ 𝑥 ‖ 𝐴 = i n f 𝜆 > 0 1 𝜆 ( 1 + 𝜌 ( 𝜆 𝑥 ) ) , ∀ 𝑥 ∈ 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) . ( 1 . 1 5 )

By combining special case of 𝑢 𝑘 and 𝑣 𝑗 , we get the following modular spaces: if 𝑢 𝑘 = 1 / ( 𝑘 + 1 ) and 𝑣 𝑗 = 1 for all 𝑘 , 𝑗 ∈ ℕ , then the space 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) reduces to the modular space c e s ( 𝑝 ) (see [7]) normed by ‖ 𝑥 ‖ 𝐿 , c e s ( 𝑝 ) ⎧ ⎪ ⎨ ⎪ ⎩ 𝑥 = i n f 𝜆 > 0 ∶ 𝜌 𝜆 = ∞ 𝑘 = 0 1 𝑘 + 1 𝑘 𝑗 = 0 | | | 𝑥 ( 𝑗 ) 𝜆 | | | 𝑝 𝑘 ⎫ ⎪ ⎬ ⎪ ⎭ ≤ 1 . ( 1 . 1 6 ) If 𝑢 𝑘 = 1 / 𝑃 𝑘 , 𝑣 𝑗 = 𝑝 𝑗 , and 𝑃 𝑘 = ∑ 𝑘 𝑗 = 0 𝑝 𝑗 for all 𝑘 , 𝑗 ∈ ℕ , then the space 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) reduces to the modular space 𝑁 𝜌 ( 𝑝 ) (see [8]), normed by ‖ 𝑥 ‖ 𝐿 , 𝑁 𝜌 ( 𝑝 ) ⎧ ⎪ ⎨ ⎪ ⎩ 𝑥 = i n f 𝜆 > 0 ∶ 𝜌 𝜆 = ∞ 𝑘 = 0 1 𝑃 𝑘 𝑘 𝑗 = 0 𝑝 𝑗 | | | 𝑥 ( 𝑗 ) 𝜆 | | | 𝑝 𝑘 ⎫ ⎪ ⎬ ⎪ ⎭ ≤ 1 . ( 1 . 1 7 )

2. Kadec-Klee Property and Modular Structure of 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 )

In this section we will give some basic properties of the modular 𝜌 on the space 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) . Also we will investigate some relationships between the modular 𝜌 and the Luxemburg norm on 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) .

Proposition 2.1. The functional 𝜌 is a convex modular on 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) .

Proof. The proof is obvious. Hence we omitted it.

Proposition 2.2. For any 𝑥 ∈ 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) , the following assertions are satisfied: (i)If   0 < 𝑎 < 1 and | | 𝑥 | | 𝐿 > 𝑎 , then 𝜌 ( 𝑥 ) > 𝑎 𝐻 ,(ii)if 𝑎 ≥ 1 and | | 𝑥 | | 𝐿 < 𝑎 , then 𝜌 ( 𝑥 ) < 𝑎 𝐻 ,(iii)if | | 𝑥 | | 𝐿 ≤ 1 , then 𝜌 ( 𝑥 ) ≤ | | 𝑥 | | 𝐿 , (iv) | | 𝑥 | | 𝐿 = 1 if and only if 𝜌 ( 𝑥 ) = 1 .

Proof. It can be proved with standard techniques in a similar way as in [16, 20].

Proposition 2.3. Let ( 𝑥 𝑛 ) be a sequence in 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) . Then (i)if l i m 𝑛 → ∞ | | 𝑥 𝑛 | | 𝐿 = 1 , then l i m 𝑛 → ∞ 𝜌 ( 𝑥 𝑛 ) = 1 , (ii)if l i m 𝑛 → ∞ 𝜌 ( 𝑥 𝑛 ) = 0 , then l i m 𝑛 → ∞ | | 𝑥 𝑛 | | 𝐿 = 0 .

Proof. (i) Suppose that l i m 𝑛 → ∞ | | 𝑥 𝑛 | | 𝐿 = 1 and let 𝜀 ∈ ( 0 , 1 ) . Then there exists 𝑛 0 such that 1 − 𝜀 < | | 𝑥 𝑛 | | 𝐿 < 1 + 𝜀 for all 𝑛 ≥ 𝑛 0 . By Proposition 2.2(i) and (ii), for all 𝑛 ≥ 𝑛 0 , the inequality ( 1 − 𝜀 ) 𝐻 < | | 𝑥 𝑛 | | 𝐿 < ( 1 + 𝜀 ) 𝐻 implies that l i m 𝑛 → ∞ 𝜌 ( 𝑥 𝑛 ) = 1 .
(ii) Suppose that | | 𝑥 𝑛 | | 𝐿 ↛ 0 . Then there is an 𝜀 ∈ ( 0 , 1 ) and a subsequence ( 𝑥 𝑛 𝑘 ) of ( 𝑥 𝑛 ) such that | | 𝑥 𝑛 𝑘 | | 𝐿 > 𝜀 for all 𝑘 ∈ ℕ . By Proposition 2.2(i), we obtain that 𝜌 ( 𝑥 𝑛 𝑘 ) > 𝜀 𝐻 for all 𝑘 ∈ ℕ , which means that 𝜌 ( 𝑥 𝑛 𝑘 ) ↛ 0 as 𝑛 → ∞ . Hence 𝜌 ( 𝑥 𝑛 ) ↛ 0 .

Now we have the following.

Theorem 2.4. The space 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) is a Banach space with respect to the Luxemburg norm defined by ‖ 𝑥 ‖ 𝐿 𝑥 = i n f 𝜆 > 0 ∶ 𝜌 𝜆 ≤ 1 . ( 2 . 1 )

Proof. Let ( 𝑥 𝑛 ( 𝑗 ) ) be a Cauchy sequence in 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) and 𝜀 ∈ ( 0 , 1 ) . Thus, there exists 𝑛 0 such that ‖ ‖ 𝑥 𝑛 − 𝑥 𝑚 ‖ ‖ 𝐿 < 𝜀 , ( 2 . 2 ) for all 𝑚 , 𝑛 ≥ 𝑛 0 . By Proposition 2.2(iii) we have 𝜌 𝑥 𝑛 − 𝑥 𝑚 < ‖ ‖ 𝑥 𝑛 − 𝑥 𝑚 ‖ ‖ 𝐿 < 𝜀 , ( 2 . 3 ) for all 𝑛 , 𝑚 ≥ 𝑛 0 , which means that ∞ 𝑘 = 0 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | 𝑥 𝑛 ( 𝑗 ) − 𝑥 𝑚 | | ( 𝑗 ) 𝑝 𝑘 < 𝜀 , ( 2 . 4 ) for 𝑚 , 𝑛 ≥ 𝑛 0 . For fixed 𝑘 , the last inequality gives that | | 𝑥 𝑛 ( 𝑗 ) − 𝑥 𝑚 | | ( 𝑗 ) < 𝜀 , ( 2 . 5 )
Hence we obtain that the sequence ( 𝑥 𝑛 ( 𝑗 ) ) is a Cauchy sequence in ℝ . Since ℝ is complete, 𝑥 𝑚 ( 𝑗 ) → 𝑥 ( 𝑗 ) as 𝑚 → ∞ . Therefore, for fixed 𝑘 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | 𝑥 𝑛 | | ( 𝑗 ) − 𝑥 ( 𝑗 ) < 𝜀 ( 2 . 6 ) as 𝑚 → ∞ and for all   𝑛 ≥ 𝑛 0 .
It remains to show that the sequence ( 𝑥 ( 𝑗 ) ) is an element of 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) . From the inequality (2.3), we can write ∞ 𝑘 = 0 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | 𝑥 𝑛 ( 𝑗 ) − 𝑥 𝑚 ( | | 𝑗 ) 𝑝 𝑘 < 𝜀 ( 2 . 7 ) for all 𝑚 , 𝑛 ≥ 𝑛 0 . So we obtain 𝜌 𝑥 𝑛 − 𝑥 𝑚 𝑥 ⟶ 𝜌 𝑛 − 𝑥 , ( 2 . 8 ) as 𝑚 → ∞ for all 𝑛 ≥ 𝑛 0 . Since for all 𝑛 ≥ 𝑛 0 , ∞ 𝑘 = 0 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | 𝑥 𝑛 ( 𝑗 ) − 𝑥 𝑚 ( | | 𝑗 ) 𝑝 𝑘 ⟶ ∞ 𝑘 = 0 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | 𝑥 𝑛 ( | | 𝑗 ) − 𝑥 ( 𝑗 ) 𝑝 𝑘 , ( 2 . 9 ) as 𝑚 → ∞ , then by (2.3) we have 𝜌 ( 𝑥 𝑛 − 𝑥 ) < ‖ 𝑥 𝑛 − 𝑥 ‖ 𝐿 < 𝜀 for all 𝑛 ≥ 𝑛 0 . This means that 𝑥 𝑛 → 𝑥 as 𝑛 → ∞ . So we have ( 𝑥 𝑛 0 − 𝑥 ) ∈ 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) . Since 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) is a linear space, we have 𝑥 = 𝑥 𝑛 0 − ( 𝑥 𝑛 0 − 𝑥 ) ∈ 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) . Therefore the sequence space 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) is a Banach space with respect to Luxemburg norm. This completes the proof.

Now, we give a proposition concerning Kadec-Klee property of the space 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) .

Proposition 2.5. Let 𝑥 ∈ 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) and ( 𝑥 𝑛 ) ⊆ 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) . If 𝜌 ( 𝑥 𝑛 ) → 𝜌 ( 𝑥 ) as 𝑛 → ∞ and 𝑥 𝑛 ( 𝑗 ) → 𝑥 ( 𝑗 ) as 𝑛 → ∞ for all 𝑗 ∈ ℕ , then 𝑥 𝑛 → 𝑥 as 𝑛 → ∞ .

Proof. Let 𝜀 > 0 . Since ∑ 𝜌 ( 𝑥 ) = ∞ 𝑘 = 0 ( ∑ 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | 𝑥 ( 𝑗 ) | ) 𝑝 𝑘 < ∞ , we have ∞ 𝑘 = 𝑘 0 + 1 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | | | 𝑥 ( 𝑗 ) 𝑝 𝑘 < 𝜀 6 𝐾 , ( 2 . 1 0 ) where 𝐻 = s u p 𝑝 𝑘 , 𝐾 = m a x { 1 , 2 𝐻 − 1 } .
Since 𝜌 ( 𝑥 𝑛 ∑ ) − 𝑘 0 𝑘 = 0 ( ∑ 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | 𝑥 𝑛 ( 𝑗 ) | ) 𝑝 𝑘 ∑ → 𝜌 ( 𝑥 ) − 𝑘 0 𝑘 = 0 ( ∑ 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | 𝑥 ( 𝑗 ) ) | ) 𝑝 𝑘 as 𝑛 → ∞ and 𝑥 𝑛 ( 𝑗 ) → 𝑥 ( 𝑗 ) as 𝑛 → ∞ for all 𝑗 ∈ ℕ , there exists 𝑛 0 ∈ ℕ such that | | | | | ∞ 𝑘 = 𝑘 0 + 1 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | 𝑥 𝑛 | | ( 𝑗 ) 𝑝 𝑘 − ∞ 𝑘 = 𝑘 0 + 1 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | | | 𝑥 ( 𝑗 ) 𝑝 𝑘 | | | | | < 𝜀 3 𝐾 ( 2 . 1 1 ) for all 𝑛 ≥ 𝑛 0 . Also since 𝑥 𝑛 ( 𝑗 ) → 𝑥 ( 𝑗 ) for all 𝑗 ∈ ℕ , we have 𝑘 0 𝑘 = 0 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | 𝑥 𝑛 ( | | 𝑗 ) − 𝑥 ( 𝑗 ) 𝑝 𝑘 < 𝜀 3 ( 2 . 1 2 ) for all 𝑛 ≥ 𝑛 0 . It follows from (2.3), (2.10), and (2.11) that for 𝑛 ≥ 𝑛 0 , 𝜌 𝑥 𝑛 = − 𝑥 ∞ 𝑘 = 0 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | 𝑥 𝑛 ( | | 𝑗 ) − 𝑥 ( 𝑗 ) 𝑝 𝑘 = 𝑘 0 𝑘 = 0 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | 𝑥 𝑛 | | ( 𝑗 ) − 𝑥 ( 𝑗 ) 𝑝 𝑘 + ∞ 𝑘 = 𝑘 0 + 1 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | 𝑥 𝑛 | | ( 𝑗 ) − 𝑥 ( 𝑗 ) 𝑝 𝑘 < 𝜀 3 ⎡ ⎢ ⎢ ⎣ + 𝐾 ∞ 𝑘 = 𝑘 0 + 1 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | 𝑥 𝑛 ( | | 𝑗 ) 𝑝 𝑘 + ∞ 𝑘 = 𝑘 0 + 1 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | | | 𝑥 ( 𝑗 ) 𝑝 𝑘 ⎤ ⎥ ⎥ ⎦ = 𝜀 3 ⎡ ⎢ ⎢ ⎣ 𝜌 𝑥 + 𝐾 𝑛 − 𝑘 0 𝑘 = 0 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | 𝑥 𝑛 ( | | 𝑗 ) 𝑝 𝑘 + ∞ 𝑘 = 𝑘 0 + 1 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | | | 𝑥 ( 𝑗 ) 𝑝 𝑘 ⎤ ⎥ ⎥ ⎦ < 𝜀 3 ⎡ ⎢ ⎢ ⎣ + 𝐾 𝜌 ( 𝑥 ) − 𝑘 0 𝑘 = 0 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | | | 𝑥 ( 𝑗 ) 𝑝 𝑘 + 𝜀 + 3 𝐾 ∞ 𝑘 = 𝑘 0 + 1 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | | | 𝑥 ( 𝑗 ) 𝑝 𝑘 ⎤ ⎥ ⎥ ⎦ = 𝜀 3 ⎡ ⎢ ⎢ ⎣ + 𝐾 ∞ 𝑘 = 𝑘 0 + 1 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | | | 𝑥 ( 𝑗 ) 𝑝 𝑘 + 𝜀 + 3 𝐾 ∞ 𝑘 = 𝑘 0 + 1 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | | | 𝑥 ( 𝑗 ) 𝑝 𝑘 ⎤ ⎥ ⎥ ⎦ < 𝜀 3 + 𝜀 3 + 𝜀 3 = 𝜀 . ( 2 . 1 3 ) This shows that 𝜌 ( 𝑥 𝑛 − 𝑥 ) → 0 as 𝑛 → ∞ . Hence by Proposition 2.3(ii), we have | | 𝑥 𝑛 − 𝑥 | | 𝐿 → 0 as 𝑛 → ∞ , that is, 𝑥 𝑛 → 𝑥 . This completes the proof.

Now, we give the main results of this paper involving geometric properties of the space 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) .

Theorem 2.6. The space 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) has the Kadec-Klee property.

Proof. Let 𝑥 ∈ 𝑆 ( 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) ) and ( 𝑥 𝑛 ) ⊆ 𝐵 ( 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) ) such that | | 𝑥 𝑛 | | 𝐿 → 1 and 𝑥 𝑛 𝑤 → 𝑥 as 𝑛 → ∞ . From Proposition 2.2(iv), we have 𝜌 ( 𝑥 ) = 1 , so it follows from Proposition 2.3(i) that 𝜌 ( 𝑥 𝑛 ) → 𝜌 ( 𝑥 ) as 𝑛 → ∞ . Since 𝑥 𝑛 𝑤 → 𝑥 and the   𝑖 t h -coordinate mapping 𝜋 𝑗 ∶ 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) → ℝ defined by 𝜋 𝑗 ( 𝑥 ) = 𝑥 ( 𝑗 ) is continuous linear function on 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) , it follows that 𝑥 𝑛 ( 𝑗 ) → 𝑥 ( 𝑗 ) as 𝑛 → ∞ for all 𝑗 ∈ ℕ . Thus, by Proposition 2.5, 𝑥 𝑛 → 𝑥 as 𝑛 → ∞ .

3. Uniform Opial Property of 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 )

In this section, we give some topological properties of 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) and investigate its uniform opial property.

We introduce the notation 𝑙 𝐴 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) = ( 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) , | | ⋅ | | 𝐴 ) for brevity.

Theorem 3.1. 𝑆 ( 𝑙 𝐴 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) ) is a closed subspace of 𝑙 𝐴 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) .

Proof. Let us recall the definitions of 𝑆 ( 𝑙 𝐴 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) ) and 𝑙 𝐴 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) , that is, 𝑆 𝑙 𝐴 𝜌 = ( 𝑢 , 𝑣 ; 𝑝 ) 𝑥 ∈ ℓ 0 , 𝑙 ∶ 𝜌 ( 𝜆 𝑥 ) < ∞ ∀ 𝜆 > 0 𝐴 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) = 𝑥 ∈ ℓ 0 . ∶ 𝜌 ( 𝜆 𝑥 ) < ∞ f o r s o m e 𝜆 > 0 ( 3 . 1 ) It is easy to see that 𝑆 ( 𝑙 𝐴 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) ) is a subspace of 𝑙 𝐴 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) . Next we must prove that 𝑆 ( 𝑙 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) ) is closed in 𝑙 𝐴 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) . In order to establish this fact, we show that if 𝑥 𝑛 ⊂ 𝑆 ( 𝑙 𝐴 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) ) for each 𝑛 ∈ ℕ and 𝑥 𝑛 → 𝑥 ∈ 𝑙 𝐴 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) , then 𝑥 ∈ 𝑆 ( 𝑙 𝐴 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) ) .
Take any 𝑐 > 0 . Since 𝑥 𝑛 → 𝑥 ∈ 𝑙 𝐴 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) , by Proposition 1.1, | | 𝑥 𝑛 − 𝑥 | | 𝐴 → 0 and 𝜌 ( 2 𝑐 ( 𝑥 𝑛 − 𝑥 ) ) < ∞ for some 𝑐 > 0 . Besides, since 𝑥 𝑛 ∈ 𝑆 ( 𝑙 𝐴 𝜌 ( 𝑢 , 𝑣 ; 𝑝 ) ) , 𝜌 ( 2 𝑐 𝑥 𝑛 ) < ∞ for every 𝑐 > 0 . We must show that 𝜌 ( 𝑐 𝑥 ) < ∞ for every 𝑐 > 0 . We put 𝜌 ( 𝑥 ) = ∞ 𝑘 = 0 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | | | 𝑥 ( 𝑗 ) 𝑝 𝑘 , ( 3 . 2 ) and take the sequence 𝑐 𝑥 ( 𝑗 ) such that 𝑥 𝑐 𝑥 ( 𝑗 ) = 2 𝑐 ( 𝑗 ) − 𝑥 𝑛 ( 𝑗 ) 2 + 2 𝑐 𝑥 𝑛 ( 𝑗 ) 2 . ( 3 . 3 )
Thus 𝜌 ( 𝑐 𝑥 ) = ∞ 𝑘 = 0 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | | | 𝑐 𝑥 ( 𝑗 ) 𝑝 𝑘 = ∞ 𝑘 = 0 1 2 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | 𝑥 2 𝑐 𝑛 𝑥 ( 𝑗 ) − 𝑥 ( 𝑗 ) − 2 𝑐 𝑛 | | ( 𝑗 ) 𝑝 𝑘 ≤ ∞ 𝑘 = 0 1 2 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | 𝑥 2 𝑐 𝑛 | | + 1 ( 𝑗 ) − 𝑥 ( 𝑗 ) 2 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | 𝑥 2 𝑐 𝑛 | | ( 𝑗 ) 𝑝 𝑘 ≤ 𝐾 2 ∞ 𝑘 = 0 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | 𝑥 2 𝑐 𝑛 | | ( 𝑗 ) − 𝑥 ( 𝑗 ) 𝑝 𝑘 + 𝐾 2 ∞ 𝑘 = 0 𝑘 𝑗 = 0 𝑢 𝑘 𝑣 𝑗 | | 𝑥 2 𝑐 𝑛 | | ( 𝑗 ) 𝑝 𝑘 ≤ 𝐾 2 𝜌 𝑥 2 𝑐 𝑛 + 𝐾 − 𝑥 2 𝜌 2 𝑐 𝑥 𝑛 , ( 3 . 4 ) where 𝐾 = m a x ( 1 , 2 𝐻 − 1 ) . Since 𝜌 ( 2 𝑐 ( 𝑥 𝑛 − 𝑥 ) ) < ∞ and 𝜌 ( 2 𝑐 𝑥 𝑛 ) < ∞ for every 𝑐 > 0 , we obtain 𝜌 ( 𝑐 𝑥 ) < ∞ for every 𝑐 > 0 . So 𝑥 ∈ 𝑆 ( 𝑙