Modulus of Convexity, the Coeffcient 𝑅(1,𝑋), and Normal Structure in Banach Spaces

Abstract

Let 𝛿_X(𝜖) and 𝑅(1,𝑋) be the modulus of convexity and the Domínguez-Benavides coefficient, respectively. According to these two geometric parameters, we obtain a sufficient condition for normal structure, that is, a Banach space 𝑋 has normal structure if 2𝛿_𝑋(1+𝜖)>max{(𝑅(1,𝑥)−1)𝜖,1−(1−𝜖/𝑅(1,𝑋)−1)} for some 𝜖∈[0,1] which generalizes the known result by Gao and Prus.

1. Introduction

Let 𝑋 be a Banach space. Throughout the paper, denote by 𝑆_𝑋,𝐵_𝑋 the unit sphere and unit ball of 𝑋, respectively. Recall that a Banach space 𝑋 is said to have normal structure (resp., weak, normal, structure) if for every closed bounded (resp., weakly compact) convex subset 𝐶 of 𝑋 with diam 𝐶>0, there exists 𝑥∈𝐶 such that sup{‖𝑥−𝑦‖∶𝑦∈𝐶}<diam𝐶, where diam 𝐶=sup{‖𝑥−𝑦‖∶𝑥,𝑦∈𝐶}. For a reflexive Banach space, the normal structure and weak normal structure are the same. Recently a good deal of investigations have focused on finding the sufficient conditions with various geometrical constants for a Banach space to have normal structure (see, e.g., [1–5]). The geometric condition sufficient for normal structure in terms of the modulus of convexity is given by Goebel [6], who proved that 𝑋 has normal structure provided that 𝛿_𝑋(1)>0. Here the function 𝛿_𝑋(𝜖)∶[0,2]→[0,1], defined by Clarkson [7] as𝛿_𝑋(𝜖)=inf{1−‖𝑥+𝑦‖2∶𝑥,𝑦∈𝐵_𝑋,‖𝑥−𝑦‖≥𝜖}, (1.1)is called the modulus of convexity of 𝑋. Later Gao and Prus generalized the above results as the following (see [2, 8]).

Theorem 1.1. A Banach space 𝑋 has normal structure provided that 𝛿_𝑋(1+𝜖)>𝜖/2 for some 𝜖∈[0,1].

In this paper, we obtain a class of Banach spaces with normal structure, which involves the coefficient 𝑅(1,𝑋). This coefficient is defined by Domínguez Benavides [9] as𝑅(1,𝑋)=sup{𝑙𝑖𝑚𝑖𝑛𝑓_𝑛→∞‖𝑥+𝑥_𝑛‖}, (1.2)where the supremum is taken over all 𝑥∈𝑋 with ‖𝑥‖≤1 and all weakly null sequence (𝑥_𝑛) in 𝐵_𝑋 such that𝐷[(𝑥_𝑛)]∶=𝑙𝑖𝑚𝑠𝑢𝑝_𝑛→∞(𝑙𝑖𝑚𝑠𝑢𝑝_𝑚→∞‖𝑥_𝑛−𝑥_𝑚‖)≤1. (1.3)Obviously, 1≤𝑅(1,𝑋)≤2.

2. Main Results

Let us begin this section with a sufficient condition for a Banach space 𝑋 having weak normal structure and the idea in the following proof is due to [5, Lemma 5].

Lemma 2.1. Let 𝑋 be a Banach space for which 𝐵_𝑋^∗ is 𝑤^∗-sequentially compact. If 𝑋 does not have weak normal structure, then for any 𝜂>0, there exist 𝑥₁,𝑥₂∈𝑆_𝑋 and 𝑓₁,𝑓₂∈𝑆_𝑋^∗, such that
(1)|‖𝑥₁−𝑥₂‖−1|<𝜂;(2)|𝑓_𝑖(𝑥_𝑗)|<𝜂 for 𝑖≠𝑗 and 𝑓_𝑖(𝑥_𝑖)=1,𝑖,𝑗=1,2;(3)‖𝑥₁+𝑥₂‖≤𝑅(1,𝑋)(1+𝜂).

Proof. Assume that 𝑋 does not have weak normal structure. It is well known that (see, e.g., [10]) there exists a sequence {𝑥_𝑛} in 𝑋 satisfying
(1)𝑥_𝑛 is weakly convergent to 0;(2)diam ({𝑥_𝑛}^∞_𝑛=1)=1=lim_𝑛‖𝑥_𝑛−𝑥‖ for all 𝑥∈clco{𝑥_𝑛}^∞_𝑛=1. Since 𝐵_𝑋^∗ is 𝑤^∗-sequentially compact, we can find {𝑓_𝑛} in 𝑆_𝑋^∗ satisfying
(3)𝑓_𝑛(𝑥_𝑛)=‖𝑥_𝑛‖ for all 𝑛;(4)𝑓−−→_𝑛^𝑤∗𝑓 for some 𝑓∈𝐵_𝑋^∗.
Let 𝜂∈(0,1) sufficiently small and 𝜖=𝜂/3. Then, by the properties of the sequence (𝑥_𝑛), we can choose 𝑛₁∈ℕ such that|𝑓(𝑥_𝑛1𝜖)|<2,1−𝜖≤‖𝑥_𝑛1‖≤1. (2.1)Note that the sequence {𝑥_𝑛} is weakly null and verifies 𝐷[{𝑥_𝑛}]=1. It follows from the definition of 𝑅(1,𝑋) thatliminf_𝑛‖𝑥_𝑛+𝑥_𝑛1‖≤𝑅(1,𝑋). (2.2)The rest of the proof is similar to that of [5, Lemma 5].

Theorem 2.2. A Banach space 𝑋 has normal structure provided that 𝛿_𝑋(1+𝜖)>𝑓(𝜖) for some 𝜖∈[0,1], where the function 𝑓(𝜖) is defined as𝜖𝑓(𝜖)∶=(𝑅(1,𝑋)−1)21,0≤𝜖≤,1𝑅(1,𝑋)2(1−1−𝜖1𝑅(1,𝑋)−1),𝑅(1,𝑋)<𝜖≤1. (2.3)

Proof. Observe that 𝑋 is uniformly nonsquare [11] and then reflexive. Therefore normal structure and weak normal structure coincide.
Assume first that 𝑋 fails to have weak normal structure. Fix 𝜂>0 sufficiently small and 𝜖∈[0,1]. It follows that there exist 𝑥₁,𝑥₂∈𝑆_𝑋 and 𝑓₁,𝑓₂∈𝑆_𝑋^∗, satisfying the condition in Lemma 2.1. Next, denote by 𝑅∶=𝑅(1,𝑋) and consider two cases for 𝜖∈[0,1].
Case 1 (𝜖∈[0,1/𝑅]). Now let us put𝑥𝑥=¹−𝑥₂1+𝜂,𝑦=(1−(𝑅−1)𝜖)𝑥₁+𝜖𝑥₂,1+𝜂 (2.4)and so 𝑥∈𝐵_𝑋,𝜖‖𝑦‖=‖1+𝜂(𝑥₁+𝑥₂)+1−𝑅𝜖𝑥1+𝜂¹‖≤𝑅𝜖+(1−𝑅𝜖)=1, (2.5)and also that‖𝑥−𝑦‖=‖(𝑅−1)𝜖𝑥1+𝜂¹−1+𝜖𝑥1+𝜂²‖≥1+𝜖𝑓1+𝜂²(𝑥₂)−(𝑅−1)𝜖𝑓1+𝜂²(𝑥₁)≥1+𝜖−𝜂,1+𝜂‖𝑥+𝑦‖=‖(2−(𝑅−1))𝜖𝑥1+𝜂¹−1−𝜖𝑥1+𝜂²‖≥(2−(𝑅−1))𝜖𝑓1+𝜂¹(𝑥₁)−1−𝜖𝑓1+𝜂¹(𝑥₂)≥(1−2𝜂1+𝜂)(2−(𝑅−1)𝜖). (2.6)By the definition of modulus of convexity,(1−2𝜂1+𝜂)(2−(𝑅−1)𝜖)≤‖𝑥+𝑦‖≤2(1−𝛿_𝑋(‖𝑥−𝑦‖))≤2(1−𝛿_𝑋(1+𝜖−𝜂1+𝜂)), (2.7)or equivalently,(2−(𝑅−1)𝜖)≤2(1−𝛿_𝑋(1+𝜖−𝜂1+𝜂))(1+2𝜂1−𝜂). (2.8)Letting 𝜂→0, we have2𝛿_𝑋(1+𝜖)≤(𝑅−1)𝜖, (2.9)which contradicts our hypothesis.
Case 2 (𝜖∈(1/𝑅,1]). In this case 𝑅>1, otherwise 𝜖>1. Let𝑥^′=𝑥₂−𝑥₁1+𝜂,𝑦^′=(1−(𝑅−1)𝜖^′)𝑥₁+𝜖^′𝑥₂,1+𝜂 (2.10)where 𝜖^′=1−(𝑅−1)𝜖∈[0,1/𝑅). It follows from Case 1 that 𝑥,𝑦∈𝐵_𝑋,‖𝑥−𝑦‖≥(1−2𝜂1+𝜂)(2−(𝑅−1)𝜖^′),‖𝑥+𝑦‖≥1+𝜖^′−𝜂.1+𝜂 (2.11)This implies that𝛿_𝑋(2−(𝑅−1)𝜖^′1)≤2(1−𝜖^′), (2.12)which is equivalent to𝛿_𝑋1(1+𝜖)≤2(1−1−𝜖𝑅−1). (2.13)This is a contradiction.

Remark 2.3. (1) It is readily seen that 𝑓(𝜖)≤𝜖/2 for any 𝜖∈[0,1] and Theorem 2.2 is therefore a generalization of Theorem 1.1. Moreover this generalization is strict whenever 𝑋 is the space with 𝑅(1,𝑋)<2.
(2) Consider the space 𝑋=ℝ² with the norm ‖(𝑥,𝑦)‖∶=max(|𝑥|,|𝑦|,|𝑥−𝑦|). It is known that 𝛿_𝑋(𝜖)=max{0,(𝜖−1)/2} [8] and 𝑅(1,𝑋)=1, then 𝑋 has normal structure from Theorem 2.2, but lies out of the scope of Theorem 1.1.

Corollary 2.4. Let 𝑋 be a Banach space with 𝑅(1,𝑋)=1 and 𝛿_𝑋(1+𝜖)>0forsome𝜖∈[0,1], then 𝑋 has normal structure.

Corollary 2.5. If 𝑋 is a Banach space with𝛿_𝑋1(1+1𝑅(1,𝑋))>21(1−𝑅(1,𝑋)), (2.14)then 𝑋 has normal structure.

Remark 2.6. Corollary 2.5 is equivalent to [4, Corollary 24].

Acknowledgments

The authors would like to express their sincere thanks to the referees for their valuable suggestions on this paper. This work is supported by NSF of Education Department of Henan Province (2008A110012).