Some Inequalities Concerning the Weakly Convergent Sequence Coefficient in Banach Spaces

Abstract

We establish two inequalities concerning the weakly convergent sequence coefficient and other parameters, which enable us to obtain some sufficient conditions for normal structure.

1. Introduction

Throughout this paper, we denote by 𝐵_𝑋 and 𝑆_𝑋 the unit ball and the unit sphere of a Banach space 𝑋, respectively. Let 𝐶 be a nonempty bounded subset of 𝑋. The numbers diam(𝐶)=sup{‖𝑥−𝑦‖∶𝑥,𝑦∈𝐶} and 𝑟(𝐶)=inf{sup{‖𝑥−𝑦‖∶𝑦∈𝐶}∶𝑥∈𝐶} are, respectively, called the diameter and the Chebyshev radius of 𝐶. A Banach space 𝑋 is said to have normal structure (resp., weak normal structure) ifdiam(𝐶)𝑟(𝐶)>1(1.1)for every closed bounded (resp., weakly compact) convex subset 𝐶 of 𝑋 with diam(𝐶)>0. A Banach space 𝑋 is said to have uniform normal structure (resp., weak uniform normal structure) ifinf{diam(𝐶)𝑟(𝐶)}>1,(1.2)where the infimum is taken over all bounded closed (resp., weakly compact) convex subsets 𝐶 of 𝑋 with diam(𝐶)>0. It is clear that normal structure and weak normal structure coincide when 𝑋 is reflexive.

Throughout this paper, we assume that 𝑋 does not have the Schur property. The weakly convergent sequence coefficient WCS(𝑋), introduced by Bynum [1], is reformulated by Sims and Smyth [2] as the following equivalent form:WCS(𝑋)=inf{lim_{𝑛,𝑚,𝑛≠𝑚}‖𝑥_𝑛−𝑥_𝑚‖∶𝑥−→_𝑛^𝑤0,‖𝑥_𝑛‖=1,lim_{𝑛,𝑚,𝑛≠𝑚}‖𝑥_𝑛−𝑥_𝑚‖exists}.(1.3)Obviously 1≤WCS(𝑋)≤2 and it is known that WCS(𝑋)>1 implies that 𝑋 has weak normal structure.

There are many geometric conditions implying a Banach space to have normal structure (see, e.g., [2–8]), among them𝐶_NJ1(𝑋)<1+(𝜇(𝑋))²1or𝐽(𝑋)<1+𝜇(𝑋).(1.4)Here, the coefficient 𝜇(𝑋) [9] is defined as the infimum of the set of real numbers 𝑟>0 such thatlimsup_𝑛→∞‖𝑥+𝑥_𝑛‖≤𝑟limsup_𝑛→∞‖𝑥−𝑥_𝑛‖(1.5)for all 𝑥∈𝑋 and all weakly null sequence (𝑥_𝑛) in 𝑋. The aim of this paper is to state some estimates concerning the weakly convergent sequence coefficient. By these estimates, we get sufficient conditions for normal structure in terms of the generalized James and von Neumann-Jordan constants and thus generalize the above results.

2. Generalized James Constant

The James constant, or the nonsquare constant𝐽(𝑋)=sup{‖𝑥+𝑦‖‖𝑥−𝑦‖∶𝑥,𝑦∈𝑆_𝑋}(2.1)is introduced by Casini [10] and Gao and Lau [11] and generalized by Dhompongsa et al. [4] in the following sense:𝐽(𝑎,𝑋)=sup{‖𝑥+𝑦‖‖𝑥−𝑧‖∶𝑥,𝑦,𝑧∈𝐵_𝑋,‖𝑦−𝑧‖≤𝑎‖𝑥‖},(2.2)where 𝑎 is a nonnegative parameter. Obviously, 𝐽(0,𝑋)=𝐽(𝑋), since it is known that in the definition of 𝐽(𝑋), 𝑆_𝑋 can be replaced by 𝐵_𝑋.

Theorem 2.1. Let 0≤𝑎≤1 and let 𝑋 be a Banach space without the Schur Property. Then𝑊𝐶𝑆(𝑋)≥1+(1+𝑎)/min(2,𝜇(𝑋)+𝑎)𝐽(𝑎,𝑋).(2.3)

Proof. . If 𝐽(𝑎,𝑋)=2, our estimate is trivial since WCS(𝑋)≥1 and 𝜇(𝑋)≥1.
Suppose that 𝐽(𝑎,𝑋)<2. Then 𝑋 is uniformly nonsquare and therefore reflexive (see [4]). Let {𝑥_𝑛} be a weakly null sequence in 𝑆_𝑋. Assume that 𝑑=lim_{𝑛,𝑚,𝑛≠𝑚}‖𝑥_𝑛−𝑥_𝑚‖ exists and consider a normalized functional sequence {𝑥^∗_𝑛} such that 𝑥^∗_𝑛(𝑥_𝑛)=1. Note that the reflexivity of 𝑋 guarantees, by passing to a subsequence, if necessary, that there exists 𝑥^∗∈𝑋^∗ such that 𝑥−→^∗_𝑛^𝑤𝑥^∗. Let 0<𝜖<1 and choose 𝑁 large enough so that |𝑥^∗(𝑥_𝑁)|<𝜖/2 and𝑑−𝜖<‖𝑥_𝑁−𝑥_𝑚‖<𝑑+𝜖(2.4)for all 𝑚>𝑁. By the definition of 𝜇(𝑋),limsup_𝑚→∞‖𝑥_𝑚+𝑥_𝑁𝑑+𝜖‖≤𝜇(𝑋)limsup_𝑚→∞‖𝑥_𝑁−𝑥_𝑚𝑑+𝜖‖≤𝜇(𝑋).(2.5)Then we can choose 𝑀>𝑁 large enough such that
(1)|𝑥^∗_𝑁(𝑥_𝑀)|<𝜖;(2)|(𝑥^∗_𝑀−𝑥^∗)(𝑥_𝑁)|<𝜖/2;(3)‖(𝑥_𝑁+𝑥_𝑀)/(𝑑+𝜖)‖≤𝜇(𝑋)+𝜖. Hence|𝑥^∗_𝑀(𝑥_𝑁)|≤|(𝑥^∗_𝑀−𝑥^∗)(𝑥_𝑁)|+|𝑥^∗(𝑥_𝑁)|<𝜖.(2.6)Let us put 𝑥=(𝑥_𝑁−𝑥_𝑀)/(𝑑+𝜖),𝑦=(1+𝑎)𝑥_𝑁+𝑥_𝑀𝑥(𝑑+𝜖)(𝜇+𝑎+𝜖),𝑧=^𝑁+(1+𝑎)𝑥_𝑀(𝑑+𝜖)(𝜇+𝑎+𝜖)(2.7)(for short 𝜇=𝜇(𝑋)). It follows that 𝑥,𝑦,𝑧∈𝐵_𝑋,‖𝑦−𝑧‖≤𝑎‖𝑥‖ and(𝑑+𝜖)‖𝑥+𝑦‖=‖(1+1+𝑎𝜇+𝑎+𝜖)𝑥_𝑁1−(1−𝜇+𝑎+𝜖)𝑥_𝑀‖≥(1+1+𝑎𝜇+𝑎+𝜖)𝑥^∗_𝑁(𝑥_𝑁1)−(1−𝜇+𝑎+𝜖)𝑥^∗_𝑁(𝑥_𝑀)≥1+1+𝑎𝜇+𝑎+𝜖−𝜖.(2.8)Also(𝑑+𝜖)‖𝑥−𝑧‖=‖(1+1+𝑎𝜇+𝑎+𝜖)𝑥_𝑀1−(1−𝜇+𝑎+𝜖)𝑥_𝑁‖≥(1+1+𝑎𝜇+𝑎+𝜖)𝑥^∗_𝑀(𝑥_𝑀1)−(1−𝜇+𝑎+𝜖)𝑥^∗_𝑀(𝑥_𝑁)≥1+1+𝑎𝜇+𝑎+𝜖−𝜖.(2.9)This together with the definition of 𝐽(𝑎,𝑋) gives that(𝑑+𝜖)𝐽(𝑎,𝑋)≥1+1+𝑎𝜇+𝑎+𝜖−𝜖.(2.10)Since the sequence {𝑥_𝑛} and 𝜖 are arbitrary, we getWCS(𝑋)≥𝜇+1+2𝑎𝐽(𝑎,𝑋)(𝜇+𝑎).(2.11)
Moreover, if we put 𝑥=(𝑥_𝑁−𝑥_𝑀)/(𝑑+𝜖),𝑦=(1+𝑎)𝑥_𝑁+(1−𝑎)𝑥_𝑀2(𝑑+𝜖),𝑧=(1−𝑎)𝑥_𝑁+(1+𝑎)𝑥_𝑀2(𝑑+𝜖).(2.12)It follows that 𝑥,𝑦,𝑧∈𝐵_𝑋,‖𝑦−𝑧‖=𝑎‖𝑥‖ and(𝑑+𝜖)‖𝑥+𝑦‖=‖(1+1+𝑎2)𝑥_𝑁−(1−1−𝑎2)𝑥_𝑀‖≥(1+1+𝑎2)𝑥^∗_𝑁(𝑥_𝑁)−(1−1−𝑎2)𝑥^∗_𝑁(𝑥_𝑀)≥1+1+𝑎2−𝜖.(2.13)Also(𝑑+𝜖)‖𝑥−𝑧‖=‖(1+1+𝑎2)𝑥_𝑀−(1−1−𝑎2)𝑥_𝑁‖≥(1+1+𝑎2)𝑥^∗_𝑀(𝑥_𝑀)−(1−1−𝑎2)𝑥^∗_𝑀(𝑥_𝑁)≥1+1+𝑎2−𝜖.(2.14)This together with the definition of 𝐽(𝑎,𝑋) gives that(𝑑+𝜖)𝐽(𝑎,𝑋)≥1+1+𝑎2−𝜖.(2.15)Since the sequence {𝑥_𝑛} and 𝜖 are arbitrary, we getWCS(𝑋)≥3+𝑎2𝐽(𝑎,𝑋).(2.16)Adding up (2.11) and (2.16) yields (2.3) as desired.

Corollary 2.2. Let 𝑋 be a Banach space with𝐽(𝑎,𝑋)<1+1+𝑎min(2,𝜇(𝑋)+𝑎),(2.17)for some 0≤𝑎≤1. Then 𝑋 has normal structure.

Corollary 2.3. Let 𝑋 be a Banach space with1𝐽(𝑋)<1+min(2,𝜇(𝑋)).(2.18)Then 𝑋 has normal structure.

Remark 2.4. Corollary 2.3 includes [5, Theorem 2].

3. Generalized Von Neumann-Jordan Constant

The von Neumann-Jordan constant is introduced by Clarkson [12] and reformulated by Kato et al. [6] in the following way:𝐶_NJ(𝑋)=sup{‖𝑥+𝑦‖²+‖𝑥−𝑦‖²2(‖𝑥‖²+‖𝑦‖²)∶𝑥,𝑦∈𝑋,‖𝑥‖+‖𝑦‖≠0}.(3.1)The generalized version of this constant is given by Dhompongsa et al. [3] as𝐶_NJ(𝑎,𝑋)=sup{‖𝑥+𝑦‖²+‖𝑥−𝑧‖²2‖𝑥‖²+‖𝑦‖²+‖𝑧‖²∶‖𝑥‖+‖𝑦‖+‖𝑧‖≠0,‖𝑦−𝑧‖≤𝑎‖𝑥‖},(3.2)where 𝑎 is a nonnegative parameter. Obviously, 𝐶_NJ(𝑋)=𝐶_NJ(0,𝑋).

Theorem 3.1. Let 0≤𝑎≤1 and let 𝑋 be a Banach space without the Schur Property. Then[𝑊𝐶𝑆(𝑋)]²≥1+(1+𝑎)²/𝑚𝑖𝑛(𝜇(𝑋)+𝑎,2)²𝐶_𝑁𝐽(𝑎,𝑋).(3.3)

Proof. . If 𝐶_NJ(𝑎,𝑋)=2, then (3.3) is trivial.
Suppose that 𝐶_NJ(𝑎,𝑋)<2. Then 𝑋 is uniformly nonsquare and therefore reflexive (see [3]). Let {𝑥_𝑛} be a weakly null sequence in 𝑆_𝑋 and assume that 𝑑=lim_{𝑛,𝑚,𝑛≠𝑚}‖𝑥_𝑛−𝑥_𝑚‖ exists and let 𝑥_𝑁,𝑥_𝑀,𝑥^∗_𝑁,𝑥^∗_𝑀 be chosen as in Theorem 2.1.
Now let us put 𝑥=(𝑥_𝑁−𝑥_𝑀)/(𝑑+𝜖),𝑦=(1+𝑎)((1+𝑎)𝑥_𝑁+𝑥_𝑀)(𝑑+𝜖)(𝜇+𝑎+𝜖)²,𝑧=(1+𝑎)(𝑥_𝑁+(1+𝑎)𝑥_𝑀)(𝑑+𝜖)(𝜇+𝑎+𝜖)²(3.4)(for short 𝜇=𝜇(𝑋)). It is easy to check that ‖𝑦−𝑧‖≤𝑎‖𝑥‖,𝑥∈𝐵_𝑋: ‖𝑦‖≤(1+𝑎)(𝜇+𝑎+𝜖),‖𝑧‖≤(1+𝑎)(𝜇+𝑎+𝜖),(3.5)and also that(𝑑+𝜖)‖𝑥+𝑦‖=‖(1+(1+𝑎)²(𝜇+𝑎+𝜖)²)𝑥_𝑁−(1−1+𝑎(𝜇+𝑎+𝜖)²)𝑥_𝑀‖≥(1+(1+𝑎)²(𝜇+𝑎+𝜖)²)𝑥^∗_𝑁(𝑥_𝑁)−(1−1+𝑎(𝜇+𝑎+𝜖)²)𝑥^∗_𝑁(𝑥_𝑀)≥(1+(1+𝑎)²(𝜇+𝑎+𝜖)²)(1−𝜖),(𝑑+𝜖)‖𝑥−𝑧‖=‖(1+(1+𝑎)²(𝜇+𝑎+𝜖)²)𝑥_𝑀−(1−1+𝑎(𝜇+𝑎+𝜖)²)𝑥_𝑁‖≥(1+(1+𝑎)²(𝜇+𝑎+𝜖)²)𝑥^∗_𝑀(𝑥_𝑀)−(1−1+𝑎(𝜇+𝑎+𝜖)²)𝑥^∗_𝑀(𝑥_𝑁)≥(1+(1+𝑎)²(𝜇+𝑎+𝜖)²)(1−𝜖).(3.6)By the definition of 𝐶_NJ(𝑎,𝑋),𝐶_NJ(𝑎,𝑋)≥(1−𝜖)𝑑+𝜖²(1+(1+𝑎)²(𝜇+𝑎+𝜖)²).(3.7)Since the sequence {𝑥_𝑛} and 𝜖 are arbitrary, we get[WCS(𝑋)]²𝐶_NJ(𝑎,𝑋)≥1+(1+𝑎)𝜇+𝑎².(3.8)
Moreover, if we put 𝑥=(𝑥_𝑁−𝑥_𝑀)/(𝑑+𝜖),𝑦=(1+𝑎)((1+𝑎)𝑥_𝑁+(1−𝑎)𝑥_𝑀)4(𝑑+𝜖),𝑧=(1+𝑎)((1−𝑎)𝑥_𝑁+(1+𝑎)𝑥_𝑀)4(𝑑+𝜖),(3.9)it follows that 𝑥∈𝐵_𝑋,‖𝑦−𝑧‖≤𝑎‖𝑥‖ and‖𝑦‖≤1+𝑎≤2(𝑑+𝜖)1+𝑎2,‖𝑧‖≤1+𝑎≤2(𝑑+𝜖)1+𝑎2,(3.10)and also that(𝑑+𝜖)‖𝑥+𝑦‖=‖(1+(1+𝑎)²4)𝑥_𝑁−(1−1−𝑎²4)𝑥_𝑀‖≥(1+(1+𝑎)²4)𝑥^∗_𝑁(𝑥_𝑁)−(1−1−𝑎²4)𝑥^∗_𝑁(𝑥_𝑀)≥(1+(1+𝑎)²4)(1−𝜖),(𝑑+𝜖)‖𝑥−𝑧‖=‖(1+(1+𝑎)²4)𝑥_𝑀−(1−1−𝑎²4)𝑥_𝑁‖≥(1+(1+𝑎)²4)𝑥^∗_𝑀(𝑥_𝑀)−(1−1−𝑎²4)𝑥^∗_𝑀(𝑥_𝑁)≥(1+(1+𝑎)²4)(1−𝜖).(3.11)This together with the definition of 𝐶_NJ(𝑎,𝑋) gives that𝐶_NJ(𝑎,𝑋)≥(1+(1+𝑎)²4)(1−𝜖)𝑑+𝜖².(3.12)Since the sequence {𝑥_𝑛} and 𝜖 are arbitrary, we get[WCS(𝑋)]²𝐶_NJ(𝑎,𝑋)≥1+(1+𝑎2)².(3.13)Adding up (3.8) and (3.13) yields the inequality (3.3) as desired.

Corollary 3.2. Let 𝑋 be a Banach space with𝐶_𝑁𝐽(𝑎,𝑋)<1+(1+𝑎)²min(𝜇(𝑋)+𝑎,2))²(3.14)for some 𝑎∈[0,1]. Then 𝑋 has normal structure.

Corollary 3.3. Let 𝑋 be a Banach space with𝐶_𝑁𝐽1(𝑋)<1+min(2,𝜇(𝑋))².(3.15)Then 𝑋 has normal structure.

Remark 3.4. Corollary 3.3 includes [5, Theorem 1].

Acknowledgments

The author would like to express his sincere thanks to the anonymous referees for their careful review of the manuscript and the valuable comments. This paper is supported by the National Natural Science Foundation of China (10671057), Basic Research Foundation for Department of Education of Henan Province (2008A110003), and the Natural Science Foundation of Henan Institute of Science and Technology (06055).