# Some Inequalities Concerning the Weakly Convergent Sequence Coefficient in Banach Spaces

Authors: Hongwei Jiao; Yunrui Guo
Journal: Abstract and Applied Analysis
 (2008)

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## 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.


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## Body

## 1. Introduction

Throughout this paper, we denote byBX and SX 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) if(1.1)diam(C)r(C)>1for 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) if(1.2)inf{diam(C)r(C)}>1,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 thatX 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:(1.3)WCS(X)=inf{limn,m,n≠m∥xn−xm∥:xn→w0,∥xn∥=1,limn,m,n≠m∥xn−xm∥exists}.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 them(1.4)CNJ(X)<1+1(μ(X))2orJ(X)<1+1μ(X).Here, the coefficient μ(X) [9] is defined as the infimum of the set of real numbers r>0 such that(1.5)limsupn→∞∥x+xn∥≤rlimsupn→∞∥x−xn∥for all x∈X and all weakly null sequence (xn) 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 constant(2.1)J(X)=sup{∥x+y∥⋀∥x−y∥:x,y∈SX}is introduced by Casini
[10] and Gao and Lau
[11] and generalized by
Dhompongsa et al. [4]
in the following sense:(2.2)J(a,X)=sup{∥x+y∥⋀∥x−z∥:x,y,z∈BX,∥y−z∥≤a∥x∥},where a is a nonnegative parameter. Obviously, J(0,X)=J(X),
since it is known that in the definition of J(X), SX can be replaced by BX.Theorem 2.1.
Let0≤a≤1 and let X be a Banach space without the Schur
Property. Then(2.3)WCS(X)≥1+(1+a)/min(2,μ(X)+a)J(a,X).Proof. .
IfJ(a,X)=2,
our estimate is trivial since WCS(X)≥1 and μ(X)≥1.
Suppose thatJ(a,X)<2.
Then X is uniformly nonsquare and therefore reflexive
(see [4]). Let {xn} be a weakly null sequence in SX.
Assume that d=limn,m,n≠m∥xn−xm∥ exists and consider a normalized functional
sequence {xn*} such that xn*(xn)=1.
Note that the reflexivity of X guarantees, by passing to a subsequence, if
necessary, that there exists x*∈X* such that xn*→wx*. Let 0<ϵ<1 and choose N large enough so that |x*(xN)|<ϵ/2 and(2.4)d−ϵ<∥xN−xm∥<d+ϵfor all m>N. By the definition of μ(X),(2.5)limsupm→∞∥xm+xNd+ϵ∥≤μ(X)limsupm→∞∥xN−xmd+ϵ∥≤μ(X).Then we can choose M>N large enough such  that
(1)
|xN*(xM)|<ϵ;(2)
|(xM*−x*)(xN)|<ϵ/2;(3)
∥(xN+xM)/(d+ϵ)∥≤μ(X)+ϵ. Hence(2.6)|xM*(xN)|≤|(xM*−x*)(xN)|+|x*(xN)|<ϵ.Let us put x=(xN−xM)/(d+ϵ),(2.7)y=(1+a)xN+xM(d+ϵ)(μ+a+ϵ),z=xN+(1+a)xM(d+ϵ)(μ+a+ϵ)(for short μ=μ(X)). It follows that x,y,z∈BX,∥y−z∥≤a∥x∥ and(2.8)(d+ϵ)∥x+y∥=∥(1+1+aμ+a+ϵ)xN−(1−1μ+a+ϵ)xM∥≥(1+1+aμ+a+ϵ)xN*(xN)−(1−1μ+a+ϵ)xN*(xM)≥1+1+aμ+a+ϵ−ϵ.Also(2.9)(d+ϵ)∥x−z∥=∥(1+1+aμ+a+ϵ)xM−(1−1μ+a+ϵ)xN∥≥(1+1+aμ+a+ϵ)xM*(xM)−(1−1μ+a+ϵ)xM*(xN)≥1+1+aμ+a+ϵ−ϵ.This together with the
definition of J(a,X) gives that(2.10)(d+ϵ)J(a,X)≥1+1+aμ+a+ϵ−ϵ.Since the sequence {xn} and ϵ are arbitrary, we get(2.11)WCS(X)≥μ+1+2aJ(a,X)(μ+a).
Moreover, if we putx=(xN−xM)/(d+ϵ),(2.12)y=(1+a)xN+(1−a)xM2(d+ϵ),z=(1−a)xN+(1+a)xM2(d+ϵ).It follows that x,y,z∈BX,∥y−z∥=a∥x∥ and(2.13)(d+ϵ)∥x+y∥=∥(1+1+a2)xN−(1−1−a2)xM∥≥(1+1+a2)xN*(xN)−(1−1−a2)xN*(xM)≥1+1+a2−ϵ.Also(2.14)(d+ϵ)∥x−z∥=∥(1+1+a2)xM−(1−1−a2)xN∥≥(1+1+a2)xM*(xM)−(1−1−a2)xM*(xN)≥1+1+a2−ϵ.This together with the
definition of J(a,X) gives that(2.15)(d+ϵ)J(a,X)≥1+1+a2−ϵ.Since the sequence {xn} and ϵ are arbitrary, we get(2.16)WCS(X)≥3+a2J(a,X).Adding up (2.11) and (2.16) yields (2.3)
as desired.Corollary 2.2.
LetX be a Banach space
with(2.17)J(a,X)<1+1+amin(2,μ(X)+a),for some 0≤a≤1. Then X has normal structure.Corollary 2.3.
LetX
be a Banach space with(2.18)J(X)<1+1min(2,μ(X)).Then X has normal structure.Remark 2.4.
Corollary2.3 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:(3.1)CNJ(X)=sup{∥x+y∥2+∥x−y∥22(∥x∥2+∥y∥2):x,y∈X,∥x∥+∥y∥≠0}.The generalized version of this
constant is given by Dhompongsa et al. [3] as(3.2)CNJ(a,X)=sup{∥x+y∥2+∥x−z∥22∥x∥2+∥y∥2+∥z∥2:∥x∥+∥y∥+∥z∥≠0,∥y−z∥≤a∥x∥},where a is a nonnegative parameter. Obviously, CNJ(X)=CNJ(0,X).Theorem 3.1.
Let0≤a≤1 and let X be a Banach space without the Schur
Property. Then(3.3)[WCS(X)]2≥1+(1+a)2/min(μ(X)+a,2)2CNJ(a,X).Proof. .
IfCNJ(a,X)=2,
then (3.3) is trivial.
Suppose thatCNJ(a,X)<2.
Then X is uniformly nonsquare and therefore reflexive
(see [3]). Let {xn} be a weakly null sequence in SX and assume that d=limn,m,n≠m∥xn−xm∥ exists and let xN,xM,xN*,xM* be chosen as in Theorem 2.1.
Now let us putx=(xN−xM)/(d+ϵ),(3.4)y=(1+a)((1+a)xN+xM)(d+ϵ)(μ+a+ϵ)2,z=(1+a)(xN+(1+a)xM)(d+ϵ)(μ+a+ϵ)2(for short μ=μ(X)). It is easy to check that ∥y−z∥≤a∥x∥,x∈BX:
(3.5)∥y∥≤(1+a)(μ+a+ϵ),∥z∥≤(1+a)(μ+a+ϵ),and also that(3.6)(d+ϵ)∥x+y∥=∥(1+(1+a)2(μ+a+ϵ)2)xN−(1−1+a(μ+a+ϵ)2)xM∥≥(1+(1+a)2(μ+a+ϵ)2)xN*(xN)−(1−1+a(μ+a+ϵ)2)xN*(xM)≥(1+(1+a)2(μ+a+ϵ)2)(1−ϵ),(d+ϵ)∥x−z∥=∥(1+(1+a)2(μ+a+ϵ)2)xM−(1−1+a(μ+a+ϵ)2)xN∥≥(1+(1+a)2(μ+a+ϵ)2)xM*(xM)−(1−1+a(μ+a+ϵ)2)xM*(xN)≥(1+(1+a)2(μ+a+ϵ)2)(1−ϵ).By the definition of CNJ(a,X),(3.7)CNJ(a,X)≥(1−ϵd+ϵ)2(1+(1+a)2(μ+a+ϵ)2).Since the sequence {xn} and ϵ are arbitrary, we get(3.8)[WCS(X)]2CNJ(a,X)≥1+(1+aμ+a)2.
Moreover, if we putx=(xN−xM)/(d+ϵ),(3.9)y=(1+a)((1+a)xN+(1−a)xM)4(d+ϵ),z=(1+a)((1−a)xN+(1+a)xM)4(d+ϵ),it follows that x∈BX,∥y−z∥≤a∥x∥ and(3.10)∥y∥≤1+a2(d+ϵ)≤1+a2,∥z∥≤1+a2(d+ϵ)≤1+a2,and also that(3.11)(d+ϵ)∥x+y∥=∥(1+(1+a)24)xN−(1−1−a24)xM∥≥(1+(1+a)24)xN*(xN)−(1−1−a24)xN*(xM)≥(1+(1+a)24)(1−ϵ),(d+ϵ)∥x−z∥=∥(1+(1+a)24)xM−(1−1−a24)xN∥≥(1+(1+a)24)xM*(xM)−(1−1−a24)xM*(xN)≥(1+(1+a)24)(1−ϵ).This together with the
definition of CNJ(a,X) gives that(3.12)CNJ(a,X)≥(1+(1+a)24)(1−ϵd+ϵ)2.Since the sequence {xn} and ϵ are arbitrary, we get(3.13)[WCS(X)]2CNJ(a,X)≥1+(1+a2)2.Adding up (3.8) and (3.13) yields the
inequality (3.3) as desired.Corollary 3.2.
LetX be a Banach space with(3.14)CNJ(a,X)<1+(1+a)2min(μ(X)+a,2))2for some a∈[0,1]. Then X has normal structure.Corollary 3.3.
LetX
be a Banach space with(3.15)CNJ(X)<1+1min(2,μ(X))2.Then X has normal structure.Remark 3.4.
Corollary3.3 includes [5, Theorem 1].



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*Source: 840387-2008-10-12.xml*