volume 7, issue 5, article 161, 2006.
Received 21 September, 2006;
accepted 13 October, 2006.
Communicated by:S.S. Dragomir
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Journal of Inequalities in Pure and Applied Mathematics
SOME ESTIMATES ON THE WEAKLY CONVERGENT SEQUENCE COEFFICIENT IN BANACH SPACES
FENGHUI WANG AND HUANHUAN CUI
Department of Mathematics Luoyang Normal University Luoyang 471022, China.
EMail:wfenghui@163.com
c
2000Victoria University ISSN (electronic): 1443-5756 241-06
Some Estimates on the Weakly Convergent Sequence Coefficient in Banach Spaces Fenghui Wang and Huanhuan Cui
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Abstract
In this paper, we study the weakly convergent sequence coefficient and obtain its estimates for some parameters in Banach spaces, which give some sufficient conditions for a Banach space to have normal structure.
2000 Mathematics Subject Classification:46B20.
Key words: Weakly convergent sequence coefficient; James constant; Von Neumann-Jordan constant; Modulus of smoothness.
The authors would like to thank the referee for his helpful suggestions.
Contents
1 Introduction. . . 3 2 Main Results . . . 5
References
Some Estimates on the Weakly Convergent Sequence Coefficient in Banach Spaces Fenghui Wang and Huanhuan Cui
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1. Introduction
A Banach space X said to have (weak) normal structure provided for every (weakly compact) closed bounded convex subset C ofX withdiam(C) > 0, contains a nondiametral point, i.e., there existsx0 ∈Csuch thatsup{kx−x0k: x∈C}<diam(C).It is clear that normal structure and weak normal structure coincides whenXis reflexive.
The weakly convergent sequence coefficientW CS(X), a measure of weak normal structure, was introduced by Bynum in [3] as the following.
Definition 1.1. The weakly convergent sequence coefficient ofXis defined by (1.1) W CS(X)
= inf
diama({xn})
ra({xn}) :{xn}is a weakly convergent sequence
, where diama({xn}) = lim supk→∞{kxn−xmk : n, m≥ k}is the asymptotic diameter of{xn}andra({xn}) = inf{lim supn→∞kxn−yk:y ∈co({x¯ n})is the asymptotic radius of{xn}.
One of the equivalent forms ofW CS(X)is W CS(X) = inf
n,m,n6=mlim kxn−xmk:xn→w 0,kxnk= 1
and lim
n,m,n6=mkxn−xmkexists
.
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Obviously,1≤W CS(X)≤2,and it is well known thatW CS(X)>1implies thatXhas a weak normal structure.
The constantR(a, X), which is a generalized García-Falset coefficient [10], was introduced by Domínguez [7] as: For a given real numbera >0,
(1.2) R(a, X) = supn
lim inf
n→∞ kx+xnko ,
where the supremum is taken over allx∈ Xwithkxk ≤aand all weakly null sequences{xn} ⊆BX such that
(1.3) lim
n,m,n6=mkxn−xmk ≤1.
We shall assume throughout this paper that BX and SX to denote the unit ball and unit sphere ofX, respectively. xn→w xstands for weak convergence of sequence{xn}inXto a pointxinX.
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2. Main Results
The James constant, or the nonsquare constant, was introduced by Gao and Lau in [8] as
J(X) = sup{kx+yk ∧ kx−yk:x, y ∈SX}
= sup{kx+yk ∧ kx−yk:x, y ∈BX}.
A relation between the constant R(1, X)and the James constant J(X)can be found in [6,12]:
R(1, X)≤J(X).
We now state an inequality between the James constantJ(X)and the weakly convergent sequence coefficientW CS(X).
Theorem 2.1. LetX be a Banach space with the James constantJ(X).Then
(2.1) W CS(X)≥ J(X) + 1
(J(X))2 .
Proof. IfJ(X) = 2, it suffices to note thatW CS(X)≥1.Thus our estimate is a trivial one.
If J(X) < 2, thenX is reflexive. Let {xn} be a weakly null sequence in SX. Assume thatd = limn,m,n6=mkxn−xmkexists and consider a normalized functional sequence {x∗n}such thatx∗n(xn) = 1. Note that the reflexivity ofX guarantees, by passing through the subsequence, that there existsx∗ ∈X∗ such thatx∗n →w x∗.Let0< <1and chooseN large enough so that|x∗(xN)|< /2 and
d− <kxN −xmk< d+
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for allm > N.Note that
n,m,n6=mlim
xn−xm d+
≤1 and
xN d+
≤1.
Then by the definition ofR(1, X), we can choose M > N large enough such that
xN +xM d+
≤R(1, X) +≤J(X) +, |(x∗M −x∗)(xN)|< /2, and|x∗N(xM)|< .Hence
|x∗M(xN)| ≤ |(x∗M −x∗)(xN))|+|x∗(xN)|< . Putα=J(X),
x= xN −xM
d+ , and y= xN +xM (d+)(α+). It follows thatkxk ≤1,kyk ≤1,and also that
kx+yk= 1 (d+)(α+)
(α+ 1 +)xN −(α−1 +)xM
≥ 1
(d+)(α+) (α+ 1 +)x∗N(xN)−(α−1 +)x∗N(xM)
≥ α+ 1− (d+)(α+),
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kx−yk= 1 (d+)(α+)
(α+ 1 +)xM −(α−1 +)xN
≥ 1
(d+)(α+) (α+ 1 +)x∗M(xM)−(α−1 +)x∗M(xN)
≥ α+ 1− (d+)(α+).
Thus, from the definition of the James constant,
J(X)≥ α+ 1−
(d+)(α+) = J(X) + 1− (d+)(J(X) +). Letting→0,we get
d ≥ J(X) + 1 (J(X))2 .
Since the sequence{xn}is arbitrary, we get the inequality (2.1).
As an application of Theorem2.1, we can obtain a sufficient condition for X to have normal structure in terms of the James constant.
Corollary 2.2 ([4, Theorem 2.1]). Let X be a Banach space with J(X) <
(1 +√
5)/2.ThenX has normal structure.
The modulus of smoothness [14] ofXis the functionρX(τ)defined by ρX(τ) = sup
kx+τ yk+kx−τ yk
2 −1 :x, y ∈SX
.
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It is readily seen that for anyx, y ∈SX,
kx±yk ≤ kx±τ yk+ (1−τ) (0< τ ≤1), which implies thatJ(X)≤ρX(τ) + 2−τ.
In [2], Baronti et al. introduced a constantA2(X), which is defined by A2(X) = ρX(1) + 1 = sup
kx+yk+kx−yk
2 :x, y ∈SX
. It is worth noting thatA2(X) = A2(X∗).
We now state an inequality between the modulus of smoothnessρX(τ)and the weakly convergent sequence coefficientW CS(X).
Theorem 2.3. LetXbe a Banach space with the modulus of smoothnessρX(τ).
Then for any0< τ ≤1,
(2.2) W CS(X)≥ ρX(τ) + 2
(ρX(τ) + 1)(ρX(τ)−τ + 2). Proof. Let0< τ ≤1.IfρX(τ) =τ, it suffices to note that
ρX(τ) + 2
(ρX(τ) + 1)(ρX(τ)−τ+ 2) = τ + 2
2(τ + 1) ≤1.
Thus our estimate is a trivial one.
IfρX(τ)< τ, thenXis reflexive. Let{xn}be a weakly null sequence inSX. Assume that d= limn, m, n6=mkxn−xmkexists and consider a normalized
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functional sequence {x∗n} such that x∗n(xn) = 1. Note that the reflexivity of X guarantees that there exists x∗ ∈ X∗ such that x∗n →w x∗. Let > 0 and xM, xN, xandyselected as in Theorem2.1. Similarly, we get
kx±τ yk ≥ α(τ) +τ− (d+)(α(τ) +),
whereα(τ) = ρX(τ) + 2−τ.Then by the definition ofρX(τ), we obtain ρX(τ)≥ α(τ) +τ −
(d+)(α(τ) +)−1.
Letting→0,
ρX(τ) + 1≥ α(τ) +τ
dα(τ) = ρX(τ) + 2 d(ρX(τ)−τ + 2), which gives that
d≥ ρX(τ) + 2
(ρX(τ) + 1)(ρX(τ)−τ + 2).
Since the sequence{xn}is arbitrary, we get the inequality (2.2).
It is known that ifρX(τ)< τ /2for someτ >0,thenXhas normal structure (see [9]). Using Theorem2.3, We can improve this result in the following form:
Corollary 2.4. LetX be a Banach space with ρX(τ)< τ −2 +√
τ2 + 4 2
for some τ ∈ (0,1].Then X has normal structure. In particular, if A2(X) <
(1 +√
5)/2,thenX and its dualX∗ have normal structure.
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In connection with a famous work of Jordan-von Neumann concerning inner products, the Jordan-von Neumann constant CNJ(X)of X was introduced by Clarkson (cf. [1,11]) as
CNJ(X) = sup
kx+yk2+kx−yk2
2(kxk2+kyk2) :x, y ∈Xand not both zero
. A relationship betweenJ(X)andCNJ(X)is found in ([11] Theorem 3):J(X)≤ p2CNJ(X).
In [5], Dhompongsa et al. proved the following inequality (2.3). We now re- state this inequality without the ultra product technique and the factCNJ(X) = CNJ(X∗).
Theorem 2.5 ([5] Theorem 3.8). LetXbe a Banach space with the von Neumann- Jordan constantCNJ(X).Then
(2.3) (W CS(X))2 ≥ 2CNJ(X) + 1
2(CNJ(X))2 .
Proof. If CNJ(X) = 2, it suffices to note that W CS(X) ≥ 1. Thus our esti- mates is a trivial one.
IfCNJ(X)< 2, thenX is reflexive. Let{xn}be a weakly null sequence in SX. Assume thatd = limn,m,n6=mkxn−xmkexists and consider a normalized functional sequence {x∗n}such thatx∗n(xn) = 1. Note that the reflexivity ofX gurantees that there esistsx∗ ∈X∗such thatx∗n→w x∗.Let >0and chooseN large enough so that|x∗(xN)|< /2and
d− <kxN −xmk< d+
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for allm > N.Note that
n,m,n6=mlim
xn−xm
d+
≤1 and
xN
d+
≤1.
Then by the definition ofR(1, X), we can choose M > N large enough such that
xN −xM d+
≤R(1, X) +≤p
2CNJ(X) +, |(x∗M −x∗)(xN)|< /2, and|x∗N(xM)|< .Hence
|x∗M(xN)|<|(x∗M −x∗)(xN))|+|x∗(xN)|< . Putα =p
2CNJ(X), x =α2(xN −xM), y =xN +xM.It follows thatkxk ≤ α2(d+),kyk ≤(α+)(d+), and also that
kx+yk=k(α2+ 1)xN −(α2−1)xMk
≥(α2+ 1)x∗N(xN)−(α2−1)x∗N(xM)
≥α2+ 1−3,
kx−yk=k(α2+ 1)xM −(α2−1)xNk
≥(α2+ 1)x∗M(xM)−(α2−1)x∗M(xN)
≥α2+ 1−3.
Thus, from the definition of the von Neumann-Jordan constant, CNJ(X)≥ 2(α2+ 1−3)2
2(α4(d+)2+ (α+)2(d+)2)
= 1
(d+)2 · (α2+ 1−3)2 α4+ (α+)2 .
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Sinceis arbitrary andα =p
2CNJ(X), we get
CNJ(X)≥ 1 d2
1 + 1
α2
= 2CNJ(X) + 1 d2·2CNJ(X), which implies that
d2 ≥ 2CNJ(X) + 1 2(CNJ(X))2 .
Since the sequence{xn}is arbitrary, we obtain the inequality (2.3).
Using Theorem2.5, we can get a sufficient condition forX to have normal structure in terms of the von Neumann-Jordan constant.
Corollary 2.6 ([6, Theorem 3.16], [13, Theorem 2]). LetXbe a Banach space withCNJ(X)<(1 +√
3)/2.ThenX and its dualX∗ have normal structure.
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