PYTHAGOREAN PARAMETERS AND NORMAL STRUCTURE IN BANACH SPACES

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Pythagorean Parameters Hongwei Jiao and Bijun Pang

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PYTHAGOREAN PARAMETERS AND NORMAL STRUCTURE IN BANACH SPACES

HONGWEI JIAO BIJUN PANG

Department of Mathematics Department of Mathematics

Henan Institute of Science and Technology Luoyang Teachers College Xinxiang 453003, P.R. China. Luoyang 471022, P.R. China.

EMail:hongwjiao@163.com

Received: 16 August, 2007

Accepted: 15 February, 2008 Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 46B20.

Key words: Uniform non-squareness; Normal structure.

Abstract: Recently, Gao introduced some quadratic parameters, such asE(X)andf(X).

In this paper, we obtain some sufficient conditions for normal structure in terms of Gao’s parameters, improving some known results.

Acknowledgements: The author would like to thank the anonymous referees for their helpful sugges- tions on this paper.

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1. Introduction

There are several parameters and constants which are defined on the unit sphere or the unit ball of a Banach space. These parameters and constants, such as the James and von Neumann-Jordan constants, have been proved to be very useful in the descriptions of the geometric structure of Banach spaces.

Based on a Pythagorean theorem, Gao introduced some quadratic parameters recently [1, 2]. Using these parameters, one can easily distinguish several important classes of spaces such as uniform non-squareness or spaces having normal structure.

In this paper, we are going to continue the study in Gao’s parameters. Moreover, we obtain some sufficient conditions for a Banach space to have normal structure.

LetXbe a Banach space andX^∗its dual. We shall assume throughout this paper thatB_X andS_X denote the unit ball and unit sphere ofX, respectively.

One of Gao’s parametersE(X)is defined by the formula

E(X) = sup{kx+yk²+kx−yk² :x, y ∈S_X},

whereis a nonnegative number. It is worth noting thatE(X)was also introduced by Saejung [3] and Yang-Wang [5] recently. Let us now collect some properties related to this parameter (see [1,4,5]).

(1) Xis uniformly non-square if and only ifE(X)<2(1+)²for some∈(0,1].

(2) X has uniform normal structure ifE(X)<1 + (1 +)² for some∈(0,1].

(3) E(X) =E(X),e whereXe is the ultrapower ofX.

(4) E(X) = sup{kx+yk²+kx−yk² :x, y ∈B_X}.

It follows from the property (4) that E(X) = inf

kx+yk²+kx−yk²

max(kxk²,kyk²) :x, y ∈X,kxk+kyk 6= 0

.

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Now let us pay attention to another Gao’s parameterf(X), which is defined by the formula

f(X) = inf{kx+yk²+kx−yk² :x, y ∈S_X}, whereis a nonnegative number.

We quote some properties related to this parameter (see [1,2]).

(1) Iff(X)>2for some∈(0,1],thenX is uniformly non-square.

(2) X has uniform normal structure iff₁(X)>32/9.

Using a similar method to [4, Theorem 3], we can also deduce that f(X) = f(X),e whereXe is the ultrapower ofX.

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2. Main Results

We start this section with some definitions. Recall that X is called uniformly non- square if there exists δ > 0, such that if x, y ∈ S_X then kx+yk/2 ≤ 1−δ or kx−yk/2 ≤ 1−δ. In what follows, we shall show that f(X) also provides a characterization of the uniformly non-square spaces, namelyf1(X)>2.

Theorem 2.1. X is uniformly non-square if and only iff1(X)>2.

Proof. It is convenient for us to assume in this proof thatdimX <∞.The extension of the results to the general case is immediate, depending only on the formula

f(X) = inf{f(Y) :Y subspace ofX and dimY = 2}.

We are going to prove that uniform non-squareness impliesf₁(X)> 2.Assume on the contrary thatf₁(X) = 2. It follows from the definition off(X)that there exist x, y ∈S_X so that

kx+yk²+kx−yk² = 2.

Then, sincekx+yk+kx−yk ≥2,we have

kx±yk² = 2− kx∓yk² ≤2−(2− kx±yk)²,

which implies thatkx±yk= 1. Now let us putu=x+y, v=x−y, thenu, v ∈S_X andku±vk= 2.This is a contradiction. The converse of this assertion was proved by Gao [2, Theorem 2.8], and thus the proof is complete.

Consider now the definitions of normal structure. A Banach space X is said to have (weak) normal structure provided that every (weakly compact) closed bounded convex subsetC ofXwithdiam(C)>0,contains a non-diametral point, i.e., there existsx₀ ∈C such thatsup{kx−x₀k :x ∈C}<diam(C).It is clear that normal

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structure and weak normal structure coincides whenXis reflexive. A Banach space Xis said to have uniform normal structure ifinf{diam(C)/rad(C)}>1, where the infimum is taken over all bounded closed convex subsetsCofXwithdiam(C)>0.

To study the relation between normal structure and Gao’s parameter, we need a sufficient condition for normal structure, which was posed by Saejung [4, Lemma 2]

recently.

Theorem 2.2. LetX be a Banach space with E(X)<2 +²+√

4 +²

for some∈(0,1],thenX has uniform normal structure.

Proof. By our hypothesis it is enough to show thatXhas normal structure. Suppose thatXlacks normal structure, then by [4, Lemma 2], there existex1,ex2,ex3 ∈S_X_e and fe₁,fe₂,fe₃ ∈S

Xf^∗ satisfying:

(a) kxe_i−xe_jk= 1andfe_i(xe_j) = 0for alli6=j.

(b) fe_i(ex_i) = 1fori= 1,2,3and (c) kxe₃−(ex₂+ex₁)k ≥ kxe₂+xe₁k.

Let2α() =√

4 +² + 2−and consider three possible cases.

CASE 1. kex₁ +xe₂k ≤ α().In this case, let us put xe = xe₁ −xe₂ and ey = (xe₁ + ex₂)/α().It follows thatex,ye∈B_X_e,and

kxe+yke =k(1 + (/α()))xe₁−(1−(/α()))xe₂k

≥(1 + (/α()))fe₁(xe₁)−(1−(/α()))fe₁(ex₂)

= 1 + (/α()),

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kex−yke =k(1 + (/α()))xe₂−(1−(/α()))xe₁k

≥(1 + (/α()))fe₂(xe₂)−(1−(/α()))fe₂(ex₁)

= 1 + (/α()).

CASE 2. kxe1 +xe2k ≥ α() andkxe3 +xe2 −xe1k ≤ α(). In this case, let us put ex=xe2−xe3 andye= (ex3+ex2−ex1)/α().It follows thatx,e ye∈B_X_e,and

kxe+yke =k(1 + (/α()))xe₂−(1−(/α()))xe₃−(/α())ex₁k

≥(1 + (/α()))fe₂(xe₂)−(1−(/α()))fe₂(ex₃)−(/α())fe₂(ex₁)

= 1 + (/α()),

kxe−yke =k(1 + (/α()))xe₃−(1−(/α()))xe₂−(/α())ex₁)k

≥(1 + (/α()))fe₃(xe₃)−(1−(/α()))fe₃(ex₂)−(/α())fe₃(ex₁)

= 1 + (/α()).

CASE 3. kxe₁ +xe₂k ≥ α() andkxe₃ +xe₂ −xe₁k ≥ α(). In this case, let us put ex=xe₃−xe₁ andye=xe₂.It follows thatx,e ye∈S

Xe,and kex+yke =kxe₃+ex₂−ex₁k

≥ kxe₃+xe₂−xe₁k −(1−)

≥α() +−1,

kxe−yke =kxe₃−(ex₂+ex₁)k

≥ kxe3−(xe2+ex1)k −(1−)

≥α() +−1.

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Then, by definition ofE(X)and the factE(X) = E(X),e E(X)≥2 min{1 + (/α()), α() +−1}²

= 2 +²+√ 4 +². This is a contradiction and thus the proof is complete.

Remark 1. It is proved thatE(X)<1 + (1 +)² for some∈(0,1]implies thatX has uniform normal structure. So Theorem2.2is an improvement of such a result.

Theorem 2.3. LetX be a Banach space with

f(X)>((1 +²)²+ 2(1−²))(2 +²−√

4 +²) for some∈(0,1],thenX has uniform normal structure.

Proof. By our hypothesis it is enough to show thatX has normal structure. Assume that X lacks normal structure, then from the proof of Theorem 2.2 we can find ex,ey∈B_X_e such that

kxe±yk ≥e 1 + (/α()) =α() +−1 =: β().

Putue= (ex+y)/β()e andev = (xe−y)/β().e It follows thatkeuk,kvek ≥1,and keu+evk=

1

β()((1 +)ex+(1−)ey)

≤ (1 +) +(1−)

β() ,

keu−evk= 1

β()((1−)ex+(1 +)ey)

≤ (1−) +(1 +)

β() .

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Hence, by the definition off(X)and the factf(X) =f(X), we havee f(X)≤ ((1 +) +(1−))²+ ((1−) +(1 +))²

β²()

= ((1 +²)²+ 2(1−²))(2 +²−√

4 +²),

which contradicts our hypothesis.

Remark 2. Letting = 1, one can easily get that iff₁(X) > 4(3 −√

5), then X has uniform normal structure. So this is an extension and an improvement of [2, Theorem 5.3].

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References

[1] J. GAO, Normal structure and Pythagorean approach in Banach spaces, Period.

Math. Hungar., 51(2) (2005), 19–30.

[2] J. GAO, A Pythagorean approach in Banach spaces, J. Inequal. Appl., (2006), 1-11. Article ID 94982

[3] S. SAEJUNG, On James and von Neumann-Jordan constants and sufficient con- ditions for the fixed point property, J. Math. Anal. Appl., 323 (2006), 1018–1024.

[4] S. SAEJUNG, Sufficient conditions for uniform normal structure of Banach spaces and their duals, J. Math. Anal. Appl., 330 (2007), 597–604.

[5] C. YANG AND F. WANG, On a new geometric constant related to the von Neumann-Jordan constant, J. Math. Anal. Appl., 324 (2006), 555–565.