Banach space geometric constants and their nature 

Author  YangXiangZuo 
Tutor  WeiWenZhan 
School  Guangxi Normal University 
Course  Basic mathematics 
Keywords  geometric constant normal structure uniformly normal structure fixedpoint 
CLC  O177.2 
Type  Master's thesis 
Year  2010 
Downloads  18 
Quotes  0 
Banach space geometric structure and the fixed point property are mainly throughthe geometric constants and moduli to carry out research, so the geometric structure andgeometric constants of the intrinsic link between the research became a hot issue. Recently,many research work mainly by using di?erent geometric constant, to explore the Banachspace has normal structure （uniform normal structure） of the suffcient condition.Moduli or geometric constants of Banach space are mainly introduced in the thesis.Their properties and the relationship among uniform nonsquareness,normal structureand uniform normal structure are studied. The paper is organized as follows:First, U^{（β）}（ε）, the generalization of the modulus of Uconvexity, is investigated. Thefunctional properties of this modulus, such as monotonity and continuity, are also obtained.Moreover, we get some suffcient conditions for X to have normal structure, that is, thereexist aε, 0≤ε≤2, if U_{X}^{（β）}（ε） > max {0, （1β）（ε1）} then X has uniform normalstructure;Also, if U_{X}^{（β）}（ε） > max{0, （22β）（ε1）},then X and X* has uniformly normalstructure. Moreover,some suffcient conditions for which a Banach space has normal structure by the modulus and Benavides coe?cient are obtained,which generalize some resultsof Gao and Zuo. which generalized a known result of J. Gao.Next, lower bounds for the weakly convergent sequence coeffcient in term of parameterized James cnstant and generalized James constant is established. Some suffcientconditions which imply normal structure are obtained by the bounds, and therefore aBanach space X has fixed point property.Finally, A new geometric constant E_{p}（t, X） is introduced,it is a generalization of Gaoconstant E（t, X）. Monotonicity, continuity and convexity of this new constant are discussed. Some suffcient conditions for which a Banach space has normal structure bymeans of the constant and weak orthogonal coe?cient are obtained.