The Generalized Von Neumann-jordan Constant

The geometric theory of Banach spaces is an important research direction offunctional analysis, which has widely used in many fields of modern mathematics,such as fixed point theory, cybernetics, approximation theory, theory of martingale,harmonic analysis and so on. Moreover, the geometric constants and their estimatesof Banach spaces have been one of the research direction of the geometric theory ofBanach spaces. Because geometric constants of Banach spaces have a very detaileddescription for Banach spaces, the studies of geometric constants have veryimportant theoretical significance and practical value in the geometric theory ofBanach spaces. For many years, the scholars have introduced many geometricconstants, such as, the modulus of smooth to describe the uniform smooth, Jamesconstant to describe the uniformly non-square space, and the normal structurecoefficient to describe the normal structure and so on. Furthermore, the geometriccharacteristics have been studied widely and deeply, and a lot of good results areobtained.In this thesis, a generalized von Neumann-Jordan constant is introduced andestimated, and the necessary and sufficient conditions for the uniformly non-squareproperty are given by the generalized von Neumann-Jordan constant, and therelationship between the generalized von Neumann-Jordan constant and thenon-square constant (i.e., James constant) are studied. Secondly, the generalized vonNeumann-Jordan constant is calculated accurately in some specific spaces like thespace L r[0,1]and a two-dimensional Lorentz sequence space.At the same time, we also give a sufficient condition of which a Banach spacehas the uniform formal structure and discuss the relationship between thegeneralized von Neumann-Jordan constant and the weakly convergent sequencecoefficient. Finally, the state of the generalized von Neumann-Jordan constant forthe space without weakly normal structure is shown.