U-Property And U-Convex Module In Banach Spaces

In 1978, Lau Ka-Sing introduced the U-property to Banach spaces during his study of the Chebyshev subsets of them. Henceforth, Lau Ka-Sing and Gao Ji introduced the conception of U-space and depicted its properties in 1991. For example, U-space is uniformly regular and which makes it has fixed point property, U-space is uniformly non-square and thus super-reflexive, uniformly convex space and uniformly smooth space are U-spaces, and an Banach space is an U-space iff its dual space is U-space, etc. In1990s, a lot of work had been done on U-space theory, e.g., Tingfu Wang and Donghai Ji introduced the concepts of pre U-property and nearly U-property. Under the structure of Orlicz space, they made systematic investigation of these properties, and gave the criteria for an Orlicz space to have U-property. On the basis of their work, we further investigate the U-property. We make a study of the relationship between U-property and convexies, and some other relevant geometric properties of Banch spaces. The connection between U-property and uniformly convexity, uniformly locally mid-points convexity are constructed, The U-property of the generalized lp-space and its quotient space is given. Geometric constants of the space is a quantitative index of its corresponding geometric property. From investigating geometric property of space to compute geometric constant is a furture progress from "qualitative" to "quantitative". In the second chapter, the conception, equivalent representation and estimation in Hilbert spaces of U-convex module are introduced. At the same time, its relations with smooth module and nonsquare constant are given. We also discuss the geometric properties of the common Banach spaces by the calculation, estimate and denotation.