Ball-Covering Properties And Smoothness In Banach Spaces

The whole Banach space geometry is a geometry about the unit ball and unit sphere of Banach spaces. Even among other knowledge branches, the direct uses of " ball " to study other aspects of knowledge became important parts of the corresponding branches. For instance, the Mazur intersection property which belongs to Banach space geometry;the measure of non-compactness with respect to topological degree in non-linear analysis;the packing sphere problem of unit balls in optimization theory and so on. Professor Lixin Cheng starts with a different view point to study on " how many balls which do not contain the origin can the unit sphere of a Banach space be covered by ". So does this paper starts from to study the ball-covering properties of Banach spaces.The space X is said to have ball-covering property(denoted by BCP), if it admits a ball-covering consisting of countably many balls off the origin. This paper, by constructing equivalent norms on l~∞, shows that ball-covering property is not invariant under isomorphic mappings;presents that this property of X is not heritable by its closed subspaces;and the property is not preserved under quotient mappings. ([24]: Published in Science in China Series A: Mathematics 2007 No.7) Also, it shows that the product of countably many Banach space X_n(n∈N) with X_n having BCP has BCP in the sense of supremum norm. It is well-known that modulus of convexity, modulus of smoothness and uniform regular structure constants have played an important part in depicting the fixed quantity of geometry property. Chapter 3 gives a geometry constant similar to modulus of smoothness,and gets a equivalent condition of uniformly smooth.