A Study On The Several Kinds Of Convexity In Z-space

In recent forty years, the development of theoretical study on Banach space is rapid. Excellent its strict convexity, k-strict convexity, uniform convexity, k-lmifonn convexity and Closed convexity theory research have achieved some good results. On the contrary, the theoretical study on the Z-space, which is widely spread in Banach space, develops slowly. In this paper. The author who in-troduces the definition of k-strict convexity, k-unifonn convexity.and extends to Z-space about the relations of several convexities in Banach space, achieves good results about necessary and sufficient conditions, this paper is divided into three chapters.Let X be a non-empty set, Z a integer additive group,‖·‖a norm from X to Z,(X,+,θ) a Abel group,(X,‖·‖) a normed linear space,(X,+,θ,‖·‖) a sub-normed z-linear space or a Z-space.Chapter1:The knowledge of preparation.Chapter2:We give necessary and sufficient conditions of strict convexity in Z-space and then we make k-strict convexity in Banach space into Z-space which achives necessary and sufficient conditions of k-strict convexity in Z-space.Chapter3:We give necessary and sufficient conditions of uniform convexity in Z-space and then we make k-uniform convexity in Banach space into Z-space which achives necessary and sufficient conditions of k-uniform convexity in Z-spaee.