Properties And Applications Of Geometric Constants In Banach Space

In the 1930s,Banach,a Polish mathematician,founded a basic discipline of modern mathematics--functional analysis theory.Since the emergence of functional analysis theory,its application is very extensive,and gradually become an important theoretical basis for the study of modern mathematics.It is of great significance to study the Banach geometric space theory supported by functional analysis theory.Because the geometric constants in Banach space can well describe the geometric properties of Banach space,more and more mathematical workers explore the geometric structure of Banach space by studying the geometric constants in Banach space.Therefore,the research on the geometric constants in Banach space has been a hot issue that people pay more attention to.Therefore,the generalized optical sliding mode,Maluta constant,k-convexity mode and k-smoothness mode in Banach Space are studied in this article.Firstly,this article describes the development of generalized modulus of smoothness,Maluta constant,k-convexity mode and k-smoothness mode in Banach space,and introduces some conclusions related to these geometric constants,which provide a correct direction for the research work of this paper.Secondly,after giving the definition of the generalized smooth mode and Maluta constant,analyze the relationship between the smoothed mode and the generalized smooth mode,apply the monotonicity and continuity of the generalized smoothed mode,and fully consider the parameter range The relationship between the generalized smooth mode and Maluta constant is given and proved.Furthermore,the inequality between the smooth mode and Maluta constant is extended.Finally,after briefly introducing the definitions of k-convexity modules and k-smoothness modules,the definitions and some basic properties of generalized k-convexity modules and generalized k-smoothness modules are given.The relationship between generalized k-convexity modules and generalized k-smoothness modules is also given.As applications,sufficient and necessary conditions are given for the product space to be uniformly convex space and uniformly smooth space.