Some Monotonicity With Generalized Normed In Orlicz Spaces

Nowadays,the research of Banach space theory occupies a particularly important position in modern functional analysis,which has received great attention in operations and cybernetics,operator theory,biological mathematics and financial mathematics.As a special kind of Banach space,Orlicz space is used as a channel to study Banach space.With more than a century of efforts by domestic and foreign mathematicians,the theory of Orlicz space has been increasingly perfected.Orlicz spaces with generalized norm can be regarded as an extension of Orlicz space.Classical Orlicz space and various generalized Orlicz space are included in it,so its theory and practical application in life are more universal.The in-depth study of this special space not only enriches the content of Orlicz space theory,but also provides an objective way of thinking for Banach space.This paper discusses the monotony of Orlicz space with generalized norm and the condition of norm equivalence in Orlicz space.Its work content is summarized as follows:First of all,this paper expounds the source of this topic and the practical value of studying Orlicz space,introduces the development process of Orlicz space and the important research results of Orlicz space theory,and shows the existing research content and results.Secondly,we can calculate that the Luxemburg norm and Orlicz norm can be regarded as equivalent.In this paper we give the best constant that makes the Luxemburg norm equivalent to Orlicz norm.At the same time the equivalent conditions of(?)and(?)are given.Finally,we study the geometric properties of Orlicz function space with F norm(mainly monotone points).The study of monotone in Banach space has a wide range of practical value,which is closely related to martingale theory and complex Banach space geometry theory.And the monotonicity point problem is more common and important than the monotony of space.Many mathematicians have explored the monotone points of the Spaces endowed with Luxemburg norm,Orlicz norm and Mazur-Orlicz F norm for a long time,and got many interesting conclusions.In this paper completes the equivalent characterization of monotone points in Orlicz space with Mazur-Orlicz F norm,which can be divided into four parts:lower(upper)strict monotone point,lower(upper)locally consistent monotone point.