Some Geometric Properties Of Orlicz Spaces Equipped With The P-Amemiya Norm

Geometric theory of Banach spaces is a great part of modern functional analysis.Orlicz space is a generalization of classical Lebesgue(L_p)space.As a kind of specific Banach space,it is widely used in equation theory,approximation theory,fixed point theory,operator theory,probability theory and other theories.It is well known that the p-Amemiya norm formula provides formal unity between the Orlicz norm formula and the Luxemburg norm formula,Orlicz space endowed with the p-Amemiya norm is an extension of Orlicz space endowed with the Orlicz and Luxemburg norm.In Orlicz spaces endowed with the p-Amemiya norm,a lot of geometric properties obtained general criteria.So,it is of great theoretical significance and practical value to research geometric properties of Orlicz spaces endowed with the p-Amemiya norm.In this dissertation,smoothness,strict convexity and exposedness of Orlicz spaces endowed with the p-Amemiya norm are studied.Five chapters are consisted in this thesis,the main contents are as follows:Firstly,we define a new sort of norms in the dual spaces of Orlicz function spaces endowed with the p-Amemiya norm.The new sort of norm formulas will realize unity formally about bounded linear functional f’s norm calculation formulas.The norm calculation formulas of bounded linear functional in these spaces are provided.Furthermore,the precise form of support functional for the point on the unit sphere in Orlicz function spaces endowed with the p-Amemiya norm is presented.The characteristic of smooth points is given by using the explicit form of support functional of points on the unit sphere.Finally,general criteria for smoothness of Orlicz function spaces endowed with the p-Amemiya norm are obtained by using the results of pointwise property.Secondly,the norm calculation formula about bounded linear functional f of Orlicz sequence spaces endowed with the p-Amemiya norm is given.By using the norm calculation formula of bounded linear functional,we obtain the specific form of supporting functional on unit spherical point in Orlicz sequence spaces endowed with the p-Amemiya norm,and the question of characteristic about smooth points is settled.From the result of pointwise property,we got the criteria for smoothness of Orlicz sequence spaces endowed with the p-Amemiya norm.Thirdly,we employ a new technique to obtain the criteria for strict convexity of Orlicz sequence spaces endowed with the p-Amemiya norm.This technique simplifies the corresponding conclusions of Orlicz sequence spaces endowed with the Orlicz and Luxemburg norm.Finally,we take advantage of the specific form of supporting functional about a point on the unit sphere to gain the criteria for exposed points of Orlicz spaces endowed with the p-Amemiya norm.The criteria for exposed points have displayed that the criteria for exposed points of Orlicz spaces endowed with the p-Amemiya(1<p<∞)norm are weaker than that of Orlicz spaces endowed with the Orlicz and Luxemburg norm.We obtain criteria for exposedness of these spaces by the aid of characterization of exposed points.