【作者】 韩彦昌；

【导师】 崔云安；

【作者基本信息】 哈尔滨理工大学 ， 基础数学， 2003， 硕士

【摘要】 一致凸是Banach空间的重要几何概念，其在逼近论、控制论、变分不等式等领域有着重要的应用。根据各种学科发展和应用的需要，Orlicz空间作为L_p空间的推广及作为Orlicz空间的各种不同形式的推广中较为常见的一种——Musielak-Orlicz空间。本文主要围绕一致凸这一重要的的几何性质展开讨论。本文引进依测度一致凸点(μ-URP))的概念。让X是Banach空间，x∈S(X)被称作B(X)的依测度一致凸点(μ-URP)是指：如果x_n∈B(X)且‖x_n+x‖→2(?)x_n(?)x。X称为依测度一致凸空间(μ-UR)是指：单位球面上的每一点都是依测度一致凸点。一致凸点是局部一致凸性的精细化，点态化。从宏观性质到点态性质的研究工作是Orlicz空间几何学发展过程中的一个质的飞跃。全文共分为三部分，主要工作总结如下：首先，回顾了Orlicz空间和Musielak-Orlicz空间理论的发展历程，阐述了本文各部分所讨论的内容、背景和意义。本文在第二章分别就赋Orlicz范数和Luxemberg范数给出了依测度一致凸点的判据。作为推论给出了Orlicz空间依测度局部一致凸的判据。本文在第三章，由于Musielak-Orlicz空间的复杂性，只给出了该空间局部一致凸的判别准则，本文给出了Musielak-Orlicz空间的一致凸点和弱一致凸点的判别准则。更多还原

【Abstract】 Uniform rotundity is an important geometric property in Banach spaces. There are many important applications in approximate theory, control theory and variation inequalities etc. According to the requirement of various subjects and applications,Orlicz spaces is extended from the classical spaces Lp. There are also manygeneralizations of Orlicz spaces, one of them is Musielak-Orlicz space. In the paper, we mainly investigate pointwise geometric properties in Orlicz spaces or Musielak-Orlicz spaces. For convenient to studying the geometric properties, we introduce a new definition, namely uniformly locally convex point in measure( -URP, for short). Let X be a Banach space. An element x S(X) is called a uniformly locally convex point in measure if xn B(X) and ||xn + x|| - 2= xn -- x. X is called to be uniformly local convex in measure if each point inthe unit sphere of X is uniformly local convex point in measure. It is well known that uniformly locally convex point (URP, for short) is precise and point wise property of uniformly local convex. Studying works from macroproperty into microproperty is a qualitative leap in development of Orlicz spaces geometric theory. The paper is consist of four parts. The main results of this paper are summarized as following:First it reviewed that a survey of the theory of Orlicz and Musielak-Orlicz spaces. We show the background and significance of the content of each part in this paper.Secondly the criteria for that uniformly locally convex point in measure of Orlicz spaces equipped with the Orlicz norm and Luxemberg norm are given in the chapter 2. Meanwhile we get necessity and sufficient condition for that Orlicz spaces equipped with both norms are uniformly locally covex in measure as corollaries.Last we give criteria of uniform rotundity point and weak uniform rotundity point in Musielak-Orlicz sequence spaces in chapter 3. For Musielak-Orlicz function spaces case, we only give necessity and sufficient condition of the spaces are uniformly local convex since the case is very complex.更多还原