Uniform Rotundity Of Generalized Orlicz Spaces

The geometry of Banach space is an important content in the space theory. The uniform rotundity of spaces is one of the most important geometry properties. The mian results of the thsis are summarized as follows:The author discussed the characterization of the uniform convexity of N-functions and proved 16 equivalent propositions for the uniform rotundity of Banach spaces;The thesis amended Chen Shutao's theorem about the uniform rotundity of Musielak-Orlicz spaces with Luxemburg norm;The most important work was that the author first got the characterization of rotundity of Orlicz-Lorentz spaces with Orlicz norm by improving some theorems of these spaces and rearrangements,and then gave the equivalence conditions of the uniform rotundity of Orlicz-Lorentz spaces with Orlicz norm.The author studied the sufficient and necessary conditions for the uniform rotundity of Orlicz-Lorentz spaces with Orlicz norm whenγ=∞:(1)φ∈△2;(2)φis uniformly convex;(3)ω(t) is regular.The thesis consists of four parts:Chapter 1,Uniform convexity of N-functions:The thesis first showed the uniform convexity of N-functions as it had close relationship with the Orlicz functions. The author mainly studied the uniform convexity near origin and gave sufficient and necessary conditions for the uniform convexity of N-functions.The relationship between strict convexity and uniform convexity was given.An easy critertion for the uniform convexity of N-functions was also introduced.Chapter 2,Sufficient and necessary conditions for the uniform rotundity of Banach spaces:The author proved 16 equivalent propositions for the uniform rotundity of Banach space.Chapter 3,Uniform rotundity of Musielak-Orlicz spaces:The uniform rotundity of Musielak-Orlicz spaces with Luxemburg norm has been shown by Chen Shutao.But there was some mistakes in his Lemma 2.This thesis will amend the mistake(weaken the condition of Lemma 2 ) and give another proof of his theorem.Chapter 4,Uniform convexity of Orlicz-Lorentz space with Orlicz norm:Kaminska proved equivalence conditions for the uniform rotundity of Orlicz-Lorentz spaces with Luxemburg norm.In 1999,Wu Congxin and Ren Liwei gave the basic frame of Orlicz-Lorentz space with Orlicz norm and the characterization of the rotundity whenγ＜∞.But there were few results about this norm in geometric theory.As a result, the author devoted to showing the equivalent conditions for the uniform rotundity of Orlicz-Lorentz spaces with Orlicz norm whenγ=∞.