Complex Rotundity In Musielak-orlicz Spaces Equipped With The P-amemiya Norm

In recent years, geometric theory of complex Banach spaces has receivedgreat attention from local and foreign experts in mathematics. Research forgeometric properties of complex spaces originated from the related properties ofvector-valued analytic functions, when the difference in these properties ofvector-valued analytic functions between complex spaces and real spaces wasdiscovered, the interest in the study of this aspect was arouse quickly, and a newunderstanding on geometric properties of complex spaces was obtained. Hence,research on geometric theory of complex Banach spaces has become more active.Geometry of complex Banach spaces not only contains quite abundant contents,but also has applications in various branches of knowledge, among others inmartingales theory, operator theory, harmonic analysis, Banach algebras, C-algebras, differential equations, and quantum mechanics theory. The convexitywhich not only has clearly geometrical significance but also is closely withcontrol theory, best approximation theory and fixed point theory, is an importantpart of geometric theory of Banach spaces. This thesis which emphaticallydiscusses the complex convexity of Musielak-Orlicz function spaces andMusielak-Orlicz sequence spaces equipped with the p-Amemiya norm.The mainresults are as follows:Firstly, the general situation of Orlicz Spaces theories and their developmentat home and abroad are introduced. Then the significance of the research of thispaper is given.Secondly, the complex convexity of Musielak-Orlicz function spacesequipped with the p-Amemiya norm is studied. On one hand, it is proved that forany Musielak-Orlicz function space equipped with the p-Amemiya norm,complex strongly extreme points of the unit ball coincide with complex extremepoints of the unit ball. In addition, criteria for them in Musielak-Orlicz functionspaces equipped with the p-Amemiya norm are also given. Criteria for complexrotundity and complex mid-point locally uniform convexity of above spaces are also deduced. On the other hand, criteria for complex strongly extreme points ofthe unit ball and complex mid-point locally uniform convexity of Musielak-Orlicz function spaces equipped with the p-Amemiya norm when p aregiven respectively.Finally, the complex convexity of Musielak-Orlicz sequence spacesequipped with the p-Amemiya norm are discussed. Criteria for complex extremepoints and complex strongly extreme points of the unit ball in Musielak-Orliczsequence spaces equipped with the p-Amemiya norm are given. Moreover,criteria for complex rotundity and complex mid-point locally uniform convexityof above spaces are also deduced. Criteria for complex strongly extreme points ofthe unit ball and complex mid-point locally uniform convexity of Musielak-Orlicz sequence spaces equipped with the p-Amemiya norm when p areshown respectively.