Complex Convexity And Several Geometric Properties In Banach Spaces

In recent years, geometric theory of complex Banach spaces has received great at-tention from local and foreign experts in mathematics. Research for geometric propertiesof complex spaces originated from that for related properties of vector-valued analyticfunctions, when the diference in these properties of vector-valued analytic functions be-tween complex spaces and real spaces was discovered, the interest in the study of thisaspect was arouse quickly, and a new understanding on geometric properties of complexspaces was obtained. Hence, research on geometric theory of complex Banach spaces hasbecome more active. Geometry of complex Banach spaces not only contains quite abun-dant contents, but also has applications in various branches of knowledge, among othersin martingales theory, operator theory, harmonic analysis, Banach algebras, C-algebras,diferential equations, hydrodynamics theory and quantum mechanics theory. The con-vexity which not only has clearly geometrical significance but also is closely with controltheory, best approximation theory and fixed point theory, is an important part of geometrictheory of Banach spaces. This thesis which emphatically discusses the complex convex-ity of Orlicz function spaces and Orlicz sequence spaces equipped with the p-Amemiyanorm and Orlicz-Bochner function spaces equipped with the Luxemburg norm, consistsof five chapters. The main results are as follows:Firstly, the concepts of complex strongly extreme points are introduced, and twoequivalent definitions of which in general complex Banach spaces are shown by meansof new modulus functions. Moreover, the complex convexity of Orlicz function spacesequipped with the p-Amemiya norm are studied. On the one hand, it is proved that forany Orlicz function space equipped with the p-Amemiya norm(1≤p <∞, p is odd),complex strongly extreme points of the unit ball coincide with complex extreme pointsof the unit ball. In addition, criteria for them in Orlicz function spaces equipped with thep-Amemiya norm are also given. Criteria for complex rotundity and complex mid-pointlocally uniform convexity of above spaces are also deduced. On the other hand, criteria forcomplex strongly extreme points of the unit ball and complex mid-point locally uniformconvexity of Orlicz function spaces equipped with the p-Amemiya norm when p=∞aregiven respectively. Secondly, the complex convexity of Orlicz sequence spaces equipped with the p-Amemiya norm are discussed. Criteria for complex extreme points and complex stronglyextreme points of the unit ball in Orlicz sequence spaces equipped with the p-Amemiyanorm are given. Moreover, criteria for complex rotundity and complex mid-point locallyuniform convexity of above spaces are also deduced. Criteria for complex strongly ex-treme points of the unit ball and complex mid-point locally uniform convexity of Orliczsequence spaces equipped with the p-Amemiya norm when p=∞are shown respec-tively.Thirdly, criteria for complex extreme points of the unit ball of Orlicz-Bochner func-tion spaces equipped with the Luxemburg norm are given, and criteria for complex rotun-dity and complex uniform convexity of the whole spaces are also shown respectively.Finally, the concept of ellipsoid in general Banach spaces is introduced, some def-initions of geometric properties in the sense of ellipsoid such as the strict convexity, theuniform convexity and the modulus of convexity are given. It is proved that the Banachspace which is uniformly convex in the sense of ellipsoid is reflexive, and a sufcient con-dition for Banach-Saks property in Banach spaces is obtained. The diference between theellipsoid and the unit ball in point-wise geometric properties is shown clearly by the spe-cific example. Moreover, it is also proved that the Banach space which is strictly convexis strictly convex in the sense of ellipsoid.