RESEARCH ON STRUCTURAL THEORY OF BANACH SPACES AND ITS APPLICATIONS

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In this research we studied the structural theory of Banach spaces. We mainly investigated some geometrical and probabilistic properties of Banach spaces in connection with classical norm inequalities and their generalizations. We also investigated some constants related to geometrical properties of Banach spaces and clarified the relations between these constants and properties.The main results are stated as follows :1. Clarkson type inequalities and geometry of Banach spaces :We consider some multi-dimensional Clarkson type inequalities for Banach spaces, and prove the exact relations between such inequalities and the concepts of type and cotype. We also consider some weighted Clarkson type inequalities for Banach spaces, and prove the exact relations between such inequalities and the concepts of p-uniform smoothness and q-uniform convexity.2. Von Neumann-Jordan, James constants and geometry of Banach spaces X :We estimate the von Neumann-Jordan constant C_<NJ> (X) from below and above by the James constant J (X), and the normal structure coefficient N (X) by C_<NJ> (X). Furthermore, we introduce James type constants J_t (X), and investigate the relations between such constants and geometrical properties of Banach spaces. The estimate of N (X) by J_t (X) is also given.3. Hlawka type inequalities and their generalizations :We give some generalizations of Hlawka type inequalities in connection with type and cotype. An integral form of the Hlawka inequality is also considered, and weighted Djokovic inequality is proved. A new interpretation of Djokovic inequality is also given.4. Absolute norms and direct sums of Banach spaces :Absolute norms on C^2 and C^n are considered, and fundamental properties of these norms are shown. We also consider absolute norms on direct sums of two Banach spaces, and the strict convexity and uniform smoothness of such spaces are investigated.