RESEARCH ON STRUCTURE THEORY OF BANACH SPACES AND NORM INEQUALITIES WITH APPLICATIONS

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In this research we considered some generalizations and refinements of classical norm inequalities and geometric constants, and investigated geometric properties of Banach spaces X in connection with those inequalities and constants. In particular, we considered refined generalizations of Hanner inequality and Schaffer constant, and characterized geometric properties of X in terms of these inequalities and constants. We also investigated geometric properties of ψ-direct sums of Banach spaces.The main results are stated as follows :1. Norm inequalities and geometry of Banach spacesWe consider some refined generalizations of Hanner's inequality for Banach spaces X, and characterize geometric properties of X such as uniform non-squareness, p-uniform smoothness and quniform convexity in terms of these inequalities. We also consider extensions of strong random Clarkson inequalities, and characterize Banach spaces of strong type p in terms of those inequalities.2. Refined generalizations of Schaffer constant and geometry of Banach spacesWe introduce new geometric constants (or functions) φx(τ) for Banach spaces X as refined generalizations of the Schaffer constant S(X), and investigate some geometric properties of X in terms of these constants (or functions). In particular, the normal structure coefficients N(X) of X can be estimated by φx(τ). Some examples of concrete Banach spaces X with the calculation of φx(τ) are also given.3. Geometric properties of ψ-direct sums of Banach spacesWe consider the ψ-direct sum (X_1 【symmetry】 X_2 【symmetry】 【triple bond】 【symmetry】 X_n)_ψ of Banach spaces X_1, X_2, 【triple bond】, X_n, and investigate geometric properties such as uniform non-squareness, uniform convexity, uniform non-l^n_1-ness (B_n-convexity) and fixed point properties. These properties of such spaces can be described in terms of the function ψ and Banach spaces X_1,X_2,【triple bond】, X_n.