RESEARCH ON NORM INEQUALITIES IN BANACH SPACES AND ITS APPLICATIONS

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In this research we investigated some geometrical properties of Banach spaces X in connection with classical norm inequalities and their generalizations. We also considered some generalizations of James and Schaffer constants, and showed that some geometrical properties of X can be described in terms of these constants. The uniform convexity and non-squareness of ψ-direct sums of Banach spaces were also investigated.The main results are stated as follows :1. Norm inequalities and geometry of Banach spaces :We consider some generalizations of the classical norm inequalities for Banach spaces X , and characterize some geometrical properties of X in terms of these inequalities. In particular, we prove the exact relations between weighted Clarkson type inequalities and the concepts of p-uniform smoothness and q-uniform convexity, and give the optimal 2-uniform convexity inequalities for concrete Banach spaces. Some extensions of Hanner type and Hlawka type inequalities are also given.2. James, Schaffer type constants and geometry of Banach spaces :We introduce the James type constant J_<X, t>(τ) and the Schaffer type constant S_<X, t>(τ) for Banach spaces X, and investigate fundamental properties of these constants. We show that some properties of X such as uniform convexity, smoothness and non-squareness can be described in terms of the constants J_<X, t>(τ) and S_<X, t>(τ). Some examples of concrete Banach spaces with the calculation of these constants are also given.3. Absolute norms and ψ-direct sums of Banach spaces :We consider the ψ-direct sum X【symmetry】 _ψ Y of Banach spaces X and Y, and show that X【symmetry】_ψY is uniformly convex (locally uniformly convex) if and only if X, Y are uniformly convex (locally uniformly convex) and ψ is strictly convex. Similar characterizations of strict convexity and uniform non-squareness for X 【symmetry】TJ_ψ Y are also given.