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

本文考虑了经典的范数不等式和几何常数的推广和加细，并结合这些不等式和几何常数研究了Banach空间X的几何性质。特别地，我们考虑了Hanner不等式和Schaffer常数的精化推广，并利用这些不等式和常数刻画了X的几何性质。我们还研究了Banach空间的n-直和的几何性质，主要结果如下：1。Banach空间的范数不等式与几何考虑了Banach空间X上Hanner不等式的一些精化推广，并利用这些不等式刻画了X的一致非平方性、p-一致光滑性和q-一致凸性等几何性质。我们还考虑了强随机克拉克森不等式的推广，并利用这些不等式刻画了强p型Banach空间. Schaffer常数的精化推广与Banach空间的几何我们在Banach空间X中引入了新的几何常数（或函数）φx（τ）作为Schaffer常数S（X）的精化推广，并利用这些常数（或函数）研究了X的一些几何性质.特别地，X的正规结构系数N（X）可以通过φx（τ）来估计。给出了具体Banach空间X中φx（τ）的计算实例.本文研究了Banach空间X_1，X_2，[三键]，X_n的n-直和（X_1 [对称性] X_2 [对称性] [三键] [对称性] X_n）_n的一致非方性，一致凸性，一致非l ^n_1-凸性（B_n-凸性）和不动点性质。这类空间的这些性质可以用函数λ和Banach空间X_1，X_2，[三键]，X_n来描述。

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.