RESEARCH ON NORM INEQUALITIES IN BANACH SPACES AND ITS APPLICATIONS

本文研究了Banach空间X与经典范数不等式及其推广的一些几何性质。我们还考虑了James和Schaffer常数的一些推广，并证明了X的一些几何性质可以用这些常数来描述。研究了Banach空间的齐凸性和非平方性。主要研究结果如下：1。Banach空间的范数不等式与几何：我们考虑了Banach空间X的经典范数不等式的一些推广，并利用这些不等式刻画了X的一些几何性质。特别地，我们证明了加权Clarkson型不等式与p-一致光滑和q-一致凸性概念之间的确切关系，并给出了具体Banach空间的最优2-一致凸性不等式。并给出了Hanner型不等式和Hlawka型不等式的一些推广。Banach空间的James， Schaffer型常数和几何：我们引入了Banach空间X的James型常数J_<X, t>(τ)和Schaffer型常数S_<X, t>(τ)，并研究了这些常数的基本性质。我们证明了X的一些性质如均匀凸性、光滑性和非平方性可以用常数J_<X, t>(τ)和S_<X, t>(τ)来描述。最后给出了具体Banach空间的一些算例，并给出了这些常数的计算。Banach空间的绝对范数与ψ-直和：我们考虑Banach空间X和Y的ψ-直和X【对称】_ψY，证明X【对称】_ψY是一致凸（局部一致凸）当且仅当X、Y是一致凸（局部一致凸）且ψ是严格凸。给出了X【对称】TJ_ψ Y的严格凸性和一致非方性的类似性质。

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.