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RESEARCH ON STRUCTURAL THEORY OF BANACH SPACES AND ITS APPLICATIONS

Banach空间结构理论及其应用研究

基本信息

批准号：
11640177
负责人：
TAKAHASHI Yasuji
金额：
$ 0.96万
依托单位：
OKAYAMA PREFECTURAL UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640177/
关键词：
Clarkson type inequality Hlawka type inequality Rademacher type-cotype von Neumann-Jordan constant James type constant Uniformly convex space Absolute norm Geometry of Banach spaces クラークソン型不等式ラフカ型不等式ラデマツハータイプ・コタイプジョルダン・フォンノイマン定数 /

项目摘要

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.

本文研究了Banach空间的结构理论。我们主要研究了Banach空间的一些几何和概率性质，这些性质与经典范数不等式及其推广有关。我们还研究了与Banach空间几何性质有关的一些常数，并阐明了这些常数与性质的关系。主要研究结果如下：1。Clarkson型不等式与Banach空间的几何：我们考虑了Banach空间的一些多维Clarkson型不等式，并证明了这些不等式与型和共型概念之间的确切关系。我们还考虑了Banach空间的一些加权Clarkson型不等式，并证明了这些不等式与p-一致光滑性和q-一致凸性概念之间的确切关系。Banach空间X的Von Neumann-Jordan、James常数和几何：我们用James常数J (X)从下往上估计Von Neumann-Jordan常数C_<NJ> (X)，用C_<NJ> (X)估计正常结构系数N (X)。进一步，我们引入了James型常数J_t (X)，并研究了这些常数与Banach空间几何性质的关系。给出了J_t (X)对N (X)的估计。Hlawka型不等式及其推广：我们给出了Hlawka型不等式在类型和共型方面的一些推广。考虑了Hlawka不等式的积分形式，并证明了加权Djokovic不等式。本文还对德约科维奇不等式给出了新的解释。Banach空间的绝对范数与直接和：讨论C^2和C^n上的绝对范数，并给出这些范数的基本性质。我们还考虑了两个Banach空间的直接和的绝对范数，并研究了这类空间的严格凸性和一致光滑性。