RESEARCH ON NORM INEQUALITIES IN BANACH SPACES AND ITS APPLICATIONS

Banach空间中的范数不等式及其应用研究

基本信息

批准号：
13640188
负责人：
TAKAHASHI Yasuji
金额：
$ 2.18万
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：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640188/
关键词：
Clarkson type inequality Hanner type inequality Norm inequality James type constant Schaffer type constant Uniformly convex space Geometry of Banach spaces Direct sums of Banach spaces ラフカ型不等式タイプ・コタイプ絶対ノルムノイマン・ジョルダン定数

项目摘要

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.

在这项研究中，我们研究了Banach空间X的一些几何特性，该特性与经典规范不平等及其概括有关。我们还考虑了James和Schaffer常数的一些概括，并表明可以用这些常数描述X的一些几何特性。还研究了BANACH空间的ψ导向总和的均匀凸度和非方向。主要结果表示如下：1。 Banach空间的规范不平等和几何形状：我们考虑了Banach空间X的经典规范不等式的某些概括，并以这些不平等的方式表征了X的一些几何特性。特别是，我们证明了加权的克拉克森型不平等与P-均匀平滑度和Q-均匀凸的概念之间的确切关系，并给出了混凝土Banach空间的最佳2-均匀凸度不平等。还给出了Hanner类型和Hlawka类型不等式的一些扩展。2。 James，Schaffer类型常数和Banach空间的几何形状：我们介绍了Banach Spaces X的James类型常数J_ <X，T>（τ）和Schaffer型常数S_ <X，T>（τ），并研究这些常数的基本属性。我们表明，X的某些属性，例如均匀的凸度，平滑度和非方格，可以用常数J__ <x，t>（τ）和S_ <x，t>（τ）来描述。还给出了一些用这些常数计算的混凝土巴拉克空间的例子。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.还给出了X [对称性]TJ_ψY的严格凸度和均匀非方面的类似字符。