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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的一些与经典范数不等式及其推广有关的几何性质。我们还考虑了James常数和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ψ-直和的一致凸性和非平方性质，主要结果如下：1.范数不等式与Banach空间的几何：我们考虑了Banach空间X的一些经典范数不等式的推广，并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-一致光滑性和q-一致凸性的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。还给出了Hanner型和Hlawka型不等式的一些推广。James，Schaffer型常数与Banach空间的几何：我们引入Banach空间X的James型常数J_&lt；X，t&gt；(τ)和Schaffer型常数S_&lt；X，t&gt；(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，如一致凸性、光滑性和非方性，可以用常数J_&lt；X，t&gt；(τ)和S_&lt；X，t&gt；(τ)来描述。给出了几个具体的Banach空间的例子，并计算了这些常数。Banach空间的绝对范数与ψ直和：我们研究了Banach空间X和Y的ψ直和X[对称性]_ψY，证明了X[对称性]_ψY一致凸(局部一致凸)当且仅当X，Y一致凸(局部一致凸)，ψ严格凸。给出了X[对称性]TJ_ψY的严格凸性和一致非方性质的类似刻划。