Structure Theory of Infinite Dimensional Banach Spaces and a Gaussian Correlation Problem

无限维Banach空间的结构理论和高斯相关问题

• 批准号: 9706828
• 负责人: Thomas Schlumprecht
• 金额: $ 6.41万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706828&HistoricalAwards=false
• 关键词: Structure; Theory; Infinite; Dimensional; Banach

9706828 Schlumprecht The concept of distortion played a crucial role in recent discoveries of infinite dimensional Banach space theory. It tries to specify the following problem: Given the norm topology of a Banach space, how different are the equivalent norms generating this topology? Our main question is the following: Are there any Banach spaces which are distortable but not arbitrarily distortable? Directly connected to the concept of distortion is the concept of stabilization and the spectrum of Lipschitz functions on the sphere of an infinite dimensional Banach space. It was introduced by V. Milman in 1971. The author intends to discuss the question whether, after stabilization, the spectrum of a Lipschitz function must be a connected set. Furthermore it is intended to work on several problems concerning the asymptotic structure of Banach spaces. They could be summed up as follows: How much information about the infinite dimensional properties of a Banach space can be deduced from its asymptotic properties? The second part of this proposal treats the following problem about Gaussian measures: Are two n-dimensional convex symmetric sets positively correlated with respect to the standard Gaussian measure? Banach spaces were introduced to provide a frame in which solutions of Differential Equations exist, i.e. equations which model dynamic systems. The author intends to continue his study about their structure. Roughly speaking, one might summarize the problems as follows: What are the building blocks of Banach spaces, are there "atomic" spaces? The second part of the proposal treats a group of questions concerning multi-dimensional normal distributions. The following question is a generic problem in statistics: One has tables for standard normal distr ibutions, but no tables for multi-dimensional normal distributions (there are simply too many of them). So, if we are dealing with a multi-dimensional problem for which the data can be approximated by a multi-dimensional normal, how can we deal with it? In order to find estimates of confidence regions the author intends to find lower estimates for probability of multi-dimensional events using the one dimensional standard distribution.

9706828 施伦普雷希特 扭曲的概念在 在最近发现的无限维 Banach空间理论它尝试指定以下内容 问题：给定Banach空间的范数拓扑，如何 生成这个拓扑的等价范数是不同的吗 我们的主要问题是：是否存在巴拿赫 空间是可扭曲的但不是任意扭曲的？ 与扭曲的概念直接相关的是 上Lipschitz函数的稳定性和谱 无穷维Banach空间的球面。介绍了 1971年，V. Milman。作者打算讨论 问题是，在稳定之后， Lipschitz函数必须是连通集。更是 打算工作的几个问题有关的渐近 Banach空间的结构这些问题可归纳如下： 有多少关于无限维度的信息 一个Banach空间的性质可以从它的渐近 属性？本提案的第二部分涉及以下内容 关于高斯测度的问题：两个n维凸的 对称集与标准正相关 高斯测度？ Banach空间的引入提供了一个框架， 微分方程的解存在，即方程， 模型动态系统作者打算继续他的研究 关于他们的结构。粗略地说，人们可以总结一下 问题如下：什么是Banach空间的积木， 是否存在“原子”空间？建议的第二部分涉及 关于多维正态分布的一组问题。 下面的问题是统计学中的一个普遍问题： 标准正态分布的表格，但没有 多维正态分布（有简单的 太多了）。所以，如果我们面对的是一个多维度的 数据可以用多维近似的问题 正常，我们该如何应对？为了找到 作者打算找到较低估计的置信区域 多维事件的概率使用一维 标准分布