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Mathematical Sciences: Structure of Infinite-Dimensional Banach Spaces

数学科学：无限维 Banach 空间的结构

基本信息

批准号：
8903197
负责人：
Haskell Rosenthal
金额：
$ 27.09万
依托单位：
University of Texas at Austin
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
1989
资助国家：
美国
起止时间：
1989-06-15 至 1993-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=8903197&HistoricalAwards=false
关键词：
Mathematical Sciences Structure Infinite Dimensional

项目摘要

Professors Rosenthal and Odell will work on a variety of problems in the theory of Banach spaces. One central problem concerns identifying the complemented subspaces of the space of absolutely integrable functions on an interval. This is related to the structure of Banach spaces and convex sets failing the Radon-Nikodym property. The question of the existence of subspaces isomorphic to the space of null or absolutely summable sequences in a hereditarily non-reflexive Banach space will be pursued. The approach Rosenthal and Odell will use involves analysis of the structure of spaces of Baire functions. Their recently invented notion of governance of a class of Banach spaces by a function will also play an important role. The theory of Banach spaces is about analysis in infinitely many dimensions. For each finite dimension n, there is only one n-dimensional space, in most respects as well understood in general as in the familiar cases when n is 1, 2, or 3. These are used to keep track of situations in which there are only finitely many degrees of freedom. When the number of degrees of freedom becomes infinite, as would for instance be necessary to describe the configuration of a piece of string, there are many possible reasonable choices for the ambient space in which to perform analysis. The work supported by this award is largely concerned with the ways in which these spaces fit into another, a line of investigation that lays bare a good deal of their structure.

罗森塔尔和奥德尔教授将研究巴纳赫空间理论中的各种问题。一个中心问题涉及确定区间上绝对可积函数空间的补子空间。这与不符合 Radon-Nikodym 性质的 Banach 空间和凸集的结构有关。我们将探讨遗传性非自反巴拿赫空间中是否存在与零序列或绝对可和序列空间同构的子空间的问题。罗森塔尔和奥德尔将使用的方法涉及对贝尔函数空间结构的分析。他们最近发明的通过函数治理一类 Banach 空间的概念也将发挥重要作用。巴纳赫空间理论是关于无限多维的分析。对于每个有限维 n，只有一个 n 维空间，在大多数方面，就像在 n 为 1、2 或 3 时的熟悉情况一样，一般都很好理解。这些用于跟踪仅存在有限多个自由度的情况。当自由度的数量变得无限时，例如描述一根绳子的配置所必需的，对于执行分析的环境空间有许多可能的合理选择。该奖项支持的工作主要关注这些空间如何融入另一个空间，一系列的调查揭示了它们的结构。