Mathematical Sciences: Unconditional Structure in Banach Spaces

数学科学：Banach 空间中的无条件结构

• 批准号: 9500125
• 负责人: Nigel Kalton
• 金额: $ 14.45万
• 依托单位: University of Missouri-Columbia
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-05-01 至 1999-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9500125&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Unconditional; Structure; Banach

Kalton 9500125 The principal investigator plans to study a number of fundamental questions in the theory of unconditional structure in Banach spaces. Among the questions to be considered is the longstanding problem of whether a complemented subspace of a Banach space with an unconditional basis also has an unconditional basis, and a similar problem for complemented subspaces of Banach lattices. The resolution of these problems could have a profound effect on the theory of Banach spaces. Another question to be studied is whether a Banach space in which every closed subpsace has an unconditional basis (or just local unconditional structure) is necessarily a Hilbert space. A number of other problems concerning unconditional structure are also to be investigated. Unconditional bases are a natural framework for the study of the decomposition of functions (or signals) into basis elements (or simpler pure components) as in the modern theory of wavelets. This proposal is concerned with studying situations where such decompositions exist and where they fail to exist. By studying the possible unconditional bases of a certain space or class of functions one is able to obtain important information concerning the space. ***

Kalton 9500125 首席研究员计划研究 Banach 空间无条件结构理论中的一些基本问题。 需要考虑的问题之一是长期存在的问题，即具有无条件基的 Banach 空间的补子空间是否也具有无条件基，以及 Banach 格子的补子空间的类似问题。 这些问题的解决可能会对巴纳赫空间理论产生深远的影响。 另一个需要研究的问题是，每个闭子空间都有无条件基（或只是局部无条件结构）的巴拿赫空间是否必然是希尔伯特空间。 有关无条件结构的许多其他问题也有待研究。 无条件基是研究将函数（或信号）分解为基元素（或更简单的纯分量）的自然框架，如现代小波理论中那样。 该提案涉及研究这种分解存在和不存在的情况。 通过研究某一空间或一类函数的可能无条件基，人们能够获得有关该空间的重要信息。 ***