Mathematical Sciences: Unconditional Structure in Banach Spaces

首席研究员卡尔顿9500125计划研究Banach空间中无条件结构理论中的一些基本问题。要考虑的问题中有一个长期存在的问题，即具有无条件基的Banach空间的可补子空间是否也有无条件基，以及Banach格的可补子空间是否也有无条件基的类似问题。这些问题的解决将对Banach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(或者只是局部无条件结构)的Banach空间是否一定是Hilbert空间。关于无条件结构的其他一些问题也有待调查。在现代小波理论中，无条件基是研究将函数(或信号)分解为基本元素(或更简单的纯分量)的自然框架。这项建议涉及研究这种分解存在和不存在的情况。通过研究某个空间或一类函数的可能的无条件基，可以获得有关该空间的重要信息。***

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. ***