Mathematical Sciences: Unconditional Structure in Banach Spaces

9500125首席研究员计划研究一些基本的 在Banach空间中的无条件结构理论的问题。 在需要考虑的问题中， 是否一个Banach空间的补子空间具有无条件 基础也有一个无条件的基础，和一个类似的问题， Banach格的补子空间 解决这些 这些问题可能对Banach空间理论产生深远的影响。 另一个要研究的问题是，是否一个Banach空间，其中每个 闭子空间有一个无条件基（或局部无条件基 结构）必然是一个希尔伯特空间。 许多其他问题 关于无条件结构也要研究。 无条件基是一个自然的研究框架， 将函数（或信号）分解成基本元素（或更简单的 纯分量），如在现代小波理论中。 这项建议是 关注研究存在这种分解的情况， 他们不存在的地方。 通过研究可能的无条件碱基 对于某个空间或函数类，人们能够获得重要的 关于空间的信息。 ***

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. ***