Twisted Sums and Unconditional Structure for Banach Spaces

Banach 空间的扭曲和和无条件结构

• 批准号: 9870027
• 负责人: Nigel Kalton
• 金额: $ 18.63万
• 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: 1998
• 资助国家: 美国
• 起止时间: 1998-05-01 至 2004-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9870027&HistoricalAwards=false
• 关键词: Twisted; Sums; Unconditional; Structure; Banach

Proposal: DMS-9870027 Principal Investigator: Nigel J. Kalton Abstract: The aim of this proposal is to study a number of problems related to the notion of an extension of one Banach space by another, and the relationships between extensions and unconditional structure. It is proposed to attempt to resolve the conjecture that if every minimal extension of a Banach space is trivial then the space has finite cotype, and to attempt a classification of those spaces so that every extension by a Hilbert space is trivial. Both these problems can be reformulated in other terms and would have considerable significance independent of the theory of extensions. The proposer will also work on the fundamental problem in the theory of unconditional structure of whether every complemented subspace of a Banach space with unconditional basis also has an unconditional basis. The theory of extensions can be considered in the following terms. Suppose we are given a centrally symmetric solid in three or more dimensions and we are allowed only to compute its cross-section by a slice in some directions and the shadow cast by the body in the perpendicular directions. This allows us to compute certain information about the solid, but not to reconstruct it completely. The aim of this project is to obtain more complete information about the body under certain additional conditions. It turns out that many questions of importance in mathematical analysis and applications, although not expressed in this form, can be visualized as problems of this nature, perhaps involving infinite dimensions. An unconditional basis in a particular class of functions is a collection of relatively simple functions with which we can approximate every member of the class; such approximations are very important for many practical situations, for example in engineering applications. One of the aims of this project is to decide general conditions under w hich a basis can be constructed. There is interaction between the theory of extensions and of unconditional bases which this project will explore further.

建议：DMS-9870027首席研究员：Nigel J. Kalton摘要：该建议的目的是研究与另一个Banach空间扩展的概念有关的许多问题，以及扩展与无条件结构之间的关系。 有人建议试图解决以下猜想：如果Banach空间的每个最小扩展是微不足道的，那么该空间具有有限的cotype，并尝试对这些空间进行分类，以使希尔伯特空间的每个扩展空间都是微不足道的。这两个问题都可以用其他术语重新校正，并且与扩展理论无关。 提议者还将在无条件结构理论的基本问题上致力于无条件结构的理论，即banach空间的每个互补的子空间是否也有无条件的基础。 扩展理论可以通过以下术语来考虑。 假设我们在三个或多个维度上给出了中央对称的固体，并且只允许我们通过某些方向的切片计算其横截面，而人体则以垂直方向施放的阴影。这使我们能够计算有关固体的某些信息，但不能完全重建它。 该项目的目的是在某些其他条件下获取有关身体的更多完整信息。事实证明，在数学分析和应用中，尽管没有以这种形式表达的许多重要性问题，但可以将其视为这种性质的问题，也许涉及无限维度。在特定类别的函数中，无条件的基础是相对简单的功能的集合，我们可以通过这些功能近似同类的每个成员。 这种近似对于许多实际情况非常重要，例如在工程应用中。该项目的目的之一是决定在基础上确定一般条件。扩展理论与无条件基础的理论之间存在相互作用，该项目将进一步探索。