Asymptotic structures in Banach spaces

Banach 空间中的渐近结构

• 批准号: 0099366
• 负责人: Edward Odell
• 金额: $ 8.85万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0099366&HistoricalAwards=false
• 关键词: Asymptotic; structures; Banach; spaces

The geometry of a separable infinite dimensional Banach spaceX can be studied and better understood through a study of its spreadingmodels and asymptotic structure (those finite dimensional bases thatcan be found inside X, but arbitrarily spread out). Knowledge of thesestructures does not always pass through to infinite dimensional informationabout X but sometimes, surprisingly, it does. The author will study anumber of open problems of this nature using the tools of infinitecombinatorics, analysis and logic. For example if a Banach space has onlyone spreading model, must it contain a copy of one of the classical Banachspaces? This project concerns the study of the geometry of normed linearspaces. The easiest example of such a space is ordinary three dimensionalEuclidean space. However one may, even in two or three dimensions, haveother geometries than Euclidean. For example in the "taxicab" spacedistances between points in the plane are computed by traveling theshortest route along roads that run only horizontally or vertically. Inthis geometry the set of all points equidistant from a fixed point isdiamond shaped rather than a circle. Applications of such alternategeometries are numerous in physics, engineering, signal processing andmany other sciences. The state of a system or a signal may be given by asequence of numbers and one may have to have a way of computing thedistance between two states or signals to see how close they are. And inthese applications one has to often use finite dimensional spaces oflarger dimension than three or even spaces of infinite dimension. Theauthor will be exploring the latter case by studying the "asymptotic" structures of these alternate geometries. This is a method oflinking finite and infinite dimensional structure. The techniques to be used are a combination of analysis, infinitary combinatorics andlogic. The problems that arise could also impact and motivate developmentin these latter areas as well.

可分无限维Banach SpaceX的几何可以通过研究它的扩散模型和渐近结构(可以在X内部找到但任意展开的有限维基)来研究和更好地理解。关于这些结构的知识并不总是传递到关于X的无限维信息中，但有时，令人惊讶的是，它确实如此。作者将利用无限组合数学、分析和逻辑的工具来研究这类性质的若干公开问题。例如，如果一个Banach空间只有一个扩散模型，它必须包含一个经典Banach空间的副本吗？这个项目涉及赋范线性空间的几何的研究。这种空间最简单的例子是普通的三维欧几里得空间。然而，即使是在二维或三维空间，人们也可能有欧几里得以外的其他几何。例如，在“出租车”中，平面上点之间的空间距离是通过沿仅水平或垂直的道路行驶最短路线来计算的。在该几何中，距某一固定点等距的所有点的集合是菱形的，而不是圆形的。这种交替分类在物理学、工程学、信号处理和许多其他科学中有着广泛的应用。系统或信号的状态可以由一系列数字给出，人们可能必须有一种方法来计算两个状态或信号之间的距离，以确定它们有多近。在这些应用中，人们不得不经常使用比三维甚至无限维空间更大的有限维空间。作者将通过研究这些交替几何的“渐近”结构来探索后一种情况。这是一种将有限维结构和无限维结构相结合的方法。所使用的技术是分析、无限组合和逻辑的组合。出现的问题也可能影响和推动后两个领域的发展。