Infinite-Dimensional Banach and Operator Spaces

无限维 Banach 和算子空间

• 批准号: 9801153
• 负责人: Edward Odell
• 金额: $ 7.28万
• 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: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2001-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9801153&HistoricalAwards=false
• 关键词: Infinite; Dimensional; Banach; Operator; Spaces

ABSTRACTOdell This project involves several open problems in geometric functional analysis. 1) Does there exist a distortable Banach space of bounded distortion? 2) If all spreading models of any block basis of a given basis for X are 1-equivalent to some l_p space does X contain l_p? 3) Must every Banach space admit a "nice" spreading model? 4) The study of the ordinal l_1(+) index of a Banach space. In the usual geometry of space one can measure the distance between two points by means of a carpenter's tape measure (distance is measured "as the crow flies"). There are however other natural distances which a mathematician encounters. For example the "taxicab distance" between two points on this surface would be the distance between the points as measured by a taxi that could only travel horizontally or vertically (imagine all roads run N-S or E-W). Mathematicians must study the geometry of these different notions of distance in order to solve those problems from whence the geometry arose. Furthermore many mathematical problems require one to work in spaces of higher than three dimensions, even infinite dimensional space (for example in quantum mechanics differential equations,...). This project concerns such problems. For example the author and Th. Schlumprecht proved that ordinary (tape measure) infinite dimensional space could have its geometry altered ("distorted") so as to be that one could not recapture the original geometry up to any given multiple in any infinite dimensional "slice". This is false for the taxicab geometry. But it remains a famous open problem as to whether or not there exists a geometry which can be altered but not too badly. Other problems of this project involve determining infinite dimensional substructures of a geometry given limited "asymptotical" (less than infinite dimensional but more than finite dimensional) information.

摘要该项目涉及几何泛函分析中的几个开放性问题。 1）是否存在有界扭曲的可扭曲Banach空间？ 2) 如果 X 的给定基的任何块基的所有扩展模型都是 1-等价于某个 l_p 空间，那么 X 是否包含 l_p？ 3）每个Banach空间都必须承认一个“好的”传播模型吗？ 4) Banach空间序数l_1(+)索引的研究。 在通常的空间几何中，人们可以用木匠的卷尺测量两点之间的距离（距离是“直线测量”的）。 然而，数学家还遇到其他自然距离。 例如，该表面上两点之间的“出租车距离”将是由只能水平或垂直行驶的出租车测量的点之间的距离（想象所有道路都是南北或东西走向）。 数学家必须研究这些不同距离概念的几何，以便解决几何产生的问题。 此外，许多数学问题需要在高于三维空间，甚至无限维空间中工作（例如在量子力学微分方程中，......）。 本项目涉及此类问题。 例如作者和Th。施伦普雷希特证明，普通（卷尺）无限维空间的几何形状可能会改变（“扭曲”），以致人们无法在任何无限维“切片”中重新捕获原始几何形状直至任何给定的倍数。 对于出租车几何形状来说，这是错误的。但是否存在可以改变但不会太严重的几何形状仍然是一个著名的悬而未决的问题。 该项目的其他问题涉及在给定有限的“渐近”（小于无限维但大于有限维）信息的情况下确定几何的无限维子结构。