Mathematical Sciences: Structure of Infinite-Dimensional Banach Spaces

数学科学：无限维 Banach 空间的结构

• 批准号: 9500874
• 负责人: Haskell Rosenthal
• 金额: $ 25.18万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-06-01 至 1999-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9500874&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Structure; Infinite; Dimensional

9500874 Rosenthal The proposers intend to work on several fundamental problems in the structure theory of infinite dimensional Banach spaces. Specifically, Odell will study whether there are Banach spaces of bounded distortion, or of arbitrary distortion but not biorthogonal distortion. These questions are strongly motivated by Odell's recent joint discovery with Schlumprecht, that Hilbert space is distortable in the strongest sense. He also intends to study whether certain almost isometric asymptotic assumptions on a Banach space insure that it contains one of the classical spaces lp for 1 or = p infinity or c subscript 0. Odell will also investigate whether the Huff criterion characterizes spaces isomorphic to those with the uniform Kadec-Klee property, and whether the Point of Continuity Property is basically determined. Rosenthal will study further possible characterizations of spaces containing c subscript 0, related to his recent characterization that a non-reflexive space contains c subscript 0 provided all its subspaces have weakly sequentially complete duals. For example, he will study the question of whether an script L infinity saturated space contains c subscript 0, and whether a Banach subspace X of C(K) contains c subscript 0 provided the complement of X in X** contains functions whose alpha-th oscillation is bounded, for all countable ordinals alpha. Rosenthal also wises to investigate Banach spaces with separable duals, for example the problem of whether such spaces are isomorphic to subspaces of quotients of C(alpha) (for alpha a countable ordinal), with particular attention given to the Hilbert space case. Rosenthal will also study the long-standing open problem of whether complemented subspaces of C(K) spaces are again isomorphic to a C(K)-space. The study of structural properties of infinite-dimensional Banach spaces is one of the most profound, fundamental research areas i n modern analysis. These spaces underlie the mathematical foundations of quantum mechanics, and understanding and penetrating their structure is basic for modern research in functional analysis and its applications to physics. Recently there have been spectacular advances in this area. Deep long standing open problems have been solved, and both proposers have been heavily involved in these advances. Of course there are many more fundamental open problems remaining and the proposers intend to tackle some of these basic questions. ***

[9500874] Rosenthal提议者打算研究无限维巴拿赫空间结构理论中的几个基本问题。具体来说，Odell将研究是否存在有界畸变的Banach空间，或者是否存在任意畸变但不存在双正交畸变的Banach空间。Odell最近与Schlumprecht的联合发现强烈地激发了这些问题，希尔伯特空间在最强烈的意义上是可扭曲的。他还打算研究Banach空间上的某些几乎等距渐近假设是否保证它包含经典空间lp中的一个，p为1或= p无穷或c下标0。Odell还将研究Huff准则是否与具有一致Kadec-Klee性质的空间同构，以及连续性点是否基本确定。Rosenthal将进一步研究包含c下标0的空间的可能表征，这与他最近的表征有关，即一个非自反空间包含c下标0，前提是它的所有子空间都具有弱序完全对偶。例如，他将研究一个符号L无穷饱和空间是否包含下标0，以及c (K)的Banach子空间X是否包含下标0，如果X**中的X的补包含其α -振荡是有界的函数，对于所有可数序数α。Rosenthal还研究了具有可分对偶的巴拿赫空间，例如，这些空间是否同构于C（α）商的子空间（对于α是可数数列）的问题，特别注意了希尔伯特空间的情况。Rosenthal还将研究C(K)空间的补子空间是否同构于C(K)空间这一长期存在的开放问题。对无限维巴拿赫空间结构性质的研究是现代分析中最深刻、最基础的研究领域之一。这些空间构成了量子力学的数学基础，理解和穿透它们的结构是现代功能分析研究及其在物理学中的应用的基础。最近在这个领域有了惊人的进展。长期悬而未决的深层次问题得到了解决，两位提案人都积极参与了这些进步。当然，还有许多更根本的悬而未决的问题，提议者打算解决其中的一些基本问题。＊＊＊