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建议者打算在无限尺寸Banach空间的结构理论中处理一些基本问题。 具体而言，Odell将研究是否有界限扭曲的Banach空间，还是任意失真但不是生物扭曲的失真。这些问题是由奥德尔（Odell）最近与Schlumprecht的联合发现的强烈动机，即希尔伯特（Hilbert）空间在最强的意义上是令人失望的。他还打算研究Banach空间上的某些几乎等轴测渐近假设是否包含1或= P Infinity或C下标的经典空间LP之一。ODELL还将调查霍夫标准还将调查与统一的Kadec-Klee属性以及基本确定属性的统一属于统一的kadec-Klee属性的空间同构的特征。罗森塔尔（Rosenthal）将研究包含C下标0的空间的进一步表征，这与他最近的表征相关，即非反射空间包含C下标0，前提是其所有子空间都具有较弱的顺序完整的双重偶。例如，他将研究脚本l无限饱和空间是否包含C下标0的问题，以及C（k）的Banach子空间X是否包含C下标0提供了x **中x的补充，其中包含alpha-th振动界的函数，对于所有可计数序列alpha。罗森塔尔（Rosenthal）还明智地研究了具有可分离二元的Banach空间，例如，此类空间与C（Alpha）商的子空间同构（对于Alpha）（对于Alpha是可计数的序数）的问题，特别注意了Hilbert Space案例。 Rosenthal还将研究长期以来的开放问题，即C（K）空间的补充子空间是否再次与C（K）空间同构。 无限二维Banach空间的结构特性的研究是现代分析中最深刻的基础研究领域之一。 这些空间是量子力学的数学基础以及理解和穿透结构的基础，这是功能分析及其在物理学上的应用中的基础。 最近，该领域取得了惊人的进步。 已经解决了深刻的开放问题，两个建议者都参与了这些进步。 当然，还有更多的基本开放问题，建议者打算解决其中一些基本问题。 ***