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. ***

小行星9500874 提议者打算在2004年解决几个基本问题。 无限维Banach空间的结构理论。 具体地说，奥德尔将研究是否存在Banach空间 有界畸变，或任意畸变但非双正交 畸变这些问题的强烈动机是奥德尔最近的联合 与Schlumprecht发现，希尔伯特空间是扭曲的最强 道理啊他还打算研究是否某些几乎等距渐近 在Banach空间上的假设确保它包含一个经典的 空间 LP 为 1 或= p 无穷大或c下标0。奥德尔会 也研究了Huff准则是否刻画了空间同构 对于那些具有统一的Kadec-Klee属性的人来说， 连续性基本确定。罗森塔尔将进一步研究 包含c下标0的空间的可能特征，与 他最近对非自反空间的描述 c下标0 只要它的所有子空间都有弱序列完备的子空间。为 例如，他将研究是否一个脚本L无穷饱和的问题 空间包含 c下标0，以及是否一个Banach子空间 X 的 C（K） 包含 c下标0，条件是X** 中X的补数包含 函数的 α次振荡是有界的，对所有可数序数 α的罗森塔尔还明智地调查Banach空间与可分的， 例如，这样的空间是否同构于 C（alpha）的导数（对于 alpha是可数序数），特别是 注意希尔伯特空间的情况。罗森塔尔还将研究 C（K）空间的可补子空间是否是一个长期悬而未决的问题 也同构于C（K）-空间。 无限维Banach空间结构性质的研究 是现代分析学中最深刻、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， 了解和深入了解它们的结构是现代研究的基础 泛函分析及其在物理学中的应用 近来 在这一领域取得了惊人的进步。 长期存在的深层次问题 这两个问题都得到了解决，两位提案人都积极参与了这些进步。 当然，还有许多根本性的问题尚未解决， 提议者打算解决其中一些基本问题。 ***