Mathematical Sciences: Functional Analysis

DMS提案#9622841 PI：Dr. Maria Girardi 南卡罗来纳州@哥伦比亚大学 吉拉迪的最终目标是获得更好的 对功能分析的理解;她的方法 是研究其各种 分领域;在这个时候，她的主要推力是在 Banach空间理论的分支。 她将继续 她的工作，她已经取得了进展， 过去几年，在几个（相关） Banach空间理论中的长期问题， 都集中在拉东-尼科迪姆地产附近 她还将继续检查Bochner-Lebesgue 空间和开始一个新的项目， 泛函分析与同调代数。 一般来说，Banach空间是一个集合。 这些点有一种方法， 测量任意两点之间的“距离”。 一个例子是三维欧几里得（普通）空间。 另一个例子是“适当的好”空间 函数（所以在这种类型的Banach空间中的一个点 是一个函数），其中距离被适当地阐明。 Banach空间理论的一个实际应用是 光盘编码;声音信号大致是 一个合适的Banach空间中的函数。 不幸的是声音信号包含太多 存储在CD上的信息， 用Banach空间中的元素逼近 这是接近真实的元素（这是 距离的概念进入）。 了解 Banach空间的结构有助于上述，并且 其他实际应用。

DMS Proposal #9622841 PI: Dr. Maria Girardi University of South Carolina @ Columbia Girardi's ultimate goal is to gain a better understanding of Functional Analysis; her approach is to examine the interplay between its various sub-areas; at this time, her main thrust is in the branch of Banach Space Theory. She will continue her work, in which she has made progress over the past years, on several (related) long-standing problems in Banach space theory that are centered around the Radon-Nikodym Property. She will also continue examining the Bochner-Lebesgue space and begin a new project that ties together Functional Analysis and Homological Algebra. Roughly speaking, a Banach space is a collection of points that has, among other things, a way to measure the "distance" between any two points. An example is the 3-D Euclidean (ordinary) space. Another example is the space of "suitably nice" functions (so a point in this type of Banach space is a function) with distance suitably clarified. A practical application of Banach space theory is compact disc coding; a sound signal is, roughly speaking, a function in a suitable Banach space. Unfortunately the sound signal contains too much information to store on the CD so the signal is approximated by an element from the Banach space that is close to the true element (this is where the notion of distance enters). Understanding the structure of Banach spaces aids in the above, and other, practical applications.