Mathematical Sciences: Geometry of Classical Banach Spaces

Alspach教授将继续研究连续函数空间的补子空间和p次方可积函数的补子空间。长期目标是将这类空间的补子空间分类到同构。舒特教授将研究可积函数空间的有限维子空间，并研究Banach空间的凸几何的某些特征。Banach空间的理论是关于无限多个维度的分析。对于每个有限维n，只有一个n维空间，在大多数方面，就像在n为1、2或3的常见情况下一般被理解的那样。这些空间用于跟踪只有有限多个自由度的情况。当自由度数变得无限时，例如描述一段弦的构型将是必要的，对于在其中执行分析的环境空间有许多可能的合理选择。该奖项支持的工作主要涉及这些空间如何融入另一个空间，这是一条暴露了它们结构的大量调查路线。

Professor Alspach's part of this project will continue his investigation of the complemented subspaces of spaces of continuous functions and of functions whose p-th powers are integrable. The long-term goal is to classify up to isomorphism the complemented subspaces of such spaces. Professor Schutt will study finite-dimensional subspaces of the space of integrable functions and investigate certain features of the convex geometry of Banach spaces. The theory of Banach spaces is about analysis in infinitely many dimensions. For each finite dimension n, there is only one n-dimensional space, in most respects as well understood in general as in the familiar cases when n is 1, 2, or 3. These are used to keep track of situations in which there are only finitely many degrees of freedom. When the number of degrees of freedom becomes infinite, as would for instance be necessary to describe the configuration of a piece of string, there are many possible reasonable choices for the ambient space in which to perform analysis. The work supported by this award is largely concerned with the ways in which these spaces fit into another, a line of investigation that lays bare a good deal of their structure.