Mathematical Sciences: Geometry of Classical Banach Spaces

Alspach教授将继续研究连续函数空间和p次幂可积函数空间的补子空间。长期目标是将这些空间的互补子空间分类到同构。Schutt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。巴拿赫空间的理论是关于无限多维的分析。对于每一个有限维n，只有一个n维空间，在大多数情况下，当n为1、2或3时，一般情况下都很容易理解。它们用于跟踪只有有限多个自由度的情况。当自由度的数量变得无限时，例如描述一根弦的结构是必要的，对于执行分析的环境空间有许多可能的合理选择。该奖项支持的作品主要关注的是这些空间如何与另一个空间相适应，这是一条揭示其结构的调查线。

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.