Banach空间的非线性商

The study of nonlinear quotients of Banach spaces is an important part of nonlinear geometric Banach space theory. In this project we study nonlinear quotients of Banach spaces in three different categories: Lipschitz category, uniform category and coarse category. The main concern is twofold: (1) linear characterization of nonlinear quotients of some classical Banach spaces; (2) identifying those Banach space properties that are stable under nonlinear quotient maps. We will improve existing techniques and develop new tools so as to demonstrate the connection and difference between the theory of nonlinear quotient and the theory of nonlinear homeomorphism (embedding).

对Banach空间的非线性商的研究是Banach空间非线性几何理论的重要组成部分之一。本课题在三个不同的范畴下研究Banach空间的非线性商：Lipschitz范畴、一致范畴和粗范畴。研究内容主要包括两个方面：（1）一些经典Banach空间的非线性商的线性刻画；（2）寻找在非线性商映射下保持不变的性质。我们将对该领域中已有的技术进行推广并发展新的工具，以此来诠释非线性商理论与非线性同胚（嵌入）理论之间的联系和差异。

本项目研究了Banach空间之间的非线性商映射及其相关性质。我们主要关注一些渐近性质在非线性商映射下的稳定性。特别地，我们证明了Rolewicz引入的(β)性质在一致商映射和粗商映射下都是保持的。证明的方法基于我们得到的一个关于(β)性质的新的度量刻画。