Banach Spaces: Theory and Applications

Banach空间及其几何学为研究物理学、信号处理和大型数据集的分析提供了一个重要的框架。这个关于Banach空间的研究项目将有两个主要方向。第一部分是Banach空间坐标系的研究。如果我们使用某个Banach空间对物理或信号分析中的给定问题进行建模，我们还需要该空间的“适当坐标系”，即我们希望用一个数字序列来表示该空间的元素。什么是“适当的坐标系”当然取决于具体问题，但通常目标是用最少的坐标来近似Banach空间的给定元素，并以最小的误差和最少的工作量从给定的坐标序列重建元素。并不是每个Banach空间都允许具有极小性质的坐标系;人们希望坐标系满足。因此，人们还需要找到一个给定的Banach空间的标准，以将其嵌入到一个具有坐标系的空间中，而不会失去原始空间的拓扑和几何性质。度量空间在计算机科学中经常被用来模拟大型数据集，我们的第二个目标是研究度量空间到Banach空间的嵌入，并且一方面从度量空间嵌入某些Banach空间的性质获得关于度量空间的结构的信息，另一方面推导Banach空间的几何性质，在这个项目中研究的许多问题要么来自于数学的其他领域，要么与之相关，如描述集理论，调和分析，度量几何和逼近理论。所采用的技术将涉及分析，几何，无限组合学和逻辑的组合。 考虑的问题之一是一个旧的一个从调和分析，它要求是否空间的p-可积函数和其他功能空间有一个Schauder基础所形成的翻译只有一个元素。 另一个突出的问题是嵌入一致凸Banach空间到这样的空间的基础或有限维分解。连同他的同事Sivakumar和他们的联合学生，基顿哈姆，调查人员打算研究的表示和近似的元素，某些功能类使用冗余坐标系。本研究也追求一个新的方向，探讨度量几何中的问题。本文利用度量空间的可嵌入性来刻画Banach空间的几何和拓扑性质，如自反性。

Banach spaces together with their geometry provide an important framework for studying problems in physics, signal processing, and the analysis of large data sets. This research project on Banach spaces will have two main directions. The first one is the study of coordinate systems of Banach spaces. If we model a given problem in physics or signal analysis using a certain Banach space we will also need an "appropriate coordinate system" for that space, i.e. we want to represent the elements of this space by a sequence of numbers. What constitutes an "appropriate coordinate system" will of course depend on the specific problem, but generally the goal is to approximate a given element of a Banach space as well as possible with the least amount of coordinates, and to reconstruct the element from the given sequence of coordinates with the smallest possible error and the least amount of effort. Not every Banach space admits coordinate systems which have the minimality properties; one would like a coordinate system to satisfy. Therefore one also needs to find for a given Banach space criteria to embed it into a space with coordinate systems, without losing the topological and geometrical properties of the original space. Metric spaces are often used in Computer Science to model large data sets, and our second objective is to investigate embeddings of metric spaces into Banach spaces, and to obtain on the one hand information about the structure of the metric space, from the property that it embeds in certain Banach spaces, and on the other hand deduce geometric properties of a Banach space, from the property that certain metrics embed or do not embed in it. Many of the problems under study in this project either originate from, or are related to, other areas of mathematics such as descriptive set theory, harmonic analysis, metric geometry, and approximation theory. The techniques to be employed will involve a combination of analysis, geometry, infinite combinatorics, and logic. One of the problems considered is an old one from harmonic analysis, which asks whether the space of p-integrable functions and other function spaces have a Schauder basis formed by translates of only one element. Another prominent problem is the embedding of uniformly convex Banach spaces into such spaces with a basis or a finite dimensional decomposition. Together with his colleague Sivakumar and their joint student, Keaton Hamm, the investigator intends to study the representation and approximation of elements of certain function classes using redundant coordinate systems. The research also pursues a new direction and investigates problems in metric geometry. This work aims to characterize geometric and topological properties of Banach space like reflexivity by the embeddability of certain metric spaces.