Banach Spaces and Applications

Banach空间是称为向量的对象的集合，这些对象可以相加或乘以数字来形成其他向量。有一个矢量之间的距离的概念，它类似于我们居住的三维世界中熟悉的点之间的距离的概念。数学家们已经发现，Banach空间提供了正确的框架，可以在其中建立泛函分析和偏微分方程等主要数学领域的公式。科学家和工程师还使用Banach空间对流体力学、信号处理和金融等应用领域的问题进行建模。Banach空间有无限的种类，它们可以通过光滑性和凸性等几何性质相互区分。属于Banach空间的单个向量由称为系数的无限数字串来标识。数据压缩中的一个重要问题是找到一种过程，有时被称为贪婪算法，用于选择最重要的系数，使得得到的有限串向量距离原始向量较短，因此是原始向量的良好近似值。这个项目将研究Banach空间理论的基本问题以及在其他领域的应用。所采用的方法将是泛函分析的那些方法，以及针对每个特定问题的新见解。这些问题包括Banach旋转问题，该问题询问Hilbert空间是否是唯一具有传递等距群的可分无限维Banach空间。还将研究渐近中点凸性的新概念。一个有待解决的问题是，这个性质的同构版本是否等价于已知的渐近一致凸性的概念。在非线性Banach空间理论领域需要解决的另一个问题是，在一致商映射下，$p$-凸性是否保持。在其他领域中的应用包括关于贪婪算法在Banach空间中的收敛的公开问题，包括Lebesgue空间的重要情况，以及相关的关于无条件和贪婪收敛的公开问题，包括任意Banach空间中拟贪婪序列的存在性和Elton常数的有界性的公开问题。其他应用包括具有受限等距性质的矩阵的改进的显式构造，以及基本系统和冗余系统的系数量化性质。

A Banach space is a collection of objects called vectors which can be added together or multiplied by numbers to form other vectors. There is a concept of distance between vectors which is analogous to the familiar notion of distance between the points in the three-dimensional world which we inhabit. Mathematicians have found that Banach spaces provide the correct framework in which to formulate major areas of mathematics such as Functional Analysis and Partial Differential Equations. Banach spaces are also used by scientists and engineers to model problems in applied areas such as fluid mechanics, signals processing, and finance. There are an infinite variety of Banach spaces which can be distinguished from each other by geometrical properties such as smoothness and convexity. An individual vector belonging to a Banach space is identified by an infinite string of numbers called coefficients. An important problem in data compression is to find a procedure, sometimes called a greedy algorithm, for selecting the most significant coefficients so that the resulting finite string vector is a short distance from, and hence a good approximation to, the original vector. This project will investigate fundamental problems in Banach space theory and applications to other areas. The methods employed will be those of Functional Analysis together with new insights specific to each particular problem. These problems include the Banach Rotation Problem which asks whether Hilbert space is the only separable infinite-dimensional Banach space with a transitive isometry group. The new notion of asymptotic midpoint convexity will also be investigated. An open question to be solved is whether the isomorphic version of this property is equivalent to the known concept of asymptotic uniform convexity. Another problem to be solved in the area of nonlinear Banach space theory is whether $p$-convexity is preserved under uniform quotient mappings. Applications to other areas to be investigated include open questions on the convergence of greedy algorithms in Banach spaces, including the important case of Lebesgue spaces, and related open questions on unconditionality and greedy convergence, including the open problems of the existence of quasi-greedy sequences in arbitrary Banach spaces and of the boundedness of the Elton constants. Other applications include improved explicit constructions of matrices with the Restricted Isometry Property, and coefficient quantization properties for bases and redundant systems.