Banach Spaces and Applications

巴拿赫空间是称为向量的对象的集合，可以将它们相加或乘以数字以形成其他向量。有一个矢量之间距离的概念，类似于我们所居住的三维世界中点之间距离的熟悉概念。数学家们发现，巴拿赫空间提供了一个正确的框架，在这个框架中，可以表述一些重要的数学领域，如泛函分析和偏微分方程。科学家和工程师也使用巴拿赫空间来模拟流体力学、信号处理和金融等应用领域的问题。巴拿赫空间有无穷多种，它们可以通过平滑性和凹凸性等几何性质来相互区分。属于巴拿赫空间的单个向量由称为系数的无限数字串标识。数据压缩中的一个重要问题是找到一个过程，有时称为贪心算法，用于选择最重要的系数，使得到的有限字符串向量与原始向量的距离很近，因此很接近原始向量。本项目将研究巴拿赫空间理论中的基本问题及其在其他领域的应用。所采用的方法将是功能分析的方法，并结合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离无限维Banach空间。本文还研究了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的概念。在非线性巴那赫空间理论中，另一个需要解决的问题是在一致商映射下$p$-凸性是否保持。在其他有待研究的领域的应用包括贪心算法在Banach空间中的收敛性的开放问题，包括Lebesgue空间的重要情况，以及相关的关于无条件和贪心收敛的开放问题，包括任意Banach空间中拟贪心序列的存在性和Elton常数的有界性的开放问题。其他应用包括改进的矩阵的显式构造，具有限制等距性质，以及基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.