Banach Spaces and Applications

University Of South Carolina At Columbia, Columbia SC

Investigators

Abstract

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.