Bornologies and Lipschitz Analysis

Metric spaces are studied using bornologies,which are families of subsets that contain singletons,are stable under finite unions,and are stable under taking subsets of their members. The power set is the largest bornology,while the bornology of finite subsets is the smallest. Functions are connected to bornologies,such as local Lipschitzness and Bourbaki boundedness. Variational notions of strong uniform continuity and convergence are discussed,leading to the bornology of UC-subsets and UC-spaces. Lipschitz and locally Lipschitz functions are studied,and it is shown that a function between metric spaces has a particular metric property if and only if it is followed by a real-valued Lipschitz function. Bornological convergence of nets of closed subsets is considered,and topologies of uniform convergence for continuous linear operators between normed spaces are explained. Finally,a bornological extension of a topological space is presented.

Format: Hardback
Length: 232 pages
Publication date: 27 April 2023
Publisher: Taylor & Francis Ltd

This comprehensive monograph presents, for the first time in book form, a comprehensive exploration of the large structure of metric spaces as captured by bornologies. Bornologies are families of subsets that encompass the singletons, exhibit stability under finite unions, and maintain stability when subsets of their members are taken. The power set of the space forms the largest bornology, while the bornology of its finite subsets represents the smallest. In between these extremes, various other bornologies arise, including metrically bounded subsets, relatively compact subsets, totally bounded subsets, and Bourbaki bounded subsets.

Classes of functions play a crucial role in the study of bornologies. For instance, a function is locally Lipschitz if and only if its restriction to each relatively compact subset is Lipschitz. Similarly, a subset is Bourbaki bounded if and only if every uniformly continuous function on the space is bounded when restricted to the subset. Extensive attention is devoted to variational notions of strong uniform continuity and strong uniform convergence with respect to bornology members, resulting in the development of the bornology of UC-subsets and UC-spaces. Spaces where the uniformly continuous real-valued functions are stable under pointwise product are characterized in terms of the coincidence of Bourbaki bounded subsets with a typically larger bornology.

Special emphasis is placed on Lipschitz and locally Lipschitz functions. Various uniformly dense subclasses of locally Lipschitz functions within real-valued continuous functions, Cauchy continuous functions, and uniformly continuous functions are presented. It is demonstrated quite generally that a function between metric spaces possesses a specific metric property if and only if whenever it is followed in a composition by a real-valued Lipschitz function, the composition also exhibits the property.

Bornologi.

Weight: 624g
Dimension: 181 x 261 x 21 (mm)
ISBN-13: 9780367497873