Modern analysis and topology
The purpose of the author is to give an introduction into the theory of uniform spaces and its application in modern analysis. The first part begins with a survey on metric spaces including metrization theorems (Nagata-Smirnov, Arkhangel'skij). Then uniform spaces are introduced (as spaces equipped with a collection of uniform coverings) and their classical theory is developed; besides convergence of nets, clustering of transfinite sequences is also used. Compactifications, paracompactifications and realcompactifications of completely regular spaces are studied as completions of suitable uniform spaces. The second part gives an introduction to measure theory and Lebesgue integration. Then the theory of Haar measure is generalized for a suitable class of locally compact uniform spaces, together with a kind of differentiation giving a constructive representation for the Radon-Nikodym derivative under rather general conditions. A chapter is devoted to the theory of Hilbert and Banach spaces and the illustration of the usefulness of uniformities in the theory of function spaces. Besides classical results, a rich collection of modern theories is thoroughly discussed, including a large number of the author's results. The lack of a list of references is compensated by extended historical surveys at the beginning of each section. It is only to hope that the reader will not be discouraged by some minor inaccuracies (e.g. the proof of Lemma 1.1 is wrong, the Lebesgue integral is defined for \(f\geq 0\) but used for complex valued functions, etc.).
scientific article; zbMATH DE number 785441