Topological Uniform Structures (Dover Books on Mathematics)

"Proofs are detailed and carefully done … there is a lot of fine material in this book." — Bulletin of the American Mathematical Society.
While many sources offer partial coverage of uniform spaces, topological groups, topological vector spaces, topological algebras, and abstract harmonic analysis, this graduate-level text was the first to give a thorough and fully detailed account of all these topics. It also contains a section on vector-valued measure spaces as well as numerous problems and examples.
Chief among the salient features of the work is its overall unifying theme of topologies compatible with increasingly enriched algebraic structures. The book gradually deepens the reader’s appreciation of the rich interplay among diverse areas of mathematics. Each section begins with a brief summary of the important points to appear, and the theory unfolds smoothly through an abundance of relevant commentary and well-chosen examples.
In addition, the remarks that follow most theorems and examples are replete with useful information, counterexamples, and other points worth noting. Finally, each section ends with a set of problems (more than 220, with several subparts in each).
Largely self-contained and accessible to students with a background in elementary analysis, linear algebra, and point set topology, the book nevertheless covers a good deal of advanced material. However, because of its readability, leisurely pace, and highly illustrative contents, it is an ideal reference for advanced undergraduates and beginning graduate students.