With many new concrete examples and historical notes, Topological Vector Spaces, Second Edition provides one of the most thorough and up-to-date treatments of the Hahn-Banach theorem. This edition explores the theorem's connection with the axiom of choice, discusses the uniqueness of Hahn-Banach extensions, and includes an entirely new chapter on vector-valued Hahn-Banach theorems. It also considers different approaches to the Banach-Stone theorem as well as variations of the theorem. The book covers locally convex spaces; barreled, bornological, and webbed spaces; and reflexivity. It traces the development of various theorems from their earliest beginnings to the present day, providing historical notes to place the results in context. The authors also chronicle the lives of key mathematicians, including Stefan Banach and Eduard Helly. Features, Provides extensive coverage of the Hahn-Banach and Banach-Stone theorems, Discusses the evolution of the Hahn-Banach theorem and Eduard Helly's considerable contribution to it, Presents historical notes on the development of many important theorems and the people who discovered and proved them, including The Scottish Café group, Includes numerous end-of-chapter exercises, a broad spectrum of examples, and detailed proofs, Suitable for both beginners and experienced researchers, this book contains an abundance of examples, exercises of varying levels of difficulty with many hints, and an extensive bibliography and index.