Designed for a one-year course in topological vector spaces, this text is geared toward advanced undergraduates and beginning graduate students of mathematics. The subjects involve properties employed by researchers in classical analysis, differential and integral equations, distributions, summability, and classical Banach and Frechét spaces. Optional problems with hints and references introduce non-locally convex spaces, Köthe-Toeplitz spaces, Banach algebra, sequentially barrelled spaces, and norming subspaces. Extensive introductory chapters cover metric ideas, Banach space, topological vector spaces, open mapping and closed graph theorems, and local convexity. Duality is the treatment's central theme, highlighted by a presentation of completeness theorems and special topics such as inductive limits, distributions, and weak compactness. More than 30 tables at the end of the book allow quick reference to theorems and counterexamples, and a rich selection of problems concludes each section.