This book, designed for a one-year course at the beginning graduate level, displays those properties of topological vector spaces which are used by researchers in classical analysis, differential and integral equations, distributions, summability, and classical Banach and Frechet spaces. In addition, optional examples and problems (with hints and references) will set the reader's foot on numerous paths such as non-locally convex (e.g., ultrabarrelled) spaces, Kothe-Toeplitz spaces, Banach algebra, sequentially barrelled spaces, and norming subspaces.