PRELIMINARY TEXT: DO NOT USE This book is intended to be a systematic text on topological vector spaces and presupposes familiarity with the elements of general topology and linear algebra. Each of the chapters is preceded by an introduction and followed by exercises. These exercises are devoted to further results and supplements, in particular, to examples and counter-examples. Hints have been given where it seemed appropriate. This second edition has been thoroughly revised and includes a new chapter on C^* and W^* algebras.