This text covers a first course in bilinear maps and tensor products intending to bring the reader from the beginning of functional analysis to the frontiers of exploration with tensor products. Tensor products, particularly in infinite-dimensional normed spaces, are heavily based on bilinear maps. The author brings these topics together by using bilinear maps as an auxiliary, yet fundamental, tool for accomplishing a consistent, useful, and straightforward theory of tensor products. The author's usual clear, friendly, and meticulously prepared exposition presents the material in ways that are designed to make grasping concepts easier and simpler. The approach to the subject is uniquely presented from an operator theoretic view. An introductory course in functional analysis is assumed. In order to keep the prerequisites as modest as possible, there are two introductory chapters, one on linear spaces (Chapter 1) and another on normed spaces (Chapter 5), summarizing the background material required for a thorough understanding. The reader who has worked through this text will be well prepared to approach more advanced texts and additional literature on the subject.

Preface.- 1. Linear-Space Results.- 2. Bilinear Maps: Algebraic Aspects.- 3. Algebraic Tensor Product.- 4. Interpretations.- 5. Normed-Space Results.- 6. Bounded Bilinear Maps.- 7. Norms on Tensor Products.- 8. Operator Norms.- 9. Tensor Product Operators.- References.- List of Symbols.- Index.