Abstract: This chapter presents some background results about operator spaces and establishes some notations which will be used throughout this book. It can serve as a mini-course on the basics of operator space theory for readers with less mathematical maturity. The lengthy proofs usually belong to very well-known results (such as Ruan's theorem, or the extension/characterization theorems for completely positive or completely bounded maps). Topics covered include notation and conventions, completely positive maps, operator space duality, operator space tensor products, and duality and tensor products. Notes and historical remarks are presented at the end of the chapter.

Keywords: Hilbert space, operator spaces, space duality, space tensor products,