This book is an introductory text written with minimal prerequisites. The plan is to impose a distance structure on a linear space, exploit it fully, and then introduce additional features one by one, only when we cannot get any further without them. The book naturally falls into two parts, and each of them is developed independently of the other. The first part deals with normed spaces, their completeness and continuous linear maps on them, and includes the theory of compact operators. A much shorter second part treats inner product spaces, their completeness, and leads to the spectral theorem for compact self-adjoint operators. Four appendices on Fixed Points, Extreme Points, Sturm-Liouville Problems and Unbounded Operators indicate further areas of development. Emphasis is given on examples which illustrate abstract concepts, and on applications of results proved in the text. In addition to proving existence and uniqueness of the solution of a problem, an approximate construction of the solution is pointed out. Problems of varying degrees of difficulty are given at the end of each of the 24 sections. The statement of each problem contains its answer.