A primer of Hilbert space

This book deals with the very basic theory of Hilbert space. Chapter 1 deals with the fundamental theory of vector spaces. The notion of a vector space is recalled, together with related techniques. Key definitions and theorems concerning vector spaces in general and linear independence are then reviewed. In chapter 2 the notions of a inner product, normed space and metric space are examined, together with their mutual relationships. Finally, in chapter 3 the very basic tools in the theory of Hilbert space are studied, introducing the natural generalisation of the customary vector space notions to the infinite dimensional framework. In particular, the matter of expanding a vector in terms of a (not necessarily finite) orthogonal basis is introduced. Several examples, problems and exercises are proposed.