Inner product spaces

Abstract

Abstract In this chapter we consider ways of ‘multiplying’ two vectors in a vector space V to give a scalar. Such a product is called an inner product, or scalar product, and the theory is based initially on a familiar example (sometimes called the dot product) on three-dimensional vectors in ℝ3. To begin with, we consider real vector spaces, but we will consider complex vector spaces later on. The scalar product, inner product, or dot product, of two vectors in ℝ2 or ℝ3 is rather well known. We start this chapter with its definition and description of its main properties. Throughout this section, llvll will denote the length of the vector v in ℝ2 or ℝ3