# Inner Product Spaces

## Abstract

As discussed in the previous chapter, vector spaces are analogous to three-dimensional Euclidean space \(\mathbf {V}_3\) of geometric vectors (directed line segments). However, such important concepts as the length of a vector and the angle between two vectors were not introduced for abstract spaces. In three-dimensional Euclidean space, using the lengths of two vectors and the angle between them, we can calculate the inner product (the dot product) of these vectors. Many geometric problems in the space \(\mathbf {V}_3\) are solved with help of the dot product. The concept of an inner product on an abstract space will be introduced axiomatically in this chapter. After that, the concepts of the length of a vector and the angle between two vectors will be introduced based on the concept of the inner product. Then we will investigate the concept of orthogonal bases. Some important examples of orthogonal bases in finite-dimensional spaces, particularly in polynomial spaces, will be constructed. The basic properties of subspaces of unitary spaces will be described. We begin our considerations with inner products on the spaces \({\mathbb {R}}^n\) and \({\mathbb {C}}^n\).