Norms and Inner Products

Abstract

In this chapter, we generalize to more abstract settings two key properties of ${\mathbb{R}^n}$: the Euclidean norm of a vector and the dot product of two vectors. The generalizations, norms and inner products, respectively, are set in a general vector space. Many of our applications will be set in this framework.

The basic notions of topology go through with almost no change in the definitions. However, some theorems can be quite different. For example, being closed and bounded is not sufficient to imply compactness in an infinite-dimensional normed vector space, such as C[a,b].