Linear Algebra with Applications, 10th Edition

5.4 Inner Product Spaces

Scalar products are useful not only in ${ℝ}^n$, but also in a wide variety of contexts. To generalize this concept to other vector spaces, we introduce the following definition.

Definition and Examples

Definition

An inner product on a vector space V is an operation on V that assigns, to each pair of vectors x and y in V, a real number $\langle \mathbf{x},\mathbf{y}\rangle$ satisfying the following conditions:

1. $\langle \mathbf{x},\mathbf{x}\rangle \ge 0$ with equality if and only if $\mathbf{x}=0$

2. $\langle \mathbf{x},\mathbf{y}\rangle =\langle \mathbf{y},\mathbf{x}\rangle$ for all x and y in V

3. $\langle \alpha \mathbf{x}+\beta \mathbf{y},\mathbf{z}\rangle =\alpha \langle \mathbf{x},\mathbf{z}\rangle +\beta \langle \mathbf{y},\mathbf{z}\rangle$ for all x, y, z in V and all scalars $\alpha$ and $\beta$

A vector space V with an inner product is called an inner product space.

The Vector Space ${ℝ}^n$

The standard inner product for ${ℝ}^n$ is the scalar product

$$\langle \mathbf{x},\mathbf{y}\rangle ={\mathbf{x}}^T\mathbf{y}$$

Given a vector w with positive entries, ...