Linear Algebra, 5th Edition

6.1 Inner Products and Norms

Many geometric notions such as angle, length, and perpendicularity in ${\text{R}}^2$ and ${\text{R}}^3$ may be extended to more general real and complex vector spaces. All of these ideas are related to the concept of inner product.

Definition.

Let V be a vector space over F. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F, denoted $\langle x,y\rangle$, such that for all x, y, and z in V and all c in F, the following hold:

1. (a) $\langle x+z,y\rangle =\langle x,y\rangle +\langle z,y\rangle .$

2. (b) $\langle cx,y\rangle =c\langle x,y\rangle .$

3. (c) $\overline{\langle x,y\rangle }=\langle y,x\rangle$, where the bar denotes complex conjugation.

4. (d) If $x\ne 0$ then $\langle x,x\rangle$ is a positive real number.

Note that (c) reduces to $\langle x,y\rangle =\langle y,x\rangle$ if $F=R.$ Conditions (a) and (b) simply ...