Page 815 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

We leave it as an exercise to show that if the functions and are vectors in complex

then the following formula defines an inner product on complex :

In complex inner product spaces, as in real inner product spaces, the norm (or length) of a vector u is defined by
and the distance between two vectors u and v is defined by
It can be shown that with these definitions, Theorems 6.2.2 and 6.2.3 remain true in complex inner product spaces (Exercise
35).

EXAMPLE 4 Norm and Distance in
If and are vectors in with the Euclidean inner product, then
and

Observe that these are just the formulas for the Euclidean norm and distance discussed in Section 10.4.

EXAMPLE 5 Norm of a Function in Complex

Calculus Required

If complex has the inner product of Example 3, and if , where m is any integer, then with the help of Formula

15 of Section 10.3, we obtain

Orthogonal Sets