Page 451 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Let and be two functions in and define

(6)

This is well-defined since the functions in are continuous. We shall show that this formula defines an inner product on

by verifying the four inner product axioms for functions , , and in :

1.

which proves that Axiom 1 holds.
2.

which proves that Axiom 2 holds.
3.

which proves that Axiom 3 holds.

4. If is any function in , then for all x in ; therefore,

Further, because and is continuous on , it follows that if and
if and only if . This
only if for all x in . Therefore, we have

proves that Axiom 4 holds.

Calculus Required

EXAMPLE 10 Norm of a Vector in relative to this inner

If has the inner product defined in the preceding example, then the norm of a function (7)
product is , which on squaring yields

and the unit sphere in this space consists of all functions f in that satisfy the equation