Page 229 - A Course in Linear Algebra with Applications

7.2: Inner Product Spaces 213

Example 7.2.5
Find the norm of the function sin mx in the inner product
space C[0, IT] of Example 7.2.4.
Once again we have to compute an integral:

2
|| sin rax|| 2 = / sin mx dx = / -(1 — cos 2mx)dx = ir/2.
Jo Jo 2
Hence || sin mx\\ = ^/(TT/2). It follows that the functions

2~
sin mx, m = 1, ,...,
2
n
form a set of mutually orthogonal unit vectors. Such sets are
called orthonormal and will be studied in 7.3.
There is an important inequality relating inner product
and norm which has already been encountered for Euclidean
spaces.
Theorem 7.2.1 (The Cauchy - Schwartz Inequality)
Let u and v be vectors in an inner product space. Then

| < u, v > | < ||u|| ||v||.

Proof
We can assume that v ^ 0 or else the result is obvious. Let
t denote an arbitrary real number. Then, using the defining
properties of the inner product, we find that < u—tv, u—tv >
equals

2
< u, u > - < u, v > t- < v, u > t+ < v, v > t ,
which reduces to

2 2
||u|| 2 - 2 < u,v > t+ \\v\\ t > 0.