Page 291 - Matrix Analysis & Applied Linear Algebra

5.3 Inner-Product Spaces 287

• If V is the vector space of real-valued continuous functions defined on the
interval (a, b), then
b
f|g = f(t)g(t)dt
a
is an inner product on V.

n×1
Just as the standard inner product for C defines the euclidean norm on
√
n×1
∗
C by x 2 = x x, every general inner product in an inner-product space
V defines a norm on V by setting

= . (5.3.3)
It’s straightforward to verify that this satisfies the first two conditions in (5.2.3)
on p. 280 that define a general vector norm, but, just as in the case of euclidean
norms, verifying that (5.3.3) satisfies the triangle inequality requires a generalized
version of CBS inequality.

General CBS Inequality

If V is an inner-product space, and if we set = , then

| x y | ≤ x y for all x, y ∈V. (5.3.4)
2
Equality holds if and only if y = αx for α = x y / x .

2
Proof. Set α = x y / x (assume x = 0, for otherwise there is nothing to
prove), and observe that x αx − y =0, so
2
0 ≤ αx − y = αx − y αx − y
=¯α x αx − y − y αx − y (see Exercise 5.3.2)
2 2
y x − x y y x
= − y αx − y = y y − α y x = .
2
x
2
Since y x = x y , it follows that x y y x = | x y | , so
2 2 2
y x −| x y |
0 ≤ =⇒| x y | ≤ x y .
2
x
Establishing the conditions for equality is the same as in Exercise 5.1.9.

Let’s now complete the job of showing that = is indeed a vector
norm as defined in (5.2.3) on p. 280.