Measure, Integration & Real Analysis, 2021a

Section 8A Inner Product Spaces 215
8.5 Example norms on inner product spaces
In each of the following examples, the inner product is the standard inner product
as defined in Example 8.2.
• If n ∈ Z + and (a 1 ,...,a n ) ∈ F n , then
√
‖(a 1 ,...,a n )‖ = |a 1 | 2 + ···+ |a n | 2 .
Thus the norm on F n associated with the standard inner product is the usual
Euclidean norm.
• If (a 1 , a 2 ,...) ∈ l 2 , then
‖(a 1 , a 2 ,...)‖ =
( ∞
∑ |a k | 2) 1/2
.
k=1
Thus the norm associated with the inner product on l 2 is just the standard norm
‖·‖ 2 on l 2 as defined in Example 7.2.
• If μ is a measure and f ∈ L 2 (μ), then
(∫
‖ f ‖ =
| f | 2 dμ) 1/2.
Thus the norm associated with the inner product on L 2 (μ) is just the standard
norm ‖·‖ 2 on L 2 (μ) as defined in 7.17.
The definition of an inner product (8.1) implies that if V is an inner product space
and f ∈ V, then
• ‖f ‖≥0;
• ‖f ‖ = 0 if and only if f = 0.
The proof of the next result illustrates a frequently used property of the norm on
an inner product space: working with the square of the norm is often easier than
working directly with the norm.
8.6 homogeneity of the norm
Suppose V is an inner product space, f ∈ V, and α ∈ F. Then
‖α f ‖ = |α|‖f ‖.
Proof
We have
‖α f ‖ 2 = 〈α f , α f 〉 = α〈 f , α f 〉 = αα〈 f , f 〉 = |α| 2 ‖ f ‖ 2 .
Taking square roots now gives the desired equality.
Measure, Integration & Real Analysis, by Sheldon Axler