Measure, Integration & Real Analysis, 2021a

244 Chapter 8 Hilbert Spaces
8.61 Definition orthonormal basis
An orthonormal family {e k } k∈Γ in a Hilbert space V is called an orthonormal
basis of V if
span {e k } k∈Γ = V.
In addition to requiring orthonormality (which implies linear independence), the
definition above differs from the definition of a basis by considering the closure of
the span rather than the span. An important point to keep in mind is that despite the
terminology, an orthonormal basis is not necessarily a basis in the sense of 6.54. In
fact, if Γ is an infinite set and {e k } k∈Γ is an orthonormal basis of V, then {e k } k∈Γ is
not a basis of V (see Exercise 9).
8.62 Example orthonormal bases
• For n ∈ Z + and k ∈{1, . . . , n}, let e k be the element of F n all of whose
coordinates are 0 except the k th coordinate, which is 1:
e k =(0,...,0,1,0,...,0).
Then {e k } k∈{1,...,n} is an orthonormal basis of F n .
• Let e 1 = ( 1 √3 ,
1 √3 ,
1 √3
)
, e2 = ( − 1 √
2
,
1 √2 ,0 ) , and e 3 = ( 1 √6 ,
1 √6 , − 2 √
6
)
. Then
{e k } k∈{1,2,3} is an orthonormal basis of F 3 , as you should verify.
• The first three bullet points in 8.51 are examples of orthonormal families that are
orthonormal bases. The exercises ask you to verify that we have an orthonormal
basis in the first and second bullet points of 8.51. For the third bullet point
(trigonometric functions), see Exercise 11 in Section 10D or see Chapter 11.
The next result shows why orthonormal bases are so useful—a Hilbert space with
an orthonormal basis {e k } k∈Γ behaves like l 2 (Γ).
8.63 Parseval’s identity
Suppose {e k } k∈Γ is an orthonormal basis of a Hilbert space Vand f , g ∈ V. Then
(a)
f = ∑ 〈 f , e k 〉e k ;
k∈Γ
(b) 〈 f , g〉 = ∑ 〈 f , e k 〉〈g, e k 〉;
k∈Γ
(c) ‖ f ‖ 2 = ∑ |〈 f , e k 〉| 2 .
k∈Γ
Proof The equation in (a) follows immediately from 8.58(b) and the definition of an
orthonormal basis.
Measure, Integration & Real Analysis, by Sheldon Axler