Measure, Integration & Real Analysis, 2021a

Section 8C Orthonormal Bases 241
If n > m, then 8.52 implies that
‖g n − g m ‖ 2 =
∑
j∈Ω n \Ω m
|α j | 2 < 1 m 2 .
Thus g 1 , g 2 ,...is a Cauchy sequence and hence converges to some element g of V.
Temporarily fixing m ∈ Z + and taking the limit of the equation above as n → ∞,
we see that
‖g − g m ‖≤ 1 m .
To show that ∑ k∈Γ α k e k = g, suppose ε > 0. Let m ∈ Z + be such that
m 2 < ε.
Suppose Ω ′ is a finite set with Ω m ⊂ Ω ′ ⊂ Γ. Then
∥ ∥
∥
∥∥ ∥
∥∥
∥g − ∑ α j e j ≤‖g − gm ‖ + ∥g m − ∑ α j e j
j∈Ω ′ j∈Ω ′
≤ 1 m + ∥ ∥∥
∥ ∥∥
∑ α j e j
j∈Ω ′ \Ω m
= 1 m + (
∑
j∈Ω ′ \Ω m
|α j | 2) 1/2
< ε,
where the third line comes from 8.52 and the last line comes from 8.55. Thus
∑ k∈Γ α k e k = g, completing the proof.
8.56 Example a convergent unordered sum need not converge absolutely
Suppose {e k } k∈Z + is the orthogonal family in l 2 defined by setting e k equal to
the sequence that is 0 everywhere except for a 1 in the k th slot. Then by 8.54, the
unordered sum
∑
k∈Z + 1
k
e k
converges in l 2 (because ∑ k∈Z + 1 k 2 < ∞) even though ∑ k∈Z +‖ 1 k e k‖ = ∞. Note
that ∑ k∈Z + 1 k e k =(1, 1 2 , 1 3 ,...) ∈ l2 .
Now we prove an important inequality.
8.57 Bessel’s inequality
Suppose {e k } k∈Γ is an orthonormal family in an inner product space V and f ∈ V.
Then
∑ |〈 f , e k 〉| 2 ≤‖f ‖ 2 .
k∈Γ
Measure, Integration & Real Analysis, by Sheldon Axler