Measure, Integration & Real Analysis, 2021a

Section 8C Orthonormal Bases 239
Proof Suppose {α j } j∈Ω is a family in F. Standard properties of inner products
show that
as desired.
∥ ∥∥
2 〈 〉
∥ ∑ α j e j = ∑ α j e j , ∑ α k e k
j∈Ω
j∈Ω k∈Ω
= ∑ α j α k 〈e j , e k 〉
j,k∈Ω
= ∑ |α j | 2 ,
j∈Ω
Suppose Ω is a finite set and {e j } j∈Ω is an orthonormal family in an inner product
space. The result above implies that if ∑ j∈Ω α j e j = 0, then α j = 0 for every j ∈ Ω.
Linear algebra, and algebra more generally, deals with sums of only finitely many
terms. However, in analysis we often want to sum infinitely many terms. For example,
earlier we defined the infinite sum of a sequence g 1 , g 2 ,...in a normed vector space
to be the limit as n → ∞ of the partial sums ∑ n k=1 g k if that limit exists (see 6.40).
The next definition captures a more powerful method of dealing with infinite sums.
The sum defined below is called an unordered sum because the set Γ is not assumed
to come with any ordering. A finite unordered sum is defined in the obvious way.
8.53 Definition unordered sum; ∑ k∈Γ f k
Suppose { f k } k∈Γ is a family in a normed vector space V. The unordered sum
∑ k∈Γ f k is said to converge if there exists g ∈ V such that for every ε > 0, there
exists a finite subset Ω of Γ such that
∥
∥ ∥∥
∥g − ∑ f j < ε
j∈Ω ′
for all finite sets Ω ′ with Ω ⊂ Ω ′ ⊂ Γ. If this happens, we set ∑ k∈Γ f k = g. If
there is no such g ∈ V, then ∑ k∈Γ f k is left undefined.
Exercises at the end of this section ask you to develop basic properties of unordered
sums, including the following:
• Suppose {a k } k∈Γ is a family in R and a k ≥ 0 for each k ∈ Γ. Then the unordered
sum ∑ k∈Γ a k converges if and only if
}
sup{
∑ a j : Ω is a finite subset of Γ < ∞.
j∈Ω
Furthermore, if ∑ k∈Γ a k converges then it equals the supremum above. If
∑ k∈Γ a k does not converge, then the supremum above is ∞ and we write
∑ k∈Γ a k = ∞ (this notation should be used only when a k ≥ 0 for each k ∈ Γ).
Measure, Integration & Real Analysis, by Sheldon Axler