Measure, Integration & Real Analysis, 2021a

240 Chapter 8 Hilbert Spaces
• Suppose {a k } k∈Γ is a family in R. Then the unordered sum ∑ k∈Γ a k converges
if and only if ∑ k∈Γ |a k | < ∞. Thus convergence of an unordered summation in
R is the same as absolute convergence. As we are about to see, the situation in
more general Hilbert spaces is quite different.
Now we can extend 8.52 to infinite sums.
8.54 linear combinations of an orthonormal family
Suppose {e k } k∈Γ is an orthonormal family in a Hilbert space V. Suppose {α k } k∈Γ
is a family in F. Then
(a)
the unordered sum ∑ α k e k converges
k∈Γ
⇐⇒ ∑ |α k | 2 < ∞.
k∈Γ
Furthermore, if ∑ k∈Γ α k e k converges, then
(b)
∥
∥∑
k∈Γ
∥ ∥∥
2
α k e k = ∑ |α k | 2 .
k∈Γ
Proof First suppose ∑ k∈Γ α k e k converges, with ∑ k∈Γ α k e k = g. Suppose ε > 0.
Then there exists a finite set Ω ⊂ Γ such that
∥
∥
∥∥
∥g − ∑ α j e j < ε
j∈Ω ′
for all finite sets Ω ′ with Ω ⊂ Ω ′ ⊂ Γ. IfΩ ′ is a finite set with Ω ⊂ Ω ′ ⊂ Γ, then
the inequality above implies that
∥ ∥∥
‖g‖−ε < ∥ ∑ α j e j < ‖g‖ + ε,
j∈Ω ′
which (using 8.52) implies that
‖g‖−ε <
(
∑
j∈Ω ′ |α j | 2) 1/2
< ‖g‖ + ε.
Thus ‖g‖ = ( ∑ k∈Γ |α k | 2) 1/2 , completing the proof of one direction of (a) and the
proof of (b).
To prove the other direction of (a), now suppose ∑ k∈Γ |α k | 2 < ∞. Thus there
exists an increasing sequence Ω 1 ⊂ Ω 2 ⊂···of finite subsets of Γ such that for
each m ∈ Z + ,
8.55 ∑
j∈Ω ′ \Ω m
|α j | 2 < 1 m 2
for every finite set Ω ′ such that Ω m ⊂ Ω ′ ⊂ Γ. For each m ∈ Z + , let
g m = ∑
j∈Ω m
α j e j .
Measure, Integration & Real Analysis, by Sheldon Axler