Measure, Integration & Real Analysis, 2021a

Section 8C Orthonormal Bases 249
Recall that a subset Γ of a set V can be thought of as a family in V by considering
{e f } f ∈Γ , where e f = f . With this convention, a subset Γ of an inner product space
V is an orthonormal subset of V if ‖ f ‖ = 1 for all f ∈ Γ and 〈 f , g〉 = 0 for all
f , g ∈ Γ with f ̸= g.
The next result characterizes the orthonormal bases as the maximal elements
among the collection of orthonormal subsets of a Hilbert space. Recall that a set
Γ ∈Ain a collection of subsets of a set V is a maximal element of A if there does
not exist Γ ′ ∈Asuch that Γ Γ ′ (see 6.55).
8.74 orthonormal bases as maximal elements
Suppose V is a Hilbert space, A is the collection of all orthonormal subsets of V,
and Γ is an orthonormal subset of V. Then Γ is an orthonormal basis of V if and
only if Γ is a maximal element of A.
Proof First suppose Γ is an orthonormal basis of V. Parseval’s identity [8.63(a)]
implies that the only element of V that is orthogonal to every element of Γ is 0. Thus
there does not exist an orthonormal subset of V that strictly contains Γ. In other
words, Γ is a maximal element of A.
To prove the other direction, suppose now that Γ is a maximal element of A. Let
U denote the span of Γ. Then
U ⊥ = {0}
because if f is a nonzero element of U ⊥ , then Γ ∪{f /‖ f ‖} is an orthonormal subset
of V that strictly contains Γ. Hence U = V (by 8.42), which implies that Γ is an
orthonormal basis of V.
Now we are ready to prove that every Hilbert space has an orthonormal basis.
Before reading the next proof, you may want to review the definition of a chain (6.58),
which is a collection of sets such that for each pair of sets in the collection, one of
them is contained in the other. You should also review Zorn’s Lemma (6.60), which
gives a way to show that a collection of sets contains a maximal element.
8.75 existence of orthonormal bases for all Hilbert spaces
Every Hilbert space has an orthonormal basis.
Proof Suppose V is a Hilbert space. Let A be the collection of all orthonormal
subsets of V. Suppose C⊂Ais a chain. Let L be the union of all the sets in C. If
f ∈ L, then ‖ f ‖ = 1 because f is an element of some orthonormal subset of V that
is contained in C.
If f , g ∈ L with f ̸= g, then there exist orthonormal subsets Ω and Γ in C such
that f ∈ Ω and g ∈ Γ. Because C is a chain, either Ω ⊂ Γ or Γ ⊂ Ω. Either way,
there is an orthonormal subset of V that contains both f and g. Thus 〈 f , g〉 = 0.
We have shown that L is an orthonormal subset of V; in other words, L ∈A.
Thus Zorn’s Lemma (6.60) implies that A has a maximal element. Now 8.74 implies
that V has an orthonormal basis.
Measure, Integration & Real Analysis, by Sheldon Axler