Measure, Integration & Real Analysis, 2021a

Section 8C Orthonormal Bases 251
EXERCISES 8C
1 Verify that the family {e k } k∈Z as defined in the third bullet point of Example
8.51 is an orthonormal family in L 2( (−π, π] ) . The following formulas should
help:
(sin x)(cos y) =
(sin x)(sin y) =
(cos x)(cos y) =
sin(x + y)+sin(x − y)
,
2
cos(x − y) − cos(x + y)
,
2
cos(x + y)+cos(x − y)
.
2
2 Suppose {a k } k∈Γ is a family in R and a k ≥ 0 for each k ∈ Γ. Prove 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, prove that if ∑ k∈Γ a k converges then it equals the supremum above.
3 Suppose {e k } k∈Γ is an orthonormal family in an inner product space V. Prove
that if f ∈ V, then {k ∈ Γ : 〈 f , e k 〉 ̸= 0} is a countable set.
4 Suppose { f k } k∈Γ and {g k } k∈Γ are families in a normed vector space such that
∑ k∈Γ f k and ∑ k∈Γ g k converge. Prove that ∑ k∈Γ ( f k + g k ) converges and
∑ ( f k + g k )=∑ f k + ∑ g k .
k∈Γ
k∈Γ k∈Γ
5 Suppose { f k } k∈Γ is a family in a normed vector space such that ∑ k∈Γ f k converges.
Prove that if c ∈ F, then ∑ k∈Γ (cf k ) converges and
∑ (cf k )=c ∑ f k .
k∈Γ
k∈Γ
6 Suppose {a k } k∈Γ is a family in R. Prove that the unordered sum ∑ k∈Γ a k
converges if and only if ∑ k∈Γ |a k | < ∞.
7 Suppose { f k } k∈Z + is a family in a normed vector space V and f ∈ V. Prove
that the unordered sum ∑ k∈Z + f k equals f if and only if the usual ordered sum
∑ ∞ k=1 f p(k) equals f for every injective and surjective function p : Z+ → Z + .
8 Explain why 8.58 implies that if Γ is a finite set and {e k } k∈Γ is an orthonormal
family in a Hilbert space V, then span{e k } k∈Γ is a closed subspace of V.
9 Suppose V is an infinite-dimensional Hilbert space. Prove that there does not
exist a basis of V that is an orthonormal family.
Measure, Integration & Real Analysis, by Sheldon Axler