Measure, Integration & Real Analysis, 2021a

252 Chapter 8 Hilbert Spaces
10 (a) Show that the orthonormal family given in the first bullet point of Example
8.51 is an orthonormal basis of l 2 .
(b) Show that the orthonormal family given in the second bullet point of Example
8.51 is an orthonormal basis of l 2 (Γ).
(c) Show that the orthonormal family given in the fourth bullet point of Example
8.51 is not an orthonormal basis of L 2( [0, 1) ) .
(d) Show that the orthonormal family given in the fifth bullet point of Example
8.51 is not an orthonormal basis of L 2 (R).
11 Suppose μ is a σ-finite measure on (X, S) and ν is a σ-finite measure on (Y, T ).
Suppose also that {e j } j∈Ω is an orthonormal basis of L 2 (μ) and { f k } k∈Γ is an
orthonormal basis of L 2 (ν) for some countable set Γ. Forj ∈ Ω and k ∈ Γ,
define g j,k : X × Y → F by
g j,k (x, y) =e j (x) f k (y).
Prove that {g j,k } j∈Ω, k∈Γ is an orthonormal basis of L 2 (μ × ν).
12 Prove the converse of Parseval’s identity. More specifically, prove that if {e k } k∈Γ
is an orthonormal family in a Hilbert space V and
‖ f ‖ 2 = ∑ |〈 f , e k 〉| 2
k∈Γ
for every f ∈ V, then {e k } k∈Γ is an orthonormal basis of V.
13 (a) Show that the Hilbert space L 2 ([0, 1]) is separable.
(b) Show that the Hilbert space L 2 (R) is separable.
(c) Show that the Banach space l ∞ is not separable.
14 Prove that every subspace of a separable normed vector space is separable.
15 Suppose V is an infinite-dimensional Hilbert space. Prove that there does not
exist a translation invariant measure on the Borel subsets of V that assigns
positive but finite measure to each open ball in V.
[A subset of V is called a Borel set if it is in the smallest σ-algebra containing
all the open subsets of V. A measure μ on the Borel subsets of V is called
translation invariant if μ( f + E) =μ(E) for every f ∈ V and every Borel set
E of V.]
∫ 1 ∣
16 Find the polynomial g of degree at most 4 that minimizes ∣x 5 − g(x) ∣ 2 dx.
17 Prove that each orthonormal family in a Hilbert space can be extended to
an orthonormal basis of the Hilbert space. Specifically, suppose {e j } j∈Ω is
an orthonormal family in a Hilbert space V. Prove that there exists a set Γ
containing Ω and an orthonormal basis { f k } k∈Γ of V such that f j = e j for every
j ∈ Ω.
18 Prove that every vector space has a basis.
Measure, Integration & Real Analysis, by Sheldon Axler
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