Measure, Integration & Real Analysis, 2021a

236 Chapter 8 Hilbert Spaces
16 Suppose V is a Hilbert space and P : V → V is a linear map such that P 2 = P
and ‖Pf‖≤‖f ‖ for every f ∈ V. Prove that there exists a closed subspace U
of V such that P = P U .
17 Suppose U is a subspace of a Hilbert space V. Suppose also that W is a Banach
space and S : U → W is a bounded linear map. Prove that there exists a bounded
linear map T : V → W such that T| U = S and ‖T‖ = ‖S‖.
[If W = F, then this result is just the Hahn–Banach Theorem (6.69) for Hilbert
spaces. The result here is stronger because it allows W to be an arbitrary
Banach space instead of requiring W to be F. Also, the proof in this Hilbert
space context does not require use of Zorn’s Lemma or the Axiom of Choice.]
18 Suppose U and W are subspaces of a Hilbert space V. Prove that U = W if and
only if U ⊥ = W ⊥ .
19 Suppose U and W are closed subspaces of a Hilbert space. Prove that P U P W = 0
if and only if 〈 f , g〉 = 0 for all f ∈ U and all g ∈ W.
20 Verify the assertions in Example 8.46.
21 Show that every inner product space is a subspace of some Hilbert space.
Hint: See Exercise 13 in Section 6C.
22 Prove that if V is a Hilbert space and T : V → V is a bounded linear map such
that the dimension of range T is 1, then there exist g, h ∈ V such that
for all f ∈ V.
Tf = 〈 f , g〉h
23 (a) Give an example of a Banach space V and a bounded linear functional ϕ
on V such that |ϕ( f )| < ‖ϕ‖‖f ‖ for all f ∈ V \{0}.
(b) Show there does not exist an example in part (a) where V is a Hilbert space.
24 (a) Suppose ϕ and ψ are bounded linear functionals on a Hilbert space V such
that ‖ϕ + ψ‖ = ‖ϕ‖ + ‖ψ‖. Prove that one of ϕ, ψ is a scalar multiple of
the other.
(b) Give an example to show that part (a) can fail if the hypothesis that V is a
Hilbert space is replaced by the hypothesis that V is a Banach space.
25 (a) Suppose that μ is a finite measure, 1 ≤ p ≤ 2, and ϕ is a bounded
linear functional on L p (μ). Prove that there exists h ∈ L p′ (μ) such that
ϕ( f )= ∫ fhdμ for every f ∈ L p (μ).
(b) Same as part (a), but with the hypothesis that μ is a finite measure replaced
by the hypothesis that μ is a measure.
[See 7.25, which along with this exercise shows that we can identify the dual of
L p (μ) with L p′ (μ) for 1 < p ≤ 2. See 9.42 for an extension to all p ∈ (1, ∞).]
26 Prove that if V is an infinite-dimensional Hilbert space, then the Banach space
B(V, V) is nonseparable.
Measure, Integration & Real Analysis, by Sheldon Axler