Measure, Integration & Real Analysis, 2021a

234 Chapter 8 Hilbert Spaces
Suppose that μ is a measure and
1 < p ≤ ∞. In7.25 we considered the
natural map of L p′ (μ) into ( L p (μ) ) ′ , and
Frigyes Riesz (1880–1956) proved
8.47 in 1907.
we showed that this maps preserves norms. In the special case where p = p ′ = 2,
the Riesz Representation Theorem (8.47) shows that this map is surjective. In other
words, if ϕ is a bounded linear functional on L 2 (μ), then there exists h ∈ L 2 (μ) such
that
∫
ϕ( f )=
fhdμ
for all f ∈ L 2 (μ) (take h to be the complex conjugate of the function given by 8.47).
Hence we can identify the dual of L 2 (μ) with L 2 (μ). In9.42 we will deal with other
values of p. Also see Exercise 25 in this section.
EXERCISES 8B
1 Show that each of the inner product spaces in Example 8.23 is not a Hilbert
space.
2 Prove or disprove: The inner product space in Exercise 1 in Section 8A is a
Hilbert space.
3 Suppose V 1 , V 2 ,...are Hilbert spaces. Let
V =
Show that the equation
{
( f 1 , f 2 ,...) ∈ V 1 × V 2 ×···:
〈( f 1 , f 2 ,...), (g 1 , g 2 ,...)〉 =
∞
∑
k=1
∞
∑
k=1
}
‖ f k ‖ 2 < ∞ .
〈 f k , g k 〉
defines an inner product on V that makes V a Hilbert space.
[Each of the Hilbert spaces V 1 , V 2 ,...may have a different inner product, even
though the same notation is used for the norm and inner product on all these
Hilbert spaces.]
4 Suppose V is a real Hilbert space. The complexification of V is the complex
vector space V C defined by V C = V × V, but we write a typical element of V C
as f + ig instead of ( f , g). Addition and scalar multiplication are defined on
V C by
( f 1 + ig 1 )+(f 2 + ig 2 )=(f 1 + f 2 )+i(g 1 + g 2 )
and
(α + βi)( f + ig) =(α f − βg)+(αg + β f )i
for f 1 , f 2 , f , g 1 , g 2 , g ∈ V and α, β ∈ R. Show that
〈 f 1 + ig 1 , f 2 + ig 2 〉 = 〈 f 1 , f 2 〉 + 〈g 1 , g 2 〉 +(〈g 1 , f 2 〉−〈f 1 , g 2 〉)i
defines an inner product on V C that makes V C into a complex Hilbert space.
Measure, Integration & Real Analysis, by Sheldon Axler