Measure, Integration & Real Analysis, 2021a

Section 8A Inner Product Spaces 223
18 Suppose μ is a measure. For f , g ∈ L 2 (μ), define 〈 f , g〉 by
(a) Using the inequality
∫
〈 f , g〉 =
f g dμ.
| f (x)g(x)| ≤ 1 2
( | f (x)| 2 + |g(x)| 2) ,
verify that the integral above makes sense and the map sending f , g to 〈 f , g〉
defines an inner product on L 2 (μ) (without using Hölder’s inequality).
(b) Show that the Cauchy–Schwarz inequality implies that
‖ fg‖ 1 ≤‖f ‖ 2 ‖g‖ 2
for all f , g ∈ L 2 (μ) (again, without using Hölder’s inequality).
19 Suppose V 1 ,...,V m are inner product spaces. Show that the equation
〈( f 1 ,..., f m ), (g 1 ,...,g m )〉 = 〈 f 1 , g 1 〉 + ···+ 〈 f m , g m 〉
defines an inner product on V 1 ×···×V m .
[Each of the inner product spaces V 1 ,...,V m may have a different inner product,
even though the same inner product notation is used on all these spaces.]
20 Suppose V is an inner product space. Make V × V an inner product space
as in the exercise above. Prove that the function that takes an ordered pair
( f , g) ∈ V × V to the inner product 〈 f , g〉 ∈F is a continuous function from
V × V to F.
21 Suppose 1 ≤ p ≤ ∞.
(a) Show the norm on l p comes from an inner product if and only if p = 2.
(b) Show the norm on L p (R) comes from an inner product if and only if p = 2.
22 Use inner products to prove Apollonius’s identity:
In a triangle with sides of length a, b, and c, let d
be the length of the line segment from the midpoint
of the side of length c to the opposite vertex. Then
a 2 + b 2 = 1 2 c2 + 2d 2 .
Measure, Integration & Real Analysis, by Sheldon Axler