Measure, Integration & Real Analysis, 2021a

222 Chapter 8 Hilbert Spaces
9 The angle between two vectors (thought of as arrows with initial point at the
origin) in R 2 or R 3 can be defined geometrically. However, geometry is not as
clear in R n for n > 3. Thus the angle between two nonzero vectors a, b ∈ R n
is defined to be
〈a, b〉
arccos
‖a‖‖b‖ ,
where the motivation for this definition comes from the previous exercise. Explain
why the Cauchy–Schwarz inequality is needed to show that this definition
makes sense.
10 (a) Suppose f and g are elements of a real inner product space. Prove that f
and g have the same norm if and only if f + g is orthogonal to f − g.
(b) Use part (a) to show that the diagonals of a parallelogram are perpendicular
to each other if and only if the parallelogram is a rhombus.
11 Suppose f and g are elements of an inner product space. Prove that ‖ f ‖ = ‖g‖
if and only if ‖sf + tg‖ = ‖tf + sg‖ for all s, t ∈ R.
12 Suppose f and g are elements of an inner product space and ‖ f ‖ = ‖g‖ = 1
and 〈 f , g〉 = 1. Prove that f = g.
13 Suppose f and g are elements of a real inner product space. Prove that
〈 f , g〉 = ‖ f + g‖2 −‖f − g‖ 2
.
4
14 Suppose f and g are elements of a complex inner product space. Prove that
〈 f , g〉 = ‖ f + g‖2 −‖f − g‖ 2 + ‖ f + ig‖ 2 i −‖f − ig‖ 2 i
.
4
15 Suppose f , g, h are elements of an inner product space. Prove that
‖h − 1 2 ( f + g)‖2 = ‖h − f ‖2 + ‖h − g‖ 2
2
− ‖ f − g‖2
.
4
16 Prove that a norm satisfying the parallelogram equality comes from an inner
product. In other words, show that if V is a normed vector space whose norm
‖·‖ satisfies the parallelogram equality, then there is an inner product 〈·, ·〉 on
V such that ‖ f ‖ = 〈 f , f 〉 1/2 for all f ∈ V.
17 Let λ denote Lebesgue measure on [1, ∞).
(a) Prove that if f : [1, ∞) → [0, ∞) is Borel measurable, then
(∫ ∞
1
) 2 ∫ ∞
f (x) dλ(x) ≤ x 2( f (x) ) 2 dλ(x).
1
(b) Describe the set of Borel measurable functions f : [1, ∞) → [0, ∞) such
that the inequality in part (a) is an equality.
Measure, Integration & Real Analysis, by Sheldon Axler