Measure, Integration & Real Analysis, 2021a

Section 8A Inner Product Spaces 221
EXERCISES 8A
1 Let V denote the vector space of bounded continuous functions from R to F.
Let r 1 , r 2 ,...be a list of the rational numbers. For f , g ∈ V, define
〈 f , g〉 =
∞
∑
k=1
Show that 〈·, ·〉 is an inner product on V.
f (r k )g(r k )
2 k .
2 Prove that if μ is a measure and f , g ∈ L 2 (μ), then
‖ f ‖ 2 ‖g‖ 2 −|〈f , g〉| 2 = 1 2
∫ ∫
| f (x)g(y) − g(x) f (y)| 2 dμ(y) dμ(x).
3 Suppose f and g are elements of an inner product space and
‖ f + g‖ 2 = ‖ f ‖ 2 + ‖g‖ 2 .
(a) Prove that if F = R, then f and g are orthogonal.
(b) Give an example to show that if F = C, then f and g can satisfy the
equation above without being orthogonal.
4 Find a, b ∈ R 3 such that a is a scalar multiple of (1, 6, 3), b is orthogonal to
(1, 6, 3), and (5, 4, −2) =a + b.
5 Prove that
( 1
16 ≤ (a + b + c + d)
a + 1 b + 1 c + 1 )
d
for all positive numbers a, b, c, d, with equality if and only if a = b = c = d.
6 Prove that the square of the average of each finite list of real numbers containing
at least two distinct real numbers is less than the average of the squares of the
numbers in that list.
7 Suppose f and g are elements of an inner product space and ‖ f ‖≤1 and
‖g‖ ≤1. Prove that
√
√1 −‖f ‖ 2 1 −‖g‖ 2 ≤ 1 −|〈f , g〉|.
8 Suppose a and b are nonzero elements of R 2 . Prove that
〈a, b〉 = ‖a‖‖b‖ cos θ,
where θ is the angle between a and b (thinking of a as the vector whose initial
point is the origin and whose end point is a, and similarly for b).
Hint: Draw the triangle formed by a, b, and a − b; then use the law of cosines.
Measure, Integration & Real Analysis, by Sheldon Axler