Measure, Integration & Real Analysis, 2021a

Section 8A Inner Product Spaces 219
8.14 Example Cauchy–Schwarz inequality for L 2 (μ)
Suppose μ is a measure and f , g ∈ L 2 (μ). Applying the Cauchy–Schwarz
inequality with the standard inner product on L 2 (μ) to | f | and |g| gives the inequality
∫
(∫
| fg| dμ ≤
) 1/2 (∫
| f | 2 dμ
|g| 2 dμ) 1/2.
The inequality above is equivalent to
In 1859 Viktor Bunyakovsky
Hölder’s inequality (7.9) for the special
(1804–1889), who had been
case where p = p ′ = 2. However,
Cauchy’s student in Paris, first
the proof of the inequality above via the
proved integral inequalities like the
Cauchy–Schwarz inequality still depends
one above. Similar discoveries by
upon Hölder’s inequality to show that the
Hermann Schwarz (1843–1921) in
definition of the standard inner product
1885 attracted more attention and
on L 2 (μ) makes sense. See Exercise 18
led to the name of this inequality.
in this section for a derivation of the inequality
above that is truly independent of Hölder’s inequality.
If we think of the norm determined by an
inner product as a length, then the triangle inequality
has the geometric interpretation that the
length of each side of a triangle is less than the
sum of the lengths of the other two sides.
8.15 triangle inequality
Suppose f and g are elements of an inner product space. Then
‖ f + g‖ ≤‖f ‖ + ‖g‖,
with equality if and only if one of f , g is a nonnegative multiple of the other.
Proof
8.16
8.17
We have
‖ f + g‖ 2 = 〈 f + g, f + g〉
= 〈 f , f 〉 + 〈g, g〉 + 〈 f , g〉 + 〈g, f 〉
= 〈 f , f 〉 + 〈g, g〉 + 〈 f , g〉 + 〈 f , g〉
= ‖ f ‖ 2 + ‖g‖ 2 + 2Re〈 f , g〉
≤‖f ‖ 2 + ‖g‖ 2 + 2|〈 f , g〉|
≤‖f ‖ 2 + ‖g‖ 2 + 2‖ f ‖‖g‖
=(‖ f ‖ + ‖g‖) 2 ,
where 8.17 follows from the Cauchy–Schwarz inequality (8.11). Taking square roots
of both sides of the inequality above gives the desired inequality.
Measure, Integration & Real Analysis, by Sheldon Axler