Page 282 - Matrix Analysis & Applied Linear Algebra

278 Chapter 5 Norms, Inner Products, and Orthogonality

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5.1.12. The classical form of H¨older’s inequality states that if p> 1 and
q> 1 are real numbers such that 1/p +1/q =1, then
1/p 1/q

n n n
p q

|x i y i |≤ |x i | |y i | .
i=1 i=1 i=1
Derive this inequality by executing the following steps:
(a)By considering the function f(t)=(1 − λ)+ λt − t λ for 0 <λ< 1,
establish the inequality
λ 1−λ
α β ≤ λα +(1 − λ)β
for nonnegative real numbers α and β.
(b)Let ˆ x = x/ x and ˆ y = y/ y , and apply the inequality of part (a)
p q
to obtain
n n n
1 p 1 q
y
y
|ˆx i ˆ i |≤ |ˆx i | + |ˆ i | =1.
p q
i=1 i=1 i=1
(c)Deduce the classical form of H¨older’s inequality, and then explain why
this means that
|x y|≤ x y .
∗
p q
5.1.13. The triangle inequality x + y ≤ x + y for a general p-norm
p p p
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is really the classical Minkowski inequality, which states that for
p ≥ 1,
1/p 1/p 1/p

n n n
p p p

|x i + y i | ≤ |x i | + |y i | .
i=1 i=1 i=1
Derive Minkowski’s inequality. Hint: For p> 1, let q be the number
such that 1/q =1 − 1/p. Verify that for scalars α and β,
p p/q p/q p/q
|α + β| = |α + β||α + β| ≤|α||α + β| + |β||α + β| ,
and make use of H¨older’s inequality in Exercise 5.1.12.

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Ludwig Otto H¨older (1859–1937) was a German mathematician who studied at G¨ottingen and
lived in Leipzig. Although he made several contributions to analysis as well as algebra, he is
primarily known for the development of the inequality that now bears his name.
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Hermann Minkowski (1864–1909) was born in Russia, but spent most of his life in Germany
as a mathematician and professor at K¨onigsberg and G¨ottingen. In addition to the inequality
that now bears his name, he is known for providing a mathematical basis for the special theory
of relativity. He died suddenly from a ruptured appendix at the age of 44.