186 CHAPTER 8. NORMED LINEAR SPACES
which implies||z||− ||w|| ≤ ||z−w||
and now switching z and w, yields
||w||− ||z|| ≤ ||z−w||
which implies 8.3.19.Any normed vector space is a metric space, the distance given by
d (x,y)≡ ∥x−y∥
This satisfies all the axioms of a distance. Therefore, any normed linear space is a metricspace with this metric and all the theory of metric spaces applies.
Definition 8.3.7 When X is a normed linear space which is also complete, it is called aBanach space.
8.3.4 The p NormsExamples of norms are the p norms on Cn for p ̸= 2. These do not come from an innerproduct but they are norms just the same.
Definition 8.3.8 Let x ∈ Cn. Then define for p≥ 1,
||x||p ≡
(n
∑i=1|xi|p
)1/p
The following inequality is called Holder’s inequality.
Proposition 8.3.9 For x,y ∈ Cn,
n
∑i=1|xi| |yi| ≤
(n
∑i=1|xi|p
)1/p( n
∑i=1|yi|p
′)1/p′
The proof will depend on the following lemma shown later.
Lemma 8.3.10 If a,b≥ 0 and p′ is defined by 1p +
1p′ = 1, then
ab≤ ap
p+
bp′
p′.
Proof of the Proposition: If x or y equals the zero vector there is nothing to prove.
Therefore, assume they are both nonzero. Let A= (∑ni=1 |xi|p)1/p and B=
(∑
ni=1 |yi|p
′)1/p′
.