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′

.