11.4.4 The p Norms
Examples of norms are the p norms on ℂn. These do not come from an inner product but they are norms
just the same.
Definition 11.4.8 Let x ∈ ℂn. Then define for p ≥ 1,
The following inequality is called Holder’s inequality.
Proposition 11.4.9 For x,y ∈ ℂn,
The proof will depend on the following lemma.
Lemma 11.4.10 If a,b ≥ 0 and p′ is defined by
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 =
. Then using Lemma
Theorem 11.4.11 The p norms do indeed satisfy the axioms of a norm.
Proof: It is obvious that
does indeed satisfy most of the norm axioms. The only one that is not
clear is the triangle inequality. To save notation write
in place of
in what follows. Note also that
Then using the Holder inequality,
so dividing by
It only remains to prove Lemma 11.4.10.
Proof of the lemma: Let p′ = q to save on notation and consider the following picture:
Note equality occurs when ap = bq. ■
Alternate proof of the lemma: First note that if either a or b are zero, then there is nothing to
show so we can assume b,a > 0. Let b > 0 and let
Then the second derivative of f is positive on
so its graph is convex. Also
. Then a short computation shows that there is only one critical point, where f
minimized and this happens when a
is such that ap
. At this point,
0 for all a
and this proves the lemma. ■
Another example of a very useful norm on Fn is the norm
You should verify that this satisfies all the axioms of a norm. Here is the triangle inequality.
It turns out that in terms of analysis (limits of sequences, completeness and so forth), it makes absolutely
which norm you use. There are however, significant geometric
differences. This will be
explained later. First is the notion of an orthonormal basis.