3.3.4 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 3.3.7 Let x ∈ ℂn. Then define for p ≥ 1,
The following inequality is called Holder’s inequality.
Proposition 3.3.8 For x,y ∈ ℂn,
The proof will depend on the following lemma.
Lemma 3.3.9 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 3.3.9
Theorem 3.3.10 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 3.3.9.
Proof of the lemma: Let p′ = q to save on notation and consider the following
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
is minimized and this happens when a
is such that ap
. At this
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, it makes absolutely no difference
which norm you
use. This will be explained later. First is a short review of the notion of orthonormal
bases which is not needed directly in what follows but is sufficiently important to