We use norms for error analysis. The natural question arises: which norm should we use?
Example: Given 
in meters and
in meters. Then 
is a good approximation to 
because the rel. error
And 
is considered a bad approximation as
But suppose the first component of each vector is measured in kms. In this case,
. The relative error measured
as:
If we use a different norm, say the 1-norm, the results would be different.
Definiton: Let B be real linear space,
:
is an inner product if
if 
.
Example: Over 
, 
Lemma: Cauchy-Schwartz: 
.
Lemma: 
is a norm.
Definition: 
are orthogonal.
Lemma: Let 
and 
be two norms on 
, then
constants 
such that
.
For some particular norms, the following results are useful:
