Proof.
Let
. We prove the proprieties in order.
- Since
for
. Then
The positive branch of the square root function is an increasing function so we have
- Note that
is equivalent to for
. Of course, this quickly implies
There is a subtle point here. We also need to show
implies that for
. There are many reasonable candidates for norms that fail this property. It is often easier to prove the
contrapositive. Recall the contrapositive of ( implies ) is ( implies ). These expressions are logically equivalent. Prove one and you have proved the other. So we assume
. This means there is an index such that
. Since
has only one root at , then we know
. Furthermore
for the remaining components of
. Hence,
Using the increasing property of the square root function, we get
. (This may seems like too many words to show something that is obvious, but this obvious property fails enough mathematicians had to create the concept of semi-norm).
- Using the fact that
, quickly get this property.
- The triangle inequality is also the problematic part of showing a function is a norm. Let
. Then