The norm is formally defined as
The norm has several special cases that supposedly arise often in linear algebra, numerical analysis, and machine learning.
The norm, commonly called the “taxicab” norm:
The norm, commonly called the “Euclidean” norm:
It can be shown that this definition of the norm is equivalent to taking the limit as of the norm:
Proof. Let and . We wish to show that .
We have that
We arrive at the last item because for every (because for each ). Thus we have
so, taking a limit as , we have
In other words, we have sandwiched between and , implying equality. Therefore .
Consider also this proof:
Proof. Let . Observe that for all we have that
however, , so
or rather, that .