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:

  • The 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

and that


but then,


however, , so

or rather, that .