All norms are equivalent in the sense that for any two norms $a$ and $b$, there exist constants $c_1$ and $c_2$ such that

This result is known as the equivalence of norms and means that we can often generalize results for any one norm to all norms.

For proofs involving specific examples, see L1-L2 inequality and Max norm inequality.

Proof of the general case is beyond the scope of this course, but here is one way of doing it.