### Functional Equations VII: The *p*-Norms

#### Posted by Tom Leinster

The $p$-norms have a nice multiplicativity property:

$\|(A x, A y, A z, B x, B y, B z)\|_p = \|(A, B)\|_p \, \|(x, y, z)\|_p$

for all $A, B, x, y, z \in \mathbb{R}$ — and similarly, of course, for any numbers of arguments.

Guillaume Aubrun and Ion Nechita showed that this condition completely characterizes the $p$-norms. In other words, *any* system of norms that’s multiplicative in this sense must be equal to $\|\cdot\|_p$ for some $p \in [1, \infty]$. And the amazing thing is, to prove this, they used some nontrivial probability theory.

All this is explained in this week’s functional equations notes, which start on page 26 here.

Posted at March 22, 2017 2:12 AM UTC