The n-Category Café

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.