### Characterizing the *p*-Norms

#### Posted by Tom Leinster

Some mathematical objects acquire a reputation for being important. We know they’re important because our lecturers told us so when we were students, and because we’ve observed that they’re treated as important by large groups of research mathematicians. If you stood up in
public and asked exactly what was so important about them, you might fear getting
laughed at as an ignoramus… but perhaps no one would have a
really good answer. There’s only a *social* proof of importance.

I have a soft spot for theorems that take a mathematical object known
*socially* to be important and state a precise *mathematical* sense in which
it’s important. This might, for example, be a universal property
(‘it’s the *universal* thing with these good properties’) or a unique
characterization (‘it’s the *unique* thing with these good properties’).

Previously I’ve enthused about theorems that do this for the category $\Delta$, the topological space $[0, 1]$, and the Banach space $L^1$. Today I’ll enthuse about a theorem that does it for the $p$-norms $\Vert\cdot\Vert_p$. The theorem is from a recent paper of Guillaume Aubrun and Ion Nechita.

The statement is beautifully simple.

First here’s some notation.

- For any finite set $I$, we have the real vector space $\mathbf{R}^I$.
- For any injection $f: I \to J$ between finite sets, there is the induced linear map $f_*: \mathbf{R}^I \to \mathbf{R}^J$, got by reindexing as dictated by $f$ and padding out with $0$s.
- For any $x \in \mathbf{R}^I$ and $y \in \mathbf{R}^J$, there is an element $x \otimes y \in \mathbf{R}^{I \times J}$, whose $(i, j)$-coordinate is $x_i y_j$. (I call it $x \otimes y$ because if you identify $\mathbf{R}^{I \times J}$ with $\mathbf{R}^I \otimes \mathbf{R}^J$, that’s what it is.)

A **norm system** is a sensible way of assigning a norm to each vector space
$\mathbf{R}^I$, where $I$ ranges over finite sets. In other words, it consists
of a
specified norm
$\Vert\cdot\Vert$ on $\mathbf{R}^I$ for each finite set $I$, such that whenever
$f: I \to J$ is an injection of finite sets and $x \in \mathbf{R}^I$, then
$\Vert f_*(x) \Vert = \Vert x \Vert$. All that says is that if you pad out $x$
with some zeros, and switch the order of the coordinates round, it doesn’t
change the norm.

**Example:** For each $p \in [1, \infty]$ there’s a norm system
$\Vert\cdot\Vert_p$ given by the usual formulas: if $p \lt \infty$ then
$\Vert x \Vert_p = (\sum_{i \in I} |x_i|^p )^{1/p}$
(for finite sets $I$ and $x \in \mathbf{R}^I$), and $\Vert x \Vert_\infty =
\max_{i \in I} |x_i|$.

A norm system is **multiplicative** if $\Vert x \otimes y \Vert = \Vert x
\Vert \Vert y \Vert$ for all finite sets $I$ and $J$, $x \in
\mathbf{R}^I$, and $y \in \mathbf{R}^J$.

**Example:** For each $p \in [1, \infty]$, the norm system
$\Vert\cdot\Vert_p$ is multiplicative.

Theorem (Aubrun and Nechita)The only multiplicative norm systems are $\Vert\cdot\Vert_p$ ($p \in [1, \infty]$).

I find this amazing, both in itself and because it wasn’t known half a century ago. In fact it’s only the first of two theorems in their paper; the second concerns the $L_p$ norms (as opposed to what we’ve just been discussing, the $\ell_p$ norms). Anyway, I’ll stick to the first one.

I’ll say a bit more later about why I find this amazing, but first I should point out that my phrasing of Aubrun’s and Nechita’s theorem is a bit different from theirs.

**How Aubrun and Nechita put it**
Someone reading my phrasing might think: That’s a bit extravagant. A norm on
$\mathbf{R}^I$ for *every* finite set $I$? Well, the axioms imply that if
$I \cong J$ then the norm on $\mathbf{R}^I$ determines the norm on
$\mathbf{R}^J$, so we might as well just consider one finite set of each
cardinality. That is, instead of taking this huge system of norms, we take
just
one permutation-invariant norm on each of the spaces $\mathbf{R}^n$ ($n
\in \mathbf{N}$), such that $\Vert (x_1, \ldots, x_n, 0) \Vert = \Vert (x_1,
\ldots, x_n) \Vert$.

That’s fine, but there’s a small price to pay: in order to state the multiplicativity axiom, you have to choose a bijection between $\{1, \ldots, n\} \times \{1, \ldots, m\}$ and $\{1, \ldots, n m\}$ for each $n$ and $m$.

But you still might find that extravagant. The axioms imply that the norm
on $\mathbf{R}^{n + 1}$ determines the norm on $\mathbf{R}^n$, so you might as
well just work with $c_{00}$, the space of real sequences $(x_n)_{n =
1}^\infty$ that are $0$ in all but finitely many places. So instead of having
a whole family of norms, you have just *one* norm; it’s a
permutation-invariant norm on
$c_{00}$.

That’s still fine, but there’s again a price to pay: in order to state the multiplicativity axiom, you have to choose a bijection between $\mathbf{Z}^+ \times \mathbf{Z}^+$ and $\mathbf{Z}^+$ (where $\mathbf{Z}^+$ is the set of positive integers). It doesn’t matter which you choose, in the sense that if multiplicativity holds for one choice then it holds for all of them. But you do have to choose one.

That, then, is what Aubrun and Nechita do: they characterize $\Vert\cdot\Vert_p$ ($p \in [1, \infty]$) as the only norms on $c_{00}$ that are permutation-invariant and multiplicative. They don’t mention what I’ve called ‘norm systems’.

**Something impressive** Here’s a consequence of Aubrun and Nechita’s
theorem that is, I think, quite non-obvious. At least, I completely failed to
prove it without the aid of their theorem; maybe you can do better.

The $p$-norms have a special property — which, in what follows, I will
call the **special property**. Given
$x^1 \in \mathbf{R}^{n_1}, \ldots, x^k \in \mathbf{R}^{n_k},$
write
$x^1; \ldots; x^k \in \mathbf{R}^{n_1 + \cdots + n_k}$
for their concatenation. The special property is that
$\Vert x^1; \ldots; x^k \Vert_p$
is determined by $\Vert x^1 \Vert_p, \ldots, \Vert x^k \Vert_p$ alone.

How? By the formula $\Vert x^1; \ldots; x^k \Vert_p = \Vert (\Vert x^1 \Vert_p, \ldots, \Vert x^k \Vert_p) \Vert_p.$ One perspective on the formula is this: it says that there’s a monoid structure on $[0, \infty)$ whose $k$-fold multiplication is $(y_1, \ldots, y_k) \mapsto \Vert (y_1, \ldots, y_k) \Vert_p$ ($y_i \in [0, \infty)$). This is kind of obvious when $p \lt \infty$, since the operation $y \mapsto \Vert y\Vert_p$ is what you get when you transport addition — itself a monoid structure on $[0, \infty)$ — across the bijection $( )^p: [0, \infty) \to [0, \infty)$.

The Aubrun–Nechita theorem implies that any multiplicative norm system
has this special property (because it must be one of the $p$-norms). But can
you prove this *directly*, without the aid of their theorem? I couldn’t.
I made a bit of progress, but the progress I made more or less reproduced the
beginning of their proof of the theorem itself. (It didn’t reproduce the end,
which is an application of Cramér’s large deviation theorem.)

Actually, I think that if you *can* prove that any multiplicative norm
system has the special property, then you can probably build an alternative
proof of the Aubrun–Nechita theorem, as follows.

There’s a fairly ancient body of
work on ‘generalized means’, for which the classic text is Hardy,
Littlewood and Pólya’s 1934 book *Inequalities*. A generalized mean of numbers $x_1, \ldots, x_n$ is
something like the arithmetic mean, $\sum (1/n) x_i$, or the harmonic mean,
$(\sum (1/n) x_i^{-1})^{-1}$, or more generally $(\sum (1/n)
x_i^p)^{1/p}$ for any real $p$. The limiting case $p \to 0$ is the geometric
mean, $\prod x_i^{1/n}$. You can also change the uniform weighting $1/n, \ldots, 1/n$ to
some non-uniform weighting.

Part of that ancient body of work is a collection of theorems characterizing generalized means. Obviously generalized means and $p$-norms are closely related, and when I first saw Aubrun and Nechita’s paper I thought I could deduce their result from these classical theorems. But the step in the deduction that I couldn’t fill in was what I just mentioned: showing directly that a multiplicative norm system has the special property.

Incidentally, I first saw news of this paper at John’s blog. We were discussing characterizations of the Rényi entropies, which are closely related to generalized means and $p$-norms. Mark Meckes mentioned this paper. It would be nice to use Aubrun and Nechita’s characterization of the $p$-norms to produce new characterizations of generalized means and Rényi entropies.

## Re: Characterizing the p-Norms

I have a dim recollection of seeing F. J. Linton’s name attached to a talk or preprint or paper about functorial aspects of l^p-spaces. Unfortunately I have no idea what the result was, but I wonder if it might be related to your take on Aubrun and Nechita’s paper.