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 , the topological space , and the Banach space . Today I’ll enthuse about a theorem that does it for the -norms . 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 , we have the real vector space .
- For any injection between finite sets, there is the induced linear map , got by reindexing as dictated by and padding out with s.
- For any and , there is an element , whose -coordinate is . (I call it because if you identify with , that’s what it is.)
A norm system is a sensible way of assigning a norm to each vector space , where ranges over finite sets. In other words, it consists of a specified norm on for each finite set , such that whenever is an injection of finite sets and , then . All that says is that if you pad out with some zeros, and switch the order of the coordinates round, it doesn’t change the norm.
Example: For each there’s a norm system given by the usual formulas: if then (for finite sets and ), and .
A norm system is multiplicative if for all finite sets and , , and .
Example: For each , the norm system is multiplicative.
Theorem (Aubrun and Nechita) The only multiplicative norm systems are ().
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 norms (as opposed to what we’ve just been discussing, the 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 for every finite set ? Well, the axioms imply that if then the norm on determines the norm on , 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 (), such that .
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 and for each and .
But you still might find that extravagant. The axioms imply that the norm on determines the norm on , so you might as well just work with , the space of real sequences that are 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 .
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 and (where 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 () as the only norms on 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 -norms have a special property — which, in what follows, I will call the special property. Given write for their concatenation. The special property is that is determined by alone.
How? By the formula One perspective on the formula is this: it says that there’s a monoid structure on whose -fold multiplication is (). This is kind of obvious when , since the operation is what you get when you transport addition — itself a monoid structure on — across the bijection .
The Aubrun–Nechita theorem implies that any multiplicative norm system has this special property (because it must be one of the -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 is something like the arithmetic mean, , or the harmonic mean, , or more generally for any real . The limiting case is the geometric mean, . You can also change the uniform weighting to some non-uniform weighting.
Part of that ancient body of work is a collection of theorems characterizing generalized means. Obviously generalized means and -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 -norms. Mark Meckes mentioned this paper. It would be nice to use Aubrun and Nechita’s characterization of the -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.