nLab G-norm

Idea

A G-norm is like a norm (or better, an F-norm) on a vector space, but on an abelian group instead. Just as the topological structure of any topological vector space may be specified by a family of F-pseudonorms, so the topological structure of any topological abelian group may be specified by a family of G-pseudonorms. A G-norm on an abelian group is equivalent to a translation-invariant metric on it.

Definitions

Let $G$ be an abelian group.

A G-pseudonorm, or G-seminorm, on $G$ is a function $\rho\colon G \to \mathbb{R}$ that satisfies the following conditions:

• $\rho(0) \leq 0$, where $0$ is the identity element of $G$;
• $\rho(x + y) \leq \rho(x) + \rho(y)$ for all $x, y$ in $G$; and
• $\rho(-x) \leq \rho(x)$ for all $x$ in $G$.

From these axioms, the following simple conditions (sometimes included in the definition) follow:

• $\rho(x) \geq 0$ for all $x$ in $G$
• $\rho(0) = 0$;
• $\rho(-x) = \rho(x)$; and
• $\rho(n x) \leq {|n|} \rho(x)$ for every integer $n$ and all $x$ in $G$.

A G-pseudonorm is definite (or postive-definite, but we have positivity anyway) if in addition:

• $\rho(x) \ne 0$ for all $x \ne 0$ in $G$.

And of course, from that follows:

• $\rho(x) \gt 0$ for all $x \ne 0$ in $G$.

A G-norm is a definite G-pseudonorm.

A G-(pseudo)norm is homogeneous (or $\mathbb{Z}$-homogeneous) if we have:

• $\rho(n x) \geq {|n|} \rho (x)$ for infinitely many integers $n$, for all $x$ in $G$.

It then follows that

• $\rho(n x) = {|n|} \rho (x)$ for every integer $n$ and all $x$ in $G$.

While homogeneous G-norms are particularly nice, we need general G-pseudonorms if we wish to describe arbitrary topological abelian groups.

Examples

Given any translation-invariant (pseudo)metric $d$ on $G$, we get a G-(pseudo)norm $\rho$ on $G$ by $\rho(x) \coloneqq d(0,x)$. Conversely, given any G-(pseudo)norm $\rho$, we get a translation-invariant (pseudo)metric $d$ by $d(x,y) \coloneqq \rho(y - x)$. These operations are inverses. (We could add a ‘quasi-’ in here if we drop the rule that $\rho(-x) = \rho(x)$. But note that G-quasinorms and translation-invariant quasimetrics on an abelian monoid do not correspond.)

If $G$ happens to be the underlying abelian group of a real (or complex) vector space $V$, then any F-(pseudo)norm on $V$ is a G-(pseudo)norm on $G$, but not conversely. Similarly, any (pseudo)norm on $V$ is a homogeneous G-(pseudo)norm on $G$, but not conversely.

The obvious norms on $\mathbb{Z}^n$, seen as a subset of $\mathbb{R}^n$ with one of its usual structures as a Banach space, is a homogeneous G-norm.

Any abelian group has a G-norm given by $\rho(x) = 1$ whenever $x \ne 0$ (and of course $\rho(x) = 0$ otherwise). Except on the trivial group, this is not homogeneous. It corresponds to the discrete metric on $G$.

Given any topological abelian group $G$ and any neighbourhood $N$ of $0$, let $B \coloneqq N \cap -N = \{ x \;|\; x \in N,\; -x \in N \}$; then $B$ is also a neighbourhood of $0$. Let $B_0 \coloneqq G$, let $B_1 \coloneqq B$, and recursively choose $B_{n^+}$ so that $x + y + z \in B_n$ whenever $x, y, z \in B_{n^+}$ (which is possible since addition is continuous). Then

$$\rho (x) \coloneqq \inf \Big\{ \sum_{i = 1}^m \inf \{ 2^{-n} \;|\; x_i \in B_n \} \;\Big|\; x = \sum_{i = 1}^m x_i \Big\}$$

defines a G-pseudonorm on $G$ such that, given any net $(x_\nu)$ in $G$, we have that $\rho(x_\nu)$ converges to $0$ if and only if $x_\nu \in N$ holds eventually.

Families of G-pseudonorms

Any family of G-pseudonorms on an abelian group $G$ makes $G$ into a topological abelian group (TAG), and every TAG structure on $G$ arises in this way. However, different collections of G-pseudonorms may determine the same topological structure.

In one direction, let $G$ be an abelian group, and suppose that we equip $G$ with an arbitrary collection $D$ of G-pseudonorms. Every G-pseudonorm determines a pseudometric, and these pseudometrics generate a gauge space structure on $G$ and thence a topological structure. Because the pseudometrics involved are translation-invariant, we have in fact made $G$ into a TAG. In more detail: A subset $U$ of $G$ is open if and only if, for every $x$ in $U$, for some list $\rho_1,\ldots,\rho_n$ from $D$ and some real number $\epsilon \gt 0$, for every $y$ in $G$, if $\rho_i(y - x) \lt \epsilon$ for every $i$, then $y \in U$. Then you can check that these subsets form a topology relative to which the group operations are continuous.

It may also be nice to look at the uniform space structure on $G$; the gauge and the TAG structure determine the same uniform structure. Explicitly, a binary relation $\sim$ on $G$ is an entourage if and only if, for some list $\rho_1,\ldots,\rho_n$ from $D$ and some real number $\epsilon \gt 0$, for every $x$ and $y$ in $G$, if $\rho_i(y - x) \lt \epsilon$ for every $i$, then $x \sim y$. Then these entourages form the uniform structure on $G$ which is compatible with the group structure and whose underlying topological structure is the one above.

Conversely, let $G$ be a TAG. Then the collection of all continuous G-pseudonorms $\rho\colon G \to \mathbb{R}$ generates the topological structure on $G$. The proof is complicated, but essentially it amounts to this: applying the final example from the Examples section above to each neighbourhood of $0$ (or at least to each neighbourhood in a neighbourhood base), check that the G-pseudonorms defined are continuous, and check that there are enough of them to generate a topology at least as strong as the actual topology on $G$; the converse is immediate. In other words, the hard thing that has to be checked is that there are enough continuous G-pseudonorms.

In weak foundations

The only really tricky part is the proof that there are enough continuous G-pseudonorms to generate the topology on any TAG $G$. The development seems to be predicative over $\mathbb{R}$; you can't speak of the collection of all continuous G-pseudonorms if you are being strongly predicative, but you really only need one for each neighbourhood in a neighbourhood base of $G$. However, it is not constructive or predicative over $\mathbb{N}$, because the infima may not exist. Also, we use dependent choice.

The same problems arise in proving that every topological vector space structure is generated by some family of F-pseudonorms; on the other hand, locally convex spaces (which are generated by pseudonorms) are better behaved. There may be a similar theory of locally convex TAGs based on homogeneous G-pseudonorms, but I haven't looked into this.

It would be natural, in constructive mathematics, to attempt the development with localic groups. Even in classical mathematics, however, there may be (and are) sober TAGs which cannot be interpreted as localic groups, such as the additive group of rational numbers with its topology as a subspace of the real line. Probably we have to start by requiring a G-pseudonorm to be a continuous map from the localic group $G$ to the locale of real numbers, rather than starting with a discrete abelian group, but I haven't looked into this further.

References

HAF, Chapters 22 and 26. See especially Section 26.29 for the last Example.