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.
Let be an abelian group.
A G-pseudonorm, or G-seminorm, on is a function that satisfies the following conditions:
From these axioms, the following simple conditions (sometimes included in the definition) follow:
A G-pseudonorm is definite (or postive-definite, but we have positivity anyway) if in addition:
And of course, from that follows:
A G-norm is a definite G-pseudonorm.
A G-(pseudo)norm is homogeneous (or -homogeneous) if we have:
It then follows that
While homogeneous G-norms are particularly nice, we need general G-pseudonorms if we wish to describe arbitrary topological abelian groups.
Given any translation-invariant (pseudo)metric on , we get a G-(pseudo)norm on by . Conversely, given any G-(pseudo)norm , we get a translation-invariant (pseudo)metric by . These operations are inverses. (We could add a ‘quasi-’ in here if we drop the rule that . But note that G-quasinorms and translation-invariant quasimetrics on an abelian monoid do not correspond.)
If happens to be the underlying abelian group of a real (or complex) vector space , then any F-(pseudo)norm? on is a G-(pseudo)norm on , but not conversely. Similarly, any (pseudo)norm on is a homogeneous G-(pseudo)norm on , but not conversely.
The obvious norms on , seen as a subset of with one of its usual structures as a Banach space, is a homogeneous G-norm.
Given any topological abelian group and any neighbourhood of , let ; then is also a neighbourhood of . Let , let , and recursively choose so that whenever (which is possible since addition is continuous). Then
defines a G-pseudonorm on such that, given any net in , we have that converges to if and only if holds eventually.
Any family of G-pseudonorms on an abelian group makes into a topological abelian group (TAG), and every TAG structure on arises in this way. However, different collections of G-pseudonorms may determine the same topological structure.
In one direction, let be an abelian group, and suppose that we equip with an arbitrary collection of G-pseudonorms. Every G-pseudonorm determines a pseudometric, and these pseudometrics generate a gauge space structure on and thence a topological structure. Because the pseudometrics involved are translation-invariant, we have in fact made into a TAG. In more detail: A subset of is open if and only if, for every in , for some list from and some real number , for every in , if for every , then . 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 ; the gauge and the TAG structure determine the same uniform structure. Explicitly, a binary relation on is an entourage if and only if, for some list from and some real number , for every and in , if for every , then . Then these entourages form the uniform structure on which is compatible with the group structure and whose underlying topological structure is the one above.
Conversely, let be a TAG. Then the collection of all continuous G-pseudonorms generates the topological structure on . The proof is complicated, but essentially it amounts to this: applying the final example from the Examples section above to each neighbourhood of (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 ; the converse is immediate. In other words, the hard thing that has to be checked is that there are enough continuous G-pseudonorms.
The only really tricky part is the proof that there are enough continuous G-pseudonorms to generate the topology on any TAG . The development seems to be predicative over ; 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 . However, it is not constructive or predicative over , 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 to the locale of real numbers, rather than starting with a discrete abelian group, but I haven't looked into this further.
HAF, Chapters 22 and 26. See especially Section 26.29 for the last Example.