Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
I see that you added the condition to the entry normed ring.
Just noting that the previous form of the definition was following the reference given, Def. 1.2.1 in Berkovich 2009. (?)
Maybe worth commenting on and referencing different definitions?
Thanks. The comment is fine, but best would be to add a reference which suports your claim. (Not that I am doubting it, but it would still be good to add a reference.)
It looks like Remark 2.2 is also false unless this axiom is added, since a normed group includes the axiom $\rho(g^{-1}) = \rho(g)$.
I wonder if Berkovich just made a typo?
added the original reference:
The extra axiom in not stated there, either.
In Bosch, Güntzer & Remmert 1984 the axiom is also not stated, but they prove that it follows from their axioms (Prop. 3).
Bosch–Güntzer–Remmert is about non-archimedean norms though (prop. 3 uses the ultrametric inequality), and Gelfand’s first axiom for normed rings is that they’re Banach spaces (so the paper is actually about Banach algebras)
This looks like it gives the definition you are after:
(first page)
After the definition (here), I have replaced the pointer to Berkovich 2009, def. 1.2.1 and the claim that Berkovich is missing an axiom (instead the issue seems to be that Berkovich discusses the non-Archimedean case) by pointer to Ozaki, Kashiwagi & Tsuboi 1953.
1 to 9 of 9