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started to split off Banach ring from Banach algebra. But need to interrupt now.
I added the example of $\mathbb{Z}_p$.
I have added the actual definition to normed ring. Added the remark that the traditional definition is a bit stricter that that of a commutative monoid internal to the category of normed abelian groups.
(This seems to be an important subtle point. If anyone has any further insight to offer, please do.)
The same issue applies to Banach algebra. It's just a question what the morphisms in your category of normed abelian groups (or of Banach spaces) are. You are taking the morphisms to be bounded additive (or linear) maps, but this gives a too general notion of isomorphism (even though it is the one traditionally called ‘isomorphism’ by Banach theorists). If you take the morphisms to be the short additive (or linear) maps, then you get both the right notion of isomorphism (the one called ‘isometric isomorphism’ by Banach theorists) and the right notion of Banach ring (or algebra).
BTW, is there a reason for having normed ring separate from Banach ring? Are there incomplete normed rings of interest in their own right?
Anyway, I would like to edit the remark to simply state that a (complete) (semi)normed ring is precisely a monoid in the monoidal category of (complete) (semi)normed abelian groups with short group homomorphisms as morphisms and the projective tensor product, but then that would destroy the reference to Bassat–Kremnitzer, which maybe you want to keep for some other reason.
Please feel free to edit.
re #5:
I suppose it’s true that whenever one runs into an incomplete normed ring one tends to next go and complete it.
But why not have an entry on the concept itself. There are classical monographs titled “Normed rings” (e.g. Naimark 59), so then why not an $n$Lab entry.
Or if one goes to any review such as Jarden 11, chapter 2 and searches for the string “not complete” one gets enough hits that would seem to justify to discuss the concept in its own right.
There are two examples of ‘not complete’ normed rings in Jarden 11, chapter 2; but even if there weren't, I would certainly want to cover normed rings, but I'm questioning whether it's worth having separate pages for normed rings in general and for complete normed rings. For comparison, we cover normed vector spaces and even seminormed vector spaces, but these are covered within the article on Banach spaces.
So my question in #5 is not really phrased well. It's not so much whether there are any worthwhile examples of incomplete normed rings (although Jarden does have an interesting one: the ring of those formal power series (over a given non-Archimedean valued field) that converge somewhere other than at $0$). The real question is whether the subjects are different enough that there are different things to say about them. If we are just going to repeat everything about normed rings and the same things about complete normed rings, then I wouldn't want separate articles; otherwise I would.
It's quite possible that normed ring is a better title than Banach ring, especially if most of the work is done by algebraists, who are liable to favour terminology such as ‘complete normed ring’. (By the way, Jarden has some extra conditions on what it means to be a normed ring, which I think amount to making it non-Archimedean. That doesn't seem to affect the one place that you cite it, however.)
I made the edit suggested in comment #6, although in such a way as to leave the point made by Bassat–Kremnitzer (just in case it matters).
If you are energetic and feel like improving exposition by merging entries, I won’t object.
On the other hand, if two concepts overlap a lot, one good way to go is to keep two seperate pages such as to make hyperlinking be more to-the-point, but have one of the two be very brief and have the other one keep all the discussion.
I do feel like doing that, and I can easily do it for normed group, normed ring, and Banach algebra (to cite the names that I would want to use; ‘Banach group’ in particular is not available, being used for at least two other things, and I am a little mixed up on the normed fields).
But there may be a larger issue here. You write:
On the other hand, if two concepts overlap a lot, one good way to go is to keep two seperate pages such as to make hyperlinking be more to-the-point, but have one of the two be very brief and have the other one keep all the discussion.
I see you do this, and I don't particularly care for it, because then I find that I have to make two clicks: one to the very brief page and then another to the page with all of the content. And then I'm never quite sure if the material on the page with the content applies to the subject of the brief page. I'd rather have the page with all of the content explicitly cover both concepts, so that people who write in it will feel the need to specify the correct range of application of the properties that they are writing about.
On the other hand, when there are rather different things to say on the different pages, then they should be separate. It's quite possible that pages that once were not separate should become so, that is (or should be thought of as) normal around here.
I find that I have to make two clicks:
No, wait, this is a misunderstanding. What I am proposing is meant to reduce the amount of searching. If I want to point a reader specifically to the definition of a norm on a ring, then, if the entry exists, I point to “normed ring” and with one click the reader has the intended information.
If on the other hand I have to send the reader to “Banach ring” when trying to point to just the definition of a norm on a ring, then I have to leave the reader alone with figuring out which bit of information on a long page he is meant to take note of.
If however you do want to point the reader to comprehensive discussion, you point to “Banach ring” right away. Again, it’s just one click.
(I'll change from rings to algebras in your example so that it will better fit the way the pages are right now.)
So if you want send the reader to the definition of normed algebra or of a norm on an algebra, then you link to normed algebra; if you want to send them to a broad discussion, then you link to Banach algebra. But if you want to send the reader to the definition of a Banach algebra, then you still have send them to a long page. However, if you're careful, then you can send them to the specific part of the long page with the definition of Banach algebra, normed algebra, or norm on an algebra.
Currently, this can only be done if Banach algebra is organized more carefully, but that should be done regardless.
Sure, agreed!
It is true that with carefully organized entries we may always point to the respective subsections. But this alone is often not good enough as can be seen by taking this idea to its limit: then we could just have the whole $n$Lab in one single page with suitable subsections.
This is incidentally the kind of issue that I was asking for reaction to over in another thread, namely this is a problem with the entries valuation and absolute value. The former entry effectively tells the reader that chances are he is really looking for the latter entry, but most nLab entries who by this logic should point to the latter entry to in fact point to the former.
The point being that typing
[[X]]
should produce a link that sends the reader to an entry which does not make ado before saying what “X” is.
But I think we both agree on this. Certainly I am not making the claim that anything on the $n$Lab, let alone the entries we are currently talking about, are optimal.
What got us into this specific discussion here was that for my own benefit I wanted to lay out an organization of concepts as in analytic geometry ingredients – table and in the course of this I created a few stubs to fill that entry.
Whereas I would have made some redirects to fill that entry, at least to begin with.
I am not too clear on the difference between the two notions of valuation myself; I can read the definitions, but I don't the grasp the point of the distinction.
But I think that both of our versions will achieve our shared goal of having [[X]]
send the reader to a place where they will quickly find out what ‘X’ means. The Definition section at Banach space seems to me to do this pretty well: we lay out the first few properties under the definition of ‘pseudonorm’, then put in the next to define ‘norm’, then define (in this context) ‘complete’, then ‘Banach space’. Any direct definition of ‘Banach space’ would have to cover the same ground; the only difference is that we throw out a few other terms along the way. (Actually, this could be improved; we should define ‘normed vector space’ earlier, so that people can get that defined before reading irrelevant stuff about completeness.)
Where the difference comes in is that people following a link to normed vector space (which redirects to Banach space) currently get sent to the page that not only defines the term but covers the basic material about it; whereas people following a link to normed algebra (which does not redirect to Banach algebra) get pretty much only the definition and have to follow another link to get more in-depth material.
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