Definition talk:Norm on Division Ring

After careful consideration, I think this definition may be generalised to a ring without zero divisors. This is good, as then it will incorporate the absolute value on $\Z$. --Lord_Farin 05:30, 9 December 2011 (CST)
 * I believe you can accomplish that by just dropping to ring. Positive definite + multiplicative will then prove it to have no proper zero divisors. That distinction will only come up if you decide to define a seminorm for rings. --Dfeuer (talk) 18:06, 17 January 2013 (UTC)

Wikipedia, at Berkovich space, says this:
 * A seminorm on a ring A is a non-constant function f→|f| from A to the non-negative reals such that |0| = 0, |1| = 1, |f + g| ≤ |f| + |g|, |fg| ≤ |f||g|. It is called multiplicative if |fg| = |f||g| and is called a norm if |f| = 0 implies f = 0.

I have no idea if that's right, of course. However, suppose $R$ is a division ring with a multiplicative seminorm.

Then since the seminorm is not constant, there is an $x$ such that $|x|≠0$. Let $y≠0_R$.

Then $|x|=|x \circ y^{-1}\circ y| \le |x \circ y^{-1}||y|$, so $|y|≠0$.

Thus a multiplicative seminorm on a division ring is a norm. --Dfeuer (talk) 20:53, 17 January 2013 (UTC)


 * Having discovered a definition of that site, it is now important that you find some corroborative evidence elsewhere to back it up. Not only is Wikipedia a tertiary source but it is well-known as being laughably unreliable. --prime mover (talk) 21:33, 17 January 2013 (UTC)

Change suggestion
The pages "Normed Division Ring" and "Norm (Division Ring)" are structured differently to the similarly named pages "Normed Vector Space" and "Norm (Vector Space)". The page "Normed Division Ring" is simply a redirect to "Norm (Division Ring)". Whereas "Normed Vector Space" actually defines a normed vector space which links to "Norm (Vector Space)" which in turn transcludes the definition for a normed vector space from "Normed Vector Space". Should the pages "Normed Division Ring" to "Norm (Division Ring)" be structured similar to the pages "Normed Vector Space" and "Norm (Vector Space)" or should the redirect page "Normed Division Ring" be removed? I'm happy to make the necessary changes to do either, or any other preferred approach. --Leigh.Samphier (talk) 08:44, 7 October 2018 (EDT)


 * I moved the above comment from the redirect page onto here because it makes no sense there. --prime mover (talk) 04:27, 7 October 2018 (EDT)


 * It's complicated. There is already a transclusion of Norm (Division Ring) in Norm (Vector Space) which itself also has a page Normed Vector Space where the assumption appears to be that Normed Division Ring is a specialisation of Normed Vector Space. I don't know, myself, as I have never studied norm theory and so I don't understand either its motivation or direction.


 * A recent contributor greatly enjoyed starting jobs of moving things around and refactoring stuff to suit his personal idea of how things should be, but appeared not to be such a huge fan of finishing of what he started. Consequently there is a lot of stuff which has been left in an inconsistent state, and sometimes with a plentiful number of broken links as a result of an incoherent renaming strategy. This is I believe one of those areas.


 * The top level page should be "norm" and it should then be subpaged as "Norm/Vector Space" and "Norm/Division Ring" and so on, for whatever objects are appropriate. Whichever one is the most general should be at the top, and more specialised contexts should be at the bottom (as, for example, we have pages structured Topology $\to$ Metric Spaces $\to$ Real Spaces and Complex Spaces, etc. according to the concept being defined. The reason for the subpage name structure is that if you name something Concept/Subconcept, then at the top of the page you automatically get a link back to Concept, which can be useful. If you transclude pages which are not so named, you don't get the link and so if you want it you have to build it specifically.


 * Bottom line: as long as it's consistent it does not matter really, except to note that:
 * A page named "Norm (Division Ring)" defines the norm that has been imposed on a division ring.
 * A page named "Normed Division Ring" defines a division ring that has had a norm imposed on it.
 * Whether it is worth maintaining a separate page for both concepts is a philosophical position -- in general, certain have preferred in the past to separate the concepts onto their own pages, while other prefer to merge them. I myself am of the former persuasion.


 * Bottom bottom line: feel free to take this area on and make it consistent and aesthetic. --prime mover (talk) 09:08, 7 October 2018 (EDT)