Definition talk:Trivial Norm

a) Worth linking to the Definition:Discrete Metric? It's very similar in concept.

b) Is there a reason why a norm can be defined only on a division ring? I'm currently (while thinking of something else) vaguely wondering if there's a reason why we can't impose a norm on the integers. (there's probably an obvious one which I can't think of.) --prime mover 14:51, 1 December 2011 (CST)
 * a) Yes, surely. It is the associated metric ($d(x,y) = |x-y|$).


 * b) Not sure. One needs that there are no zero divisors (otherwise N1+N2 gives a contradiction as $\R$ has none), but indeed the requirement for inverses seems fatuous (even the unity seems not necessary). I will think some more. If this works, I think we can also define more general vector spaces... just for the sake of it. --Lord_Farin 16:42, 1 December 2011 (CST)


 * This may also correlate with some of the work Linus44 was doing early summer on various subtle refinements of the Integral Domain. Specifically a Definition:Euclidean Domain has a Euclidean Valuation which may be a precursor to a Norm - but there are subtleties and unfortunately I have other directions I'm going at the moment. --prime mover 16:52, 1 December 2011 (CST)