User talk:Lord Farin/Backup/Definition:Norm

I wonder whether it might be more aesthetically pleasing to use $|\!|x|\!|$ instead of $||x||$ for the symbol defining the norm, in the same way we use $\left[\!\left[{a}\right]\!\right]_m$ for equivalence classes.

I know it's fiddly to enter, but if you save a template on your home page it's straightforward to cut and paste it in. Thoughts? --Matt Westwood 07:44, 12 January 2009 (UTC)

I actually prefer $||x||$. --Cynic (talk) 20:04, 12 January 2009 (UTC)

Doesn't really matter to me which ones are used, but I do like the ones that are closer together. So long as you can tell that they are two separate lines. --Joe (talk) 22:42, 12 January 2009 (UTC)

I've just found that on Cauchy-Schwarz Inequality the notation $\| x\|$ is used, which seems standard. So we can start using that, if that's a good idea. --Matt Westwood 21:47, 17 January 2009 (UTC)

Incidentally, taking a closer look at this, it seems that $X$ can't just be "any" set. It needs to be a structure with two operations $+$ and $\times$, and needs to have the element $0$ in it. Hence it seems to probably need to be at least a ring or something. This needs to be checked.--Matt Westwood 21:53, 17 January 2009 (UTC)

This isn't how a norm is defined! There need be no binary $\times$ operation over the set, the correct property 2 should read $\| c x \| = |c| \|x\|$ for any scalar $c$ and $x \in X$. Furthermore, over at Definition:Norm (Linear Space), it is not an "abuse of language" to say a set is a NLS (normed linear space) without mentioning the norm, as for a set to obtain NLS status it merely has to have some norm, any norm, possible on it, and so that norm need not be specified. I'm correcting this. Colors

Okay, wait. So apparently there are norms outside of linear algebra. I should have remembered that because I've read about UFD's somewhere. Anyway, why are we calling this one Norm and the other one Norm (linear space), when clearly the latter is by far the most default definition of a norm. Reverting my edit. Colors

There's a problem with the definition of the norm on this page, as I pointed out above. It's mentioned in terms of a general set, but it seems to me that $X$ must at least be a ring, as there are two operations defined on this set (times and plus). I abandoned trying to make sense of this page some time ago. The "Linear Space" version is more coherent. --Matt Westwood 20:32, 20 April 2010 (UTC)

FWIW I think it's confusing to call anything except a norm on a vector space a norm. On a field the definition on this page is an absolute value, and may as well be called as such on a ring or some set with less structure. Admittedly, norms of ideals in algebraic number theory satisy this definition, but in this case one speaks of the norm - there is only one such norm, it isn't part of a larger family of norms for which a general definition is needed, and it is defined in terms of vector spaces anyway. I think three pages (two existing):

• Field Norm
• Norm (Linear Space)
• Absolute value

would be clearer. Linus44 10:23, 20 February 2011 (CST)

Category listing

This page is currently appearing in the category Definitions/Mapping Theory. You might want to temporarily comment that category out so it doesn't. Same may apply to any other pages you've got backed up. No big deal. --prime mover 02:26, 20 January 2012 (EST)

Thanks. I will check. --Lord_Farin 03:03, 20 January 2012 (EST)