Definition talk:P-Product Metric

There are other useful/important metrics on products of metrics. One, whose name I cant remember at the moment, defines $d(p,q) = \max \{d(p_i,q_i)\}$, and then there are others that work for countably infinite products. I kind of think this page should be renamed, new pages added, and perhaps a category created for metrics on product spaces. I'll try to work on that a bit tonight/tomorrow. --Dfeuer (talk) 02:13, 17 December 2012 (UTC)


 * The metric you mentioned is already mentioned; it is denoted $d_{\infty}$ on this page.


 * I don't see why this page should be renamed. Could you please explain?


 * Yes, maybe the subpage should be renamed in due course. There is already a  message at the bottom which basically reiterates some of what you are saying. I think you can feel free to make additions, but I don't really know what other contributors have in mind for this page now.


 * The category you appear to be thinking of seems to already exist as Category:Product Space; you might think that a rename is desirable. --abcxyz (talk) 02:56, 17 December 2012 (UTC)


 * The page should be renamed because it's not a definition; it's a list of related definitions. Category:Product space is too broad for my purpose: I was thinking of a subcategory for metrics on product spaces. I suspect it will be a small subcategory, but why not? Sorry I missed the d_\infty you mention. --Dfeuer (talk) 03:15, 17 December 2012 (UTC)


 * This type of merging several related definitions into a single page, with transclusions, is practiced all over this site. Examples include Definition:Set Union, Definition:Continuity, Definition:Homomorphism (Abstract Algebra), Definition:Increasing, and many others. Unless I'm missing something, there is basically equal merit in changing this page as in changing many of those. If you have an objection to this which is compelling enough (as it will be a large-scale change if it ever takes place), it probably belongs to the main talk page. As for me, I'm sorry, but I can't see anything wrong with the way it currently is.


 * The categories Category:Product Spaces and Category:Product Space do seem rather strange (and maybe potentially confusing). Perhaps we should replace these with Category:Product Spaces (Topology) and Category:Product Spaces (Metric Spaces)? I'd like some additional feedback before any action is taken. --abcxyz (talk) 03:35, 17 December 2012 (UTC)


 * They're only sort of related. They are all metrics inducing product topologies on topological products of spaces whose topologies are induced by metrics. There is, as far as I know, no notion of a product of metric spaces, per se. Unlike a number of other pages with related definitions, these definitions don't lead to isomorphic structures. I'm pretty new to PW, though, so I don't really know how things are usually done, but I have noticed that prime.mover likes to separate rather than combine whenever possible.--Dfeuer (talk) 03:47, 17 December 2012 (UTC)


 * Well then, I'm not sure. Comments on this from someone with access to the source works?
 * So what do you propose: have this particular page refer specifically to pairwise/finite products and a separate page for countably infinite products? --abcxyz (talk) 04:03, 17 December 2012 (UTC)


 * I was thinking more of having it refer specifically to $r$-products, and having other pages for various other metrics on products. Dfeuer (talk) 05:57, 17 December 2012 (UTC)

What can $r$ be?
The Wikipedia article on the Minkowski metric indicates it can be any real number greater than 0. This page says a real number at least 1. The proof of equivalence of generalized Euclidean metrics says strictly positive integer. Mess. I'm a fan of generality, so if $\R_{>0}$ will do o the trick, I think we should go with that. Howsoever, I think it would also be neat to attempt a proof that the generalized Euclidean metrics are equivalent for integral $r$ that doesn't bring in the monster derivative from you-know-where. --Dfeuer (talk) 22:58, 15 January 2013 (UTC)


 * I understand you're a fan of generalisation, but please understand that there is still a place for the special, i.e. the application to only numbers. There are many studying mathematics, in particular topology and metric spaces, who have not encountered the full breadth of abstract algebra to whom a proof based on the most general objects possible will be incomprehensible.


 * You yourself expressed haughty indignation at another contributor who expressed a theorem in categorical terms (at the end of the spectrum of generality) purely because, at the stage of study at which you now are, you were not up to speed. So please respect the various levels of learning that may or may not have been achieved by the readers of these pages.


 * And I would thank you not to refer to pages that other people have written as a "mess" until you yourself are capable of writing perfect pages in one go without crying for help. A suggested plan made to you many years ago (or was it only a month? Goodness me what a horribly tedious month it has been, sweet death, come my way) was that if you are unsure of what you are expecting to put on a page then you do it in your sandbox rather than make 30 or so edits on a page before you can finally get it right. It makes it incredibly tedious to follow a page's history with such a large number of page edits. --prime mover (talk) 23:07, 15 January 2013 (UTC)


 * I think the "mess" referred to is the fact that there tacitly seem to be various degrees of generality. That's not good. --Lord_Farin (talk) 23:13, 15 January 2013 (UTC)


 * Prime.mover, how would someone be reading a proof that a certain function is a metric if they don't know what a metric is? Yes, there's loads of room for number stuff—you'll note that I recently added a proof of something or other for reals using the mean value theorem when I already had a perfectly good general version. But once you're proving that the Generalized Euclidean metric is a metric, you must know what a metric is, and that the usual metric on the real line is one such. --Dfeuer (talk) 23:39, 15 January 2013 (UTC)


 * Based on the Minkowski inequality, it looks like it'll have to be at least one to be a metric. Not sure what Wikipedia is talking about. --Dfeuer (talk) 00:01, 16 January 2013 (UTC)