Talk:P-Product Metrics on Real Vector Space are Topologically Equivalent

It looks to me like the original author managed to do the hard part of this proof and then got stuck on the easy part, thinking zie had to prove more than was actually necessary. I replaced it with something far simpler that I think does work. However, I would really appreciate if other folks could check that I didn't do something totally stupid. --Dfeuer (talk) 18:32, 15 January 2013 (UTC)


 * There is nowhere where it is shown that $d_r \left({x,y}\right) \ge d_\infty \left({x, y}\right)$, or anything similar, as far as I can see. The step that shows why $d_r \left({x, y}\right) \ge d_{r+1} \left({x, y}\right)$ should be correct. --Anghel (talk) 19:06, 15 January 2013 (UTC)


 * Good point. I don't think that was proved in any of the material I erased either. --Dfeuer (talk) 19:09, 15 January 2013 (UTC)


 * But that follows trivially from the fact that the $n$th root is increasing, doesn't it? I'm thinking the whole proof of the chain of inequalities might well belong on a separate page to which this one need not link. Unless I'm thinking entirely bone-headedly, which is always a possibility. --Dfeuer (talk) 19:15, 15 January 2013 (UTC)


 * Yes, it does. So we just need to add that argument. Good job. --Anghel (talk) 19:34, 15 January 2013 (UTC)

I see your comment on "Nothing here seems to require the underlying factor space to be the real line; any metric space should do", but if you do see fit to expand it, please do it on another page. --prime mover (talk) 22:36, 15 January 2013 (UTC)


 * I have already posted this generalization for binary products on Product Space Metric Induces Product Topology. --abcxyz (talk) 02:38, 16 January 2013 (UTC)

Improve template
Yes, the proof here really doesn't need to deal with integers. But if you come up with a method that doesn't require the horrible derivative, I think it'd still be good to give that derivative its own page. I'm sure it must come up somewhere else. --Dfeuer (talk) 03:03, 16 January 2013 (UTC)


 * Proving $d_{\infty} \ge n^{-1/r} d_r$ is pretty similar to proving the special case $d_{\infty} \ge n^{-1} d_1$. --abcxyz (talk) 05:04, 16 January 2013 (UTC)


 * I don't have time tonight, but I'll try to work on it on Thursday. Do you have any ideas for a name for the function whose derivative is calculated here? &mdash;Dfeuer (talk) 05:50, 16 January 2013 (UTC)