Combining Norms

SlipEternal

MHF Helper
Nov 2010
3,728
1,571
The p-adic norms are all of the possible norms on the rationals (any other norm is equivalent to one of the p-adic norms, where p is prime or infinity). Given two norms on a space (say the p-adic and q-adic norms), is there a way to combine them? For example, given \(\displaystyle r \in \mathbb{Q}\), would \(\displaystyle \sqrt{|r|_p^2 + |r|_q^2}\) be a norm on the rationals? If not, is there an intuitive way to "combine" norms?
 

chiro

MHF Helper
Sep 2012
6,608
1,263
Australia
Hey SlipEternal.

What do you want to do? Do you want to show that some new combination of norms meets the requirements for a norm in some space? In other words, do you want to show that your expression involving the square root meets the criteria for a norm?
 

SlipEternal

MHF Helper
Nov 2010
3,728
1,571
Hey Chiro,

I am looking for a way to describe sequences that are Cauchy with respect to multiple norms simultaneously. For example, I want to extend the rationals by all sequences that converge with respect to both the p-adic and q-adic norms. So, I hoped each sequence that converges with respect to both of those norms might also converge with respect to a third norm (one that somehow combines the two). But, I think an easier way of dealing with it might be to avoid norms and use a metric, instead. I can define \(\displaystyle d:\mathbb{Q} \times \mathbb{Q} \to [0,\infty)\) by \(\displaystyle d(a,b) = \max\{|b-a|_p,|b-a|_q\}\). Then, I can use that metric to evaluate sequences, and extend the rationals that way.
 
Last edited: