# Combining Norms

#### SlipEternal

MHF Helper
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
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
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.

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