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?