Wikipedia defines a norm on a vector space $V$ as a function $p : V \mapsto \mathbb R$. I've seen this defined similarly elsewhere. However, it seems to me that a real codomain isn't always necessary.
Take as an example the vector space over $\mathbb Q$ with vectors $v \in \mathbb Q^n$. If we combine this with the $L^1$ norm, then the codomain is again $\mathbb Q$.
Am I missing something here? Is my example flawed?