By definition, norms are defined over some $\mathbb{R}$ or $\mathbb{C}$ vector space. Why do we only restrict ourselves to these fields when other fields give rise to interesting objects as well? (e.g. p-adic evaluation)
Is it a historic reason or because other fields would give properties so different that we‘d rather not also associate the term “norm“ with it? If so, then I assume that $\mathbb{R}$ and $\mathbb{C}$ are similar enough to make those the two fields that give rise to norms?
Best Answer
I don't think this is true. People who work in $p$-adic analysis often work with with the quantity $|\vec{x}| = \max_{1 \leq i \leq n} (|x_i|_p)$ for $\vec{x} \in \mathbb{Q}_p^n$, with $|\ |_p$ the $p$-adic norm. This is a norm in the sense that $|\vec{x}+\vec{y}| \leq \max(|\vec{x}|, |\vec{y}|)$, $|c \vec{x}| \leq |c|_p |\vec{x}|$ and $|\vec{x}|=0$ if and only if $\vec{x} = \vec{0}$. The metric induced by this norm on $\mathbb{Q}_p^n$ gives the standard product topology.
The group of matrices preserving this norm is a useful group: It is the matrices $g$ for which both $g$ and $g^{-1}$ have entries in $\mathbb{Z}_p$. It is usually denoted $GL_n(\mathbb{Z}_p)$, and plays the analogue of the orthogonal group. Indeed, Smith normal form for the PID $\mathbb{Z}_p$ says that every matrix in $GL_n(\mathbb{Q}_p)$ can be factored as $U \Sigma V$ where $U$ and $V$ are in $GL_n(\mathbb{Z}_p)$ and $\Sigma$ is diagonal with entries powers of $p$; it is valuable to think of this as a non-Archimedean analogue of singular value decomposition.
I learned this perspective from Chapter 4 of Kiran Kedlaya's book "$p$-adic differential equations" and I have seen plenty of other $p$-adic papers use it since.
I just looked at the OP's bio, and it looks like they are a young undergraduate. So the reason they haven't seen this might just be that linear algebra books written for undergraduates don't assume the reader knows what the $p$-adics are.