Natural Extensions of the $p$-Adic Norm to Higher Dimensions There are two classes of completions for $\Bbb{Q}$, we get $\Bbb{R}$ by considering  Cauchy sequences with respect to the standard Euclidean metric, and we get the $p$-adic numbers, $\Bbb{Q}_p$ when we consider Cauchy sequences with respect to the $p$-adic metric.
The standard Euclidean norm on $\Bbb{R}$ is easily generalized to $\Bbb{R}^n$, you start from $\lvert x\rvert = \sqrt{x^2}$ in $\Bbb{R}$ to $\lvert x\rvert = \sqrt{x_1^2+\cdots+x_n^2}$ in $\Bbb{R}^n$. I only point this out because I am trying to do the same with the $p$-adic metric.
Take $\lvert x\rvert_p =p^{-m}$ in $\Bbb{Q}_p$, does this have a natural generalization to $\Bbb{Q}_p^n?$ At first I thought to just use the product metric, but that isn't really the process used to move from the standard Euclidean norm in $\Bbb{R}$ to $\Bbb{R}^n$, and I am seeking an analagous generalization.
My next thought was if we had two $p$-adics:
$$
\begin{align*}
\alpha = \sum_{k=n}^\infty a_kp^k \\
\beta = \sum_{k=m}^\infty b_kp^k
\end{align*}
$$
Then if $x=(\alpha,\beta)\in\Bbb{Q}_p^2$ for instance we could define something like
$$\lvert x\rvert_p := p^{-\min(\lvert \alpha\rvert_p,\lvert \beta\rvert_p)}= p^{-\min(n,m)}
$$
This seems analagous since it combines the coordinates like how we take the sum of the squares in the Euclidean case:
$$
\begin{align*}
\alpha+\beta &= \sum_{k=n}^\infty a_kp^k + \sum_{k=m}^\infty b_kp^k \\
&:= \sum_{k=\min(m,n)}^\infty (a_k+b_k)p^k 
\end{align*}
$$
But I'm not sure if there is a more natural analogous extension.
My other thought was to look geometrically, the standard Euclidean metric has a very natural geometric picture which is perhaps why its generalization to $\Bbb{R}^n$ is so natural. Unfortunately the $p$-adic metric is much harder to visualize especially if we wanted to try and look at the space $\Bbb{Q}_p^n$!
I couldn't find very much literature on higher dimensional spaces $\Bbb{Q}_p^n$ Do you know of an analagous extension to the $p$-adic metric, or know of any resources that talk in a little more detail about $\Bbb{Q}_p^n$ and its geometry?
EDIT:
As per How does one define an inner product on the space $V=\Bbb{Q}_p^n$?, we could look at the problem as defining an inner product space at which point we would take the natural inner product on $\Bbb{Q}_p^n$
$$
\lvert\langle x,x\rangle\rvert_p = \lvert x_1^2 + \cdots + x_n^2\rvert_p
$$
Could this be the more natural definition?
 A: The obvious norm on $\Bbb{Q}_p^n$ is
$$\|v\| = \max_i |v_i|_p$$ 
The point is that it is $GL_n(\Bbb{Z}_p)$-invariant 
$\qquad$ (it is clear that $\|Av\| \le \|v\|$ thus $\|v\| = \|A^{-1} Av\| \le \|Av\|$ and $\|v\|=\|Av\|$) 
Whence $\|.\|$ is in some sense the "most natural" norm on $\Bbb{Q}_p^n$ and being invariant for the  maximal compact subgroup $GL_n(\Bbb{Z}_p)$  of $GL_n(\Bbb{Q}_p)$ it can be considered as a $p$-adic equivalent of $\Bbb{R}^n$'s Euclidean norm (which is invariant for the maximal compact subgroup $O_n(\Bbb{R})$ of $GL_n(\Bbb{R})$).
A: You might have more luck finding references if you allow your ground field to be any complete ultrametric field, or especially local field, instead of just $\Bbb Q_p$. Indeed the max-norm (after choosing a basis) is the standard norm on a finite dimensional vector space over such a field; it's also a classical result that on such finite dimensional vector spaces over complete nonarchimedean fields, all norms are equivalent i.e. induce the same topology (meaning in particular that the choice of basis for the maximum norm does not really matter). See e.g. 5.4 and 5.5 in these notes, or this quick intro. Note that the theory quickly goes to infinite-dimensional Banach spaces, as the finite dimensional case is kind of boring.
I don't have it at hand right now, but I remember that in Schikhof's Ultrametric Calculus, several sections are devoted to that, and are accessible at a beginner's level.
As to your edit, I want to emphasize the comments I put under that question and answer you link to. An "inner product" on vector spaces over $\Bbb Q_p$ makes no sense to me without siginificantly changing the definition of what an inner product is. If it is supposed to be bilinear, its codomain cannot be $\Bbb R$ but a $p$-adic field; but if its codomain is a $p$-adic field, then e.g. it makes no sense to say it has "positive" or "negative" values.
What one can actually talk about are quadratic forms or equivalently symmetric bilinear forms on $p$-adic vector spaces -- that is a rich and interesting theory -- but one of its corner stones is a classical result by Witt that most of these forms are isotropic; indeed, except in very small dimensions, for any such symmetric bilinear form $b(\cdot, \cdot)$, there will be non-zero vectors $x$ such that $b(x,x) =0$. Which to me is a good argument against calling anything like that an inner product.
