TITLE: Why are we defining the norms on certain vector spaces the way they are?
QUESTION [6 upvotes]: What's the intuition behind defining $\|x\|_{\infty} = \max_{1 \le i \le n}\{|x_i|\}$ on the space of ordered $n$-tuples of complex numbers?
I'm asking because I've been asked to find a norm on the space of complex $m \times n$ matrices that is the analogue of $\|\cdot \|_{\infty}$. I looked around and Wikipedia mentions that $$\|A\|\ _{\infty} = \max_{1 \le i \le m} \sum_{j=1}^n |a_{ij}|.$$ 
Can someone explain how/why this is the proper analogue and also answer the first question I asked?

REPLY [2 votes]: Although it's not really the spirit in which you're asking, there are some senses in which these norms are the 'only' possibilities.  They are not literally the only possibilities, and, as Studzinski pointed out in a now-deleted comment, all norms on a finite-dimensional space are topologically equivalent, so there is topologically no reason to prefer them to any random norm you can imagine; but you can impose additional conditions that single them out in the space of norms—see, for example, http://www.ams.org/mathscinet-getitem?mr=231183.
As Ethan Bolker (https://math.stackexchange.com/a/1251246/87579) and Steven Taschuk (https://math.stackexchange.com/a/1251242/87579) point out, these norms also have the property that their unit 'balls' are visually appealing—which is a subjective thing, so let's just say that they have a reasonably nice collection of symmetries.  Focussing on symmetries explains why we tend to prefer the $\ell^2$ norm:  because it has the most symmetries!  See http://www.ams.org/mathscinet-getitem?mr=1272946 .
Another paper on the symmetry-theoretic approach to norms, which shares an author with the one linked just above, is http://www.ams.org/mathscinet-getitem?mr=1763059 .