What does the norm define whenever it is written in an "untraditional way"? My question might seem very weird.
Let me explain myself. A norm is defined in this way:

a norm is a function that assigns a strictly positive length or size to each vector in a vector space

The most common norms are the $|x|$ for dimR=1 and $\sqrt{x^{2} + y^{2}}$ for dimR=2.
Yet, if I state that $||v||=5|x| + 2|y|$ with $v={x,y}$, it is also a norm, and this what I ment by "untraditional way". Which makes not much sense to me. What's the point of the norm then? What can I do with it? In general, in the x,y plan, a norm defines a distance from the point 0(or any other point), what would $||v||=5|x| + 2|y|$ define?
Secondly, a norm could be defined as $||x||_{p}=(\sum_{i=1}^{n}(|x_{i}|^{p}))^{1/p}$. And I can use this norm in any dimension. If I choose $p=3$ in dimension 2, I don't see what the result shows us.
Can someone clarify all this?
 A: A different norm will be a different way of specifying distance.  
example
If your city has only east-west and north-south streets, and a taxis charges 5 cents a mile to go east-west and $2$ cents a mile to go north-south, then
$$
\|v\| = 5|x| + 2|y|
$$
would be a norm you might want to use.  If $u$ and $v$ are two addresses in your city, then the "distance" $\|u-v\|$ is the cost of a taxi ride from one to the other.
A: In addition to @Gedgar's answer, there are other reasons to introduce these norms to students:


*

*To understand the geometry of these norms by drawing the unit ball in each case. From this, you can see that any two such norms are "equivalent" in that they generate the same topology on $\mathbb{R}^n$. This motivates and clarifies the fact that any two norms a finite dimensional space are equivalent - a very useful fact.

*To prove that the $\|\cdot\|_p$ is a norm on $\mathbb{R}^n$ is not trivial. Once you have that, you can prove that the $\ell^p$ spaces are normed linear spaces. Those spaces are quite useful since they are the simplest examples of infinite dimensional Banach spaces.

