I asked a similar question already, but thinking about the problem a bit more I believe I now know better what my actual problem is. In the other question I asked whether the definition of magnitude of vectors could somehow be proved, as it can indeed geometrically be shown to be correct for 1, 2 and 3 dimensions. But proving a definition is kind of nonsensical, therefore I realised what I actually wanted to ask, is how the definition can be proved to be correct. By this distinction I mean, that it needs to be shown, that the definition of magnitude for vectors of any dimensions actually fulfils certain properties of "the concept of length", for example:
A length of the thing times a factor is the length of the thing stretched by that factor:
$c|\vec v| = |c \vec v|$
and
Adding two things and taking their sum's length should be the same as adding the lengths of the things:
$|\vec v + \vec w| = |\vec v| + |\vec w|$
So that would be proving that the function $f : x \mapsto |x| \text{ where } x \in \mathbb{R}^n \text{ and } |x| \in \mathbb{R}$ is linear and a homomorphism.
I guess therefore my real question is, which are the properties that need to be proved to be obeyed by the definition of magnitude to be correct and is there a name for such a function?
To my original question I got the answer that the correctness could be proved by induction, which so far I cannot quite put together with my new realisations of what actually needs to be proved to show that magnitude behaves as one would intuitively expect.