Proving Correctness of Definition of Magnitude of Vectors in n-dimensions

Question

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.

Answer 1

There are various names you could use. A typical one is norm, which axiomatizes properties that a reasonable notion of the length of a vector should have. The axioms are

1. $|v| \ge 0$ for all $v$, and if $|v| = 0$ then $v = 0$.
2. $|cv| = |c| |v|$, where $c$ is a scalar and $v$ is a vector.
3. $|v + w| \le |v| + |w|$, the triangle inequality.

None of these are difficult to prove for the $\ell^2$ or Euclidean norm

$$|(x_1, \dots x_n)| = \sqrt{x_1^2 + \dots + x_n^2}$$

although the triangle inequality takes a bit of work.

But it's unclear to me whether this captures what you mean by "correct" (which I don't yet understand). Among other things, this definition is satisfied by many notions of norm other than the usual Euclidean one, for example the $\ell^1$ or "taxicab" or "Manhattan" norm

$$|(x_1, \dots x_n)| = |x_1| + \dots + |x_n|.$$

Perhaps you are looking for a set of properties which uniquely characterize the Euclidean norm.

Answer 2

Unfortunately, your requirements that $c|{v}| = |cv|$ and $|v+w|=|v|+|w|$ are not true for our usual definition of the magnitude in $\mathbb{R}^n$. Instead, with the usual definition it is easy to prove that $|c| \cdot |v| = |cv|$ and $|v+w| \leq |v| + |w|$ hold, both desirable properties of a magnitude definition.