A non-absolute norm function

Question

For vector norms: $$||x|| = || -x ||$$ Is it always the case, or there exists some norms such that a vector and its negative or the absolute (all elements with modulo) have different norm value?

Answer 1

In a normed vector space, one of the axioms on the norm is that for all scalars $\alpha$ and vectors $x$, $\|\alpha x\| = |\alpha| \|x\|$. So you will always have $$\|-x\| = |-1|\|x\| = \|x\|.$$

Answer 2

One of the axioms of a norm is that $\Vert \alpha x\Vert=|\alpha|\Vert x\Vert$ for all vectors $x$ and all scalars $\alpha$. Therefore, we will always have $$\Vert -x\Vert=|-1|\Vert x\Vert=\Vert x\Vert$$