Replacing one of the conditions of a norm Consider the definition of a norm on a real vector space X.
I want to show that replacing the condition
$\|x\| = 0 \Leftrightarrow x = 0\quad$
with
$\quad\|x\| = 0 \Rightarrow x = 0$
does not alter the the concept of a norm (a norm under the "new axioms" will
fulfill the "old axioms" as well).
Any hints on how to get started?

All you need to show is that $\|0\|=0$. Let $x$ be any element of the normed space. What is $\|0\cdot x\|$?
