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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?

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up vote 4 down vote accepted

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

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I am thinking that $\|0\cdot x\| = \|0\|\|x\|$, and so $\|0\|=\|0\|\|x\|$, how does this help? Could i get a few more hints? =) – N3buchadnezzar Sep 20 '12 at 23:40
Well, you need to distinguish the zero vector ${\bf 0}$ from plain old $0$ (the scalar). The scalar product satisfies $||a\cdot {\bf x}||=|a| \cdot ||{\bf x}||$. The zero vector satisfies $0 \cdot {\bf x}={\bf 0}$ for any ${\bf x}$ (and in particular ${\bf x}={\bf 0}$). So $||{\bf 0}||=||0\cdot {\bf 0}||=|0| \cdot ||{\bf 0}|| = 0 \cdot ||{\bf 0}|| = 0$. – mjqxxxx Sep 21 '12 at 0:07

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