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For $(x_1,x_2)$, is $G_1(x)=|x_1|$ a norm in $\mathbb{R}^2$?

To prove that a function $p$ is a norm we need to prove the following:

  • $p(av) = |a|p(v)$
  • $p(u + v) \leq p(u) + p(v)$
  • $p(v)\ge0$, and if $p(v)=0$ then $v$ is the zero vector

However, I don't know how to actually show this.

Any help will be much appreciated.

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Start with an easy example. What is the norm of $(0,1)$?... – Ludolila Feb 21 '13 at 22:17
In the last of your three requirements, you probably mean that $v$ should be the zero vector; not $p(v)$. – fuglede Feb 21 '13 at 22:19
Thanks, it's been changed :) – Levi Feb 21 '13 at 22:24
I think the norm of $(0,1)$ is $1$? – Levi Feb 21 '13 at 22:42
Think again :) $G_1(0,1)=|0|=0$... – Ludolila Feb 21 '13 at 23:30

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