Q: Consider the mapping $||*||_1:\mathbb{C}^3 \rightarrow [0,\infty)$ given by $||(z_1,z_2,z_3)||_1 = |z_1|+|z_2|+|z_3|$. Show that $||*||_1$ is a norm on $\mathbb{C}^3$.
I have shown that the mapping satisfies homogeneity and positive-definiteness but I'm struggling with the triangle inequality part i.e. showing that $||u+v|| \le ||u|| + ||v||.$ This is what I have so far:
Let $z=(z_1,z_2,z_3)$, $z^{'}=(z_1^{'},z_2^{'},z_3^{'}) \in \mathbb{C}^3$.
$||(z+z^{'})||_1 = ||(z_1+z_1^{'},z_2+z_2^{'},z_3+z_3^{'})||$
$= |z_1+z_1^{'}|+|z_2+z_2^{'}|+|z_3+z_3^{'}|$
...and I'm stuck from there. I know I have to get to $||z+z^{'}||_1 \le ||z|| + ||z^{'}||$ but I'm not really sure how.
Am I approaching this in the right way? If someone could point me in the right direction that would be great. Thanks