Triangle Inequality

The triangle inequality states that the length of any side of a triangle is less than or equal to the sum of the lengths of the other two sides, with equality occurring only when the triangle degenerates to a line. In $\mathbb{C}^N$ , this becomes

$$\zbox {\Vert\underline{u}+\underline{v}\Vert \leq \Vert\underline{u}\Vert + \Vert\underline{v}\Vert.}$$

We can show this quickly using the Schwarz Inequality:

$$\begin{eqnarray*} \Vert\underline{u}+\underline{v}\Vert^2 &=& \left<\underline{u}+\underline{v},\underline{u}+\underline{v}\right> \\ &=& \Vert\underline{u}\Vert^2 + 2\mbox{re}\left\{\left<\underline{u},\underline{v}\right>\right\} + \Vert\underline{v}\Vert^2 \\ &\leq& \Vert\underline{u}\Vert^2 + 2\left\vert\left<\underline{u},\underline{v}\right>\right\vert + \Vert\underline{v}\Vert^2 \\ &\leq& \Vert\underline{u}\Vert^2 + 2\Vert\underline{u}\Vert\cdot\Vert\underline{v}\Vert + \Vert\underline{v}\Vert^2 \\ &=& \left(\Vert\underline{u}\Vert + \Vert\underline{v}\Vert\right)^2 \\ \,\,\Rightarrow\,\,\Vert\underline{u}+\underline{v}\Vert &\leq& \Vert\underline{u}\Vert + \Vert\underline{v}\Vert \end{eqnarray*}$$