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 ${\bf C}^N$, this becomes

$$\Vert\underline{x}+\underline{y}\Vert \leq \Vert\underline{x}\Vert + \Vert\underline{y}\Vert.$$

We can show this quickly using the Schwarz Inequality:

$$\begin{eqnarray*} \Vert\underline{x}+\underline{y}\Vert^2 &=& \left<\underline{x... ...y}\Vert &\leq& \Vert\underline{x}\Vert + \Vert\underline{y}\Vert \end{eqnarray*}$$