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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

$\displaystyle \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... ...v}\Vert &\leq& \Vert\underline{u}\Vert + \Vert\underline{v}\Vert \end{eqnarray*}

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Triangle Difference Inequality
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Cauchy-Schwarz Inequality