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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}+\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*}


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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-10-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA