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Example:
Let's also look again at the vector-sum example, which is redrawn in Fig.5.6. The norm of the vector sum $ w=x+y$ is

$\displaystyle \Vert w\Vert \isdef \Vert x+y\Vert \isdef \Vert(2, 3) + (4, 1)\Vert
= \Vert(6, 4)\Vert = \sqrt{6^2 + 4^2} = \sqrt{52} = 2\sqrt{13}
$

while the norms of $ x$ and $ y$ are $ \sqrt{13}$ and $ \sqrt{17}$ , respectively. We find that $ \Vert x+y\Vert<\Vert x\Vert+\Vert y\Vert$ , which is an example of the triangle inequality. (Equality occurs only when $ x$ and $ y$ are collinear, as can be seen geometrically from studying Fig.5.6.)


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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 © 2014-04-21 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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