Norm Induced by the Inner Product

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Norm Induced by the Inner Product

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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 © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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Norm Induced by the Inner Product

We may define a norm on $\ using the inner product:

\

It is straightforward to show that properties 1 and 3 of a norm hold. Property 2 follows easily from the Schwarz Inequality which is derived in the following subsection. Alternatively, we can simply observe that the inner product induces the well known $L_2$ norm on ${\.

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