# Cauchy-Schwarz Inequality

## GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Cauchy-Schwarz Inequality

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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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## Cauchy-Schwarz Inequality

The Cauchy-Schwarz Inequality (or ''Schwarz Inequality'') states that for all and , we have

with equality if and only if for some scalar .

We can quickly show this for real vectors , , as follows: If either or is zero, the inequality holds (as equality). Assuming both are nonzero, let's scale them to unit-length by defining the normalized vectors , , which are unit-length vectors lying on the ''unit ball'' in (a hypersphere of radius ). We have

which implies

or, removing the normalization,

The same derivation holds if is replaced by yielding

The last two equations imply

The complex case can be shown by rotating the components of and such that becomes equal to .

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