**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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 .