###
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

In the complex case, let

, and define

. Then

is real and equal to

. By the same derivation as above,

Since

, the
result is established also in the complex case.

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