The Cauchy-Schwarz Inequality (or ``Schwarz Inequality'')
states that for all
and
, we have
with equality if and only if
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
The last two equations imply
In the complex case, let
Since