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
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.