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.