Norm Forms

The purpose here is to demonstrate the validity the relation

(12)

for a generic complex function, f(x).

We begin by decomposing the generic complex function into its real and imaginary components,

(1)

After introducing the following, more convenient notation,

(2)

(3)

eqn. 1 becomes

(4)

The conjugate of this function is given by

(5),

and the product of this and the original function (the left hand side of questioned statement) is

(6)
(7)
(8)

Now, the right hand side yields

(9)
(10)
(11)

Finally, combining eqn. 8 and eqn. 11 by their right hand sides provides the questioned statement.

(12)