Field Norm of Complex Number is Positive Definite

Theorem
Let $\C$ denote the set of complex numbers.

Let $N: \C \to \R_{\ge 0}$ denote the field norm on complex numbers:
 * $\forall z \in \C: \map N z = \cmod z^2$

where $\cmod z$ denotes the complex modulus of $z$.

Then $N$ is positive definite on $\C$.

Proof
First it is shown that $\map N z = 0 \iff z = 0$.

Let $z = x + i y$.

Then we have:

Hence the result by definition of positive definite.