While going through the Riez Representation theorem i am stuck with the use of positivity of linear functional. My question is If $\tau$ is a linear functional from $C(X)\to \mathbb C$ , $f\in C(X)$ . Then i didn't understand how positivity is used to write the following :
$\tau(|f|^2) \le \tau (||f||^21) =||f||^2\tau(1)$
I want to know where exactly the positivity is used and how ?
Thank you for your kind help.