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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.

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up vote 2 down vote accepted

It is used in the inequality, because we have for all $x\in X$, $|f(x)|^2\leq \lVert f\rVert_{\infty}^2\mathbf 1(x)$, where $\mathbf 1$ is the function constant equal to $1$. Then we apply $\tau$ to $\lVert f\rVert_{\infty}^2\mathbf 1-|f|^2\geq 0$, and we conclude by linearity.

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Thank you !... . – Theorem Jun 25 '12 at 15:54

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