This is Proposition 7.1 in Folland.
First some notation and definitions: Let $C_C(X)$ denote the space of continuous functions on $X$ with compact support. Furthermore, a linear functional $I$ on $C_C(X)$ is said to be positive if $I(f) \geq 0$ whenever $f \geq 0$. Lastly, let $\|\cdot\|_u$ denote the uniform norm.
The proposition states:
If $I$ is a positive linear functional on $C_C(X)$, for each compact $K \subset X$ there is a constant $C_K$ such that $|I(f)| \leq C_K \|f\|_u$ for all $f \in C_C(X)$ such that supp$(f) \subset K$.
I am having some trouble understanding parts of Folland's proof, included below:
It suffices to consider real-valued $f$.
Why is this sufficient?
Given a compact $K$, choose $\phi \in C_C(X, [0,1])$ such that $\phi = 1$ on $K$ (Urysohn's lemma). Then if supp$(f) \subset K$, we have $|f| \leq \|f\|_u\phi$...
I have two issues here. First, how do we know that the space is normal so that we can apply Urysohn's lemma? Secondly, I'm not sure where last inequality above is is coming from exactly. By definition of the uniform norm, isn't $|f| \leq \|f\|_u$? Is he defining $\phi = 1$ on $K$ so that he is able to add a factor of $\phi$ to the uniform norm?
...that is, $\|f\|_u\phi - f\geq 0$ and $\|f\|_u\phi + f \geq 0$.
How is he concluding that $\|f\|_u\phi - f\geq 0$? This is clear if $f$ is non-negative, by way of $I$ being positive. However, what if $f < 0$? Then how can this inequality be ensured (along with the one that follows this one)?
Thus $\|f\|_u I(\phi) - I(f) \geq 0$ and $\|f\|_u I(\phi) + I(f) \geq 0$, so that $|I(f)| \leq I(\phi)\|f\|_u$.
I understand the inequalities follow from linearity of $I$ since $I$ is a linear funcitonal, and so we achieve the desired result. However, why does he even consider the case $\|f\|_u I(\phi) + I(f) \geq 0$, it seems unnecessary for the conclusion of the theorem.
Also, how are we assured that $I(0) = 0$? I suspected this would be a property of linear functionals, and so I tried to look up a proof but could not find any.