> [!definition] > > Let $X$ be a [[Locally Compact Hausdorff Space|LCH]] space, and $C_c(X)$ be the space of [[Compactly Supported|compactly supported]] [[Continuity|continuous]] functions on $X$, equipped with the uniform norm. A [[Linear Functional|linear functional]] $I: C_c(X) \to \real$ is **positive** if $I(f) \ge 0$ whenever $f \ge 0$. > [!theorem] > > Let $I$ be a positive linear functional on $C_c(X)$, then for each compact set $K \subset X$, there exists $C_k$ such that $\abs{I(f)} \le C_K\norm{f}_u$ for all $f \in C_c(X)$ such that $\supp{f} \subset K$. > > *Proof*. Let $f \in C_c(X)$ supported in $K$, and let $\phi \in C_c(X, [0, 1])$ such that $\phi = 1$ on $K$ ([[Urysohn's Lemma]]). In this case, $\norm{f}_u \phi - f \ge 0$ and $\norm{f}_u \phi + f \ge 0$. Therefore > $ > \norm{f}_uI(\phi) - I(f) \ge 0 \quad \norm{f}_u I(\phi) + i(f) \ge 0 > $ > so $\abs{I(f)} \le I(\phi)\norm{f}_u$.