Proposition 14.9.2. Let $X$ be a LCH space and $I \in \hom(C_{c}(X; \real); \real)$ be a positive linear functional, then
For any $f, g \in C_{c}(X; \real)$ with $f \le g$, $\dpb{f, I}{C_c(X; \real)}\le \dpb{g, I}{C_c(X; \real)}$.
For any $K \subset X$ compact, there exists $C_{K} \ge 0$ such that for all $f \in C_{c}(X; \real)$ with $\supp{f}\subset K$, $|{\dpb{f, I}{C_c(X; \real)}}| \le \norm{f}_{u}$.
Proof. (1): $\dpb{g - f, I}{C_c(X; \real)}\ge 0$.
(2): By Urysohn’s lemma, there exists $g \in C_{c}(X; [0, 1])$ such that $g|_{K} = 1$. In which case,
\[-\norm{f}_{u}\dpn{g, I}{C_c(X; \real)}\le \dpn{f, I}{C_c(X; \real)}\le \norm{f}_{u}\dpn{g, I}{C_c(X; \real)}\]
so $C_{K} = \dpn{g, I}{C_c(X; \real)}$ is a desired constant.$\square$