In the book A Course in Functional Analysis by Conway, there is the following problem:
Problem. Let $ X $ be a completely regular topological space, and let $ C(X) $ denote the set of all continuous functions from $ X $ to $ \mathbb{F} \in \{ \mathbb{R},\mathbb{C} \} $. If $ K $ is a compact subset of $ X $, define $$ {p_{K}}(f) \stackrel{\text{df}}{=} \sup(\{ |f(x)| \in \mathbb{R}_{\geq 0} \mid x \in K \}). $$ Then prove that $$ \bigcap_{\text{$ K \subseteq X $ is compact}} \{ f \in C(X) \mid {p_{K}}(f) = 0 \} = \{ 0_{C(X)} \}. $$
Could anyone kindly help me with this problem? Thanks!
Suppose there is $f\in C(X)$ such that for every compact subset $K$, $p_K(f)=\sup_{x\in K}f(x) =0 $. For every $x\in X$, the set $\{x\}$ is compact so $p_{\{x\}}(f)=0$ which shows that $f=0$ on $X$.