Let $Q$ be an open interval of ${\mathtt R}$ and $E$ be the space of continuous and bounded functions in $Q\to \mathtt{R}$.

I call $E^*$ the set of linear functionals over $E$ and $E_+^*$ the subset of positive linear functionals.

My question is whether, for $x\in E$, the condition $\forall s\in Q,\; x(s) > 0$ is equivalent to $\forall f\in E_+^* \left[f\neq 0 \implies f(x)>0\right]$.

From right to left it is easy as it suffice to choose $f: x\mapsto x(s)$ and conclude.

But what about the other direction?

If $Q$ was compact, then I would just state that $x$ is greater than the constant $m = \min_{s\in Q} x(s)$ and thus $f(x) > m\mu_f(Q)$.

But I am interested in the case where $Q$ is open (or compact, but allowing $x$ to become zero on the border).

Does this result still hold?

What book/article can I reference for such a theorem or counter-example?

(Note that in the actual problem I am trying to solve, Q is not a real interval but a finite and disjoint union of bounded and connected open sets in $\mathtt{R}^n$. I don't think this would change much of any consequence, but it is maybe better to state it now.)

Thank you in advance for your help!