My understanding of the Hahn-Banach theorem is as follows: given a linear subspace $U$ of $V$, a linear functional $f$ defined on $U$, and a sub-linear functional $p$ which dominates $f$, there exists a linear functional $F$ on $V$ with $F(x) \leq p(x)$ for all $x \in V$ and $F(x) = f(x)$ for $x\in U$.
In Cybenko 1989, the proof of Theorem 1, the first step uses Hahn Banach. Specifically, $R$ is a proper subspace of $C(I)$ where $I$ is the unit cube, the author states that by the Hahn Banach theorem, there exists a linear functional $L$ on $C(I)$, with the property that $L\neq 0$ but $L(R) = 0$.
How does this follow from the Hahn Banach theorem? In particular, why does the Hahn Banach theorem state that there exists an $L$ is not just zero everywhere? Also, where is the dominating sublinear functional coming into play?