Let $\Omega \subset \mathbb{R}^n$ be open. Let $u,v \in L^1_\text{loc}(\Omega)$ with $u \ne 0$ a.e on a set of positive measure. Assume that $$\phi \in C_c^\infty(\Omega), \int u\phi > 0 \implies \int v \phi \ge 0$$
Show that there exists $ \lambda \ge 0$ such that $v=\lambda u$
My try: I tried constructing this problem first in the abstract setting. Let $E$ be an arbitrary vector space. Let $u$ and $v$ be functionals on $E$ with $u \ne 0$. Assume that $$\phi \in E, u(\phi) \gt 0 \implies v (\phi) \ge 0$$ Show that there exists $ \lambda \ge 0$ such that $v=\lambda u$. If I am able to show that then the original question follows by takiing $E=C_c^\infty(\Omega)$.
Since $u \ne 0$, there exists $\phi_0 \in E$ such that $u(\phi_0)=1$. For any $\epsilon \gt 0 , \phi \in E$, we have $u(\phi-\phi_0u(\phi)+\epsilon\phi_0)=\epsilon \gt 0$. So by the above property, $$v(\phi-\phi_0u(\phi)+\epsilon\phi_0) \ge 0 \implies v(\phi)-v(\phi_0)u(\phi)+\epsilon v(\phi_0) \ge 0$$. This happens for every $\epsilon \gt 0$. Hence $v(\phi) \ge u(\phi_0)u(\phi)$. But I am unable to conclude that equality holds.
How do I show that?? Thanks for the help!!