Identifying elements of $H^{-1}$

Let $U$ be a bounded open set in $\mathbb{R}^n$ for $n \geq 3$.

1. Let $f \in H^{-1}(U) \cap L^2(U)$.

I think the action of $f$ on $g \in H_0^1(U)$ must be $$\big<f,g\big>=\int_U fg$$

Indeed $f$ is linear and $$\Big|\big<f,g\big>\Big|\leq ||f||_{L^2}||g||_{L^{2}} \leq ||f||_{L^2}||g||_{H_0^1}$$

2. Let $f \in H^{-1}(U) \cap L^1(U)$.

For $g \in L^{\infty}(U)\cap H_0^1(U)$, a dense subset of $H_0^1(U)$, the same action does not work, for if, $$\big<f,g\big>=\int_U fg$$ by Holder's we only get

$$\Big|\big<f,g\big>\Big|\leq ||f||_{L^1}||g||_{L^{\infty}} $$ and we can't claim $f \in H^{-1}(U).$

Is there a way to define the action of $f \in H^{-1}(U) \cap L^1(U)$ on $H_0^1(U)$ or a dense subset of it?