I read this:
Suppose $a$ is a bounded (in $H^1_0$) coercive bilinear form and it holds that $$\langle u_t, w \rangle + a(u, w) = 0$$ for all $w \in E_M$, where $\cup_{M \in \mathbb{N}} E_M$ is dense in $H^1_0$. So by density, the equation above holds for all $w \in H^1_0$.
I don't understand why it holds for all $H^1_0$. Can someone flesh out the argument please?
The general procedure is as follows: If you want to show equality for $w\in H^1_0$, approximate it with $w^n\in E_M$, i.e. $w^n\to w$ in $H^1_0$. For all these $w^n$, equality holds, now check that you can pass to the limit in all parts of the quality and you are done.