How to extend results from space of smooth functions to Sobolev spaces?

Question

I regularly see proofs involving Sobolev spaces where the proof states it will show some result holds for, say, $u \in H_0^1(\Omega)$ where $\Omega$ is a smooth bounded domain.

Then right away it will say that it suffices to the result holds for $u \in C^\infty(\Omega)$. So we have proved that the result holds for $u \in C^\infty(\Omega)$, how then can we show rigorously that this result also holds for $u \in H_0^1(\Omega)$?

Answer 1

The result usually has the form $F(u) = G(u)$ where $F$ and $G$ are some functionals on the Sobolev space. If we establish that

1. $F$ and $G$ are continuous with respect to the Sobolev norm
2. $F(u) = G(u)$ for all $u$ in some dense subset of the Sobolev space

then the conclusion $F\equiv G$ follows in the usual way: $f,g$ continuous from $X$ to $Y$. if they are agree on a dense set $A$ of $X$ then they agree on $X$

The set of smooth compactly supported functions is dense in $H^1_0$. The continuity of $F$ and $G$ is often glossed over in proofs, as "easy to see" because $F$ and $G$ are made of things that are controlled by the Sobolev norm.