In a problem I ended up with the vector space $V=\{f\in H^1(\Omega):\int_\Omega f=0\}$. I think it can be proven (and it would be really helpful) that $||f||_{L^2(\Omega)}\le c ||\nabla f||_{L^2(\Omega)}$ because this would show norm equivalence. But how do I show that?
When I try to bound $f$ by $\nabla f$ I end up with the integral of $f$ over $\partial\Omega$ and I cannot see how this bounds the integral of $f$ in $\Omega$. Is there something I don't see or is there an other way to do it?
EDIT: $\Omega$ is a bounded subset of $\mathbb{R}^n$.