Do we have: $$ H_0^1(\Omega)=\{f\in H^1(\Omega): \gamma f=0\}, $$ while $\gamma:H^1(\Omega)\rightarrow H^{1/2}(\partial \Omega)$ is the trace map?
The teacher says that it's true in the course, without giving a proof. I have checked the textbook and searched for a proof, but failed. I understand that "$\subset$" is obvious but how about "$\supset$"? How to say some function can be approximated by $C_c^\infty$?
More genearlly, do we have: $$ H_0^m(\Omega)=\{f\in H^m(\Omega):\gamma_jf=0,\quad j=0,\cdots,m-1\} $$ while $\gamma_j:H^m\rightarrow H^{m-j-\frac{1}{2}}(\partial\Omega)$ are trace maps?