Sobolev Space Notation Question

Question

Hi my lecturer for Functional Analysis has used the following notation when talking about Sobolev spaces: $$ \overset{\circ}{W}\vphantom{W}^m(\Omega) = \overline{\mathcal{C^{\infty}_0}(\Omega)}^{W^m(\Omega)} $$ $W^m(\Omega)$ is the notation for a Sobolev space but I get lost when the above notation is introduced to define a new space. I know that it consists of functions which are zero at the boundary but I just dont understand what is happening in the above notation when defining it. Could someone please give me a rundown of what this notation actually means?

Answer 1

$V:=\mathcal C_0^\infty$ is a subspace of $W^m(\Omega)$. There is a topology on $W^m(\Omega)$ given by a norm. Then we take the closure of $V$ with respect to this norm, and this gives by definition $W^m_0(\Omega)$.