# Sobolev Space Notation 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?

## 1 Answer

$V:=\mathcal C_0^\infty$ 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 $W^m_0(\Omega)$.

• Ok I think I mostly understand this. However I am not sure what you mean by a topology given by a norm. According to all the definitions I have encountered a topology does not have an associated norm. And the definitions do not require one. I may be going down a rabbit hole here but how is a topology given by a norm? – james Feb 17 '13 at 11:12
• A norm $N$ gives a metric, namely $d(x,y):=N(x-y)$, and a metric gives a topology. – Davide Giraudo Feb 17 '13 at 16:12