Closed subspace of a Sobolev Space

[lmedin02]

Homework Statement

I am considering the space $\tilde{W}^{1,2}(\Omega)$ to be the class of functions in $W^{1,2}(\Omega)$ satisfying the property that its average value on $\Omega$ is 0. I would like to show that $\tilde{W}^{1,2}(\Omega)$ is a closed subspace of $W^{1,2}(\Omega)$.

Homework Equations

$W^{1,2}(\Omega)$ is the space of $L^2(\Omega)$ so that their distributional derivative also lie in $L^2(\Omega)$.

The Attempt at a Solution

It is clear that $\tilde{W}^{1,2}(\Omega)$ is a subspace of ${W}^{1,2}(\Omega)$. So I now consider a convergent sequence of functions $\tilde{w}_k$ in $\tilde{W}^{1,2}(\Omega)$ converging to a function $w$ in $W^{1,2}(\Omega)$. I am having trouble showing that $w$ has average value 0 and hence belongs in $\tilde{W}^{1,2}(\Omega)$. Any suggestions.

 

[lmedin02]

I am having a hard time making this conclusion. How about if the sequence is bounded?