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?