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Closed subspace of a Sobolev Space

  1. Feb 9, 2013 #1
    1. The problem statement, all variables and given/known data

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

    2. Relevant equations
    [itex]W^{1,2}(\Omega)[/itex] is the space of [itex]L^2(\Omega)[/itex] so that their distributional derivative also lie in [itex]L^2(\Omega)[/itex].


    3. The attempt at a solution

    It is clear that [itex]\tilde{W}^{1,2}(\Omega)[/itex] is a subspace of [itex]{W}^{1,2}(\Omega)[/itex]. So I now consider a convergent sequence of functions [itex]\tilde{w}_k[/itex] in [itex]\tilde{W}^{1,2}(\Omega)[/itex] converging to a function [itex]w[/itex] in [itex]W^{1,2}(\Omega)[/itex]. I am having trouble showing that [itex]w[/itex] has average value 0 and hence belongs in [itex]\tilde{W}^{1,2}(\Omega)[/itex]. Any suggestions.
     
  2. jcsd
  3. Feb 9, 2013 #2
    I am having a hard time making this conclusion. How about if the sequence is bounded?
     
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