1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook