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Math Help - Proving sets are weakly compact

  1. #1
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    Proving sets are weakly compact

    Hi,I want to prove that the sets
    B_{n} ={g : 0 ≤ g ≤n} are weakly compact in σ(L^{∞},L^{1}).
    Thank you for your help
    Last edited by mr fantastic; May 30th 2009 at 04:04 PM. Reason: Re-titled the post
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  2. #2
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    Quote Originally Posted by calme View Post
    Hi,I want to prove that the sets
    B_{n} ={g : 0 ≤ g ≤n} are weakly compact in σ(L^{∞},L^{1}).
    Thank you for your help
    The unit ball of L^\infty is weak*-compact (that is, compact in the \sigma(L^\infty,L^1)-topology). So all you have to do to prove that B_1 is weak*-compact is to show that it is weak*-closed in the unit ball of L^\infty. The result for general B_n then follows by taking scalar multiples.
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  3. #3
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    thank you a lot for your reply can you give me more details to prove that it's weakly closed can I use krein-smulitain theorem
    it's really kind of you
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