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Weakly convergent sequences are bounded

  1. Mar 21, 2012 #1
    1. The problem statement, all variables and given/known data

    I would like to show that a weakly convergent sequence is necessarily bounded.

    3. The attempt at a solution
    I would like to conclude that if I consider a sequence [itex]{Jx_k}[/itex] in [itex]X''[/itex]. Then for each [itex]x'[/itex] in [itex]X'[/itex] we have that [itex]\sup|Jx_k(x')|[/itex] over all [itex]k[/itex] is finite. I am not sure why this is the case since
    [itex]\sup|Jx_k(x')|=\sup|x'(x_k)|\leq\sup||x'||||x_k||[/itex] and [itex]\sup||x_k||[/itex] over all [itex]k[/itex] does not seem finite.

    From here I would be able to make my conclusion by applying the uniform boundedness principle or the Banach-Steinhaus Theorem.
     
  2. jcsd
  3. Mar 28, 2012 #2

    morphism

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    I don't really understand your notation.

    Shouldn't you just look at various functionals applied to your sequence - since each of these will converge, you get pointwise boundedness. But then apply Hahn-Banach to find a functional that attains the norm of your sequence at a specific point. So...
     
  4. Mar 30, 2012 #3
    Thanks for your comments. I was able to resolve my issue with this problem.
     
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