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Compactness/convergence in Banach spaces

  1. Oct 12, 2012 #1
    Been doing exercises on compactness/sequential compactness of objects in Banach spaces and some of my solutions come down to whether
    holds in

    a) arbitrary finite-dimensional Banach space
    b) lp, 1 <= p <= infinity

    Does it?
     
  2. jcsd
  3. Oct 12, 2012 #2
    No, this does not hold. Consider the sequence in [itex]\ell^2[/itex]:

    [tex]x_1=(1,0,0,0,...)[/tex]
    [tex]x_2=(0,1,0,0,...)[/tex]
    [tex]x_3=(0,0,1,0,...)[/tex]
    and so on.
     
  4. Oct 12, 2012 #3
    Doesn't hold for finite-dimensional Banach space as well?
     
  5. Oct 12, 2012 #4
    It does hold for finite dimensional spaces since those are isomorphic to [itex]\mathbb{R}^n[/itex].
     
    Last edited: Oct 12, 2012
  6. Oct 12, 2012 #5
    Thank you!
     
  7. Oct 13, 2012 #6

    Bacle2

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    Science Advisor

    In general, in metrizable spaces, compactness and sequential compactness are

    equivalent. The unit ball is compact/seq. compact in a normed v.space V iff

    V is finite-dimensional.
     
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