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Question about Banach space

  1. Feb 28, 2009 #1
    Let X be a Banach space. We show p(X) is closed in X**, where p:X-->X** is defined by p(x)=T_x and T_x:X*-->F is defined by T_x(x*)=x*(x) (F is a field).

    I think I should pick a convergent sequence {x_n} in p(X) (x_n -->x)and show that x belongs to p(X). i.e. show there exists a w in X such that p(w)=x.
    But for some reason I am not getting the answer.
     
  2. jcsd
  3. Mar 1, 2009 #2
    You can use the http://en.wikipedia.org/wiki/Hahn-Banach_theorem" [Broken] to show that p is an isometry. This then easily implies that its image is closed.
     
    Last edited by a moderator: May 4, 2017
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