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