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Math Help - unit sphere in an infinite dimensional Banach space

  1. #1
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    unit sphere[solved by myself]

    Let X be an infinite dimensional Banach space and show that the unit sphere S_X=\{||x||=1\} is a G_\delta set (i.e. a countable intersection of open sets) in the unit ball B_X=\{||x||\leq1\}, when the latter is endowed with the relative topology inherited from the weak topology of X.

    I know that the unit sphere is dense in the unit ball in the above topology, but how to show it is G_\delta? And I know that the open set in X under the weak topology of X are all unbounded, and the complement of unit ball is open since unit ball is closed under the weak topology. Thanks in advance!
    Last edited by Jameson; May 3rd 2009 at 01:16 PM. Reason: restored original question
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  2. #2
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    Quote Originally Posted by frankmelody View Post
    Let X be an infinite dimensional Banach space and show that the unit sphere S_X=\{||x||=1\} is a G_\delta set (i.e. a countable intersection of open sets) in the unit ball B_X=\{||x||\leq1\}, when the latter is endowed with the relative topology inherited from the weak topology of X.

    I know that the unit sphere is dense in the unit ball in the above topology, but how to show it is G_\delta? And I know that the open set in X under the weak topology of X are all unbounded, and the complement of unit ball is open since unit ball is closed under the weak topology. Thanks in advance!
    Do you know the theorem that a closed convex subset of a Banach space is weakly closed? That implies that for n\in\mathbb{N} the set \{x\in X:\|x\|\leqslant1-\tfrac1n\} is weakly closed, and so the set G_n = \{x\in B_X:\|x\|>1-\tfrac1n\} is weakly open. Since \textstyle S_X = \bigcap_{n\in\mathbb{N}}G_n, it follows that S_X is a G_\delta set.

    If you don't know the "closed and convex implies weakly closed" theorem then you'll need to show from the definition that the set G_n is weakly open. This follows from the Hahn–Banach theorem, which tells you that if \|x\|>1-\tfrac1n then there is a functional f in the unit sphere of the dual space such that |f(x)|>1-\tfrac1n.
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