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Extreme points

  1. Oct 28, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that if X is a locally compact space but not compact,

    then [itex] B(C_0 (X)) [/itex] has no extreme points,

    in which [itex]B(X)=\{ x | \; ||x|| \leq 1 \}[/itex] and

    [itex]C_0(X)[/itex] = all continuous function [itex]f: X \rightarrow \mathbb{F}[/itex] ( with [itex]\mathbb{F}[/itex] the complex plain or the real line) such that for all [itex]\epsilon>0[/itex], [itex]\{x \in X : \; |f(x)| \geq \epsilon \}[/itex] is compact.


    2. Relevant equations



    3. The attempt at a solution

    One desires to proof that every [itex] z \in B(C_0(X)) [/itex]

    can be written as a convex combination of [itex] y_1 , y_2 \in B(C_0(X)) [/itex], i.e. [itex] y_1+y_2=z [/itex]. The tricky part is finding such [itex] y_1,y_2 [/itex].

    The only thing I have proved so far is the existence of a converging net [itex]x_i \rightarrow x[/itex] such that [itex]f(x_i) \rightarrow 0[/itex], anybody got an idea how I can use this to prove the above.






    Thanks in advance!
     
  2. jcsd
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