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Unit Ball

  1. Apr 8, 2006 #1
    Consider the unit ball [tex]B_1=\{f:\rho_u(f,0)\leq 1\}[/tex] in the metric space [tex](C[0,1],\rho_u)[/tex] where [tex]\rho_u(f,g)=sup\{\forall x(|f(x)-g(x)|)\}.[/tex] Show that there exists a sequence [tex]g_n\in B_1[/tex] such that NO subsequence of [tex]g_n[/tex] converges in [tex]\rho_u[/tex].

    I want to know if what I'm doing is right. Suppose I define a sequence of functions that converge to a function OUTSIDE of C[0,1]=(the space of all continuous functions on 0,1), then any subsequence of such a function would not converge, right?
     
  2. jcsd
  3. Apr 9, 2006 #2

    AKG

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    Right...........
     
  4. Apr 9, 2006 #3
    But at no point did I use the fact that the function I defined doesn't converge in [tex]\rho_u[/tex].
     
    Last edited: Apr 9, 2006
  5. Apr 9, 2006 #4

    AKG

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    Well you're missing a lot of facts. I figured you would fill in the details, and you just wanted to know if your idea would work. So yes, it will work. C[0,1] is a subspace of B[0,1], the space of all bounded real-valued functions on [0,1]. You can give B[0,1] the same metric. Then there are some more easy details to work out, but you can do it. In fact, I don't know if you have to regard C[0,1] as a subspace of any other space, you can try to prove more directly that no subsequence converges.
     
  6. Apr 9, 2006 #5
    Great. Thanks.
     
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