Determine if this subset is compact

  1. 1. The problem statement, all variables and given/known data

    Let [tex](X,d) = (C[0,1], d_\infty)[/tex], [tex]S_1[/tex] is the set of constant
    functions in [tex]B(0,1)[/tex], and [tex]S_2 = \{ f \in C[0,1] | \norm{f}_\infty
    = 1\}[/tex].

    Are [tex]S_1[/tex] and [tex]S_2[/tex] compact?

    2. Relevant equations

    3. The attempt at a solution

    I am trying to use the Arzela - Ascoli theorem. For [tex]S_1[/tex], the set of functions with value in the ball (assuming that's what the question meant) [tex]B(0,1)[/tex] are bounded. They are also equicontinuous at all [tex] x \in [0,1] [/tex]. How do I show if the subset is closed or not?

    For [tex]S_2[/tex], how does the norm [tex]||f||_\infty = 1 [/tex] determine if the set is closed, bounded and equicontinuous? What is the norm [tex]||f||_\infty = 1 [/tex] defined as?
    Last edited: May 16, 2010
  2. jcsd
  3. Landau

    Landau 905
    Science Advisor

    Well, first you have to understand the notation and definitions. [itex]d_\infty[/itex] is just the metric induced by the supremum norm:

    [tex]\|f\|_\infty:=\sup_{x\in[0,1]} |f(x)|[/tex]
  4. [tex]S_1[/tex] is not closed because the function [tex]f = 0[/tex] is a limit point
    outside [tex]S_1[/tex]. Therefore [tex]S_1[/tex] is not compact.

    For [tex]S_2[/tex], the metric space [tex]d_\infty(f,g) := \norm{f -
    g}_\infty[/tex] means that it is bounded, however it does not make [tex]S_2[/tex] equicontinuous. Is the subset closed?
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