1. Not finding help here? Sign up for a free 30min 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!

Determine if this subset is compact

  1. May 16, 2010 #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. May 16, 2010 #2

    Landau

    User Avatar
    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. May 17, 2010 #3
    [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?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Determine if this subset is compact
  1. Identify compact subsets (Replies: 15)

Loading...