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complexnumber May16-10 07:04 AM

Determine if this subset is compact
 
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?

Landau May16-10 10:53 AM

Re: Determine if this subset is compact
 
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]

complexnumber May17-10 06:12 AM

Re: Determine if this subset is compact
 
Quote:

Quote by Landau (Post 2719565)
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]

[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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