(adsbygoogle = window.adsbygoogle || []).push({}); 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?

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# Determine if this subset is compact

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