Determine if this subset is compact

[complexnumber]

Homework Statement

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

Are S_1 and S_2 compact?

Homework Equations

The Attempt at a Solution

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

For S_2, how does the norm ||f||_\infty = 1 determine if the set is closed, bounded and equicontinuous? What is the norm ||f||_\infty = 1 defined as?

 

[Landau]

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

\|f\|_\infty:=\sup_{x\in[0,1]} |f(x)|

 

[complexnumber]

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

\|f\|_\infty:=\sup_{x\in[0,1]} |f(x)|

S_1 is not closed because the function f = 0 is a limit point
outside S_1. Therefore S_1 is not compact.


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