# Thread: compactness of a set

1. ## compactness of a set

Let F be a subset of C1([0,1]) with metric coming from the C1 norm fC1=f∥∞+ f’∥∞. Show that F is compact in C1 if and only if F is closed and bounded and {f’: fF} C([0,1]) is equicontinuous.

could anyone help me to solve this question ?

2. Originally Posted by jin_nzzang
Let F be a subset of C1([0,1]) with metric coming from the C1 norm fC1=f∥∞+ f’∥∞. Show that F is compact in C1 if and only if F is closed and bounded and {f’: fF} C([0,1]) is equicontinuous.

could anyone help me to solve this question ?
This is the Azela-Ascoli Theorem

A few proofs can be found here. Yay for $\frac{\epsilon }{3}$

Arzelà?Ascoli theorem - Wikipedia, the free encyclopedia

3. hi thanks for your help,

but it seems a bit different from the arzela-ascoli theorem

in the form of the norm and the original set.

how do i have to approach when i deal with C1([0,1]) ?

4. The Ascoli-Arzela can also be stated as:

A set $K \subset C([a,b])$ is compact if and only if $K$ is closed, bounded and equicontinuous. In this case, any metric would work as well as long as it's defined on that particular metric space.