Let F be a subset of C1([0,1]) with metric coming from the C1 norm ∥f∥C1=∥f∥∞+ ∥f’∥∞. Show that F is compact in C1 if and only if F is closed and bounded and {f’: f∈F} ⊂ C([0,1]) is equicontinuous.
could anyone help me to solve this question ?
Let F be a subset of C1([0,1]) with metric coming from the C1 norm ∥f∥C1=∥f∥∞+ ∥f’∥∞. Show that F is compact in C1 if and only if F is closed and bounded and {f’: f∈F} ⊂ 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 $\displaystyle \frac{\epsilon }{3}$
Arzelà?Ascoli theorem - Wikipedia, the free encyclopedia
The Ascoli-Arzela can also be stated as:
A set $\displaystyle K \subset C([a,b]) $ is compact if and only if $\displaystyle 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.