# Compactness

• Jan 18th 2010, 08:12 AM
anlys
Compactness
Hello,
I need help with the following questions:
Let B = {f in C([0,1], R) | f' is continuous on (0,1), f(0)=0, and |f'(x)| <= 1}. Show that cl(B) is compact.
I'm stuck on this problem. Appreciate anyone's help on this. Thank you in advance. Thanks :)
• Jan 18th 2010, 11:34 AM
Jose27
Quote:

Originally Posted by anlys
Hello,
I need help with the following questions:
Let B = {f in C([0,1], R) | f' is continuous on (0,1), f(0)=0, and |f'(x)| <= 1}. Show that cl(B) is compact.
I'm stuck on this problem. Appreciate anyone's help on this. Thank you in advance. Thanks :)

Just remember that for a $C^1$ function they are equivalent:

1) $f$ is Lipschitz cont. with constant $M$
2) $|f'|$ is bounded by $M$

Use this to prove that $\delta = \frac{ \epsilon }{M}$ works in the def. of (uniform) equicontinuity of $B$. Now use Arzela-Ascoli.
• Jan 18th 2010, 11:40 AM
Enrique2
Mean value theorem shows that B is equicontinuous .

Moreover, since $f(0)=0$, again mean value shows that, for $x\in (0,1)$
$|f(x)|=|f(x)-f(0)|\leq |x-0|=|x|$ for each $f\in B$. This yields that B is pointwise bounded (for x=1 is an easy consequence). Now Arzela Ascoli tells us that B is relatively compact, that is equivalent to the desired asertion.

By the way, the part of equicontinuity that I only mention above is better explained in Jose27 post
• Jan 18th 2010, 03:54 PM
anlys
Quote:

Originally Posted by Enrique2
Mean value theorem shows that B is equicontinuous .

Moreover, since $f(0)=0$, again mean value shows that, for $x\in (0,1)$
$|f(x)|=|f(x)-f(0)|\leq |x-0|=|x|$ for each $f\in B$. This yields that B is pointwise bounded (for x=1 is an easy consequence). Now Arzela Ascoli tells us that B is relatively compact, that is equivalent to the desired asertion.

By the way, the part of equicontinuity that I only mention above is better explained in Jose27 post

Thank you, Jose27 and Enrique2 for the input. I have tried using the hints that you gave me and now I can prove that the set B is equicontinuous and pointwise bounded. Thanks again :)
• Jan 19th 2010, 01:12 AM
Enrique2
Quote:

Originally Posted by anlys
Thank you, Jose27 and Enrique2 for the input. I have tried using the hints that you gave me and now I can prove that the set B is equicontinuous and pointwise bounded. Thanks again :)

Please excuse me, it is my fault, but you have to check uniform boundedness instad of pintwiese boundedness. Anyway the given argument shows that B is in the unit ball of C([0,1])