Thread: 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

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 $\displaystyle C^1$ function they are equivalent:

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

Use this to prove that $\displaystyle \delta = \frac{ \epsilon }{M}$ works in the def. of (uniform) equicontinuity of $\displaystyle B$. Now use Arzela-Ascoli.

Mean value theorem shows that B is equicontinuous .

Moreover, since $\displaystyle f(0)=0$, again mean value shows that, for $\displaystyle x\in (0,1)$
$\displaystyle |f(x)|=|f(x)-f(0)|\leq |x-0|=|x|$ for each $\displaystyle f\in B$. 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

Originally Posted by Enrique2

Mean value theorem shows that B is equicontinuous .

Moreover, since $\displaystyle f(0)=0$, again mean value shows that, for $\displaystyle x\in (0,1)$
$\displaystyle |f(x)|=|f(x)-f(0)|\leq |x-0|=|x|$ for each $\displaystyle f\in B$. 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

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])