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
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