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
Mean value theorem shows that B is equicontinuous .
Moreover, since , again mean value shows that, for
for each . 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